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Classical_Mechanics_Goldstein_3ed (GPS)

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Published textbook by Herbert Goldstein, Charles Poole and John Safko, kept in the archive's collection of downloaded physics books. Chapters cover Lagrange's equations, variational principles, central forces, rigid body motion, oscillations, special relativity, Hamilton's equations, canonical transformations, Hamilton-Jacobi theory, chaos, perturbation theory and continuous systems and fields, plus appendices on Euler angles and groups. This is a copy of a standard text, not Phil's own writing.

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CLASSICAL MECHANICS THIRD EDITION Herbert Goldstein Columbia University Charles Poole University ofSouth Carolina John Saﬂ<o University ofSouth Carolina ~<~ 4-e°H,, SanFranc1sco Boslor NewYork g Capetown Hong Kong London Madnd Mex1coCity B) Montreal Munich Pam Singapore Sydney Tokyo Toronto 0 § Q- 6W 11Ox Contents 1ISurvey oftheElementary Principles 1 1.1 Mechanics ofaParticle 1 1.2 Mechanics ofaSystem ofPa11icles 5 1.3 Constraints 12 1.4 D’Alembert’s Principle andLagrange’s Equations 16 1.5 Vel0city—Dependent Potentials andtheDissipation Function 22 1.6 Simple Applications oftheLagrangian Formulation 24 2IVariational Principles andLagrange's Equations 34 2.1 Hami1ton’s Principle 34 2.2 Some Techniques oftheCalculus ofVariations 36 2.3 Derivation ofLagrange’s Equations fromHamilton’s Prmciple 44 2.4 Extension ofHamilton’s Principle toNonholonomic Systems 45 2.5 Advantages ofaVariational Principle Formulation 51 2.6 Conservation Theorems andSymmetry Properties 54 2.7 Energy Function andtheConservation ofEnergy 60 3ITheCentral Force Problem 70 3.1 Reduction totheEquivalent One-Body Problem 70 3.2 TheEquations ofMotion andFirstIntegrals 72 3.3 TheEquivalent One-Dimensional Problem, and Classiﬁcation ofOrbits 76 3.4 TheVirial Theorem 83 3.5 TheDifferential Equation fortheOrbit, andIntegrable Power-Law Potentials 86 3.6 Conditions forClosed Orbits (Bertrand’s Theorem) 89 3.7 TheKepler Problem: Inverse-Square LawofForce 92 3.8 TheMotion inTime intheKepler Problem 98 3.9 TheLap1ace—Runge—Lenz Vector 102 3.10 Scattering inaCentral Force Field 106 3.11 Transformation oftheScattering Problem toLaboratory Coordinates 114 3.12 TheThree-B odyProblem 121 V Conterts 4ITheKinematics ofRigid Body Motion 134 4.1 TheIndependent Coordinates ofaRigid Body 134 4.2 Orthogonal TI'Zl1'lSlOl'TI'l3.llO11S 139 4.3 Formal Properties oftheTransfonnation Matrix 14-4 4.4 TheEuler Angles 150 4.5 TheCayley-Klein Parameters andRelated Quantities 154 4.6 Euler’s Theorem ontheMotion ofaRigid Body 155 4.7 Finite Rotations 161 4.8 Inﬁnitesiinal Rotations I63 4.9 RateofChange ofaVector l7l 4.10 TheCoriolis Effect 174 5ITheRigid Body Equations ofMotion 184 5.1 Angular Momentum andKinetic Energy orMotion about aPoint 184 5.2 Tensors 188 5.3 TheInertia Tensor andtheMoment ofInertia 191 5.4 TheEigenvalues oftheInertia Tensor andthePrincipal AxisTransformation 195 5.5 Solving Rigid Body Problems andtheEuler Equations of Motion 198 5.6 Torque-free Motion ofaRigid Body 200 5.7 TheHeavy Symmetrical TopwithOnePoint Fixed 208 5.8 Precession oftheEquinoxes andofSatellite Orbits 223 5.9 Procession ofSystems ofCharges inaMagnetic Field 230 6IOscillations 238 6.1 Formulation oftheProblem 238 6.2 TheEigenvalue Equation andthePrincipal AxisTransformation 241 6.3 Frequencies ofFreeVibration, andNormal Coordinates 250 6.4 FreeWbrations ofaLinear Triatomic Molecule 253 6.5 Forced Vibrations andtheEffect ofDissipative Forces 259 6.6 Beyond Small Oscillations: TheDamped Driven Pendulum andthe Josephson Junction 265 7ITheClassical Mechanics ofthe Special Theory ofRelativity 276 7.1 Basic Postulates oftheSpecial Theory 277 7.2 Lorentz Transformations 280 7.3 Velocity Addition andThomas Precession 282 7.4 Vectors andtheMetric Tensor 286 8ITheHamilton Equations ofMotionContents vii 7.5 7.6 7.7 7.8 7.9 7.10 7.11 8.1 8.2 8.3 8.4 8.5 8.61-Forms andTensors 289 Forces intheSpecial Theory; Electromagnetism 297 Relativistic Kinematics ofCollisions andMany-Particle Systems 300 Relativistic Angular Momentum 309 TheLagrangian Formulation ofRelativistic Mechanics 312 Covariant Lagrangian Formulations 318 Introduction totheGeneral Theory ofRelativity 324 334 Legendre Transformations andtheHamilton Equations ofMotion 334 Cyclic Coordinates andConservation Theorems 343 Routh’s Procedure 347 TheHamiltonian Formulation ofRelativistic Mechanics 349 Derivation ofI-lamilton’s Equations from a Variational Principle 353 ThePrinciple ofLeast Action 356 9ICanonical Transformations 368 10I9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9TheEquations ofCanonical Transformation 368 Examples ofCanonical Transformations 375 TheHarmonic Oscillator 377 TheSymplectic Approach toCanonical Transformations 381 Poisson Brackets andOther Canonical Invariants 388 Equations ofMotion, Inﬁnitesimal Canonical Transformations, and Conscrvation Theorems inthePoisson Bracket Formulation 396 TheAngular Momentum Poisson Bracket Relations 408 Symmetry Groups ofMechanical Systems 412 Liouville’s Theorem 419 Hamilton-Jacobi Theory andAction-Angle Variables 430 10.1 10.2 10.3 10.4 10.5 10.6TheHamilton-Jacobi Equation forHami1ton’s Principal Function 430 TheHarmonic Oscillator Problem asanExample ofthe Hamilton-Jacobi Method 434 TheHamilton-Jacobi Equation forI-1arni1ton’s Characteristic Function 440 Separation ofVariables intheHamilton-Jacobi Equation 444 lgnorable Coordinates andtheKepler Problem 445 Action-angle Variables inSystems ofOneDegree ofFreedom 452 11I 12I 13I Appendix AI Appendix BIContents 10.7 Action-Angle Variables forCompletely Separable Systems 457 10.8 TheKepler Problem inAction-angle Variables 466 Classical Chaos 11.1 Periodic Motion 484 11.2 Perturbations andthel(olmogorov—Amold—Moser Theorem 487 11.3 Attractors 489 11.4 Chaotic Trajectories andLiapunov Exponents 491 11.5 Poincare Maps 494 11.6 Hénon-Heiles I-Iamiltonian 496 11.7Bifurcations, Driven-clamped Harmonic Oscillator, andParametric Resonance 505 11.8 TheLogistic Equation 509 11.9 Fractals andDimcnsionality 516 Canonical Perturbation Theory 12.1 Introduction 526 12.2 Time-dependent Perturbation Theory 527 12.3 Illustrations ofTime-dependent Perturbation Theory 533 12.4 Time-independent Perturbation Theory 541 12.5 Adiabatic Invariants 549 Introduction totheLagrangian andHamiltonian Formulations forContinuous Systems andFields 13.1 TheTransition from aDiscrete toaContinuous System 558 13.2 TheLagrangian Formulation forContinuous Systems 561 13.3 TheStress-energy Tensor andConservation Theorems 566 13.4 1-lamiltonian Formulation 572 13.5 Relativistic Field Theory 577 13.6 Examples ofRelativistic FieldTheories 583 13.7 _\1oether‘s Theorem 589 Euler Angles inAlternate Conventions andCayley-Klein Parameters Groups andAlgebras Selected Bibliography Author Index Subject Index483 526 558 601 605 617 623 625 Preface totheThird Edition Theﬁrstedition ofthistextappeared in1950, anditwassowellreceived that itwent through asecond printing theverynextyear. Throughout thenextthree decades itmaintained itsposition astheacknowledged standard textfortheintro- ductory Classical Mechanics course ingraduate levelphysics cmricula through- outtheUnited States, andinmany other countries around theworld. Some major institutions alsouseditforsenior levelundergraduate Mechanics. Thirty years later,in1980, asecond edition appeared which was“athrough-going revision of theﬁrstedition.” Thepreface tothesecond edition contains thefollowing state- rnent: "Ihavetriedtoretain, asmuch aspossible, theadvantages oftheﬁrstedition while taking intoaccount thedevelopments ofthesubject itself, itsposition inthe curriculum, anditsapplications toother ﬁelds.” Thisisthephilosophy which has guided thepreparation ofthisthirdedition twenty more years later. Thesecond edition introduced oneadditional chapter onPerturbation Theory, andchanged theordering ofthechapter onSmall Oscillations. Inaddition itadded asigniﬁcant amount ofnewmaterial which increased thenumber ofpages by about 68%.Thisthirdedition addsstillonemore newchapter onNonlinear Dy- namics orChaos, butcounterbalances thisbyreducing theamount ofmaterial in several oftheother chapters, byshortening thespace allocated toappendices, by considerably reducing thebibliography, andbyomitting thelonglistsofsymbols. Thusthethirdedition iscomparable insizetothesecond. Inthechapter onrelativity wehaveabandoned thecomplex Minkowski space infavor ofthenowstandard realmetric. Twooftheauthors prefer thecomplex metric because ofitspedagogical advantages (HG) andbecause itﬁtsinwellwith Clifford Algebra formulations ofPhysics (CPP), butthedesire toprepare students whocaneasily move forward intoother areas oftheory suchasﬁeldtheory and general relativity dominated overpersonal preferences. Some modem notation suchas1-forms, mapping andthewedge product isintroduced inthischapter. Thechapter onChaos isanecessary addition because ofthecurrent interest innonlinear dynamics which hasbegtm toplayasigniﬁcant roleinapplications ofclassical dynamics. Themajority ofclassical mechanics problems andappli- cations intherealworld include nonlineanties, andit1Simportant forthestudent tohaveagrasp ofthecomplexities involved, andofthenewproperties thatcan emerge. Itisalsoimportant torealize theroleoffractal dimensionality inchaos. New sections have been added andothers combined oreliminated here and therethroughout thebook, withtheomissions toagreatextent motivated bythe desire nottoextend theoverall length beyond thatofthesecond edition. Asection ix Preface totheThird Edition wasadded ontheEuler andLagrange exact solutions tothethreebody problem. Inseveral places phase space plotsandLissajous ﬁgures wereappended toillus- Irate solutions. Thedamped driven pendulum wasdiscussed asanexample that explains theworkings ofJosephson junctions. Thesymplectic approach wasclar- iﬁedbywriting outsome ofthematrices. Theharmonic oscillator wastreated withanisotropy, andalsoinpolar coordinates. Thelastchapter oncontinua and ﬁelds wasformulated inthemodem notation introduced intherelativity chap- ter.Thesigniﬁcances ofthespecial unitary group intwodimensions SU(2) and thespecial orthogonal group inthree dimensions SO(3) were presented inmore up-to-date notation, andanappendix wasadded ongroups andalgebras. Special tables wereintroduced toclarify properties ofellipses, vectors, vector ﬁelds and 1-forms, canonical transformations, andtherelationships between thespacetime andsymplectic approaches. Several ofthenewfeatures andapproaches inthisthirdedition hadbeenmen- tinned aspossibilities inthepreface tothesecond edition, suchasproperties of group theory, tensors innon-Euclidean spaces, and“new mathematics” oftheoret- icalphysics suchasmanifolds. Thereference to“One areaomitted thatdeserves special attention—nonlinear oscillation andassociated stability questions” now constitutes thesubject matter ofournewChapter ll“Classical Chaos.” Wede- bated whether toplace thisnewchapter afterPerturbation theory where itﬁts morelogically. orbefore Perturbation theory where itismorelikely tobecovered inclass, andwechose thelatter. Thereferees whoreviewed ourmanuscript were evenly divided onthisquestion. Themathematical levelofthepresent edition isabout thesame asthatofthe ﬁrsttwoeditions. Some ofthemathematical physics, suchasthediscussions ofhermitean andunitary matrices, wasomitted because itpertains much more toquantum mechanics thanitdoestoclassical mechanics, andlittleusednota- tionslikedyadics werecurtailed. Space devoted topower lawpotentials, Cayley- Klein parameters, Routh’s procedure, timeindependent perturbation theory, and thestress-energy tensor wasreduced. Insome cases reference wasmade tothe second edition formore details. Theproblems attheendofthechapters were divided into“derivations” and“exercises,” andsome newoneswereadded. Theauthors areespecially indebted toMichael A.Unseren andForrest M. Hoffman oftheOakRidge National laboratory fortheir I993compilation of errata inthesecond edition thattheymade available ontheIntemet. Itishoped thatnottoomany newerrors haveslipped intothispresent revision. Wewishto thank thestudents whousedthistextincourses withus,andmade antunber of useful suggestions thatwereincorporated intothemanuscript. Professors Thomas Sayetta andthelateMike Schuette made helpful comments ontheChaos chapter, andProfessors Joseph Johnson andJames Knight helped toclarify ourideas onLieAlgebras. Thefollowing professors reviewed themanuscript andmade many helpful suggestions forimprovements: Yoram Alhassid, Yale University; Dave Ellis, University ofToledo; JohnGruber, SanJoseState; Thomas Handler, University ofTennessee; Daniel I-long, Lehigh University; Kara Keeter, Idaho State University; Carolyn Lee;Yannick Meurice, University ofIowa; Daniel Preface totheThird Edition Xi Marlow, Princeton University; Julian Noble, University ofVirginia; Muhammad Numan, Indiana University ofPennsylvania; Steve Ruden, University ofCalifor- nia,Irvine; JackSemura, Portland State University; Tammy AnnSmecker-Hane. University ofCalifornia, Irvine; Daniel Stump, Michigan StateUniversity; Robert Wald, University ofChicago; Doug Wells, Idaho StateUniversity. Ithasindeed beenanhonor fortwoofus(CPP andJLS)tocollaborate as co-authors ofthisthirdedition ofsuchaclassic book ﬁftyyears afteritsﬁrstap- pearance. Wehaveadmired thistextsince weﬁrststudied Classical Mechanics fromtheﬁrstedition inourgraduate student days(CPP in1953andJLSin1960), andeachofususedtheﬁrstandsecond editions inourteaching throughout die years. Professor Goldstein istobecommended forhaving written andlateren- hanced suchanoutstanding contribution totheclassic Physics literature. Above allweregister ourappreciation andacknolwedgement inthewords of Psalm 19,1: 01‘oripavot dmyoﬁvrat Eoéav @605 Flushing, New York HERBERT GOLDSTEIN Columbia, South Carolina CHARLES P.POOLE, JR. Columbia, South Carolina JOHN L.SAFKO July,2000 CHAPTER l.1ISurvey ofthe Elementary Principles Themotion ofmaterial bodies formed thesubject ofsome oftheearliest research pursued bythepioneers ofphysics. From their efforts there hasevolved avast ﬁeld known asanalytical mechanics ordynamics, orsimply, mechanics. Inthe present century thetenn“classical mechanics” hascome intowide usetodenote thisbranch ofphysics incontradistinction tothenewer physical theories, espe- cially quantum mechanics. Weshall follow thisusage, interpreting thename to include thetypeofmechanics arising outofthespecial theory ofrelativity. Itis thepurpose ofthisbook todevelop thestructure ofclassical mechanics andto outline some ofitsapphcations ofpresent-day interest inpurephysics. Basic to anypresentation ofmechanics areanumber offundamental physical concepts, such asspace, time, sirnultaneity, mass, andforce. Forthemost part, however, these concepts willnotbeanalyzred critically here; rather, they willbeassumed as undeﬁned tenns whose meanings arefamiliar tothereader. MECHANICS OFAPARTICLE Letrbetheradius vector ofaparticle from some given ongin andvitsvector velocity: .1v=-'. (1.1)dl Thelinear momentum poftheparticle isdeﬁned astheproduct oftheparticle mass anditsvelocity: p=mv. (1.2) Inconsequence ofinteractions withexternal objects andﬁelds, theparticle may experience forces ofvarious types, e.g.,gravitational orelectrodynamic; thevec- torsumofthese forces exerted ontheparticle isthetotalforce F.Themechanics oftheparticle iscontained inNewton ’ssecond lawofmotion, which states that there existframes ofreference inwhich themotion oftheparticle isdescribed by thedlfferential equation dp _=—E l. Fdt P. (3) 1 Chapter 1Sun/ey oftheElementary Principles or dF=— . 1.4 dl(rnv) () Inmostinstances, themassoftheparticle 1Sconstant andEq.(1.4)reduces to dF=m7?;=ma. (15) where aisthevector acceleration oftheparticle deﬁned by d2 a=T; (1.6) Theequation ofmotion isthusadifferential equation ofsecond order, assuming Fdoesnotdepend onhigher-order derivatives. Areference frame inwhich Eq.(1.3) isvalid iscalled aninertial orGalilean system. Evenwithin classical mechanics thenotion ofaninertial system issome- thing ofanidealization. Inpractice, however, itisusually feasible tosetupaoo- ordinate system thatcomes asclose tothedesired properties asmayberequired. Formany purposes. areference frame ﬁxedinEarth (the“laboratory system”) is asufﬁcrent approximation toaninertial system, while forsome astronomical pur- poses itmaybenecessary toconstruct aninertial system byreference tndistant galaxies. Many oftheimportant conclusions ofmechanics canbeexpressed intheform ofconservation theorems, which indicate under whatconditions various mechan- icalquantities areconstant intime. Equation (1.3) directly furnishes theﬁrstof these, the Conservation Theorem fortheLinear Momentum ofaParticle: Ifthetotalforce, F,iszero, thenp=Oandthelinear momertum, p,isconserved. Theangular momentum oftheparticle about point 0,denoted byL,isdeﬁned as L=r xp, (1.7) where ristheradius vector from Ototheparticle. Notice thattheorder ofthe factors isimportant. Wenowdeﬁne themoment offorce ortorque about Oas N=rxF. (1.8) Theequation analogous to(1.3)forNisobtained byforming thecrossproduct of rwithEq.(1.4): I‘XF=N=l'X%(mV). (1.9) 1.1Mechanics ofaParticle 3 Equation (1.9) canbewritten inadifferent form byusing thevector identity: d d+(rxmv)=vxmv+rx ——(mv), (1.10)at dt where theﬁrstterm ontheright obviously vanishes. lnconsequence ofthisiden- tity.Eq.(1.9) takes theform d dL .N=E(1'xmv)-gt—=L. (1.11) NotethatbothNandLdepend onthepoint 0,about which themoments are taken. AswasthecaseforEq.(1.3), thetorque equation, (1.11), alsoyields animme- diateconservation theorem, thistimethe Conservation Theorem fortheAngular Momentum ofaParticle: Ifthetotal torque, N,irzemthenL=0,andtheangular momentum Lisconserved. Next consider thework done bytheextemal force Fupon theparticle ingoing from point ltopoint 2.Bydeﬁnition, thiswork is 2 W11 =/F-d5. (1.12) -1 Forconstant mass (aswillbeassumed fromnowonunless otherwise speciﬁed), theintegral inEq.(1.12) reduces to dv m d 2 /F-ds_m/dt vdt_§fZ;(v )dt, W12=gag-vf). (1.13)andtherefore Thescalar quantity mvz/2 iscalled thekinetic energy oftheparticle andisde- noted byT,sothatthework done isequal tothechange inthekinetic energy: W12 =T2—T1. (1.14) Iftheforce ﬁeld issuch thatthework W12isthesame foranyphysically possible pathbetween points land2,thentheforce (andthesystem) issaidtobe conservative. Analternative description ofaconservative system isobtained by imagining theparticle being taken from point 1topoint 2byonepossible path andthenbeing returned topoint 1byanother path. Theindependence ofW12on theparticular pathimplies thattheworkdonearound suchaclosed circuit iszero, i.e. ¢F-a's=O. (1.15) Chapter 1Survey oftheElementary Principles Physically itisclearthatasystem cannot beconservative iffriction orother dis- sipation forces arepresent, because F-dsduetofriction isalways positive and theintegral cannot vanish. Byawell-known theorem ofvector analysis, anecessary andsufﬁcient condi- tionthatthework, W12, beindependent ofthephysical pathtaken bytheparticle isthatFbethegradient ofsome scalar function ofposition: F=—VV(r), (1.15) where Viscalled thepotential, orpotential energy. Theexistence ofVcanbe inferred intuitively byasimple argument. IfW12isindependent ofthepathof integration between theendpoints 1and2.itshould bepossible toexpress Wm asthechange inaquantity thatdepends onlyupon thepositions oftheendpoints. Thisquantity maybedesignated by—-V,sothatforadifferential pathlength we havetherelation F-ds=—dV OI‘ 8VFY 2 1-71 ds which isequivalent toEq.(1.16). Note thatinEq.(1.16) wecanaddtoVany quantity constant inspace, without affecting theresults. Hence thezerolevelofV isarbitrary. Foraconservative system, thework done bytheforces is W12 =V;—V2. (1.17) Combining Eq.(1.17) withEq.(1.14), wehavetheresult T1+V1: T;+V2, (1.18) which states insymbols the Energy Conservation Theorem foraParticle: Iftheforces acting onaparticle areconservative, thenthetotalenergy oftheparticle, T+V,isconserved. Theforce applied toaparticle mayinsome circumstances begiven bythe gradient ofascalar function thatdepends explicitly onboththeposition ofthe particle andthetime. However, thework done ontheparticle when ittravels a distance ds, F-ds= —alds, 8s isthennolonger thetotalchange in—Vduring thedisplacement, since Valso changes explicitly withtimeastheparticle moves. Hence, thework doneasthe 1.2I1.2 Mechanics ofaSystem ofParticles 5 particle goesfrompoint ltopoint 2isnolonger thedifference inthefunction V between those points. While atotalenergy T+Vmaystillbedeﬁned, itisnot conserved during thecourse ofthepa1ticle’s motion. MECHANICS OFASYSTEM OFPARTICLES Ingeneralizing theideas oftheprevious section tosystems ofmany particles, wemust distinguish between theexternal forces acting ontheparticles dueto sources outside thesystem. andinternal forces on,say,some particle iduetoall other particles inthesystem. Thus, theequation ofmotion (Newton’s second law) forthe1‘thparticle iswritten as Zn.+Ff”=11.. <119>J where Ff”)stands foranextemal force, andFJ,istheintemal force ontheith particle duetothejthparticle (F,,,naturally, iszero). Weshall assume thatthe Fq(liketheFf”)obeyNewton’s thirdlawofmotion initsoriginal form: thatthe forces twoparticles exert oneach other areequal andopposite. This assumption (which doesnotholdforalltypes offorces) issometimes referred toastheweak lawofaction andreaction Summed overallparticles, Eq.(1.19) takestheform dz (e)d—t2Z:m,r, =ZF, +Zr,,. (1.20)r r 1.; 1'»-‘=1' Thefirstsumonthenight issimply thetotalextemal force F“),while thesecond term vanishes, since thelawofaction andreaction states thateachpairF,j+Fy, iszero. Toreduce theleft-hand side, wedeﬁne avector Rastheaverage ofthe radii vectors oftheparticles, weighted inproportion totheirmass: R_ =%ig. (1.21) Thevector Rdeﬁnes apo1nt known asthecenter ofmass, ormore loosely asthe center ofgravity, ofthesystem (cf.Fig.1.1).With thisdeﬁnition, (1.20) reduces to Mﬁ —Elli“) =Fl‘) (122)21:2— ‘_ ’ 'l which states thatthecenter ofmass moves asifthetotal external force were acting ontheentire massofthesystem concentrated atthecenter ofmass. Purely internal forces, iftheobeyNewton’s thirdlaw,therefore havenoeffect onthe Chapter 1Survey oftheElementary Principles 9 0 ml °,. Q . ,. \Center ofmass. . I‘, R 0° I11] '1 0. FIGURE 1.1Thecenter ofmassofasystem ofparticles. inutiun ofthecenter ofmass. Anoft-quoted example 1Sthemotion Olanexploding shell—ihe center ofmass ofthefragments traveling asiftheshell were stillina single piece (neglecting airresistance). Thesame principle isinvolved injetand rocket propulsion. Inorder thatthemotion ofthecenter ofmass beunaffected, theejection oftheexhaust gases athighvelocity must becounterbalanced bythe forward motion ofthevehicle ataslower velocity. ByEq.(1.21) thetotallinear momentum ofthesystem, an anP-X:m,E-M—Zt—, (1.23) isthetotalmass ofthesystem times thevelocity ofthecenter ofmass. Conse- quently, theequation ofmotion forthecenter ofmass, (1.23), canberestated as the Conservation Theorem fortheLinear Momentum ofaSystem ofParticles: Ifthe totalexternal force iszero. thetotallinear momentum isconserved. Weobtain thetotal angular momentum ofthesystem byforming thecross product r,xp,andsumming overi.Ifthisoperation isperfonned inEq.(1.191, there results, withtheaidoftheidentity, Eq.(1.10), Zn,x13,)=Zia, xpt)=L=Zr,XFf“)+Zr,XF],-.(1.24)I i r ii?-£1} Thelastterm ontheright in(1.24) canbeconsidered asumofthepairs ofthe form 1.2Mechanics ofaSystem ofParticles 7 .0 I 4 1.”0 0.0 rl '1 l 0 FIGURE 1.2 Thevector r,-1-between theithandjthparticles. using theequality ofaction andreaction Butr,-—rJisidentical withthevector r,1-fromjtoi(cf.Fig.1.2),sothattheright-hand sideofEq.(1.25) canbewritten as 1'”xF,,. Iftheinternal forces between twoparticles, inaddition tobeing equal andoppo- site,alsoliealurig thehirejoining thepaiticles—a condition known asthestrong lawofaction andreaction—then allofthesecrossproducts vanish. Thesumover pairs iszerounder thisassumption andEq.(1.24) maybewritten intheform ‘ll’ (e)—= . 1.2dt N (6) Thetimederivative ofthetotalangular momentum isthusequal tothemoment oftheextemal force about thegiven point. Corresponding toEq.(1.26) isthe Conservation Theorem forTotal Angular Momentum: Lisconstant intimeifthe applied (external) torque iszero. (Itisperhaps worthwhile toemphasize thatthisisavector theorem; i.e.,Lz willbeconserved ifNE”)iszero,evenifN?)andN£8)arenotzero.) Notethattheconservation oflinear momentum intheabsence ofapplied forces assumes thattheweak lawofaction andreaction isvalidfortheinternal forces. Theconservation ofthetotalangular momentum ofthesystem intheabsence of applied torques requires thevalidity ofthestrong lawofaction andreaetion—-that theintemal forces inaddition becentral. Many ofthefamiliar physical forces, such asthatofgravity, satisfy thestrong form ofthelaw.Butitispossible to ﬁndforces forwhich action andreaction areequal even though theforces arenot central (seebelow). inasystem involving moving charges, theforces between thecharges predicted bytheBlot-Savart lawmayindeed violate bothforms of Chapter 1Survey oftheElementary Principles theaction andreaction law.*Equations (1.23) and(1.26), andtheircorresponding conservation theorems, arenotapplicable insuchcases, atleastintheform given here. Usually itisthenpossible toﬁndsome generalization ofPorLthatis conserved. Thus, inanisolated system ofmoving charges itisthesumofthe mechanical angular momentum andtheelectromagnetic “angular momentum" of theﬁeldthatisconserved. Equation (l.23)states thatthetotallinear momentum ofthesystem isthesame asiftheentire mass wereconcentrated atthecenter ofmassandmoving withit. Theanalogous theorem forangular momentum ismore complicated. With the origin 0asreference point, thetotalangular momentum ofthesystem is L=Zr, xp,. l LetRbetheradius vector from 0tothecenter ofmass, andlet1':betheradius vector fromthecenter ofmasstotheithparticle. Then wehave(cf.Fig.1.3) r,=1“,+R (1.27) and v,=vf+v where dRv=—dl‘ I »: Centerr' ofmass R 0 FIGURE 1.3 Thevectors involved intheshiftofreference point fortheangular momen- lllm. *Iftwocharges aremoving uniformly withparallel velocity vectors thatarenotperpendicular tothe lineJoining thecharges, thenthenetmutual forces areequal andopposite butdonotliealong the vector between thecharges. Consider, further, twocharges moving (instantaneously) soasto“cross theT,"i.e.,onecharge moving directly attheother, Wl1.lCh inturnismoving atright angles totheﬁrst Then thesecond charge exerts atnonvanishing magnetic force ontheﬁrst,without experiencing any magnetic reaction torce atthatinstant. 1.2Mechanics ofaSvstem ofParticles 9 isthevelocity ofthecenter ofmassrelative toO,and atVF; isthevelocity oftheithparticle relative tothecenter ofmass ofthesystem. Using Eq.(1.27), thetotalangular momentum takes ontheform L=ZR xm,v-l-Zr: xm,vf+ xv+Rx i i t l Thelasttwoterms inthisexpression vanish, forbothcontain thefactor 2mlrz, which, itwillberecognized, deﬁnes theradius vector ofthecenter ofmass inthe verycoordinate system whose origin isthecenter ofmass andistherefore anull vector. Rewriting theremaining terms, thetotalangular momentum about 0is L=R><Mv-i-Erjxpj. (1.22)l Inwords, Eq.(1.28) saysthatthetotalangular momentum about apoint 0is theangular momentum ofmotion concentrated atthecenter ofmass, plusthe angular momentum ofmotion about thecenter ofmass. Theform ofEq.(1.28) emphasizes thatingeneral Ldepends ontheorigin O,through thevector R.Only ifthecenter ofmass 15atrestwithrespect to0willtheangular momentum be independent ofthepointofreference. Inthiscase,theﬁrsttermin(1.28) vanishes, andLalways reduces totheangular momentum taken about thecenter ofmass. Finally, letusconsider theenergy equation. Asinthecaseofasingle paiticle, wecalculate thework donebyallforces inmoving thesystem from aninitial conﬁguration 1,toaﬁnalconﬁguration 2: 2 2 2 W);=Zfl F,-ds,=Zf1FfeJ.ds,+Zfl F],-ds,. (1.29)1 z I;v 1951 Again, theequations ofmotion canbeusedtoreduce theintegrals to 2 2 2 1 ifl F,.ds=;£ m,v,.v,d¢=;fl d(5m;v?). Hence, thework donecanstillbewritten asthedifference oftheﬁnalandinitial kinetic energies: Wiz=T2—T1- where T,thetotalkinetic energy ofthesystem, is 1T=5;:m,v,2. (1.30) Chapter ISurvey oftheHementary Principles Making useofthetransformations tocenter-of-mass coordinates, given inEq. (1.27), wemayalsowrite Tas :r=%Zi:m,(v+\':)-(v+\'f) 1 21 /2 d 1 _2Zijmm +2Z:m,ti +v dt(zijmntl , andbythereasoning already employed incalculating theangular momentum, the lasttermvanishes, leaving 1 1r=5Mv2+5;m,v{1 (1.31) Thekinetic energy, liketheangular momentum, thusalsoconsists oftwoparts: thekinetic energy obtained ifallthemasswereconcentrated atthecenter ofmass, plusthekinetic energy ofmotion about thecenter ofmass. Consider nowtheright-hand sideofEq.(1.29). Inthespecial casethatthe external forces arederivable interms ofthegradient ofapotential, theﬁrstterm canbewritten as 2 2 2 2! Fi(e)'dsI=_Ej ViVi'dsr=_ i 1 t l t 1 where thesubscript ionthedeloperator indicates thatthederivatives arewith respect tothecomponents ofr,.Iftheinternal forces arealsoconservative, then themutual forces between the1'thandjthparticles, F,-,andF,,,canbeobtained from apotential function V,1.Tosatisfy thestrong lawofaction andreaction, VU canbeafunction onlyofthedistance between theparticles: Vt,=V.,(l1't —1')l). (1-32) Thetwoforces arethenautomatically equal andopposite, F].=-—V,V,J=+VJV,J =—F,,, (1.33) andliealong thelinejoining thetwoparticles, VVU-(l1‘i —1')l) =(1's-1‘i)f, (1-34) where fissome scalar function. IfV,Iwere alsoafunction ofthedifference of some other pairofvectors associated withtheparticles, suchastheirvelocities or(tostepintothedomain ofmodem physics) theirintrinsic “spin” angular mo- menta, thentheforces would stillbeequal andopposite, butwould notnecessarily liealong thedirection between theparticles. 1.2 Mechanics ofaSystem ofParticles II When theforces areallconservative, thesecond terminEq.(1.29) canbe rewritten asasumoverpairs ofparticles, thetemis foreachpairbeing ofthe fomi 2 —fl<v.v.-,--as.+v,v.,-ds_,~>- Ifthedifference vector rt—1']isdenoted byr,J,andifV;_,-stands forthegradient withrespect to1-,,,then V,V,, =V,JV,-J=—V,V,,. and ds,—dsJ =dr, -dr’, =dr,_,-, sothatthetermfortheijpairhastheform '-f VI] ‘df”. Thetotalworkarising frominternal forces thenreduces to 1 2 1 2 -izf v,,v,,--at-,,=-52:1/,, . (135) 1.] I 1,1 -l 1%; 1%} Thefactor %appears inEq.(1.35) because insumming overboth Iandjeach member ofagiven pairisincluded twice, ﬁrstintheisummation andtheninthe _]summation. From theseconsiderations, itisclearthatittheextemal andintemal forces ate bothderivable from potentials itispossible todeﬁne atotalpotential energy, V, ofthesystem, 1V-Z:V,+2Z:l/,1. (1.36) ~41 suchthatthetotalenergy T+Visconserved, theanalog oftheconservation theorem (1.18) forasingle particle. Thesecond term ontheright inEq.(1.36) willbecalled theinternal potential energy ofthesystem. Ingeneral, itneed notbezeroand,more important, itmay varyasthesystem changes with time. Only fortheparticular class ofsystems known asrigid bodies willtheintemal potential always beconstant. Formally, arigid body cartbedeﬁned asasystem ofparticles inwhich thedistances r,J areﬁxedandcannot varywithtime. Insuchcase, thevectors a'r,-1-canonlybe perpendicular tothecorresponding 13],andtherefore totheF;1-.Therefore, ina rigidbodytheinternalforces donowork, andtheintemal potential mustremain 1G3 -Lhapter 1Survey oftheElementary Principles constant. Since thetotalpotential isinanycase uncertain towithin anadditive constant, anunvarying internal potential canbecompletely disregarded indis- cussing themotion ofthesystem. CONSTRAINTS From theprevious sections onemight obtain theimpression thatallproblems in mechanics havebeen reduced tosolving thesetofdifferential equations (1.19): "11?! 1 1 XFJI. J Onemerely substitutes thevarious forces acting upon theparticles ofthesystem, turns themathematical crank, andgrinds outtheanswers! Even from apurely physical standpoint, however, thisviewisoversimpliﬁed. Forexample, itmaybe necessary totakeintoaccount theconstraints thatlimitthemotion ofthesystem. Wehavealready metonetypeofsystem involving constraints, namely rigid bod- ies,where theconstraints onthemotions oftheparticles keepthedistances r,_, unchanged. Other examples ofconstrained systems caneasily befurnished. The beads ofanabacus areconstrained toone-dimensional motion bythesupporting wires. Gasmolecules within acontainer areconstrained bythewalls oftheves- seltomove onlyinside thecontainer. Aparticle placed onthesurface ofasolid sphere issubject totheconstraint thatitcanmove onlyonthesurface orinthe region exterior tothesphere. Constraints maybeclassiﬁed invarious ways, andweshallusethefollowing system. Iftheconditions ofconstraint canbeexpressed asequations connecting thecoordinates oftheparticles (andpossibly thetime) having thefonn f(l'1,l'2,l'3,---J) =0, (1-37) thentheconstraints aresaidtobeholonomic. Perhaps thesimplest example of holonomic constraints istherigidbody, where theconstraints areexpressed by equations ofthefomi (r,—r_,)2—c,2J =O. Aparticle constrained tomove along anycurve oronagiven surface isanother obvious example ofaholonomic constraint, withtheequations deﬁning thecurve orsurface acting astheequations ofaconstraint. Consnaints nutexpressible inthisfashion arecalled nonholonomic. Thewalls ofagascontainer constitute anonholonomic constraint. Theconstraint involved intheexample ofaparticle placed onthesurface ofasphere isalsononho1o- nomic, foritcanbeexpressed asaninequality r2—a2Z0 1.3Constraints 13 (where aistheradius ofthesphere), which isnotintheform of(1.37). Thus, in agravitational ﬁeldaparticle placed onthetopofthesphere willslide down the surface partofthewaybutwilleventually falloff. Constraints arefurther classiﬁed according towhether theequations ofcon- straint contain thetimeasanexplicit variable (rheonomous) orarenotexplicitly dependent ontime (scleronomous). Abead sliding onarigid curved wireﬁxed inspace isobviously subject toascleronomous constraint; ifthewireismoving insome prescribed fashion, theconstraint isrheonomous. Note thatifthewire moves, say,asareaction tothebead’s motion, thenthetunedependence ofthe constraint enters intheequation oftheconstraint onlythrough thecoordinates ofthecurved wire (which arenowpartofthesystem coordinates). Theoverall constraint isthenscleronomous. Constraints introduce twotypes ofdifﬁculties inthesolution ofmechanical problems. First, thecoordinates r,arenolonger allindependent, since theyare connected bytheequations ofconstraint; hence theequations ofmotion (1.19) arenotallindependent. Second, theforces ofconstraint, e.g.,theforce thatthe wireexerts onthebead(orthewallonthegasparticle), isnotfurnished apri- ori.Theyareamong theunknowns oftheproblem andmustbeobtained fromthe solution weseek. Indeed, imposing constraints onthesystem ISsimply another method ofstating thatthereareforces present intheproblem thatcannot bespec- iﬁeddirectly butareknown rather interms oftheir effect onthemotion ofthe system. Inthecaseofholonomic constraints, theﬁrstdifﬁculty issolved bytheintro- duction ofgeneralized coordinates. Sofarwehavebeenthinking implicitly in terms ofCartesian coordinates. Asystem ofNparticles. freefrom constraints, has3Nindependent coordinates ordegrees offreedom. Ifthere exist holonomic constraints, expressed inkequations intheform (1.37), thenwemayusethese equations toeliminate kofthe3Ncoordinates, andweareleftwith3N—kinde- pendent coordinates, andthesystem issaidtohave3N—kdegrees offreedom. Thiselimination ofthedependent coordinates canbeexpressed inanother way, bytheintroduction ofnew, 3N—k,independent variables q1,qg,...,q3N_k in terms ofwhich theoldcoordinates r1,r2,...,rNareexpressed byequations of theform 1‘=I'1(q1, qzt--~,¢?31v-1., I) E (1.38) rN =1'N(ql»¢l2, --'vq3N—/(1 t) containing theconstraints inthem implicitly. These aretransformation equations from thesetof(Pg)variables tothe(qt)set,oralternatively Eqs.(1.38) canbecon- sidered asparametric representations ofthe(r1)variables. Itisalways assumed thatwecanalsotransfonn backfromthe(q;)tothe(n)set,i.e.,thatEqs.(1.38) combined withthekequations ofconstraint canbeinverted toobtain anyq,asa function ofthe(rl)variable andtime. Lhapter lSurvey oftheElementary Principles Usually thegeneralized coordinates, qr,unlike theCartesian coordinates, Will notdivide intoconvenient groups ofthreethatcanbeassociated together tofonn vectors. Thus, inthecaseofaparticle constrained tomove onthesurface ofa sphere, thetwoangles expressing position onthesphere, saylatitude andlongi- tude,areobvious possible generalized coordinates. Or,intheexample ofadouble pendulum moving inaplane (two particles connected byaninextensible hght rodandsuspended byasimilar rodfastened tooneoftheparticles), satisfactory generalized coordinates arethetwoangles 01,62.(Cf.Fig.1.4.)Generalized co- ordinates, inthesense ofcoordinates other than Cartesian, areoften useful in systems without constraints. Thus, intheproblem ofaparticle moving inanex- temal central force ﬁeld (V=V(r)), there isnoconstraint involved, butitis clearly more convenient tousespherical polar coordinates thanCartesian coordi- nates. Donot,however, think ofgeneralized coordinates intenns ofconventional orthogonal position coordinates. Allsortsofquantities maybeimpressed toserve asgeneralized coordinates. Thus, theamplitudes inaFourier expansion ofrjmay beusedasgeneralized coordinates, orwemayﬁnditconvenient toemploy quan- tities withthedimensions ofenergy orangular momentum. Iftheconstraint isnonholonomic, theequations expressing theconstraint can- notbeusedtoeliminate thedependent coordinates. Anoft-quoted example of anonholonomic constraint isthatofanobject rolling onarough surface with- outslipping. Thecoordinates usedtodescribe thesystem willgenerally involve angular coordinates tospecify theorientation ofthebody. plusasetofcoordi- nates describing thelocation ofthepoint ofcontact onthesurface. Theconstraint of“rolling” connects these twosetsofcoordinates; theyarenotindependent. A change intheposition ofthepoint ofcontact inevitably means achange inits orientation. Yetwecannot reduce thenumber ofcoordinates, forthe“rolling” condition isnotexpressible asaequation between thecoordinates, inthemanner of(1.37). Rather, itisacondition onthevelocities (ie,thepoint ofcontact is stationary), adifferential condition thatcanbegiven inanintegrated form only after theproblem issolved. Q-» ~% FIGURE 1.4 Double pendulum. 1.3»constraints 15 Z J’ 4 U 0 O X FIGURE 1.5Vertical diskrolling onahorizontal plane. Asimple casewillillustrate thepoint. Consider adiskrolling onthehorizontal xyplane constrained tomove sothattheplane ofthediskisalways vertical. Thecoordinates usedtodescribe themotion might bethex.ycoordinates ofthe center ofthedisk,anangle ofrotation ¢about theaxisofthedisk,andanangle 6between theaxisofthediskandsay,thexaxis(cf.Fig1.5).Asaresult ofthe constraint thevelocity ofthecenter ofthedisk,v,hasamagnitude proportional to¢, v=a, where aistheradius ofthedisk, anditsdirection isperpendicular totheaxisof thedisk: x=vsint9 y=—vcos9. Combining these conditions, wehave twodiﬂerential equations ofconstraint: dx—asin€d¢=0, (1.39) dy+acos6414:=0. Neither ofEqs.(1.39) canbeintegrated without infactsolving theproblem; i.e., wecannot ﬁndanintegrating factor f(x,y,9,¢)thatwillturneither oftheequa- tions intoperfect differentials (cf.Derivation 4).*Hence, theconstraints cannot bereduced totheform ofEq.(1.37) andaretherefore nonholonomic. Physically wecanseethattherecanbenodirect functional relation between ¢andtheother coordinates x,y,and6bynoting thatatanypoint onitspaththediskcanbe *lnprinciple, anintegrating factor canalways befound foraﬁrst-order differential equation ofcon- straint insystems involving onlytwocoordinates andsuch constraints aretherefore holonomic. A familiar example isthetwo-dimensional motion ofacircle rolling onaninclined plane. 1.4ILhapter lSurvey oftheElementary Principles made torollaround inacircle tangent tothepathandofarbitrary radius. Atthe endoftheprocess, x,y,and9havebeenretumed totheiroriginal values, butdz haschanged byanamoint depending ontheradius ofthecircle. Nonintegrable diﬁerential constraints oftheform ofEqs.(1.39) areofcourse nottheonly typeofnonholonomic constraints. Theconstraint conditions may involve higher-order derivatives, ormayappear intheform ofinequalities, aswe have seen. Partly because thedependent coordinates canbeeliminated, problems involv- ingholonomic constraints arealways amenable toaformal solution. Butthereis nogeneral waytoattack nonholonomic examples. True, iftheconstraint isnonin- tegrable, thedifferential equations ofconstraint canbeintroduced intotheprob- lemalong withthedifferential equations ofmotion, andthedependent equations eliminated, ineffect, bythemethod ofLagrange multipliers. Weshallreturn tothismethod atalaterpoint. However, themore vicious cases ofnonholonomic constraint must betackled individually, andconsequently inthe development ofthemore formal aspects ofclassical mechanics, itisalmost invari- ablyassumed thatanyconstraint, ifpresent, isholonomic. Thisrestriction does notgreatly limittheapplicability ofthetheory, despite thefactthatmany ofthe constraints encountered ineveryday lifearenonholonomic. Thereason isthatthe entire concept ofconstraints imposed inthesystem through themedium ofwires orsurfaces orwalls isparticularly appropriate onlyinmacroscopic orlarge-scale problems. Buttoday physicists aremore interested rnatomic andnuclear prob- lems. Onthisscale allobjects, both inandoutofthesystem, consist alike of molecules, atoms, orsmaller particles, exerting deﬁnite forces, andthenotion of constraint becomes artiﬁcial andrarely appears. Constraints arethenused only asmathematical idealizations totheactual physical caseorasclassical approxi- mations toaquantum-rnechanical property, e.g.,rigid body rotations for“spin.” Such constraints arealways holonomic andﬁtsmoothly intotheframework ofthe theory. Tosurmount thesecond difﬁculty, namely, thattheforces ofconstraint are unknown apriori, weshould liketosoformulate themechanics thattheforces ofconstraint disappear. Weneed thendealonlywiththeknown applied forces. A hintastotheprocedure tobefollowed isprovided bythefactthatinaparticular system with constraints iearigid body, thework done byinternal forces (which areheretheforces ofconstraint) vanishes. Weshall follow upthisclueinthe ensuing sections andgeneralize theideascontained init. D’ALEMBERT'S PRINCIPLE AND LAGRANGI? SEQUATIONS Avirtual (inﬁnitesimali displacement ofasystem refers toachange inthecon- ﬁguration ofthesystem astheresult ofairyarbitrary inﬁnitesimal change ofthe coordinates 8r,,consistent withtheforces andconstraints imposed onthesystem atthegiven instant t.Thedisplacement iscalled virtual todistinguish itfroman actual displacement ofthesystem occurring inatime interval dt,during which 1.4 D’Alembert’s Principle andLagrange's Equations I7 theforces andconstraints maybechanging. Suppose thesystem isinequilibrium; i.e.,thetotalforceoneachparticle vanishes, F,=O.Thenclearly thedotproduct F,-8r,,which isthevirtual work oftheforce F,inthedisplacement 8r,,also vanishes. Thesumofthesevanishing products overallparticles mustlikewise be zero: Z11-8r,=0. (1.40)i Asyetnothing hasbeen saidthathasanynewphysical content. Decompose F, intotheapplied force, Ff“), andtheforce ofconstraint, f,, F,=Ff“)+r,-. (1.41) sothatEq.(1.40) becomes ZFf“).sr,+ZF, .m-,=0 (142) I I Wenowrestrict ourselves tosystems forwhich thenetvirtual work ofthe forces ofconstraint iszero.Wehaveseenthatthiscondition holds trueforrigid bodies anditisvalid foralarge number ofother constraints. Thus, ifaparticle is constrained tomove onasurface, theforce ofconstraint isperpendicular tothe surface, while thevirtual displacement must betangent toit,andhence thevirtual work vanishes. This isnolonger trueifsliding friction forces arepresent, and wemustexclude suchsystems from ourformulation. Therestriction isnotun- dulyhampering, since thefriction isessentially amacroscopic phenomenon. On theotherhand, theforces ofrolling friction donotviolate thiscondition, sincethe forces actonapoint thatismomentarily atrestandcandonoworkinaninﬁnites- imaldisplacement consistent withtherolling constraint. Note thatifaparticle is constrained toasurface thatisitself moving intime, theforce ofconstraint is instantaneously perpendicular tothesurface andthework during avirtual dis- placement isstillzeroeventhough thework during anactual displacement inthe timedtdoesnotnecessarily vanish. Wetherefore have asthecondition forequilibrium ofasystem thatthevirtual work oftheapplied forces vanishes: Zr?” -at-,=0. (1.43) Equation (1.43) isoften called theprinciple ofvirtual work. Note thatthecoef- ﬁcients of8r,cannolonger besetequal tozero; i.e.,ingeneral Ff“) 7LO,since the8r,arenotcompletely independent butarecormected bytheconstraints. In order toequate thecoefficients tozero, wemust transform theprinciple intoa form involving thevirtual displacements oftheq,,which areindependent. Equa- tion(1.43) satisﬁes ourneeds inthatitdoesnotcontain thef,,butitdeals only withstatics; wewantacondition involving thegeneral motion ofthesystem. Chapter 1Survey oftheElementary Principles Toobtain suchaprinciple, weuseadevice ﬁrstthought ofbyJames Bemoulli anddeveloped byD’Alembert. Theequation ofmotion, Ft=l.)ri canbewrittenas Fl-1.): =0» which states thattheparticles inthesystem willbeinequilibrium under aforce equal totheactual force plusa“reversed effective force” -13,-.Instead of(1.40), wecanimmediately write Err.--1'»)-er, =0. (1.44) and,making thesameresolution intoapplied torces andforces ofconstraint, there results Z~.F§“’—1'».-)-er.+Zt.-51':=0.i 1 Weagain restrict ourselves tosystems forwhich thevirtual work oftheforces of constraint vanishes andtherefore obtain Z<F,‘“’—no-tr.=0. (1.45)i which ISoften called D’Alemberl’s principle. Wehave achieved ouraim,inthat theforces ofconstraint nolonger appear. andthesuperscript (a)cannowbe dropped without ambiguity. Itisstillnotinauseful formtofumish equations ofmotion forthesystem. Wemustnowtransfomi theprinciple intoanexpression involving virtual displacements ofthegeneralized coordinates, which arethenin- dependent ofeachother(forholonomic constraints), sothatthecoefficients ofthe liq,canhesetseparately equal tozero. Thetranslation from r,toqJlanguage starts from thetransformation equations (1.38). r,=r,-(q|,qp_,...,q,,,t) (l.45') (assuming nindependent coordinates), andiscarried outbymeans oftheusual “chain rules” ofthecalculus ofpartial differentiation. Thus, v,-isexpressed in terms oftheqtbytheformula dr,- br,_8r,E—= — —_ l.6 V’ dr ;6q;,qk+ 81 (4) 1.4 L)’Alembert’s Pl‘ll1CIp1e andLagrange’s lzquations I9 Similarly, thearbitrary virtual displacement 81',canbeconnected withthevirtual displacements 8g,by drSr,=;girl), (147) Note thatnovariation oftime, 6t,isinvolved here, since avirtual displacement bydeﬁnition considers onlydisplacements ofthecoordinates. (Only thenisthe vinual displacement perpendicular totheforceofconstraint iftheconstraint itself ischanging intime.) Interms ofthegeneralized coordinates. thevirtual work oftheF,becomes 81'Zr,-an=ZF, -—‘aq, z 1,, aq-7 =ZQ_,8:1,. (1.48) 1 where theQ1arecalled thecomponents ofthegeneralized force, deﬁned as 3|Q,=2),-é. (1.49) Notethatjustastheq’sneednothavethedimensions oflength, sotheQ’sdo notnecessarily havethedimensions offorce, butQ,8qJmustalways havethe dimensions ofwork. Forexample, QJmight beatorque NJanddqjadifferential angle (19,,which makes N1d6]adifferential ofwork. Werm-nnowtotheother other term involved inEq.(1.45), which maybe written as E1‘); '81": Zmifj '8l',. 1 Expressing 8r,by(1.47), thisbecomes 6 Z1713‘; ' 1,} q] Consider nowtherelation ,,Br, d _3r, _d8r,-) _-—= — - — --- —— . 1.0 ;""" an¥141(”“" B4,)’""'dt(3qi U) Inthelastterm ofEq.(l.50)wecaninterchange thedifferentiation withrespect totandqJ,for,inanalogy to(1.46). Chapter lSurvey oftheElementary Principles d(ae)=an,=§: an,+en dt 30; Sq, kBqjdqkqk 3q)3t' _3V,' Bqj’ byEq.(1.46). Further, wealsoseefrom Eq.(1.46) that = (1-51>aqi 60; Substitution ofthese changes in(1.50) leads totheresult that .. arr d 3V, 3V, imlrl 3411'—2,:[dt (mlvl 341) mlvl 34]) , andthesecond termontheleft—hand sideofEq.(1.45) canbeexpanded into 4‘?|%l%(?%’"‘"ill"%(int)-QiltrIdentifying Z,%m,v,2 withthesystem kinetic energy T,D’Alembert’s principle (cf.Eq.(1.45)) becomes .1ar ar - 3lla(a)"@—q.l-Q1l“1=°- MNote thatinasystem ofCartesian coordinates thepartial derivative ofTwith respect toq_,vanishes. Thus, speaking inthelanguage ofdifferential geometry, thisterm arises from thecurvature ofthecoordinates qJ.Inpolar coordinates, e.g.,itisinthepartial derivative ofTwithrespect toanangle coordinate thatthe centripetal acceleration termappears. Thus far,norestriction hasbeen made onthenature oftheconstraints other thanthattheybeworkless inavirtual displacement. Thevariables qJcanbeany setofcoordinates usedtodescribe themotion ofthesystem. If,however, thecon- straints areholonomic, thenitispossible toﬁndsetsofindependent coordinates qJthatcontain theconstraint conditions implicitly inthetransformation equations (1.38). Anyvirtual displacement (Sq,isthenindependent of(iqk,andtherefore the onlywayfor(1.52) toholdisfortheindividual coefﬁcients tovanish: d(BT) 3T - T —i —Q- (1-53)drdq, liq, J There arensuchequations inall. When theforces arederivable from ascalar potential function V, F,=—V, V. 1.4D’Alembert’s Principle andLagrz-mge's Equations 21 Thenthegeneralized forces canbewritten as 8r, Br,Q : F . . : -_ V V I‘ft , J2,: I3411 i l591 which isexactly thesame expression forthepartial derivative ofafunction —V(r1,r2, ...,r~,t) withrespecttoqj: 3V=——. 1.54 Q, 8% <> Equations (1.53) canthenberewritten as dHT 'T—V dz dqj dq, Theequations ofmotion intheform (1.55) arenotnecessanly restricted toconser- vative systems, onlyifVisnotanexplicit function oftimeisthcsystem conserva- tive(cf.p.4).Asheredeﬁned, thepotential Vdoesnotdepend onthegeneralized velocities. Hence, wecaninclude aterminVinthepartial derivative withrespect tocf): d(8(T— V)) _8(T— V)=0. dz Sq] Sq, Or,deﬁning anewfunction, theLagrangian L,as L=T—V, (1.56) theEqs.(1.53) become 1- it-1=0, (1.57)drdq, Gq, expressions referred toas“Lagrange’s equations.” Notethatforaparticular setofequations ofmotion thereisnounique choice ofLagrangian such thatFaqs (1S7)lead totheequations ofmotion inthegiven generalized coordinates. Thus, inDerivations 8and10itisshown thatifL(q,4},t) isanapproximate Lagrangian andF(q,r)isanydifferentiable function ofthe generalized coordinates andtime, then . . dFL’(q,q.r)=L(q-q,I)+I (1-57') isaLagrangian alsoresulting inthesame equations ofmotion. Itisalsooften possible toﬁndalternative Lagrangians beside those constructed bythisprescrip- tion(seeExercise 20).While Eq.(1.56) isalways asuitable waytoconstruct a Lagrangian foraconservative system, itdoes notprovide theonlyLagrangian suitable forthegiven system. 1.5IChapter 1Survey oftheElementary Principles VELOCITY-DEPENDENT POTENTIALS AND THE DISSIPATION FUNCTION Lagrange’s equations canbeputintheform (1.57) even ifthere isnopotential function, V,intheusualsense, providing thegeneralized forces areobtained from afunction U(qJ,4,)bytheprescription 3U d BU - Q1="a;*a(a.-l" ‘M’Insuchcase, Eqs.(1.57) stillfollow from Eqs.(1.53) withtheLagrangian given W 1.=T-U. (1.59) Here Umaybecalled a“generalized potential,” or“velocity-dependent poten- tial.” Thepossibility ofusing sucha“potential” isnotacademic; itapplies toone veryimportant typeofforce ﬁeld, namely. theelectromagnetic forces onmoving charges. Considering itsimportance, adigression onthissubject iswellworth- while. Consider anelectric charge, q,ofmassmmoving atavelocity, v,inanother- wisecharge-free region containing bothanelectric ﬁeld.E.andamagnetic ﬁeld. B,Wl‘llCl'l maydepend upontimeandposition. Thecharge experiences aforce, called theLorentz force, given by F=q[E+(v><B)]. (1.60) BothE(t,x,y,z)andB(r,x,y,z)arecontinuous functions oftimeandpositron derivable from ascalar potential ¢(t,x,y.z)andavector potential A(t,x,y,z) by 6AE=—V -— 1.6l ¢at <=1) and B=VxA. (1.6lb) Theforce onthecharge canbederived from thefollowing velocity-dependent potential energy U=q¢—qA -v, (1.62) sotheLagrangian, L=T—U,is L=%mv2-q¢+qA-v. (1.63) 1.5Velocity-Dependent Potentials andtheDissipation Function 23 Considering justthex-component ofLagrange’s equations gives _, HA. HA), GA; (Heb dA,,)= — -—— — — — -—— . 1.64mx‘-'l”"ax +"’ax +"Zax q3x+at () Thetotaltimederivative ofA,isrelated totheparticle timederivative through dA,,8A,—— =Z -VAdz 8:+V x HA an HA an=at‘+v,,ax”+v,.-8;+vz82*. (1.65) Equation (1.6lb) gives HA.HA 8A BA <‘>‘”’*=”>'(a—§‘T’)+“Z(@—§'7§l- Combining these expressions gives theequation ofmotion inthex-direction mi?=q[Ex+(VXB)x]. (1.66) Onacomponent-by-component comparison, Eqs.(1.66) and(1.60) areidentical, showing thattheLorentz force equation isderivable from Eqs.(1.61) and(1.62). Note thatifnotalltheforces acting onthesystem arederivable from apoten- tial,thenLagrange’s equations canalways bewritten inthefonn d8L 8L _ . _i" =Qj»drBqj Hqj where Lcontains thepotential oftheconservative forces asbefore, andQJrep- resents theforces notarising from apotential. Such asituation often occurs when frictional forces arepresent. Itfrequently happens thatthefrictional force ispro- portional tothevelocity oftheparticle, sothatitsx-component hastheform Ff,\ Z—k,\/Ur. Frictional forces ofthistypemaybedenved interms ofafunction .7-',known as Rayleigh ‘sdissipation ﬁmction, anddeﬁned as 1 FZ52 (kxvgx +(C)-0,2}, +kzvizz) , I where thesummation isovertheparticles ofthesystem. From thisdeﬁnition itis clearthat SFFfx —' “Ea 1.6IChapter lSurvey oftheElementary Principles or,symbolically, Ff=—Vv.F. (1.68) Wecanalsogiveaphysical interpretation tothedissipation function. Thework done bythesystem against friction is dWf =—Ff -d1‘ =—F_f -Vdl =(kxvi +kyv; —/(21)?) dl. Hence. 2.7-"istherateofenergy dissipation duetofriction. Thecomponent ofthe generalized force resulting from theforce offriction isthen given by 3,‘ 3-Q]=Z:Fﬁ.i=_Zv,$.a_; =_ VJ-‘ ,Z " 3%by(1.51), 8.7: qr Anexample isStokes’ law,whereby asphere ofradius amoving ataspeed v,inamedium ofviscosity 17experiences thefrictional dragforce Ff=6::nav. TheLagrange equations withdissipation become dBL BL BF — ——++=0, (1-70)dtElq] Sq, Sq] sothattwoscalar functions, Land.7-',must bespeciﬁed toobtain theequations ofmotion. SIMPLE APPLICATIONS OFTHE LAGRANGIAN FORMUIATION Theprevious sections show thatforsystems where wecandeﬁne aLagrangian, i.e.,holonomic systems with applied forces derivable from anordinary orgen- eralized potential andworkless constraints, wehave averyconvenient wayof setting uptheequations ofmotion. Wewere ledtotheLagrangian formulation bythedesire toeliminate theforces ofconstraint fromtheequations ofmotion, andinachieving thisgoalwehaveobtained many otherbeneﬁts. Insetting upthe original form oftheequations ofmotion, Eqs.(1.19), itisnecessary towork with many vector forces andaccelerations. With theLagrangian method weonlydeal withtwoscalar functions, TandV,which greatly simpliﬁes theproblem. Astraightforward routine procedure cannowbeestablished forallproblems ofmechamcs towhich theLagrangian formulation isapplicable. Wehave onlyto write TandVingeneralized coordinates, form Lfrom them, andsubstitute in (1.57) toobtain theequations ofmotion. Theneeded transformation ofTandV fromCartesian coordinates togeneralized coordinates isobtained byapplying the 1.6 Simple Applications oftheLagrangian Formulation 25 transformation equations (1.38) and(l.45'). Thus, Tisgiven ingeneral by Z 12 1 Hr_8r T=Z5"""1 =25” (Za—<1i‘-”"T') ' 1 I ] Itisclear thatoncarrying outtheexpansion, theexpression forTingeneralized coordinates willhave theform .1 ..T=M0+ZM,q,+5ZM,-,,q,q,,, (1.71) J 1J< where M0,MJ,Mjkaredeﬁnite functions ofther’sandtandhence oftheq’s andt.Infact,acompaiison shows that 1 Br2M0 =Z Em: 1 I 8r, 3r,-M-= m—--—, (1.72)JZ’: iBr Eiq] and 81'; 81',-Mk= m——- . I E; !3q1 aqk Thus, thekinetic energy ofasystem canalways bewritten asthesumofthree homogeneous functions ofthegeneralized velocities, T=7i1+ T1+Tz, (1-73) where Toisindependent ofthegeneralized velocities, T1islinear inthevelocities, andT;isquadratic inthevelocities. Ifthetransformation equations donotcontain thetimeexplicitly, asmayoccur when theconstraints areindependent oftime (scleronomous), thenonlythelastterminEq.(1.71) isnonvanishing, andTis always ahomogeneous quadratic form inthegeneralized velocities. Letusnowconsider simple examples ofthisprocedure: l.Single particle inspace (a)Cartesian coordinates (b)Plane polar coordinates 2.Atwo0d’s machine 3.Time-dependent c0nstraint—bead sliding onrotating wire 1.(a)Motion ofoneparticle: using Cartesian coordinates. Thegeneralized forces needed inEq.(1.53) areobviously Fx,Fy,andFz.Then Chapter 1Survey oftheElementary Princ pies r=im(»e2+>'»’+z2). Q_8T_HT_0 Bx—8)»—32—i HT _ 3T , 8T _ $2,”-xs 5=m)’, fmza andtheequations ofmotion are d . 1, d . E(mx)=Fx.§<my>=Fy!Etna=F1. (1.14) Wearethusledback totheoriginal Newton’s equations ofmotion. (b)Motion ofoneparticle: using plane polar coordinates. Here wemust ex- press Tinterms offand0.Theequations oftransformation, i.e.,Eqs.(1.38), in thiscase aresimply x=rcos9 y=rsin0. Byanalogy to(1.46), thevelocities aregiven by :2=1‘cost? -résin6. )3=rsin6+récosél. Thekinetic energy T=%m(222 +3'12)thenreduces formally to T=%m[*2+(ré)’]. (1.15) Analtemative derivation ofEq.(1.75) isobtained byrecognizing thattheplane polar components ofthevelocity areralong r,andrélalong thedirection per- pendicular tor,denoted bytheunitvector n.Hence, thesquare ofthevelocity expressed inpolar coordinates issimply I’:+(r6')2. With theaidoftheexpression dr=f'dr+rode+iidz forthedifferential position vector, dr,incylindrical coordinates, where i’and 0areunitvectors intherand0-directions, respectively, thecomponents ofthe generalized force canbeobtained from thedeﬁnition, Eq.(1.49), a A QrZFI;:=F0rZFr, 3 ,. Q6iF0iZFOr0irFa‘ 1.6 Simple Applications oftheLagrangian Formulation 27 rA6n r(6+ A9) 0r(6) FIGURE 1.6 Derivative ofrwithrespect to9. since thederivative ofrwithrespect to6is,bythedeﬁnition ofaderivative, a vector inthedirection of6(cf.Fig.1.6).There aretwogeneralized coordinates, andtherefore lwuLagrange equations. Thederivatives occurring intherequation are 8T ,6-2 8T _ d8T ..—=', —_-=mr. ——- =mr,3r m 3r dt 81‘ andtheequation itself is mi’—mr(:)2 =F,, thesecond termbeing thecenuipetal acceleration term. Forthe6equation, we have thederivatives 1-v 1 dI d1 . d . .. .E=0, =mr2(-J, E(mr29) =mr29 +2mrr6, sothattheequation becomes 4 2. 2.. _.E(mr 9)=mr9+2mrr9 =rF9. Note thattheleftsideoftheequation isjustthetime derivative oftheangular momentum, andtherightsideisexactly theapplied torque, sothatwehavesimply rederived thetorque equation (1.26), where L=mrzé andN(0=rFg. 2.Atwood’s machine—(See Fig.1.7)anexample ofaconservative system withholonomic. scleronomous constraint (thepulley isassumed frictionless and massless). Clearly there isonlyoneindependent coordinate x,theposition of theother weight being determined bytheconstraint thatthelength oftherope between them isl.Thepotential energy is V=—M1gX —M280 —X), Chapter 1Survey oftheElementary Principles _ - 3 - x I-x l _!_FIGURE 1.7 Atwood’s machine. while thekinetic energy is T=%(M;+M3)i2. Combining thetwo,theLagrangian hastheform L=T-v=g(M1+M2)s1+ M1gx+Mgg(l—x). There isonlyoneequation ofmotion, involving thederivatives 8L X dL , F."=(M1-l"M2)X>X sothatwehave IM1+M2)55 =(M1 —M2)8. OI’ ..M1— M2x=———g,M1+M2 which isthefamiliar result obtained bymore elementary means. Thistrivial prob- lem emphasizes that theforces ofconstraint—here thetension intherope- appear nowhere intheLagrangian formulation. Bythesame token, neither can thetension intheropebefound directly bytheLagrangian method. 3.Abead (orring) sliding onauniformly rotating wireinaforce-free space. Thewireisstraight, andisrotated uniformly about some ﬁxedaxisperpendicular totheWire.Thisexample hasbeenchosen asasimple illustration ofaconstraint Derivations 29 being timedependent, withtherotation axisalong zandthewireinthexyplane. Thetransformation equations explicitly contain thetime. x=rcoswt. (co=angular velocity ofrotation) y=rsinwt. (r=distance along wirefrom rotation axis) While wecould thenﬁndT(here thesame asL)bythesame procedure used to obtain (1.71), it_issimpler totakeover (1.75) directly, expressing theconstraint bytherelation 9=cu: T=%m(22+rzwz) . NotethatTisnotahomogeneous quadratic function ofthegeneralized velocities, since thereisnowanadditional termnotinvolving r.Theequation ofmotion is then .. ') mr=mrw' =0 or .. 2 r=rw, which isthefamiliar simple hamionic oscillator equation withachange ofsign. Thesolution r=e""shows thatthebead moves exponentially outward because ofthecentripetal acceleration. Again, themethod cannot fumish theforce ofcon- straint thatkeeps thebeadonthewire. Equation (1.26) withtheangular momen- tum,L=mr-2w2e“" .provides theforce F=N/r, which produces theconstraint force, F=mrw2e“" ,acting perpendicular tothewireandtheaxisofrotation. DERIVATIONS 1.Show thatforasingle particle withconstant mass theequation ofmotion implies the following differential equation forthekinetic energy: dT_=F._dz V while ifthemass varies withtimethecorresponding equation is d(mT)i =F._dt P 2.Prove thatthemagnitude Roftheposition vector forthecenter ofmass from an arbitrary origin isgiven bytheequation 1MZRZ =Mzmlrlz —5Zm,m]r5. 1 lI Chapter 1Survey oftheElementary Principles 3. 4 5. 6. 7 8.Suppose asystem oftwoparticles isknown toobeytheequations ofmotion, Eqs. (1.22) and(1.26). From theequations ofthemotion oftheindividual particles show thattheintemal forces between particles satisfy boththeweak andthestrong laws ofaction andreaction Theargument maybegeneralized toasystem witharbitrary number ofparticles, thusproving theconverse ofthearguments leading toEqs.(1.22) and(I.26). Theequations ofconstraint fortherolling disk, Eqs.(1.39), arespecial cases ofgen- erallinear differential equations ofconstraint oftheform n Z3, (x1,...,x,,)dx, =O. i=1 Aconstraint condition ofthistype isholonomic only ifanintegrating function f(xi,...,x,,)canbefound thattums it‘moanexact differential. Clearly thefunc- tionmust besuchthat MmJ=6U&)8x] 6x, foralli¢j.Show thatnosuchintegrating ﬁactor canbefound foreither ofEqs. (1.39). Twowheels ofradius aaremounted ontheends ofacommon axleoflength bsuch thatthewheels rotate independently. Thewhole combination rollswithout slipping on aplane. Show thattherearetwononholonomic equations ofconstraint, cosQdx+sinéldy =0 sin9dx —cos9dv =%a(d¢+d¢,), (where 6,¢,and¢’havemeanings similar tothose intheproblem ofasingle vertical disk, and(x,y)arethecoordinates ofapoint ontheaxlemidway between thetwo wheels) andoneholonomic equation ofconstraint, e=c—§w-at where Cisaconstant. Aparticle moves inthexyplane under theconstraint thatitsvelocity vector isal- ways directed towards apoint onthexaxiswhose abscissa issome given function of timef(t).Show thatforf(t)differentiable, butotherwise arb.trary, theconstraint is nonholonomic. Show thatLagrange’s equations intheformofEqs.(1.53) canalsobewritten as er ar—r—2——=Q-3(1)" 31]] J These aresometimes known astheNielsen fonn oftheLagrange equations. IfLisaLagrangian forasystem ofndegrees offreedom satisfying Lagra_uge’s equa- tions, show bydirect substitution that Exercises 31 L,=L+ dF(q|,...,q,,,t) dt alsosatisﬁes Lagrange’s equations where Fisanyarbitrary, butdifferentiable, func- tionofitsarguments. 9.Theelectromagnetic ﬁeldisinvariant under agauge transformation ofthescalar and vector potential given by A—>A+Vi,lr(r, t), A81/1 ¢r¢"257' where 1/1isarbitrary (butdifferentiable). What effect does thisgauge transformation haveontheLagrangian ofaparticle moving intheelectromagnetic ﬁeld? Isthemotion affected‘? 10.Letqi,...,q,,beasetofindependent generalized coordinates forasystem ofn degrees offreedom, withaLagrangian L(q,4},t).Suppose wetransform toanother setofindependent coordinates s1,...,s,,bymeans oftransformation equations q,=q,(s],...,s,,,i), z=1,...,n. (Such atransformation iscalled ap0mt transformation.) Show thatiftheLagrangian function isexpressed asafunction ofsJ,ti’.,andtthrough theequations oftransf0i- mation. thenLsatisfies Lagrange’s equatio1s withrespect tothescoordinates: d(BL 8L__0 atas, as,‘' Inother words, theform ofLagrange’s equations isinvariant under apoint transfor- mation. EXERCISES 11.Consider auniform thindiskthatrollswithout slipping onahorizontal plane. Ahori- zontal force isapplied totheoenter ofthediskandinadirection parallel totheplane ofthedisk. (a)Derive Lagrange’s equations andﬁndthegeneralized force. (b)Discuss themotion iftheforce isnotapplied parallel totheplane ofthedisk. 12.Theescape velocity ofaparticle onEarth istheminimum velocity required atEarth’s surface inorder thattheparticle canescape fromEarth’s gravitational ﬁeld. Neglecting theresistanoe oftheatmosphere, thesystem isconservative. From theconservation theorem 1'01potential pluskinetic energy show thattheescape velocity forEarth, ignoring thepresence oftheMoon, is11.2km/s. 13.Rockets arepropelled bythemomentum reaction oftheexhaust gases expelled from thetail.Since these gases anse from thereaction ofthefuels carried intherocket, the mass oftherocket isnotconstant, butdecreases asthefuelisexpended. Show thatthe equation ofmotion forarocket projected vertically upward inauniform gravitational Chapter lSurvey oftheElementary Principles ﬁeld, neglecting atmospheric friction, is mdv ,dmi=_vi_m ‘ at at g where misthemass oftherocket andv’isthevelocity oftheescaping gases relative to therocket. Integrate thisequation toobtain vasafunction ofm,assuming acons‘ant timerateoflossofmass. Show, forarocltet starting initially from rest,withv’equal to2.1m/sandamasslosspeisecond equal to1/60th oftheinitial mass, thatinorder toreach theescape velocity theratio oftheweight ofthefueltotheweight ofthe empty rocket must bealmost 300! Twopoints ofmass mare_|0111Bd byarigid weightless rodoflength l,thecenter of which isconstrained tomove onacircle ofradius a.Express thekinetic energy in generahzed coordinates. Apointparticle moves inspace under theinﬂuence ofaforcederivable fromagener- alized potential ofthefonn U(i,v) =V(r)+u'-L. where ristheradius vector from aﬁxed point, Listheangular momentum about that point, and0isaﬁxedvector inspace. (atFindthecomponents oftheforce ontheparticle inbothCartesian andspherical polar coordinates, onthebasis ofEq.I1.58). (byShow thatthecomponents inthetwocoordinate systems arerelated toeachother asinEq.(1.49). (clObtain theequations ofmotion insphencal polar coordinates. Aparticle moves inaplane under theinﬂuence ofaforoe, acting towaiid acenter of force, whose magiiituce is 1 "2_2FrF=7<1- ,,. C- where risthedistance oftheparticle tothecenter offorce. Find thegeneralized potential thatwillresult insuch aforce, andfrom thattheLagrangian forthemotion inaplane. (Theexpression forFrepresents theforce between Lw0charges inWeber’s electrodynarnics.) Anucleus. originally atrest,decays radioactively byemitting anelectron ofinomen~ tum1.73MeV/c, andatright angles tothedirection oftheelectron aneutrino with momentum 1.00MeV/c. (The MeV, million electron volt, isaunitofenergy used inmodern physics, equal to1.60><l0_]3 J.Correspondingly, MeVlr' isauiutof linear momentum equal to5.34 ><I042 kg-m/s.l Inwhat direction does thenu- eleus recoil? What is.tsmomentum inMeV/c? Ifthemass oftheresidual nucleus is3.90Xl0'25 kgwhat isitskinetic energy. inelectron volts? ALagrangian foraarticular physical sstemcanbewritten as P Y . ... KL’=2(axz +Zbxy +cy2:l —E(axz +Zbxy +cyz) , where a,b,andcarearbitr constants butsubecttothecondition thatb2—ac 0. My J Exercises 33 What aretheequations ofmotion? Examine particularly thetwocases a=O=c andb=O,c=-a.What isthephysical system described bytheabove Lagrangian? Show thattheusual Lagrangian forthissystem asdeﬁned byEq.(1.57’) isrelated toL’byapoint transfonnation (cf.Derivation IO).What isthesigniﬁcance ofthe condition onthevalue ofb2—ac? Obtain theLagrange equations ofmotion toraspherical pendulum, i.e.,amass point suspended byarigid weightless rod. Aparticle ofmass mmoves inonedimension suchthatithastheLagrangian 2~41.='"l;‘_+mr2V(x) -i/2(1), where Vissome differentiable function ofx.Findtheequation ofmotion forx(t)and describe thephysical nature ofthesystem onthebasis ofthisequation Twomass points ofmass m1andmgareconnected byastring passing through a holeinasmooth table sothatm1rests onthetable surface andm2hangs suspended. Assuming mgmoves onlyinavertical line.what arethegeneralized coordinates for thesystem? Write theLagrange equations forthesystem and,ifpossible, discuss thephysical signiﬁcance anyofthem might have. Reduce theproblem toasingle second-order differential equation andobtain aﬁrstintegral oftheequation. What is itsphysical signiﬁcance? (Consider themotion onlyuntilmlreaches thehole.) Obtain theLagrangian andequations ofmotion forthedouble pendulum illustratec in Fig1.4,where thelengths ofthependula areI1andlgwithcorresponding masses mi andmg. Obtain theequation ofmotion foraparticle falling vertically under theinﬂuence of gravity when frictional forces obtainable from adissipation function ékvz arepresent. Integrate theequation toobtain thevelocity asafunction oftimeandshow thatthe maximum possible velocity forafallfrom restisv=mg/k. Aspring ofrestlength La(notension) isconnected toasupport atoneendandhas amass Mattached attheother. Neglect themass ofthespring, thedimension ofthe mass M,andassume thatthemotion isconﬁned toavertical plane. Also, assume that thespring onlystretches without bendnig butitcanswing intheplane. (a)Using theangular displacement ofthemass from thevertical andthelength that thestring hasstretched from itsrestlength (hanging withthemass m),ﬁndLa- g-range’s equations. (blSolve these equations forsmall stretching andangular displacements. (clSolve theequations inpart(a)tothenextorder inbothstretching andangular displacement. Thispartisamenable tohandcalculations. Using some reasonable assumptions about thespring constaiii, themass, andtherestlength, discuss the motion. Isaresonance likely under theassumptions stated intheproblem? (d)(For analytic computer programs.) Consider thespring tohave atotal mass m<<M.Neglecting thebending ofthespring, setupLagrange’s equations correctly toﬁrstorder inmandtheangular andlinear displacements. (e)(Fornumerical computer analysis.) Make setsofreasonable assumptions ofthe constants inpart(a)andmake asingle plotofthetwocoordinates asfunctions of time. CHAPTER 2.1I 34Variational Principles and Lagrange’s Equations HAMll.TON'S PRINCIPLE Thederivation ofLagrange’s equations presented inChapter lstarted from a consideration oftheinstantaneous stateofthesystem andsmall virtual displace- ments about theinstantaneous state, i.e.,from a“differential principle” such as D’Alembert’s principle. Itisalsopossible toobtain i.ag1"ange’s equations froma principle thatconsiders theentire motion ofthesystem between times 21andZ2, andsmall virtual variations ofthismotion fromtheactual motion. Aprinciple of thisnature isknown asan“integral principle." Before presenting theintegral principle, themeaning attached tothephrase “motion ofthesystem between times :1and:2”mustﬁrstbestated lI‘lmore pre- ciselanguage Theinstantaneotis conﬁguration ofasystem isdescribed bythe values ofthengeneralized coordinates q1,...,q,,,andconesponds toaparticu- larp0lntinaCartesian hyperspace where theq’sformthencoordinate axes.This n-dimensional space istherefore known asconﬁguration space. Astimegoeson, thestateofthesystem changes andthesystem point moves inconﬁguration space tracing outacurve, described as“thepathofmotion ofthesystem.” The“motion ofthesystem,” asused above, then refers tothemotion ofthesystem point along thispathinconﬁguration space. Time canbeconsidered formally asaparame- terofthecuwe; toeachpoint onthepaththere isassociated oneormore values ofthetime. Note thatconﬁguration space hasnonecessary connection withthe physical three-dimensional space, justasthegeneralized coordinates arenotnec- essarily position coordinates. Thepathofmotion inconﬁguration space hasno resemblance tothepath inspace ofanyactual particle; each point onthepath represents theentire system conﬁguration atsome given instant oftime. Theintegral Hamilton ’sprinciple describes themotion ofthose mechanical systems forwhich allforces (except theforces ofconstraint) arederivable from a generalized scalar potential thatmaybeafunction ofthecoordinates, velocities, andtime. Suchsystems willbedenoted asmonogenic. Where thepotential isan explicit function ofposition coordinates only, thenatmonogenic system isalso conservative (cf.Section 1.2). Formonogenic systems, Hamilton’s principle canbestated as Themotion ofthesystem from time:1totimetgissuchthattheline integral (called theaction ortheaction integral ), 2.1 Hamilton's Principle 35 I2 I=/l Ldt, (2.1) It where L=T-—V,hasastationary value fortheactual path ofthe motion. That is,outofallpossible paths byWl‘llCl't thesystem point could travel from itsposition attime trtoitsposition attime12,itwillactually travel along that pathforwhich thevalue oftheintegral (2.1) isstationary. Bytheterm “station- aryvalue” foralineintegral, wemean thattheintegral along thegiven pathhas thesame value towithin ﬁrst-ordcr inﬁnitcsimals asthatalong allneighboring paths (l.B.,those thatdiffer from itbyinﬁnitesimal displacements). (Cf.Fig.2.1.) Thenotion ofastationary value foralineintegral thuscorresponds inordinary function theory totheVanishing oftheﬁrstderivative. Wecansummarize Hamilton’s principle bysaying thatthemotion issuchthat thevariation ofthelineintegral Iforﬁxed 21andt2iszero: I2 5l=5f L(q1,...,q,,,<j1,...,¢_),,,z)dz=0. (2.2) H Where thesystem constraints areholonomic, Ham.ilton’s principle, Eq.(2.2), isbothanecessary andsufficient condition forLagrange’s equations, Eqs.(1.57). Thus, itcanheshown thatHami1ton’s principle follows directly fromLagrange’s equations. Instead, however, weshallprove theconverse, namely, thatLagrange’s equations follow froml-lamilton’s principle, asbeing themoreimportant theorem. That Hamilton’s principle isasufﬁcient condition forderiving theequations of motion enables ustoconstruct themechanics ofmonogenic systems fromHamil- ton’s principle asthebasic postulate rather thanNewton’s lawsofmotion. Such aformulation hasadvantages; eg,since theintegral Iisobviously invariant to thesystem ofgeneralized coordinates usedtoexpress L,theequations ofmotion mustalways havetheLagrangian fonnnomatter howthegeneralized coordinates J’l l it X FIGURE 2.1 Pathofthesystem point inconﬁguration space. 2.2IChapter 2Variational Principles andLagrange’s Equations aretransformed. More important, theformulation interms ofavariational prin- ciple IStheroute thatisgenerally followed when wetrytodescribe apparently nonmechanical systems inthemathematical clothes ofclassical mechanics, asin thetheory ofﬁelds. SOME TECHNIQUES OFTHE CALCULUS OFVARIATIONS Before demonstrating thatLagrange’s equations dofollow from (2.2), wemust ﬁrstettamine themethods ofthecalculus ofvariations, forachief problem ofthis calculus istoﬁndthecurve forwhich some given lineintegral hasastationary value. Consider ﬁrsttheproblem inanessentially one-dimensional form: Wehave a function f(y.)3,x)deﬁned onapathy=y(x) between twovalues x1andxg, where )3isthederivative ofywithrespect tox.Wewishtoﬁndaparticular path y(x)suchthatthelineintegral Jofthefunction fbetween xlandX2, ._dy>-dx, J=/x2f(y,j»,x)dx, (2.3) hasastationary value relative topaths differing inﬁnitesimally from thecorrect function y(x). Thevariable xhereplays theroleoftheparameter r,andwecon- sideronlysuchvaried paths forwhich y(x1) =y1,y(xg) =yg.(Cf.Fig.2.2.) NotethatFig.2.2doesnotrepresent conﬁguration space. Intheone-dimensional conﬁguration space, both thecorrect andvaried paths arethesegment ofthe straight lineconnecting y1andyg;thepaths differ only inthefunctional rela- tionbetween yandx.Theproblem isone-dimensional. visafunction ofxnota coordinate. y (X2-J72) 131,71) k FIGURE 2.2Varied paths ofthefunction ofy(x)intheone-dimensional extremum problem. 2.2 Some Techniques cftheCalculus ofVariations 37 Weputtheproblem inafonnthatenables ustousethefamiliar apparatus of thedifferential calculus forﬁnding thestationary points ofafunction. Since J musthaveastationary value forthecorrect pathrelative toanyneighboring path, thevariation must bezerorelative tosome particular setofneighboring paths labeled byaninﬁnitesimal parameter oz.Suchasetofpaths might bedenoted by y(x,oz),withy(x,0)representing thecorrect path. Forexample, ifweselect any function 27(x) thatvanishes atx=x1andx=xg,thenapossible setofvaried paths isgiven by >'(x.¢>1) =;v(X.0) +vm(x)- (2-4) Forsimplicity, itisassumed thatboththecorrect pathy(x)andtheauxiliary function 17(x) arewell-behaved functions—continuous andnonsingular between x1andI2,withcontinuous ﬁrstandsecond derivatives inthesame interval. For anysuchparametric family ofcurves, JinEq.(2.3)isalsoafunction ofoz: J(oz) =/x2f(y(x,a), y(x,oz),x) dx. (2.5) 1| andthecondition forobtaining astationary point isthefamiliar onethat <11 A Bytheusual methods ofdifferentiating under theintegral sign,weﬁndthat dJ f"2(BfBy BfBy) —= ——+—.-—- 41- (“-7)do: X, ByBa ByBa x 1' Consider thesecond ufthese integrals. x ~ x 2 ‘/2ﬂc;a—ydx=‘[2§lf--ii-ldx.XIByBo: xiByBxBa Integrating byparts, theintegral becomes x3 Z, X2 x I91,-a—’ax=a_f3l _f2-i(af_)aldx. (2.3)xiByBxBa ByBaxl ,1dx By Ba Theconditions onallthevaried curves arethatthey pass through thepoints (x1,yl),(x2,yg),andhence thepartial derivative ofywithrespect toozatx1and xgmust vanish. Therefore. theﬁrstlZ6ITl'l of(2.8) vanishes andEq.(2.7)reduces to QI/"’ §£_iK)"ldxdo: X] By dxBy Bo: ' Thecondition forastationary value, Eq.(2.6), istherefore equivalent totheequa- tion Chapter 2Variational Principles andl_agrange's Equations *2B dB B f(~‘~-—<>eiM-or X, By dxBy Ba 0 Now, thepartial derivative ofywithrespect tooroccurring inEq.(2.9) isa f|.lI1ClIlOl'l ofxthatisarbitrary except forcontinuity andendpoint conditions. For example, fortheparticular parametric family ofvaried paths given byEq.(2.4), itisthearbitrary function i7(x). Wecantherefore apply toEq(2.9) theso-called “fundamental lemma” ofthecalculus ofvariations, which saysif fxzM(x)i7(.r) dx=O (2.10) Xi forallarbitrary functions r7(x)continuous through thesecond derivative, then M(x)must identically vanish intheinterval (xi,J62).While aformal mathemat- icalproof ofthelemma canbefound intextsonthecalculus ofvariations, the validity ofthelemma iseasily seenintuitively. Wecanimagine constructing a function 17thatispositive intheimmediate vicinity otanychosen point inthe interval andzeroeverywhere else. Equation (2.10) canthenhold only ifM(x) vanishes atthat(arbitrarily) chosen point which shows Mmustbezerothrough- outtheinterval. From Eq.(2.9)andthefundamental lemma, ittherefore follows thatJcanhave astationary value onlyif ‘if“Q_ a-dx (a)_,)_0. (211) Thedifferential quantity, dc!EBy, (212) ‘Y0 represents theinﬁnitesimal departure ofthevaried pathfrom thecorrect path3(x) atthepoint xandthuscorresponds tothevirtual displacement introduced inChap- ter1(hence thenotation 6y).Similarly, theinﬁnitesimal variation ofJabout the correct pathcanbedesignated do:E5]. (2.13) da 0 Theassertion thatJisstationary forthecorrect pathcanthusbewritten 8]:-/.x2(g—ia—')f‘)8ydx—O. xl By dxBy requiring thaty(x) satisfy thedifferential equation (2.11). The8-notation, intro- duced through Eqs. (2.12) and(2.13), maybeused asaconvenient shorthand fortreating thevariation ofintegrals, remembering always thatitstands forthe manipulation ofparametric families ofvaried paths suchasEq.(2.4). 2.2 Some Techniques oftheCalculus ofVariations 39 Some simple examples oftheapplication ofEq.(2.11) (which clearly resembles aLagrange equation) maynowbeconsidered: 1.Shortest distance between twopoints inaplane. Anelement oflength ina plane is ds=,ldx2 +dyz andthetotallength ofanycurve going between points 1and2is 1 2 X2 d l=fds=/ x/l+(l) dx.1 xl dx Thecondition thatthecurve betheshortest pathisthatIbeaminimum. Thisis anexample oftheextremum problem asexpressed byEq.(2.3), with f=,lI+)'12. Substituting in(2.11) with Bf Bf 3"__=0s Wis By By./1+,\'>2 wehave d y _0 dx ‘/1 or 5'i =C, ./1+>>1 where cisconstant. Thissolution canbevalid onlyif 5/=H» where aisaconstant related to0by L “T/Q" Butthisisclearly theequation ofastraight line, y=ax+b, Chapter 2Variational Principles andLagrange's Equations where bisanother constant ofintegration. Strictly speaking, thestraight linehas onlybeenproved tobeanextremum path,butforthisproblem itisobviously also aminimum. Theconstants ofintegration, aandb,aredetermined bythecondition thatthecurve passthrough thetwoendpoints. (xi.yr),(£2.J/2). Inasimilar fashion wecanobtain theshortest distance between twopoints onasphere, bysetting upthearclength onthesurface ofthesphere interms of theangle coordinates ofposition onthesphere Ingeneral, curves thatgivethe shortest distance between twopoints onagiven surface arecalled thegeodesics ofthesurface. 2.Minimum surface ofrevolution. Suppose weform asurface ofrevolution bytaking some curve passing between twoﬁxedendpoints (x1,y1)and(X2,yg) deﬁning thexyplane, andrevolving itabout theyaxis(cf.Fig.2.3a). Theproblem thenistoﬁndthatcurve forwhich thesurface areaisa Theareaofa stripofthesurface is2:rxds=2:rrx\/1 +gadx,andthetotalareais 2 211'] ac,/1+j'2dx. 1 Theextremurn ofthisintegral isagain given by(2.11) where f=x-,/1-l—jP2 §£=0 ﬁ=;>"By’Bi,/1+5i1 Equation (2.11) becomes inthiscaseand J’ ... t, it, ,".~ X1-J?|) —-—i— —x Z FIGURE 2.3a Minimum surface ofrevolution. Note thatthisﬁgure isdrawn fory1and y;having thesame signrelative totherotation axis.Thisisnotassumed inthegeneral solution. 2.2 Some Techniques oftheCalculus ofVariations 41 ﬁe>1»dxt/WO1‘ _£Y_=,,,,/1+5>2 where aissome constant ofintegration clearly smaller thantheminimum value ofx.Squaring theabove equation andfactoring terms, wehave )'I2(x2 —a2)=02, orsolving, dy a F5Z7 Thegeneral solution ofthisdifferential equation, inlightofthenature ofa,is y=a/ +b=aarccosh§+b OI‘ bx=acoshy——,a which istheequation ofacatenaty. Again thetwoconstants ofintegration, aand b,aredetermined inprinciple bytherequirements thatthecurve passthrough the twogiven endpoints, asshown inFig.2.3b. Curves satisfying thepreceding equation allscale asx/aandy/awith one independent parameter b/a.Thissuggests thatwhen thesolutions areexamined indetail theyturnouttobeagreat dealmore complicated thanthese considera- Y ‘X2.2);) b- (1105) la X FIGURE 2.3b General catenary solution forminimum surface ofrevolution. Chapter 2Vanahonal Principles andLagrange’s Equations tions suggest. Forsome pairs ofendpoints, unique constants ofintegration aand bcanbefound. Butforotherendpoints, itispossible todrawtwodifferent cate- narycurves through theendpoints, while foradditional cases nopossible values canbefound foraandb.Further, recall thatEq.(2.1I)represents acondition forﬁnding curves y(x) continuous through thesecond derivative thatrender the integral stationary. Thecatenary solutions therefore donotalways represent min- imum values, butmayrepresent “inﬂection points” where thelength ofthecurve isstationary butnotminimum. Forcertain combinations ofendpoints (anexample isx1andX2both posi- tiveandbothmuch smaller thanyg—y|),theabsolute minimum inthesurface ofrevolution isprovided (cf.Exercise 8)byacurve composed ofstraight line segments—-from theﬁrstendpoint parallel tothexaxisuntiltheyaxisisreached, thenalong theyaxisuntilthepoint (0,yz)andthenoutinastraight linetothe second endpoint corresponding tothearea:r(xf +xg).Thiscurve results when a=O,forcing either x=0ory=constant. Since thiscurve hasdiscontinuous ﬁrstderivatives. weshould notexpect toﬁnditasasolution toEq.(2.11). This example isvaluable inemphasizing therestrictions thatsurround the derivation andthemeaning ofthestationary condition. Exercises '7and8exam- inetheconditions forthepathological behavior forasymmetric example. More information canbefound inmany textsonthecalculus ofvariations. 3.Thebrachistochrone problem. (SeeFig.2.4a.) Thiswell-known problem is to[indthecuwe joining twopoints, along which aparticle falling from restunder theinfluence ofgravity travels fromthehigher tothelower pointintheleasttime. Ifvisthespeed along thecurve. thenthetimerequired tofallanarelength ds isds/v, andtheproblem istoﬁndaminimum oftheintegral Zdsf]2_=j l x:r- ”\ l .FIGURE 2.4a Thebraehlstochrone problem. 2.2 Some Techniques oftheCalculus ofVariations 43 Ifyismeasured down fromtheinitial pointofrelease, theconservation theorem fortheenergy oftheparticle canbewritten as %mv2 =mgy or v=./1,7. Then theexpression forti;becomes 1~/22>’ .1+>>2-"=\/T-8)’ Theintegration ofEq.(2.11) withthisformforfisstraightforward andisleftas anexercise. Thesolution interms ofitsoneparameter, a,given by . /Ti=|_¢0$[ ]’L1 Clandfisidentiﬁed as issketched inFig.2.4bfortheﬁrstcycle (05x52rra) andthebeginning ofthe second cycle. Three cases ofsolutions areindicated. Apower-series expansion of thesolution forthelimit y<<agives )- 21. Thebrachistochrone problem isfamous inthehistory ofmathematics, forit wastheanalysis ofthisproblem byJohn Bemoulli thatledtotheformal founda- tionofthecalculus ofvariations. 11,311 VT7aV__ 2?!!! I 0 x2<<3'2 xz>>yz 20 I *2=EY2 3a )' FIGURE 2.4b Catenary solution tothebrachtstochrone problem showing positions on thecurve forthethree cases X2<<Y2,x2=%)’2. andJ62>>yg 203 -Lhapter 2Variational Principles andLagiange's Equations DERIVATION OFl.AGRANGE'S EQUATIONS FROM HAMll.TON'S PRINCIPLE Thefundamental problem ofthecalculus ofvariations iseasily generalized tothe casewhere fisafunction ofmany independent variables y,-,andtheirderivatives y,-.(Ofcourse, allthesequantities areconsidered asfunctions oftheparametric variable x.)Then avariation oftheintegral J, 51=‘iff(yi(x); yz(x), ---.ii(X); i'>2(x). ---ix)dx. (2-14)l isobtained, asbefore, byconsidering .1asafunction ofparameter atthatlabels a possible setofcurves yi(x,oz).Thus, wemayintroduce orbysetting )’l(X,01)=y1(1.0) +vHi1(X). )’2(X. 11)=)‘2(I, 0)+¢¥Ti2(X). (115) 0 - 0 . . . where yi(x,0),y2(x, 0),etc.,arethesolutions oftheextremum problem (tobe obtained) and271,172,etc.,areindependent functions ofxthatvanish attheend points andthatarecontinuous through thesecond derivative, butotherwise are completely arbitrary. Thecalculation proceeds asbefore. Thevariation ofJisgiven interms of a1 2afav, afan)-= -ea ——d a. 2.at/1°‘ x¥(ay, 801°‘+ay,80:°‘x (16) Again weintegrate byparts theintegral involved inthesecond sumofEq.(2.16): f’§1:2’a,,,=an’_f2ni('21:) d,13)‘;30!ax 65>,30!1 130¢(Ix ,3)‘; , where theﬁrsttermvanishes because allcurves passthrough theﬁxedendpoints. Substituting in(2.16), 5.1becomes 2 afaaf8.!= ———-—— »[l2,:(6)5 dx855 where, inanalogy with(2.12), thevariation 8y;is 19>’8y;= da. Since theyvariables areindependent, thevariations 8y;areindependent (e.g., thefunctions 27,(x)willbeindependent ofeach other). Hence, byanobvious extension ofthefundainental lemma (cf.Eq.(2.10)), thecondition that8]iszero)5” dx, (2.17) 2.4I1.4 |:XtE‘nSlOl1 ofHamilton's Principle toNonholonomic systems 45 requires thatthecoefﬁcients ofthe8)“;separately vanish: ___if6y, dx8)},Bf d=0, i=l,2,...,n. (2.18) Equations (2.18) represent theappropriate generalization of(2.11) toseveral variables andareknown astheEuler—Lagrange diﬁierential equations. Their so- lutions represent curves forwhich thevariation ofanintegral oftheform given in(2.14) vanishes. Further generalizations ofthefundamental variational problem areeasily possible. Thus, wecantakefasafunction ofhigher derivatives )5,'y, etc.,leading toequations different from (2.18). Orwecanextend ittocases where there areseveral parameters xJandtheintegral isthenmultiple, with falsoin- volving asvariables derivatives ofy,withrespect toeach oftheparameters xJ. Finally, itispossible toconsider variations inwhich theendpoints arenotheld ﬁxed. Forpresent purposes, what wehave derived here sufﬁces, fortheintegral in Hamiltofs principle, 2 I=f L(q,,q,.t)dt, (2.19) 1 hasJustthefomistipulated in(2.14) withthetransformation X—> I Yr—>qr f(>n,inx)—>L(q~é/it I). Inderiving Eqs.(2.18), weassumed thatthey,variables areindependent. The corresponding condition inconnection withHamilton’s principle isthatthegen- eralized coordinates q,-beindependent, which requires thattheconstraints be holonomic. TheEuler—Lagrange equations corresponding totheintegral Ithen become theLagrange equations ofmotion, dBL 8L——,——i=0, i=1,2,...,n, dt641, 64], andwehaveaccomplished ouroriginal aim,toshow thatLagrange’s equations follow fromHamilton’s principle—-for monogenic systems withholonomic con- straints. EXTENSION OFHAMll.TON'S PRINCIPLE TONONHOLONOMIC SYSTEMS Itispossible toextend Hamilton’s principle, atleastinaformal sense, tocover certain types ofnonholonomic systems. Inderiving Lagrange’s equations from Chapter 2Variational Principles andLagrange's Equations either Hamilton’s orD’Alembert’s principle, therequirement ofholonomic con- straints doesnotappear untilthelaststep,when thevariations q,areconsidered asindependent ofeachother. With nonholonomic systems thegeneralized coor- dinates arenotindependent ofeach other, anditisnotpossible toreduce them further bymeans ofequations ofconstraint oftheform f(q1, qg,...,q,,,t)=0. Hence, itisnolonger truethattheq,’sareallindependent. Another difference thatmust beconsidered intreating thevariational principle isthemanner inwhich thevaried paths areconstructed. Inthediscussion ofSec- tion2.2,wepointed outthat8y(or8q)represents avirtual displacement from a point ontheactual pathtosome point ontheneighboring varied path. But,with independent coordinates itistheﬁnalvaried paththatissigniﬁcant, nothowitis constructed. When thecoordinates arenotindependent, butsubject toconstraint relations, itbecomes important whether thevaried pathisorisnotconstructed by displacements consistent withtheconstraints. Virtual displacements, inparticular, mayormaynotsatisfy theconstraints. ltappears thatareasonably straightforward treatment ofnonholonomic sys- temsbyavariational principle ispossible onlywhen theequations ofconstraint canbeputintheform fa(q1.---,qn; éi---.12") =0- (2-20) when thiscanbedone theconstraints arecalled semi-holonomic. Theindex oz indicates thatthere maybemore thanonesuch equation. Wewillassume there aremequations inall,i.e.,oz=l,2,....m.Equation (2.20) commonly appears intherestricted form Za,,,dllk+anat=0. (2.20)k Wemight expect thatthevaried paths, orequivalently, thedisplacements con- structing thevaried path,should satisfy theconstraints ofEq.(2.20). However, it hasbeen proven thatnosuch varied pathcanbeconstructed unless Eqs. (2.20) areintegrable, inwhich casetheconstraints areactually holonomic. Avariational principle leading tothecorrect equations ofmotion cannonetheless beobtained when thevaried paths areconstructed from theactual motion byvirtual displace- ments. Theprocedure foreliminating these extra virtual displacements isthemethod ofLagrange undetermined multipliers. IfEqs.(2.20) hold, thenitisalsotruethat i1,,fa=0, (2.21)I1=l where thela,or=l,2....,m,aresome undetermined quantities, functions in general ofthecoordinates andofthetimet.Inaddition, Hamilton’s principle, 1 8/2 Ldt=O, (2.2) ii 2.4 Extension ofHamilton's Principle toNonholonomic Systems 47 isassumed toholdforthissemiholonomic system. Following thedevelopment of Section 2.3,Hamilton’s principle thenimplies that 2 atdarat _--—_ s=0. 2.22I,Z:(aw. dtBqk)q" () Thevariation cannot betaken asbefore since theqkarenotindependent; however, combining (2.21) with(2.2) gives a[2(L+i2,,fa)at=0 (2.23)1 o¢=1 Thevariation cannowbeperformed withthen8q,andmAuform+n independent variables. Forthesimplifying assumption that1,,=2t,,(z), theresulting equations =|=from8q,beoome dBL BL—— ———= , 2.24 dt(941) aqk Qk () where _ aft! _ d afﬂ _dktl aft! i.....<...)i.. iris11%;.r while the8A,,givetheequations ofconstraint (2.20). Equations (2.24) and(2.20) together constitute n+mequations forn+munknowns. Thesystem cannow beinterpreted asanm+nholonomic system withgeneralized forces Qt.The generalization toIto,=2t,,(q1, ...,q,,;()1,...,¢j,,;t)isstraightforward. Asanexample, letusconsider aparticle whose Lagrangian is L=gm(xi+)2+zz)-vet,y,1) (2.26) subject totheconstraint f(.t.$’.y)=J'rj>+ky=0 (2.27) withkaconstant. Theresulting equations ofmotion are .. ..~.3Vmx+7ty+2ty+ -5;=0, (2.28) .... -.3Vmy+7tx—k7t+Jtx-l--é;=O, (2.29) Vm?+8-=0, (2.30)Bz *1.Ray,Amer. J’.Phys. 34(406-8), 1996. Chapter 2Variational Principles andLagrange's Equations andtheequation ofconstraint, (2.20), becomes jut+Icy=0. Inthisprocess wehaveobtained more information thanwasoriginally sought. Notonly dowegettheqk’swesetouttoﬁnd, butwealsogetm2t1’s. What is thephysical signiﬁcance oftheJ11‘s?Suppose weremove theconstraints onthe system, butinstead apply extemal forces Q2insuch amanner astokeep the motion ofthesystem unchanged. Theequations ofmotion likewise remain the same. Clearly these extraapplied forces mustbeequal totheforces ofconstraint, forthey aretheforces applied tothesystem soastosatisfy thecondition of constraint. Under theinﬂuence ofthese forces Qz,theequations ofmotion are dBL 8L ,dtMk aqk Qk. (2.31) Butthese must beidentical withEqs.(2.24). Hence, wecanidentify (2.25) with Q2,thegeneralized forces ofconstraint. Inthistypeofproblem wereally donot eliminate theforces ofconstraint from theformulation. They aresupplied aspart oftheanswer. Although itisnotobvious, theversion of1-1amilton’s principle adopted here forsemiholonomic systems alsorequires thattheconstraints donowork invirtual displacements. Thiscanbemosteasily seenbyrewriting Harni1ton’s principle in theform I2 Y2 '2 Sf Ldr=6f Tdt—8f Udt=0. (2.32) T1 Ii T1 Ifthevariation oftheintegral overthegeneralized potential iscarried outbythe procedures ofSection 2.3,theprinciple takes thefonri '1 '1 av d3U]aTd= _-- -_54; 2.33fir tfa;[9qk dl(3¢1k) qkI () or,byEq.(1.58), I t 5I2Tdr=_[1ZQk6qkdr. (2.34)ti Ii1; Inthisdress, Hamilton’s principle saysthatthedifference inthetimeintegral of thekinetic energy between twoneighboring paths isequal tothenegative ofthe time integral ofthework done inthevirtual displacements between thepaths. Thework involved isthatdone onlybytheforces derivable from thegeneralized potential. Thesame Hamilton’s principle holds forbothholonomic andsemiholo- nomic systems, itmust berequired thattheadditional forces ofsemiholonomic constraints donoworkinthedisplacements 8q;,.Thisrestriction parallels theear- liercondition thatthevirtual work oftheforces ofholonomic constraint alsobe 2.4 Extension ofHamilton's Principle toNonholonomic Systems 49 zero(cf.Section 1.4).Inpractice, therestriction presents littlehandicap tothe applications, asmany problems inwhich thesem.iholonomic formalism isused relate torolling without slipping, where theconstraints areobviously workiess. Note thatEq.(2.20) isnotthemost general typeofnonholonomic constraint; e.g.,itdoesnotinclude equations ofconstraint intheform ofinequalities. On theother hand, itdoes include holonomic constraints. Aholonomic equation of constraint, f(q1iq2>q3i---iqlht) =0e isequivalent to(2.20) withnodependence onqk.Thus, theLagrange multiplier method canbeused alsoforholonomic constraints when (1)itisinconvenient to reduce alltheq’stoindependent coordinates or(2)wemight wish toobtain the forces ofconstraint. Asanother example ofthemethod, letusconsider thefollowing somewhat trivial illustration—a hoop rolling, without slipping, down aninclined plane. ln thisexample, theconstraint of“rolling” isactually holonomic, butthisfactwill beimmaterial toourdiscussion. Ontheotherhand, theholonomic constraint that thehoop beontheinclined plane willbecontained implicitly inourchoice of generalized coordinates. Thetwogeneralized coordinates arex,6,asinFig.2.5,andtheequation of rolling constraint is rd6 =dx. Thekinetic energy canberesolved intokinetic energy ofmotion ofthecenter ofmassplusthekinetic energy ofmotion about thecenter ofmass: T=%M.t2+%Mr2i’§2. Thepotential energy is V=Mg(l —x)sin¢, where listhelength oftheinclined plane andtheLagrangian is X _¢ FIGURE 2.5 Ahoop rolling down aninclined plane. Chapter 2Variational Principles andLagrange's Equations L=T—V M'2 M 2'2 =TX+%0 -Mg(l-x)sin¢. (2.36) Since there isoneequation ofconstraint, only oneLagrange multiplier Ais needed. Thecoefﬁcients appearing intheconstraint equation are: 619:7‘, ax Z ‘I. ThetwoLagrange equations therefore are M36—Mgsin¢ +A=0, (2.37) MFG‘-Ar=0, (2.38) which along withtheequation ofconstraint, ré=12, (2.39) constitutes threeequations forthreeunknowns, 6,x,2.. Differentiating (2.39) withrespect totime, wehave H5= Hence, from(2.38) M55=/X, and(2.37) becomes /_t__gsin¢_ 2_ along with hi2 and ..gSin¢ 6=i.2r Thus, thehooprollsdown theincline w1thonlyone-half theacceleration itwould have slipping down africtionless plane, andthefriction force ofconstraint is A=Mgsin¢/2. 2.5I2.5 Advantages ofaVariational Principle Formulation 51 ADVANTAGES OFAVARIATIONAI. PRINCIPLE FORMULATION Although wecanextend theoriginal formulation ofHamilton’s principle (2.2)to include some nonholonomic constraints, inpractice thisformulation ofmechan- icsismost useful when aLagrangian ofindependent coordinates canbesetup forthesystem. Thevariational principle formulation hasbeenjustly described as “elegant,” forinthecompact Hamilton’s principle iscontained allofthemechan- icsofholonomic systems withforces derivable from potentials. Theprinciple has thefurther merit thatitinvolves onlyphysical quantities thatcanbedeﬁned with- outreference toaparticular setofgeneralized coordinates, namely, thekinetic andpotential energies. Theformulation istherefore automatically invariant with respect tothechoice ofcoordinates forthesystem. From thevariational 1-lami1ton’s principle, itisalsoobvious why theLa- grangian isalways uncertain toatotal time derivative ofanyfunction ofthe coordinates andtime, asmentioned attheendofSection 1.4.Thetimeintegral ofsuch atotalderivative between points land2depends onlyonthevalues of thearbitrary function attheendpoints. Asthevariation attheendpoints iszero, theaddition ofthearbitrary timederivative totheLagrangian doesnotaffect the variational behavior oftheintegral. Another advantage isthattheLagrangian formulation canbeeasily extended todescribe systems thatarenotnormally considered indynamics—such as theelastic ﬁeld, theelectromagnetic ﬁeld, andﬁeld properties ofelementary particles. Some ofthese generalizations willbeconsidered later, butasthree simple examples ofitsapplication outside theusual framework ofmechanics, let usconsider thecases ofanRLcircuit, anLCcircuit, andcoupled circuits. Weconsider thephysical system ofabattery ofvoltage Vinseries with an inductance Landaresistance ofvalue Randchoose theelectric charge qas thedynamical variable. Theinductor actsasthekinetic energy term since the inductive effect depends uponthetimerateofchange ofthecharge. Theresistor provides adissipative termandthepotential energy isqV.Thedynamic terms in Lagrange’s equation withdissipation (1.70) are T=int’.F=hm’. andpotential energy =qV.Theequation ofmotion is v=Lij+12,;=L1‘+RI. (2.40) where 1=Qistheelectric current. Asolution forabattery connected tothe circuit attimet=0is 1=I<>(1—e-R‘/L). where I0=V/Ristheﬁnalsteady-state current ﬂow. Themechanical analog forthisisasphere ofradius aandeffective mass m’ falling inaviscous ﬂuid ofconstant density andviscosity 27under theforce of Chapter 2Variational Principles andLagrange’s Equations gravity. Theeffective mass isthedifference between theactual mass andthemass ofthedisplaced ﬂuid, andthedirection ofmotion isalong theyaxis. Forthis system, T=%m')"2, .7:= 3J'ET]aj12, andpotential energy =m’gy,where thefrictional dragforce, Ff=6::nay,called Stokes’ law,wasgiven attheendofSection 1.5. Theequation ofmotion isgiven byLagrange’s equations (1.70) as m'g=m'_')i+Gzrnay. Using v=y,thesolution (ifthemotion starts from restat1=O),is v=v,,(l—e_'/L) where r=m’/(omia) isameasure ofthetimeittakes forthesphere toreach i/eofitsterminal speed ofv0=m’g/61:nu. Another example fromelectrical circuits isaninductance, L,inseries witha capacitance, C.Thecapacitor actsasasource ofpotential energy given byq2/C where qistheelectric charge. TheLagrangian produces theequation ofmotion, ..qL —= . q+C0, (241) which hasthesolution fl=410C05W01‘, where qoisthecharge stored inthecapacitor att=0,andtheassumption isthat nocharge isﬂowing att=0.Thequantity l ‘”°-Wistheresonant frequency ofthesystem. Themechanical analog ofthissystem isthesimple harmonic oscillator de- scribed bytheLagrangian L=émirz —ékxz, which gives anequation ofmotion, mi’—l-kx=O, whose solution forthesame boundary conditions is x=X0coswot with cor)=\/kl/Tl. These twoexamples show thataninductance isaninertial term, theelectrical analog ofmass. Resistance istheanalog ofStokes’ lawtypeoffrictional drag, andthecapacitance term1/Crepresents aI-looke’s lawspring constant. Withthis 2.5 Advantages ofaVariational Principle Formulation 53 Cl RI 1., ~E‘ M12 M13 1: E2I-1 1.2% L3 ‘I-I3 Mc, R2 23R3 c3 FIGURE 2.6 Asystem ofcoupled circuits towhich theLagrangian formulation canbe applied. background, asystem ofcoupled electrical circuits ofthetypeshown inFig.2.6 hasaLagrangian oftheform 1 .1 .. <1’L=521-14? 'l'5ZM]kqjqk -Z% ‘l’Zeﬂflqil 1 J1‘ 1 JJHék andadissipation function l J where themutual inductance temis, MI/ct}Jqt,areadded totakeintoaccount the coupling between inductors. TheLagrange equations are 41 42 d1.,-2-t51§+}:M,,,-d%+R,%+(‘é-;=E,(z). (2.42) ilk where theEJ(t)terms aretheexternal emf‘s. Thisdescription oftwodifferent physical systems byLagrangians ofthesame form means thatalltheresults andtechniques devised forinvestigating oneofthe systems canbetaken overimmediately andapplied totheother. Inthisparticular case, thestudy ofthebehavior ofelectrical circuits hasbeenpursued intensely andsome special techniques have been developed; these canbedirectly applied tothecorresponding mechanical systems. Much work hasbeen done informulat- ingequivalent electrical problems formechanical oracoustical systems, andvice versa. Terms hitherto reserved forelectrical circuits (reactance, susceptance, etc.) arenowcommonly found intreatises onthetheory ofvibrations ofmechanical systems. 2.6IChapter 2Variational Principles andLagrange's Equations Additionally, onetypeofgeneralization ofmechanics isduetoasubtler form ofequivalence. Wehave seen thattheLagrangian andHamilton’s principle to- gether form acompact invariant wayofobtaining themechanical equations of motion. Thispossibility isnotreserved formechanics only; inalmost every ﬁeld ofphysics variational principles canbeusedtoexpress the“equations ofmotion,” whether theybeNewton’s equations, Maxwe]1’s equations, ortheSchrodinger equation. Consequently, when avariational principle isusedasthebasisofthefor- mulation, allsuchﬁelds willexhibit, atleasttosome degree, astructural analogy. When theresults ofexperiments show theneedforalterating thephysical content inthetheory ofoneﬁeld, thisdegree ofanalogy hasoften indicated howsimilar alterations maybecarried outinother ﬁelds. Thus, theexperiments performed early inthiscentury showed theneed forquantization ofboth electromagnetic radiation andelementary particles. Themethods ofquantization, however, were ﬁrstdeveloped forparticle mechanics, starting essentially from theLagrangian formulation ofclassical mechanics. Bydescribing theelectromagnetic ﬁeldbya Lagrangian andcorresponding Hamilton’s variational principle, itispossible to carry overthemethods ofparticle quantization toconstruct aquantum electrody- namics (cf.Sections 13.5and13.6). CONSERVATION THEOREMS AND SYMMETRY PROPERTIES Thus far,wehavebeenconcemed primarily withobtaining theequations ofmo- tion, butlittle hasbeen saidabout howtosolve them foraparticular problem once theyareobtained. Ingeneral, thisisaquestion ofmathematics. Asystem ofndegrees offreedom willhave ndifferential equations thataresecond order intime. Thesolution ofeach equation willrequire twointegrations resulting, all told, in2nconstants ofintegration. Inaspeciﬁc problem these constants willbe determined bytheinitial conditions, i.e.,theinitial values ofthenq,-’s andthe mi,’s.Sometimes theequations ofmotion willbeintegrable interms ofknown functions, butnotalways. Infact,themajority ofproblems arenotcompletely integrable. However, evenwhen complete solutions cannot beobtained, itisoften possible toextract alargeamount ofinformation about thephysical nature ofthe system motion. Indeed, suchinformation maybeofgreater interest tothephysi- cistthanthecomplete solution forthegeneralized coordinates asafunction of time. Itisimportant, therefore, toseehowmuch canbestated about themotion ofagiven system without requiring acomplete integration oftheproblem.* Inmany problems anumber ofﬁrstintegrals oftheequations ofmotion canbe obtained immediately; bythiswemean relations ofthetype _f(q1, qg,...,Q1,()2,...,t)=constant. (2.43) *Inthisandsucceeding sections llwillbeassumed, unless otherwise speciﬁed, thesystem issuchthat itsmotion iscompletely described byaHamilton"-z principle oftheform (2.2). 2.6 Conservation Theorems andSymmetry Propertnes 55 which areﬁrst-order differential equations. These ﬁrstintegrals areofinterest because theytellussomething physically about thesystem. They include, infact, theconservation lawsobtained inChapter I. Letusconsider asanexample asystem ofmass points under theinﬂuence of forces derived from potentials dependent onposition only. Then aL_aT av_aT__a 1,2,2,2 ax.=an—ax,_ax._ax;ZEm’(x'H‘+2‘) =mix: =Pix- which isthexcomponent ofthelinear momentum associated withtheith particle. This result suggests anobvious extension totheconcept ofmomentum. Thegeneralized momentum associated withthecoordinate qJshall bedeﬁned as 3L=l. 2.44 P] ad] ( ) Theterms canonical momentum andconjugate momentum areoften alsousedfor pJ.Notice thatifqJisnotaCartesian coordinate, pJdoes notnecessarily have thedimensions ofalinear momentum. Further, ifthere isavelocity-dependent potential, theneven with aCartesian coordinate qJtheassociated generalized momentum willnotbeidentical withtheusual mechanical momentum. Thus, inthecaseofagroup ofparticles inanelectromagnetic ﬁeld, theLagrangian is (cf.1.63) I_ . L=l 5m1r;2_ $ql¢(-xi) +$qlA(xl) 'rt (q,heredenotes charge) andthegeneralized momentum conjugate tox,is 81. , Pix = =mix! +111-Ax, (2-45) i.e.,mechanical momentum plusanadditional terrn. IftheLagrangian ofasystem doesnotcontain agiven coordinate qJ(although itmaycontain thecorresponding velocity 4,),thenthecoordinate issaidtobe cyclic orignorable. This deﬁnition isnotuniversal, butitisthecustomary one andwillbeusedhere.TheLagrange equation ofmotion, d8L 8L ——.--—=0.dtZlqj 8:], reduces, foracyclic coordinate, to iﬂ_0dial},- Chapter 2Variational Principles andLagrange's Equations O1’ dpj_=0, dr which mean that pI=constant. (2.46) Hence, wecanstate asageneral conservation theorem thatthegeneralized mo- mentum conjugate roacyclic coordinate isconserved. Notethatthederivation ofEq.(2.46) assumes thatq1-isageneralized coordi- nate; onethatislinearly independent ofalltheother coordinates. When equations ofconstraint exist, allthecoordinates arenotlinearly independent. Forexam- ple.theangular coordinate 6isnotpresent intheLagrangian ofahoop rolling without slipping inahorizontal plane thatwaspreviously discussed, buttheangle appears intheconstraint equations rd6=dx.Asaresult, theangular momentum, pg=mrztl, isnotaconstant ofthemotion. Equation (2.46) constitutes aﬁrstintegral oftheform (2.43) fortheequations ofmotion. Itcanbeusedfonnally toeliminate thecyclic coordinate from the problem, which canthenbesolved entirely interms oftheremaining general- izedcoordinates. Brieﬂy, theprocedure, originated byRouth, consists inmodify- ingtheLagrangian sothatitisnolonger afunction ofthegeneralized velocity corresponding tothecyclic coordinate, butinstead involves onlyitsconjugate momentum. Theadvantage insodoing isthatpIcanthenbeconsidered oneof theconstants ofintegration, andtheremaining integrations involve onlythenon- cyclic coordinates. Weshall defer adetailed discussion ofRouth’s method until theHamiltonian formulation (towhich itisclosely related) istreated. Note thattheconditions fortheconservation ofgeneralized momenta aremore general thanthetwomomentum conservation theorems previously derived. For example, theyfumish aconservation theorem foracaseinwhich thelawofac- tionandreaction isviolated, namely, when electromagnetic forces arepresent. Suppose wehave asingle particle inaﬁeldinwhich neiﬂier ¢norAdepends on x.Then xnowhere appears inLandistherefore cyclic. Thecorresponding canon- icalmomentum prmusttherefore beconserved. From (1.63) thismomentum now hastheform px=mi+qAx =constant. (2.47) Inthiscase, itisnotthemechanical linear momentum mithatisconserved but rather itssumwithqA,,.*Nevertheless, itshould stillbetruethattheconservation theorems ofChapter 1arecontained within thegeneral ruleforcyclic coordinates; withproper restrictions (2.46) should reduce tothetheorems ofSection 1.2. *ltcanbeshown from classical electrodynamics thatunder these conditions, i.e.,fl8IlLll0I.' Anor¢ depending onx,thatqA,,isexactly thex-component ottheelectromagnetic linear momentum ofthe ﬁeldassociated withthecharge q. 2.6 Conservation Theorems andSymmetry Properties 57 Weﬁrstconsider ageneralized coordinate qJ,forwhich achange dqjrepre- sents atranslation ofthesystem asawhole insome given direction. Anexample would beoneoftheCartesian coordinates ofthecenter ofmass ofthesystem. Then clearly qJcannot appear inT,forvelocities arenotaffected byashiftinthe oiigin, andtherefore thepartial derivative ofTwithrespect toq1-mustbezero. Further, wewillassume conservative systems forwhich Visnotafunction ofthe velocities. soastoeliminate suchcomplications aselectromagnetic forces. The Lagrange equation ofmotion foracoordinate sodeﬁned thenreduces to dHT 8V——_-E‘ =——E-Q. (2.48)dtHq, PJ élqj J Wewillnow show that(2.48) istheequation ofmotion forthetotal linear momentum, i.e.,thatQ1represents thecomponent ofthetotalforce along thedi- rection oftranslation ofq,,andpJisthecomponent ofthetotallinear momentum along thisdirection. Ingeneral, thegeneralized force QJisgiven byEq.(1.49): 31‘, Q=F~-—. I i laqj Since dqjcorresponds toatranslation ofthesystem along some axis, thevectors r,(qJ)andr,(qJ+dqj)arerelated asshown inFig.2.7.Bythedeﬁnition ofa derivative, wehave E=limr‘(‘U+dq’)'r‘(‘Z’)=dq’ll=n, (2.49)d([]—>0 dqJ where nistheunitvector along thedirection ofthetranslation. Hence, Q]=ZF,-n=n-F, which (aswasstated) isthecomponent ofthetotalforce inthedirection ofn.To prove theother halfofthestatement, notethatwiththekinetic energy intheform dqIn r,(q,) r,(q,+dq,) FIGURE 2.7 Change inaposition vector under translation ofthesystem. Chapter 2Variational Principles andLagrange’s Equations T=5-Zm,i'l?, theconjugate momentum is 87’ ,8r, P1"at.-3""'a_Zmv 6r, - 1r‘__@ . 341 using Eq.(1.51). Then fromEq.(2.49) P]=n‘Z: mrvz > l which again, aspredicted, isthecomponent ofthetotalsystem linear momentum along n. Suppose nowthatthetranslation coordinate qlthatwehavebeen discussing is cyclic. Then qlcannot appear inVandtherefore -31 EQl=O. ‘I1 Butthisissimply thefamiliar conservation theorem forlinear momentum—that ifagiven component ofthetotalapplied force vanishes, thecorresponding com- ponent ofthelinear momentum isconserved. Inasimilar fashion, itcanbeshown thatifacyclic coordinate qlissuchthat dqlcorresponds toarotation ofthesystem ofparticles around some axis, then theconservation ofitsconjugate momentum corresponds toconservation ofan angular momentum. Bythesame argument used above, Tcannot contain ql,for arotation ofthecoordinate system cannot affect themagnitude ofthevelocities. Hence, thepartial derivative ofTwithrespect toqlmustagain bezero,andsince Visindependent of4,-,weonce more getEq.(2.48). Butnowwewish toshow thatwithqlarotation coordinate thegeneralized force isthecomponent ofthe totalapplied torque about theaxisofrotation, andplisthecomponent ofthetotal angular momentum along thesame axis. Thegeneralized force Qlisagain given by 8r,Q Z F 'isJ $ 'aq] onlythederivative nowhasadifferent meaning. Herethechange inqlmustcor- respond toaninﬁnitesimal rotation ofthevector r,-,keeping themagnitude of thevector constant. From Fig.2.8,themagnitude ofthederivative caneasily be obtained: |dr,| =r,-sin6dql 2.6 Conservation Theorems andSymmetry Properties 59 ln ____,_- -k‘ § 1,I \I \ I, dq- \\'\ I |\ ‘- I Tx ‘ v \‘*—~% i - ll-l(q]) 75(4) +d4,) 6 FIGURE 2.8 Change ofaposition vector under rotation ofthesystem. and dll =r,sin6,Bql anditsdirection isperpendicular tobothr,andn.Clearly, thederivative canbe written invector formas ,3"=nx1-,. (2.50)341 With thisresult, thegeneralized force becomes Ql=Z11-nXr,l =ZI1-I‘,XF,, I reducing to Ql-=n-ZN,-=n-N, which proves theﬁrstpart.Asimilar manipulation ofplwiththeaidofEq.(2.50) provides proof ofthesecond partofthestatement: 8T 8rpl=—_=Em,v, a‘=Zn-r,xm,v,=n-EL,-=n-L. 341 , ‘Z1 i 2.7IChapter 2Variational Principles andLagrange’s Equations Summarizing these results, weseethatiftherotation coordinate qJiscyclic, thenQJ,which isthecomponent oftheapplied torque along n,vanishes, and thecomponent ofLalong nisconstant. Here wehave recovered theangular momentum conservation theorem outofthegeneral conservation theorem relating tocyclic coordinates. Thesigniﬁcance ofcyclic translation orrotation coordinates inrelation tothe properties ofthesystem deserves some comment atthispoint. l_fageneralized co- ordinate corresponding toadisplacement iscyclic, itmeans thatatranslation of thesystem, asifrigid, hasnoeffect ontheproblem. lnother words, ifthesystem ismvariam under translation along agiven direction, thecorresponding linear momentum isconserved. Similarly, thefactthatageneralized rotation coordinate iscyclic (and therefore theconjugate angular momentum conserved) indicates thatthesystem isinvariant under rotation about thegiven axis.Thus, ﬂiemomen- mmconservation theorems areclosely connected withthesymmetry properties ofthesystem. Ifthesystem isspherically symmetric, wecansaywithout further adothatallcomponents ofangular momentum areconserved. Or,ifthesystem is symmetric onlyabout thezaxis,thenonlyLZwillbeconserved, andsoonfor theother axes.These symmetry considerations canoften beusedwithrelatively complicated problems todetermine byinspection whether certain constants ofthe motion exist. (cf.Noether’s theorem—Sec. 13.7.) Suppose, forexample, thesystem consists ofasetofmass points moving in apotential ﬁeld generated byﬁxed sources unifonnly distributed onaninﬁnite plane, say,thez=0plane. (Thesources might beamassdistribution iftheforces weregravitational, oracharge distribution forelectrostatic forces.) Then thesym- metry oftheproblem issuch thattheLagrangian isinvariant under atranslation ofthesystem ofparticles inthex-ory-directions (butnotinthez-direction) and alsounder arotation about thezaxis. Itimmediately follows thatthex-andy- components ofthetotallinear momentum, P,andPy,areconstants ofthemotion along with LZ,thez-component ofthetotalangular momentum. However, ifthe sources wererestricted onlytothehalfplane, x3O,thenthesymmetry fortrans- lation along thexaxisandforrotation about the2axiswould bedestroyed. Inthat case, PXandLzcould notbeconserved, butP).would remain aconstant ofthe motion. Wewillencounter theconnections between theconstants ofmotion and thesymmetry properties ofthesystem several times inthefollowing chapters. ENERGY FUNCTION AND THE CONSERVATION OFENERGY Another conservation theorem weshould expect toobtain intheLagrangian for- mulation istheconservation oftotal energy forsystems where theforces are derivable frompotentials dependent onlyuponposition. Indeed, itispossible to demonstrate aconservation theorem forwhich conservation oftotalenergy repre- sents onlyaspecial case. Consider ageneral Lagrangian, which willbeafunction ofthecoordinates qIandthevelocities Q,andmayalsodepend explicitly onthe time. (Theexplicit timedependence mayarisefromthetimevariation ofexternal 2.7 Energy Function andtheConservation ofEnergy 61 potentials, orfromtime-dependent constraints.) Then thetotaltimederivative of Lis dL 8Ldq, 8Ldz}, 8L—= —— ——— ——. 2.51dz $841, dz+284, dz+3: () From Lagrange’s equations, 6L_d(BL) 3q_, dt34}, ’ and(2.51) canberewritten as dL d8L , 8Ldc}, 8L at-;.i.($)‘11+;@ .1.+at or dL_Zd.8L +6L dz_Jdzq’aq, 8z‘ Ittherefore follows that d 8L 8L— ‘—-L —=0. 2.52dz(;q’aq, )"'at () Thequantity inparentheses isoftentimes called theenergyfunczz'on* andwillbe denoted byh: . . .8Lh<qi.....q.; qi.....q..; z>=Zq,5-L. (2.53)J 1 andEq.(2.52) canbelooked onasgiving thetotaltimederivative ofh: Q3=-2. (2.54)dz 82 IftheLagrangian isnotanexplicit function oftime, i.e.,ifitdoes notappear inLexplicitly butonly implicitly through thetime variation ofqandzj,then Eq.(2.54) saysthathisconserved. Itisoneoftheﬁrstintegrals ofthemotion and issometimes referred toasJacobi’s integrall *The energy function hisidentical invalue with theHamiltonian H(SeeChapter 8)Itisgiven adifferent name andsymbol heretoemphasize thathisconsidered afunction ofnindependent variables qjandtheir time derivatives ti](along with thetime), whereas theI-lanultonian willbe treated asafunction of2nindependent vanables, qJ,[7](andpossibly thetime) l'This designation ismost otten conﬁned toaﬁrstintegral intherestncted three-body problem. How- ever, theintegral there ismerely aspecial caseoftheenergy function h,andthere issome historical precedent toapply thename Jacobi integral tothemore general situation Chapter 2Variational Principles andLagrange's Equations Under certain circumstances, thefunction histhetotal energy ofthesystem. Todetemiine what these circumstances are,werecall thatthetotalkinetic energy ofasystem canalways bewritten as T=To+T)+T2, (1.73) where T0isafunction ofthegeneralized coordinates only, T1(q,(Q)islinear inthe generalized velocities, andT2(q,zj)isaquadratic function ofthezfs.Foravery wide range ofsystems andsetsofgeneralized coordinates, theLagrangian canbe similarly decomposed asregards itsfunctional behavior intheQvariables: L(q,4»t)=Lo(q,r)+Ll(q#12»I)+L2(¢1-¢i»l‘)- (2.55) Here L2isahomogeneous function ofthesecond degree (notmerely quadratic) inzj,while L1ishomogeneous oftheﬁrstdegree inz}.There isnoreason intrinsic tomechanics thatrequires theLagrangian toconform toEq.(2.55), butinfactit doesformost problems ofinterest. TheLagrangian clearly hasthisform when the forces arederivable from apotential notinvolving thevelocities. Even withthe velocity-dependent potentials, wenotethattheLagrangian foracharged particle inanelectromagnetic ﬁeld,Eq.(1.63), satisﬁes Eq.(2.55). Now, recall thatEuler’s theorem states thatiffisahomogeneous function ofdegree ninthevariables x,, then Zr,-Q =nf. (2.56)dx, Applied tothefunction h,Eq.(2.53), fortheLagrangians oftheform (2.55), this theorem implies that h=2L2—|-L1—L=L2—L0. (2.57) Ifthetransformation equations deﬁning thegeneralized coordinates, Eqs.(1.38), donotinvolve thetimeexplicitly, thenbyEqs.(1.73) T=T2.If,further, the potential doesnotdepend onthegeneralized velocities, thenL2=TandL0= —V,sothat h=T+V=E, (2.58) andtheenergy function isindeed thetotal energy. Under these circumstances, ifVdoes notinvolve thetimeexplicitly, neither willL.Thus, byEq.(2.54), h (which isherethetotalenergy), Willbeconserved. Note thatﬂieconditions forconservation ofhareinprinciple quite distinct fromthose thatidentify hasthetotalenergy. Wecanhaveasetofgeneralized coordinates such thatinaparticular problem hisconserved butisnotthetotal energy. Ontheother hand, hcanbethetotalenergy, inthefor.m T+V,butnot beconserved. Also notethatwhereas theLagrangian isuniquely ﬁxed foreach Derivations 63 system bytheprescription L=T-—U independent ofthechoice ofgeneralized coordinates, theenergy function hde- pends inmagnitude andfunctional form onthespeciﬁc setofgeneralized co- ordinates. Foroneandthesame system, various energy functions hofdifferent physical content canbegenerated depending onhowthegeneralized coordinates arechosen. Themost common casethatoccurs inclassical mechanics isoneinwhich the kinetic energy terms areallofthefonn me}?/2orpiz/2mandthepotential energy depends onlyupon thecoordinates. Forthese conditions, theenergy function is bothconserved andisalsothetotalenergy. Finally, notethatwhere thesystem isnotconservative, buttherearefrictional forces derivable from adissipation function .7-‘,itcanbeeasily shown thatFisre- latedtothedecay rateofh.When theequations ofmotion aregiven byEq.(1.70), including dissipation, thenEq.(2.52) hastheform dh 3L 87:. z;+a-257,“ Bythedeﬁnition of.7-‘,Eq.(1.67), itisahomogeneous function ofthec}’sof degree 2.Hence, applying Euler’s theorem again, wehave dh 8L—=-2-—. .dr J: Br (259) IfLisnotanexplicit function oftime, andthesystem issuch thathisthesame astheenergy, thenEq.(2.59) saysthat2}‘istherateofenergy dissipation, clE* =—-2.7:, 2.6 dt (0) astatement proved above (cf.Sec.1.5)inlessgeneral circumstances. DERIVATIONS 1.Complete thesolution ofthebrachistochrone problem begun inSection 2.2andshow thatthedesired curve isacycloid withacuspattheinitial point atwhich theparticle isreleased. Show alsothatiftheparticle is[JI’O_]€Ct8d with aninitial kinetic energy %mv% thatthebrachistochrone isstillacycloid passing through thetwopoints witha cusp ataheight zabove theinitial point given by1%=Zgz. 2.Show thatifthepotential intheLagrangian contains velocity-dependent terms. the canonical momentum corresponding toacoordinate ofrotation 6oftheentire system Chapter 2Variational Principles andLagrange’s Equations 3. 4isnolonger themechanical angular momentum L9butisgiven by P0=Le—zﬂ-Pi XVv,U> I where Vvisthegradient operator inwhich thederivatives arewith respect tothe velocity components andnisaunitvector inthedirection ofrotation. Ifthe forces are electromagnetic incharacter. thecanonical momentum istherefore qp9=L9+Zn-r, x?'A,-. I Prove thattheshortest distance between twopoints inspace isastraight line. Show thatthegeodesics ofaspherical surface aregreat circles, i.e.,circles whose centers lieatthecenter ofthesphere. EXERCISES 5. 6 7. 8‘Aparticle issubjected tothepotential V(x) =—Fx,where Fisaconstant. The particle travels fromx=0tox=ainatimeinterval to.Assume themotion ofthe particle canbeexpressed intheformx(r)=A+BI+C:2.Findthevalues ofA,B, andCsuchthattheaction isaminimum. Find theEuler-Lagrange equation describing thebrachistochrone curve foraparticle moving inside aspherical Earth ofuniform mass density. Obtain aﬁrstintegral for thisdifferential equation byanalogy totheJacobi integral h.With thehelpofthis integral, show thatthedesired curve isahypocycloid (thecurve described byapoint onacircle rolling ontheinside ofalarger circle). Obtain anexpression forthetime oftravel along thebrachistochrone between twopoints onEarth’s surface. How long would ittaketogofrom New York toLosAngeles (assumed tobe4800 kmapart on thesurface) along abrachistochrone tunnel (assuming nofriction) andhowfarbelow thesurfacewould thedeepest point ofthetunnel be? InExample 2ofSection 2.1weconsidered theproblem ofthemmimum surface of revolution. Examine thesymmetric casex1=X2,yg=—y1 >0,andexpress the condition fortheparameter aasatranscendental equation interms ofthedimension- lessquantities k=x2/a, andoz=yg/xg. Show thatfororgreater thanacertain value 0:0twovalues ofkarepossible, foror=110onlyonevalue ofkispossible, while if or<oionorealvalue ofk(ora)canbefound, sothatnocatenary solution exists in thisregion. Findthevalue of0'0,numerically ifnecessary. Thebroken-segment solution described inthetext(cf.p.42),inwhich theareaof revolution isonlythatoftheendcircles ofradius yland_)’2,respectively, isknown as theGoldschmidt .\'0luti0n. Forthesymmetric situation discussed inExercise 7,obtain anexpression fortheratiooftheareagenerated bythecatenary solutions tothatgiven bytheGoldschmidt solution. Your result should beafunction onlyoftheparameters kandoi.Show thatforsufﬁciently Iarge values ofoiatleast oneofthecatenaries gives anareabelow thatoftheGoldschmidt solution. Ontheother hand, show thatif oi=org,theGoldschmidt solution gives alower areathanthecatenary. Exercises 65 Achain orropeofindeﬁnite length passes freely overpulleys atheights yrandY2 above theplane surface ofEarth, withahorizontal distance x;—xibetween them. If thechain orropehasauniform linear mass density, show thattheproblem ofﬁnding thecurve assumed between thepulleys isidentical withthatoftheproblem ofmini- mum surface ofrevolution. (The transition totheGoldschmidt solution astheheights y]and_)’2arechanged makes forastrikrrg lecture demonstration. SeeExercise 8.) Suppose itisknown experimentally that.aparticle fellagiven distance yoinatime £0=,/23:0/g, butthetimes offallfordistances other thanyoisnotknown. Suppose further thattheLagrangian fortheproblem isknown, butthatinstead ofsolving the equation ofmotion foryasafunction oft,itisguessed thatthefunctional form is y=at +I?t2. Iftheconstants aandbareadjusted always sothatthetime tofallyoiscorrectly given by:0,show directly thattheintegral to ILdt 0 isanextremum forrealvalues ofthecoefﬁcients onlywhen a=0andb=g/2. When twobilliard balls colhde, theinstantaneous forces between them areverylarge butactonlyinaninﬁmtesimal timeAr,insuchamanner thatthequantity fFdt Al remains ﬁmte. Such forces aredescribed asimpulsive forces, andtheintegral over Atisknown astheimpulse oftheforce. Show thatifimpulsive forces arepresent L.agrange’s equations maybetransformed into (Ml(Ml% _ . =SJ, 341f 3'11i Where thesubscripts iandfrefer tothestate ofthesystem before andafter the impulse, SJistheimpulse ofthegeneralized impulsive force corresponding toqJ, andListheLagrangian including allthenonimpulsive forces. Thetermgeneralized mechanics hascome todesignate avariety ofclassical mechan- icsinwhich theLagrangian contains timederivatives ofq,higher thantheﬁrst.Prob- lems forwhich x=f(x,>2,35,r)have been referred toas“jerky” mechanics. Such equations ofmotion haveinteresting applications inchaos theory (cf.Chapter 11).By applying themethods ofthecalculus ofvariations, show thatifthere isaLagrangian oftheform L(q,.Q,-.Q,,r),andHamilton’s principle holds withthezerovariation of bothq,andq,attheendpoints, thenthecorresponding Euler-Lagrange equations are dzat .1at aL,_- ,+=0. '=1,2,..,.<1r2(8q.) dries) sq. ‘ " Apply thisresult totheLagrangian Chapter 2Variational PI'lI'l(.Ip|(3S andLagrange’s Equations L__ m.. kg Doyourecogmze theequations ofmotion? Aheavy particle isplaced atthetopofavertical hoop. Calculate thereaction of thehoop ontheparticle bymeans oftheLagrange’s undetermined multipliers and Lagraiige’s equations. Findtheheight atwhich theparticle fallsoff. Auniform hoop ofmass mandradius rrolls without slipping onaﬁxed cylinder ofradius Rasshown intheﬁgure. Theonlyexternal force isthatofgravity. Ifthe smaller cylinder starts rolling from restontopofthebigger cylinder, usethemethod ofLagrange mulipliers toﬁndthepoint atwhich thehoop fallsoffthecylinder. AformoftheWheatstone impedance bridge has,inaddition totheusual fourresis- tances. aninductance inonearmandacapacitance intheopposite arm.SetupLand .7:fortheunbalanced budge. withthecharges intheelements ascoordinates. Using theKirchhoff junction conditions asconstraints onthecurrents, obtain theLagrange equations ofmotion, andshow thateliminating theJt’sreduces these totheusual net- work equations. Incertain bllL1Zi[10l'lS, particularly one-dimensional systems, itispossible toincorpo- ratefrictional effects without introducing thedissipation function. Asanexample, ﬁnd theequations ofmotion fortheLagrangian L=eJ/T _ 2 2 How would youdescribe thesystem? Arethere anyconstants ofmotion? Suppose a point transformation ismade oftheform s=cY'q. What istheeffective Lagrangian interms ofs?Find theequation ofmotion for.s. What dothese results sayabout theconserved quantities forthesystem? Itsometimes occurs thatthegeneralized coordinates appear separately irithekinetic energy andthepotential energy insuch amaimer thatTandVmaybewritten inthe form T=Zf.<q,>4,’ andv=ZjvaqnI I Exercises 67 Show thatLagrange’s equations thenseparate, andthattheproblein canalways be reduced toquadratures. Apoint mass isconstrained tomove onamassless hoop ofradius aﬁxed inavertical plane thatrotates about itsvertical symmetry axiswith constant angular speed w. Obtain theLagrange equations ofmotion assuming theonlyexternal forces arisefrom gravity. What aretheconstants ofmotion? Show thatifcu1Sgreater thanacritical value coo,there canbeasolution inwhich theparticle remains stationary onthehoop atapoint other thanatthebottom, butthatifw<600,theonlystationary point forthe particle isatthebottom ofthehoop. What isthevalue of£00? Aparticle moves without friction inaconservative ﬁeldofforce produced byvarious mass distributions. Ineach instance, theforce generated byavolume element ofthe distribution isderived from apotential thatisproportional tothemass ofthevolume element andisafunction onlyofthescalar distance from thevolume element. Forthe following ﬁxed, homogeneous mass distributions, statetheconserved quantities inthe motion oftheparticle: (a)Themass isunifomily distributed intheplane z=0. (h)Themass isuniformly distributed inthehalf-plane z==0,y>0. (c)Themass isunifomily distributed inacircular cylinder ofinﬁnite length, with axisalong thezaxis. (d)Themassisunifomily distributed inacircular cylinder ofﬁnitelength, withaxis along thezaxis. (e)ThemasslSumtormly distributed inan'ghtcylinder ofelliptical crosssection and inhmte length. withaxisalong thezaxis. (f)Themassisunitomily distributed inadumbbell whose axisisoriented along the zaxis. (g)Themass isintheform ofauniform wirewound inthegeometry ofaninﬁnite helical solenoid, withaxisalong thezaxis Aparticle ofmass mslides without friction onawedge ofangle Otandmass Mthatcan move without friction onasmooth horizontal surface, asshown intheﬁgure. Treating theconstraint oftheparticle onthewedge bythemethod ofLagrange multipliers, ﬁndtheequations ofmotion fortheparticle andwedge. Also obtain anexpression for theforces ofconstraint. Calculate thework done intimer bytheforces ofconstraint acting ontheparticle andonthewedge. What aretheconstants ofmotion forthe system’? Contrast theresults youhave found with thesituation when thewedge is ﬁxed. |Suggesnon: Fortheparticle youmayeithei useaCartesian coordinate system withyvertical, oronewithynormal tothewedge or.evenmore instructively, doitin bothsystems] m L /€ Chapter 2Variational Principles andLagrangt-3'5 Equations Acarriage runsalong railsonarigid beam, asshown intheﬁgure below. Thecarriage isattached tooneendofaspring ofequilibrium length r0andforce constant k,whose other endisﬁxed onthebeam. Onthecarriage, another setofrailsisperpendicular to theﬁrstalong which aparticle ofmass mmoves, heldbyaspring ﬁxed onthebeam, offorceconstant kandzeroequilibrium length. Beam, rails,springs, andcarriage are assumed tohave reromass. Thewhole system isforced tomove inaplane about the point ofattachment oftheﬁrstspring, withaconstant angular speed 0).Thelength of thesecond spring isatalltimes considered small compared tor9. (a)What istheenergy ofthesystem‘? Isitconserved? (b)Using generalized coordinates inthelaboratory system, what istheJacobi integral forthesystem? Isitconserved‘? (c)Interms ofthegenerali-zed coordinates relative toasystem rotating withtheangu- larspeed w.what istheLagrangian? What istheJacobi integral? Isitconserved? Discuss therelationship between thetwoJacobi integrals. m /ilk 1IiL/ .,~.s>K,’ /(Q I1,’ O c” Suppose aparticle moves inspace subject toaconservative potential V(r) butis constrained toalways move onasurface whose equation is0(r,t)=0.(The explicit dependence onitindicates thatthesurface maybemoving.) Theinstantaneous force of constraint istaken asalways perpendicular tothesurface. Show analytically thatthe energy oftheparticle isnotconserved ifthesurface moves intime. What physically isthereason fornonconservation oftheenergy under thiscircumstance? Consider twoparticles ofmasses mlandmg.Letm]beconﬁned tomove onacircle ofradius ainthez=0plane, centered atx=y=0.Letmlbeconﬁned tomove onacircle ofradius binthez=cplane, centered atx=y=0.Alight (massless) spring ofspring constant kisattached between thetwoparticles. (a)FindtheLagrangian forthesystem. (b)Solve theproblem using Lagrange multipliers andgiveaphysical interpretation foreachmultiplier. Theone-dimensional harmonic oscillator hastheLagrangian L=m,\':2/2 —Icxz/2. Suppose youdidnotknow thesolution tothemotion, butrealized thatthemotion must beperiodic andtherefore could bedescribed byaFouner senes oftheform x(t)=Ea} cosjrot, J=° Exercises 69 (taking r=0atatuming point) where cuisthe(unknown) angular frequency ofthe motion. This representation forx(r)deﬁnes amany-parameter pathforthesystem point inconﬁguration space. Consider theaction integral Ifortwopoints, t|andt2 separated bytheperiod T=Zn/cu. Show thatwiththisform forthesystem path, Iis anextremum fornonvanishing xonlyifaJ=0,forjgé1,andonlyiftug=klm. Adiskofradius Rrollswithout slipping inside thestationary parabola y=axz.Find theequations ofconstraint. What condition allows thedisktorollsothatittouches theparabola atoneandonlyonepoint independent ofitsposition? Aparticle otmass m1Ssuspended byamassless spring oflength L.Ithangs, without initial motion, inagravitational ﬁeldofstrength g.Itisstruck byannnpulsive hor- izontal blow, which introduces anangular velocity co.Ifnoissufﬁciently small, itis obvious thatthemass moves asasimple pendulum. Ifwissufficiently large, themass willrotate about thesupport. UseaLagrange multiplier todetermine theconditions under which thestring becomes slack atsome point inthemotion. CHAPTER 3.1I 70TheCentral Force Problem Inthischapter weshalldiscuss theproblem oftwobodies moving under thein- ﬂuence ofamutual central force asanapplication oftheLagrangian formulation. Notalltheproblems ofcentral force motion areintegrable interms ofwell-known functions. However, weshall attempt toexplore theproblem asthoroughly asis possible withthetools already developed. Inthelastsection ofthischapter we consider some ofthecomplications thatfollow bythepresence ofathirdbody. REDUCTION TOTHE EQUIVALENT ONE-BODY PROBLEM Consider amonogenic system oftwomass points, m1andmg(cf.Fig.3.1),where theonlyforces arethose cluetoaninteraction potential U.Wewillassume atﬁrst thatUisanyfunction ofthevector between thetwoparticles, F2-1'1,oroftheir relative velocity, i‘;—i'|,orofanyhigher derivatives of1'2—r1.Such asystem hassixdegrees offreedom andhence sixindependent generalized coordinates. Wechoose these tobethethree components oftheradius vector tothecenter of mass, R,plusthethree components ofthedifference vector r=1'2—r1.The Lagrangian willthenhave theform L=T(R,r)-U(r,r,...). (3.1) ml 1‘ R "'1 FIGURE 3.1 Coordinates forthetwo-body problem. 3.1 Reduction totheEquivalent One-Body Problem 71 Thekinetic energy Tcanbewritten asthesumofthekinetic energy ofthe motion ofthecenter ofmass, plusthekinetic energy ofmotion about thecenter ofmass, T’: T=%m+mnW+W with T’=%m1i"|2 +%mgi‘g. Here r’landrflaretheradii vectors ofthetwoparticles relative tothecenter of mass andarerelated torby "12I’,=-—r.mi+mz I "I 1'2= r Expressed interms ofrbymeans ofEq.(3.2), T’takes ontheform T/=1 mlml i_2 2mi+m2 andthetotalLagrangian (3.1)is L=ﬂ533W+li55L¥~Umn J mm2 2m1+m2 Itisseen thatthethree coordinates Rarecyclic, sothatthecenter ofmass iseither atrestormoving uniformly. None oftheequations ofmotion forrwill contain terms involving RorR.Consequently, theprocess ofintegration ispar- ticularly simple here. Wemerely drop theﬁrstterm from theLagrangian inall subsequent discussion. TherestoftheLagrangian isexactly what would beexpected ifwehadaﬁxed center offorce withasingle particle atadistance rfrom it,having amass mmz=--, 34 #m+m2 () where itisknown asthereduced mass. Frequently, Eq.(3.4)iswritten intheform 111_=__+_= as/1' ml m2 Thus, thecentral force motion oftwobodies about theircenter ofmass canalways bereduced toanequivalent one-body problem. 3.2IChapter 3TheCentral Force Problem THE EQUATIONS OFMOTION AND FIRST INTEGRALS Wenowrestrict ourselves toconservative central forces, where thepotential is V(r), afunction ofronly, sothattheforce isalways along r.Bytheresults of thepreceding section, weneed only consider theproblem ofasingle particle of reduced mass mmoving about aﬁxed center offorce, which willbetaken asthe origin orthecoordinate system. Since potential energy involves only theradial distance, theproblem hasspherical symmetry; i.e.,anyrotation, about anyﬁxed axis, canhave noeffect onthesolution. Hence, anangle coordinate representing rotation about aﬁxed axismust becyclic. These syrmnetry properties result ina considerable simpliﬁcation intheproblem. Since theproblem isspherically symmetric, thetotalangular momentum vec- tor, L=i-xp, isconserved. Ittherefore follows thatrisalways perpendicular totheﬁxed direc- tionofLinspace. Thiscanbetrueonlyifralways liesinaplane whose normal isparallel toL.While thisreasoning breaks down ifLiszero, themotion inthat casemust bealong astraight linegoing through thecenter offorce, forL=0 requires rtobeparallel toi",which canbesatisﬁed onlyinstraight-line motion.* Thus, central force motion isalways motion inaplane. Now, themotion ofasingle particle inspace isdescribed bythree coordinates; inspherical polar coordinates these aretheazimuth angle 0,thezenith angle (or colatitude) 1/r,andtheradial distance r.Bychoosing thepolar axistobeinthe direction ofL,themotion isalways intheplane perpendicular tothepolar axis. Thecoordinate 1//thenhasonlytheconstant value rt/2andcanbedropped from thesubsequent discussion. Theconservation oftheangular momentum vector fur- nishes three independent constants ofmotion (corresponding tothethree Carte- siancomponents). lneffect, twoofthese, expressing theconstant direction ofthe angular momentum, have been usedtoreduce theproblem from three totwode- grees offreedom. Thethird ofthese constants, corresponding totheconservation ofthemagnitude ofL,remains stillatourdisposal incompleting thesolution. Expressed nowinplane polar coordinates, theLagrangian is L=T—V =§m(r2—l—r2(§2) -V(r). (3.6) Aswasforseen, 9isacyclic coordinate, whose corresponding canonical momen- tumistheangular momentum ofthesystem: dL .pg=—.="W26. 89 *Formally i"=I-n,+rélng, hence rxi‘=0requires El=0. 3.2 TheEquations ofMotion andFirstIntegrals 73 Oneofthetwoequations ofmotion isthensimply . d -pg=E(W20) =0. (3.7) withtheimmediate integral mi-29=1. (3.8) where listheconstant magnitude oftheangular momentum. From (3.7) isalso follows that d1.E(726) =0. (3.9) Thefactor %isinserted because %r2(§ isjusttheareal vel0city—the areaswept outbytheradius vector perunittime. Thisinterpretation follows from Fig.3.2, thedifferential areaswept outintimedtbeing dA=%r(rae), andhence a'A_1r2d6 atT2at' Theconservation ofangular momentum isthusequivalent tosaying theareal velocity isconstant. Herewehavetheproof ofthewell-known Kepler’s second lawofplanetary motion: Theradius vector sweeps outequal areas inequal times. Itshould beemphasized however thattheconservation ofthearealvelocity isa general property ofcentral force motion andisnotrestricted toaninverse-square lawofforce. rdﬁI r d9 FIGURE 3.2 Theareaswept outbytheradius vector inatimedt. Chapter 3TheCentral Force Problem Theremaining Lagrange equation, forthecoordinate r.is d _. E(mr") -mr02+ =0. (3.10) Designating thevalue oftheforce along r,—8V/Br, byf(r)theequation canbe rewritten as mi‘-mi-62=f(r). (3.11) Bymaking useoftheﬁrstintegral, Eq.(3.8), élcanbeeliminated from theequa- tionofmotion, yielding asecond-order differential equation involving ronly: .. I2mr—F =f(r). (3.12) There isanother ﬁrstintegral ofmotion available, namely thetotal energy, since theforces areconservative. Onthebasis ofthegeneral energy conservation theorem, wecanimmediately statethataconstant ofthemotion is E=§m(r2+#92)+V(r), (3.13) where Eistheenergy ofthesystem. Altematively, thisﬁrstintegral could be derived again directly from theequations ofmotion (3.7) and(3.12). Thelatter canbewritten as __d 112m7'=—$ (3.14) Ifbothsides ofEq.(3.14) aremultiplied byrtheleftsidebecomes rt‘I1#1 m =— —m .at2 Theright sidesimilarly canbewritten asatotaltimederivative, forifg(r)isany function ofr,thenthetotaltimederivative ofghastheform d dgdr at“)=an Hence, Eq.(3.14) isequivalent to a1, 4 112__*-=__ V__ at<2”) dt(+2W2) 41,2112 i_E(5mi +—i+V)-0.or 2mrz 3.2 TheEquations ofMotion andFirstIntegrals 75 andtherefore 21. llimrz + +V=constant. (3.15) Equation (3.15) isthestatement oftheconservation oftotal energy, forbyus- ing(3.8) forl,themiddle term canbewritten 112 1 . 262__.=__2m1,4@2 =L.2mr- 2mr 2 and(3.15) reduces to(3.13). These ﬁrsttwointegrals giveusineffect twoofthequadratures necessary to complete theproblem. Asthere aretwovariables, rand9,atotaloffourinte- grations areneeded tosolve theequations ofmotion. Theﬁrsttwointegrations have lefttheLagrange equations astwoﬁrst-order equations (3.8) and(3.15);the tworemaining integrations canbeaccomplished (formally) inavariety ofways. Perhaps thesimplest procedure starts from Eq.(3.15). Solving for1‘,wehave 2 i= i%(E—V—#), (3.16) at:i"’.:_. (3.11)A __L,/..(EVW) Attimet=O,letrhavetheinitial value r0.Then theintegral ofbothsides ofthe equation from theinitial state tothestateattimettakes theform r ¢=f __‘1'_-_. (3.18) ’°/%(E-V— ' and Asitstands, Eq.(3.18) gives 1asafunction ofr theconstants ofintegration E,I.andrg.However, itmaybeinverted, atleastformally, togiverasafunction oftandtheconstants. Once thesolution forrisfound, thesolution 9follows immediately from Eq.(3.8), which canbewritten asOI‘ $1: Idd6= (3.19)mr Iftheinitial value of6is90,thentheintegral of(3.19) issimply 0—zf ‘Z’+0 (320)0mr2(l) 0' i 3.3IChapter 3TheCentral Force Problem Equations (3.18) and(3.20) arethetworemaining integrations, andformally theproblem hasbeen reduced toquadratures, withfourconstants ofintegration E, l,ro,60.These constants arenottheonlyonesthatcanbeconsidered. Wemight equally aswellhavetaken r0,60,fr),90,butofcourse Eandlcanalways bedeter- mined intenns ofthisset.Formany applications, however, thesetcontaining the energy andangular momentum isthenatural one.Inquantum mechanics, such constants astheinitial values ofrand9,orof1‘and9,become meaningless. but wecanstilltalkinterms ofthesystem energy orofthesystem angular momen- tum. Indeed, twosalient differences between classical andquantum mechanics appear intheproperties ofEandlinthetwotheories. Inorder todiscuss the transition toquantum theories, itistherefore important thattheclassical descrip- tionofthesystem beinterms ofitsenergy andangular momentum. THE EQUIVALENT ONE-DIMENSIONAL PROBLEM, AND CLASSIFICATION OFORBITS Although wehave solved theone-dimensional problem formally, practically speaking theintegrals (3.18) and(3.20) areusually quite unmanageable, andin anyspeciﬁc caseitisoften more convenient toperform theintegration insome other fashion. Butbefore obtaining thesolution foranyspeciﬁc force laws, let usseewhat canbelearned about themotion inthegeneral case, using onlythe equations ofmotion andtheconservation theorems, without requiring explicit solutions. Forexample, withasystem ofknown energy andangular momentum, themag- nitude anddirection ofthevelocity oftheparticle canbeimmediately determined interms ofthedistance r.Themagnitude vfollows atonce from theconservation ofenergy intheform E=émvz +V(r) v=‘iE(E—V(r)). (3.21)m Theradial velocity—the component ofi‘along theradius vector—has been given inEq.(3.16). Combined with themagnitude v,thisissufﬁcient information to furnish thedirection ofthevelocity.* These results, andmuch more, canalsobe obtained fromconsideration ofanequivalent one-dimensional problem. Theequation ofmotion inr,with6*expressed interms ofl,Eq.(3.12), involves only randitsderivatives. Itisthesame equation aswould beobtained foraO1‘ *Altematrvely, theconservation ofangular momentum fumrshes 9,theangular velocity, andthisto- gether withi-givesboththemagnitude anddirection ofi'. 3.3 TheEquivalent One-Dimensional Problem 77 ﬁctitious one-dimensional problem inwhich aparticle ofmass missubject toa force 2 f’=f+ (3.22)mi‘ Thesigniﬁcance oftheadditional termisclearifitiswritten asmréz =mug/r, which isthefamiliar centrifugal force. Anequivalent statement canbeobtained from theconservation theorem forenergy. ByEq.(3.15) themotion oftheparticle inristhatofaone-dimensional problem withaﬁctitious potential energy: V'=V+li (322’)2mr2' ' Asacheck, notethat av’ 12f'=-"aT=f(r)-F‘,mr3 which agrees withEq.(3.22). Theenergy conservation theorem (3.15) canthus alsobewritten as E=v’+gmﬁ. (3.1s') Asanillustration ofthismethod ofexamining themotion, consider aplotof V’against rforthespeciﬁc caseofanattractive inverse-square lawofforce: k f——r—2- (Forpositive k,theminus signensures thattheforce istoward thecenter offorce.) Thepotential energy forthisforce is v=-5,r andthecorresponding ﬁctitious potential is k12V’=-—— . r+2mr2 Such aplotisshown inFig.3.3;thetwodashed lines represent theseparate com- ponents k 12__ d ___’ r an 2mr2 andthesolid lineisthesumV’. Chapter 3TheCentral Force Problem [2 ‘Q7»._ ¢-a"""-"‘._ f|\_)~3"QDJ x 777 \ E‘\ \ \ \\ V’ \'~ _~ "*_____ 3C t——'~ E=0 ‘ 2 1 4* J‘- l //f E3 f. / E/ 4 // k/"=-TIT / FIGURE 3.3 Theequivalent one-dimensional potential forattractive inverse-square law offorce. Letusconsider nowthemotion ofaparticle having theenergy E1.asshown in Figs. 3.3and3.4.Clearly thisparticle cannever come closer thanrl(cf.Fig.3.4). Otherwise withr<r1,V’exceeds E1andbyEq.(3.15’)thekinetic energy would have tobenegative, corresponding toanimaginary velocity! Ontheother hand, there isnoupper limit tothepossible value ofr,sotheorbit isnotbounded. A particle willcome infrom inﬁnity, strike the“repulsive centrifugal barrier,” be repelled, andtravel backouttoinﬁnity (cf.Fig.3.5).Thedistance between Eand V’is%mr"2, i.e.,proportional tothesquare oftheradial velocity, andbecomes zero,naturally, attheturning point r1.Atthesame time,thedistance between E andVontheplotisthekinetic energy %mv2 atthegiven value ofr.Hence, the distance between theVandV’curves isémrzéz. These cuwes therefore supply themagnitude oftheparticle velocity anditscomponents foranydistance r,atthe given energy andangular momentum. Thisinformation issufﬁcient toproduce an approximate picture oftheform oftheorbit. Fortheenergy E2=0(cf.Fig.3.3),aroughly similar picture oftheorbit behavior isobtained. Butforanylower energy, suchasE3indicated inFig.3.6, wehave adifferent Story. Inaddition toalower bound r;,there isalsoamaximum value rgthatcannot beexceeded byrwithpositive kinetic energy. Themotion is then“bounded,” andtherearetwoturning points, r1andrg,alsoknown asapsidal distances. Thisdoes notnecessarily mean thattheorbits areclosed. A11thatcan besaidisthattheyarebounded, contained between twocircles ofradius r1and r2withturning points always lying onthecircles (cf.Fig.3.7). 3.3 TheEquivalent One-Dimensional Problem 79 V’II l El W-7r--—--——-%mr2 I’-——> VI FIGURE 3.4Unbounded motion atpositive energies forinverse-square lawofforce I‘ FIGURE 3.5 TheorbitforE1corresponding tounbounded motion. Chapter 3TheCentral Force Problem V’ l 7..-;_____J_, ___",-‘Q _-__-_-__-N Piulfi-m FIGURE 3.6 Theequivalent one-dimensional potential forinverse-square lawofforce, illustrating bounded motion atnegative energies. Iftheenergy isE4attheminimum oftheﬁctitious potential asshown in Fig.3.8,thenthetwobounds coincide. Insuch case, motion ispossible atonly oneradius; 2"=0,andtheorbit isacircle. Remembering thattheeffective “force” isthenegative oftheslope oftheV’curve, therequirement forcircular orbits is simply thatf’bezero, or 12 .2f(F) = =—mr6 . Wehave herethefamiliar elementary condition foracircular orbit, thattheap- plied force beequal andopposite tothe“reversed effective force” ofcentripetal /I $- ri‘H FIGURE 3.7 Thenature oftheorbits forbounded motion. 3.3 TheEquivalent One-Dimensional Problem 81 IL fl 1'7.- ‘ft FIGURE 3.8 Theequivalent one-dimensional potential ofinverse-squa.re lawofforce. illustrating thecondition forcircular orbits. acceleration.* Theproperties ofcircular orbits andtheconditions forthem will bestudied ingreater detail inSection 3.6. Note thatallofthisdiscussion oftheorbits forvarious energies hasbeen at onevalue oftheangular momentum. Changing lchanges thequantitative details oftheV’curve, butitdoes notaffect thegeneral classiﬁcation ofthetypes of orbits. Fortheattractive inverse-square lawofforce discussed above, weshall see thattheorbitforE1isahyperbola, forE2aparabola, andforE3anellipse. With other forces theorbits maynothave suchsimple forms. However, thesame general qualitative division intoopen, bounded, andcircular orbits willbetrue foranyattractive potential that(1)fallsoffslower than1/r2asr—>oo,and (2)becomes inﬁnite slower than1/r2asr—>0.Theﬁrstcondition ensures that thepotential predominates overthecentrifugal term forlarge r,while thesecond condition issuchthatforsmall ritisthecentrifugal termthatisimportant. Thequalitative nature ofthemotion willbealtered ifthepotential doesnotsat- isfythese requirements, butwemaystillusethemethod oftheequivalent poten- tialtoexamine features oftheorbits. Asanexample, letusconsider theattractive potential a _ 3V(r) =-3, with f=—7;‘-. Theenergy diagram isthenasshown inFig.3.9.Foranenergy E,there aretwo possible types ofmotion, depending upon theinitial value ofr.If7'9islessthan r1themotion willbebounded, rwillalways remain lessthanr1,andtheparticle willpassthrough thecenter offorce. Ifrisinitially greater thanr2,thenitwill *Thc caseE<E4doesnotcorrespond tophysically possible motion, forthen1'-2would havetobe negative, ori-imaginary. Chapter 3TheCentral Force Problem i \.£LZmrz \ V \ \ \ \ \ \\ E w*~—-\.\\\\_\ '7-—. ' ‘_—-—-I; / / / V’ /V I FIGURE 3.9 Theequivalent one-dimensional potential foranattractive inverse-fourth lawofforce always remain so;themotion isunbounded, andtheparticle cannever getinside the“potential” hole. Theinitial condition r1<ro<r2isagain notphysically possible. Another interesting example ofthemethod occurs foralinear restoring force (isotropic harmonic oscillator): f=—kr, v=ikrz. Forzeroangular momentum, corresponding tomotion along astraight line,V’= Vandthesituation isasshown inFig.3.10. Foranypositive energy themotion is bounded and,asweknow. simple harmonic. Ifl9k0,wehavethestateofaffairs shown inFig.3.11.Themotion thenisalways bounded forallphysically possible T i=0 VI 15 V'=V=-L 22/tr Fib- FIGURE 3.10 Effective potential forzeroangular momentum. 304 -3.4TheVirial Theorem 83 I*i0 ‘_—i> ____7.,‘Z\’/v)(//\EI,\\ .5‘-L----\1,\E l-i 2| iV—2k!‘J|'/ __|,---’|5r| fl} FIGURE 3.11 Theequivalent one-dimensional potential foralinear restoring force. energies anddoesnotpassthrough thecenter offorce. Inthisparticular case,itis easily seenthattheorbit iselliptic, foriff=—kr, thex-andy-components of theforce are fx Z --/(X, fy 1' —ky. Thetotalmotion isthustheresultant oftwosimple harmonic oscillations atright angles, andofthesame frequency, which 1ngeneral leadstoanelliptic orbit. Awell-known example isthespherical pendulum forsmall amplitudes. The familiar Lissajous ﬁgures areobtained asthecomposition oftwosinusoidal os- cillations atright angles where theratio ofthefrequencies isarational number. Fortwooscillations atthesame frequency, theﬁgure isastraight linewhen the oscillations areinphase, acircle when theyare90°outofphase, andanelliptic shape otherwise. Thus, central force motion under alinear restoring force there- foreprovides thesimplest oftheLissajous ﬁgures. THE VIRIAI. THEOREM Another property ofcentral force motion canbederived asaspecial caseofa general theorem valid foralarge variety ofsystems—the virial theorem. Itdiffers incharacter fromthetheorems previously discussed inbeing statistical innature; i.e.,itisconcemed withthetimeaverages ofvarious mechanical quantities. Consider ageneral system ofmass points withposition vectors 1',andapplied forces F,(including anyforces ofconstraint). Thefundamental equations ofmo- tionarethen 151=Fr (1-3) Weareinterested inthequantity Chapter 3TheCentral Force Problem G=ZPr '1'r~ 1' where thesummation isoverallparticles inthesystem. Thetotaltimederivative ofthisquantity is r1G . . 7t=Zr.-p.+Zp.-r.. (3-23>I I Theﬁrsttermcanbetransformed to Zfvpl I I I while thesecond temiby(1.3)is Z131 '1': =21?: ‘rt- : z Equation (3.23) therefore reduces to iipﬂrl =21.-I-zF,'l‘,. dt I I Thetimeaverage ofEq.(3.24) overatimeinterval r1sobtained byintegrating bothsides withrespect totfrom 0tor,anddividing by1:: 1’dG E_——-—__-(1E_=2 .r/(‘) drIdz T+;F’ r‘ OI" —— 1ﬁ+Zr,-r;=;[cm-0(0)]. (3.25) Ifthemotion isperiodic, i.e.,allcoordinates repeat after acertain time, andifr ischosen tobetheperiod, thentheright-hand sideof(3.25) vanishes. Asimilar conclusion canbereached evenifthemotion isnotperiodic, provided thatthe coordinates andvelocities forallparticles remain ﬁnite sothatthere isanupper bound toG.Bychoosing rsufﬁciently long,theright-hand sideofEq.(3.25) can bemade assmall asdesired. Inbothcases, itthenfollows that _ 1—iT=-5;F, -i-,. (3.26) Equation (3.26) isknown asthevirial theorem, andtheright-hand sideiscalled thevirial ofClausius. LnthisfOI'mthetheorem isimporant inthekinetic theory 3.4TheVtrialTheorem 35 ofgases sinceitcanbeusedtoderive idealgaslawforperfect gases bymeans of thefollowing brief argument. Weconsider agasconsisting ofNatoms conﬁned within acontainer ofvol- umeV.Thegasisfurther assumed tobeataKelvin temperature T(nottobe confused withthesymbol forkinetic energy). Then bytheequipartition theorem ofkinetic theory, theaverage kinetic energy ofeach atom isgiven bygkBT,kg being theBoltzmann constant, arelation thatineffect isthedeﬁnition oftemper- ature. Theleft-hand sideofEq.(3.26) istherefore %NkBT Ontheright-hand sideofEq.(3.26), theforces F;include boththeforces of interaction between atoms andtheforces ofconstraint onthesystem. Aperfect gasisdeﬁned asoneforwhich theforces ofinteraction contribute negligibly to thevirial. Thisoccurs, e.g.,ifthegasissotenuous thatcollisions between atoms occur rarely, compared tocollisions with thewalls ofthecontainer. Itisthese walls thatconstitute theconstraint onthesystem, andtheforces ofconstraint, Fe, arelocalized atthewallandcome intoexistence whenever agasatom collides withthewall.Thesumontheright-hand sideofEq.(3.26) cantherefore bere- placed intheaverage byanintegral overthesurface ofthecontainer. Theforce ofconstraint represents thereaction ofthewalltothecollision forces exerted by theatoms onthewall,i.e.,tothepressure P.Withtheusual outward convention fortheunitvector ninthedirection ofthenormal tothesurface. wecantherefore write (IF; =: OI’ 1 P5Z:F,-r,-=——2-fn-rclA. But,byGauss‘s theorem, fn-rdA=fV-rdV =3V. Thevirial theorem, Eq.(3.26), forthesystem representing aperfect gascanthere- forebewritten %N@T=%PK which, cancelling thecommon factor ofgonboth sides. isthefamiliar ideal gaslaw.Where theinterparticle forces contribute tothevirial, theperfect gas lawofcourse nolonger holds. Thevirial theorem isthentheprincipal tool,in classical kinetic theory, forcalculating theequation ofstatecorresponding tosuch imperfect gases. 3.5 IChapter 3TheCentral Force Problem Wecanfurther show thatiftheforces F,arethesumofnonfrictional forces F: andfrictional forces f,-proportional tothevelocity, thenthevirial depends only ontheFl;thereisnocontribution fromtheI}.Ofcourse, themotion ofthesystem must notbeallowed todiedown asaresult ofthefrictional forces. Energy must constantly bepumped intothesystem tomaintain themotion; otherwise alltime averages would vanish as1:increases indeﬁnitely (cf.Derivation 1.) Iftheforces arederivable from apotential, thenthetheorem becomes _ lT=- VV- 3.2 2$ rlv ( andforasingle particle moving under acentral force itreduces to _ IBV IfVisapower-law function ofr, V=arllrhl’ where theexponent ischosen sothattheforce lawgoesasr",then V :7?‘=01+l)V, andEq.(3.28) becomes T="L517. (3.29) Byanapplication ofEuler’s theorem forhomogeneous functions (cf.p.62),itis clear thatEq.(3.29) alsoholds whenever Visahomogeneous function inrof degree n+I.Forthefurther special caseofinverse-square lawforces, nis-2, andthevirial theorem takes onawell-known form: T=_tv. (3.30) THE DIFFERENTIAL EQUATION FOR THE ORBIT, AND INTEGRABLE POWER-LAW POTENTIALS Intreating speciﬁc details ofactual central force problems, achange intheorien- tation ofourdiscussion isdesirable. Hitherto solving aproblem hasmeant ﬁnding rand0asfunctions oftimewith E,Z,etc.,asconstants ofintegration. Butmost often what wereally seek istheequation oftheorbit, i.e.,thedependence ofr upon 6,eliminating theparameter t.Forcentral force problems, theelimination is particularly simple, since toccurs intheequations ofmotion onlyasavariable of differentiation. indeed, oneequation ofmotion, (3.8), simply provides adeﬁnite 3.5 TheDifferential Equation fortheOrbit 37 relation between adifferential change dtandthecorresponding change d6: 1<zt=7117240. (3.31) Thecorresponding relation between derivatives withrespect totand6is d Id dt_mrzﬁl (3.32) These relations maybeusedtoconvert theequation ofmotion (3.12) or(3.16) to adifferential equation fortheorbit. Asubstitution intoEq.(3.12) gives asecond- order differential equation, while asubstitution intoEq.(3.17) gives asimpler ﬁrst-order differential equation. Thesubstitution intoEq.(3.I2)yields 1d1at 12 an E)-7;?=1‘<'>~ 6-”) which upon substituting u=1/randexpressing theresults interms ofthepoten- tialgives 42 a1—l‘+u——E~—V(;). (3.34)402 T12<1“ Thepreceding equation issuchthattheresulting orbit issymmetric about two adjacent turning points. Toprove thisstatement, notethatiftheorbit issymmet- rical itshould bepossible toreﬂect itabout thedirection ofthefuming angle without producing anychange. Ifthecoordinates arechosen sothattheturning point occurs for9=0,then thereﬂection canbeeffected mathematically by substituting -0for9.Thedifferential equation fortheorbit, (3.34), isobviously invariant under such asubstitution. Further theinitial conditions, here u=14(0), (ﬁx =0, for0 =0, willlikewise beunaffected. Hence. theorbit equation must bethesame whether expressed intemrs of6or-6,which isthedesired conclusion. Theorbit isthere- foreinvariant under reﬂection about theapsidal vectors. Ineffect, thismeans that thecomplete orbitcanbetraced iftheportion oftheorbitbetween anytwoturning points isknown. Reﬂection ofthegiven portion about oneoftheapsidal vectors produces aneighboring stretch oftheorbit, andthisprocess canberepeated in- deﬁnitely untiltherestoftheorbit iscompleted, asillustrated inFig.3.12. Foranyparticular force law,theactual equation oftheorbit canbeobtained by eliminating tfrom thesolution (3.17) bymeans of(3.31), resulting in de= I‘I’ . (3.35)2 mrz/%(E-V(r)- Chapter 3TheCentral Force Problem _-?/II“\\ \\ \ /’/f / FIGURE 3.12 Extension oftheorbit byreﬂection ofaportion about theapsidal vectors. Withslight rearrangements, theintegral of(3.35) is I" dr 9= —-i +90, (3-36)rt,,2/241; _M_LI I2 12 or,ifthevariable ofintegration ischanged tou=I/r, " d0=90-fA-. (3.37)up M _2i _"2if)2 )2 Asinthecaseoftheequation ofmotion, Eq.(3.37), while solving theproblem formally, isnotalways apracticable solution, because theintegral oftencannot be expressed interms ofwell-known functions. Infact, onlycertain types offorce lawshave been investigated. Thernost important arethepower-law functions ofr, V=ar"'H (3.38) sothattheforce varies atthenthpower ofr.*With thispotential, (3.37) becomes ll d e=00-I (3.39)(ti) _Z_;_g£u—n—1 _u2 Thisagain isintegrable interms ofsimple functions onlyincertain cases. The particular power-law exponents forwhich theresults canbeexpressed interms of trigonometric functions are n=1, —2,—3. *'l'he casen=-1istobeexcluded from thediscussion. Inthepotential (3.38), itcorresponds toa constant potential, 1.e,noforce atallItisancqually anomalous caseiftheexponent isusedinthe force lawdirectly, since aforce varying asr‘lcorresponds toalogarithmic potential, which isnota power lawatall.Aloganthmic potential isunusual formotion about apoint, itismore characteristic ofalinesource. Further details ofthese cases aregiven inthesecond edition ofthistext. 3.6I3.6 Conditions forClosed Orbits (Bertrand’s Theorem) 89 Theresults oftheintegral for n=5,3,0,-4,-5,-7 canbeexpressed interms ofelliptic functions. These areallthepossibilities foran integer exponent where theformal integrations areexpressed interms ofsimple well-known functions. Some fractional exponents canbeshovm toleadtoelliptic functions, andmany other exponents canbeexpressed interms ofthehyperge- ometric function. Thetrigonometric andelliptical functions arespecial cases of generalized hypergeometric function integrals. Equation (3.39) canofcourse be numerically integrated foranynonpathological potential, butthisisbeyond the scope ofthetext. CONDITIONS FOR CLOSED ORBITS (BERTRAND'S THEOREM) Wehavenotyetextracted alltheinformation thatcanbeobtained from theequiv- alent one-dimensional problem orfrom theorbitequation without explicitly solv- ingforthemotion. Inparticular, itispossible toderive apowerful andthought- provoking theorem onthetypes ofattractive central forces thatleadtoclosed orbits, i.e.,orbits inwhich theparticle eventually retraces itsownfootsteps. Conditions have already been described foronekindofclosed orbit, namely a circle about thecenter offorce. Foranygiven l,thiswilloccur iftheequivalent potential V’(r)hasaminimum ormaximum atsome distance roandiftheenergy Eisjustequal toV'(r()). Therequirement thatV’haveanextremum isequiva- lenttothevamshing off’atro,leading tothecondition derived previously (cf. Section 3.3). [2 f(ro)=——,. (3-40)mro which saystheforce must beattractive forcircular orbits tobepossible. Inaddi- tion,theenergy oftheparticle mustbegiven by I2 E=V(I‘()) +——. 3.41 Zmrg () which, byEq.(3.15), corresponds totherequirement thatforacircular orbit ris zero. Equations (3.40) and(3.41) arebothelementary andfamiliar. Between them theyimply thatforanyattractive central force itispossible tohave acircular orbit atsome arbitrary radius r0,provided theangular momentum lisgiven by Eq.(3.40) andtheparticle energy byEq.(3.41). Thecharacter ofthecircular orbit depends onwhether theextremum ofV’is aminimum, asinFig.3.8,oramaximum, asifFig.3.9.Iftheenergy isslightly above thatrequired foracircular orbit atthegiven value ofZ,thenforaminimum inV’themotion, though nolonger circular, willstillbebounded. However, if Chapter 3TheCentral Force Problem V’exhibits amaximum, thentheslightest raising ofEabove thecircular value, Eq.(3.34), results inmotion thatisunbounded, with theparticle moving both through thecenter offorce andouttoinﬁnity forthepotential shown inFig.3.9. Borrowing theterminology from thecaseofstatic equilibrium, thecircular orbit arising inFig.3.8issaidtobestable; thatinFig.3.9isunstable. Thestability ofthecircular orbit isthusdetermined bythesignofthesecond derivative ofV’ attheradius ofthecircle, being stable forpositive second derivative (V’concave up)andunstable forV’concave down. Astable orbittherefore occurs if a2v' af 312 r=rn !'=rQ 0 Using Eq.(3.40), thiscondition canbewritten 3f 3f(to)5 <-T, (3.43)r=l'(] OI‘ dhlf_- - .4.’dlnr >3 (33) r=rg where f(r0)/rgisassumed tobenegative andgiven bydividing Eq.(3.40) byr0. Iftheforce behaves likeapower lawofrinthevicinity ofthecircular radius r0, f=-kr". thenthestability condition, Eq.(3.43), becomes —knr"_1 <3kr"_1 or n>-3, (3.44) where kisassumed tobepositive. Apower-law attractive potential varying more slowly than1/r2isthuscapable ofstable circular orbits forallvalues ofro. Ifthecircular orbit isstable, thenasmall increase intheparticle energy above thevalue foracircular orbitresults inonlyaslight variation ofrabout ro.Itcan beeasily shown thatforsuchsmall deviations fromthecircularity conditions, the particle executes asimple harmonic motion ml,l(E1/r)about ug: u=ug+acos ﬁ9. (3.45) Here aisanamplitude thatdepends upon thedeviation oftheenergy from the value forcircular orbits, andﬁisaquantity arising from aTaylor series expansion 3.6 Conditions forClosed Orbits (Bertrand’s Theorem) 91 oftheforcelawf(r)about thecircular orbitradius r0.Direct substitution intothe force lawgives 52=3+LE . (3.46)fdr r=I'1) Astheradius vector oftheparticle sweeps completely around theplane, ugoes through )9cycles ofitsoscillation (cf.Fig.3.13). If/-3isarational number, the ratiooftwointegers, p/q,thenafterqrevolutions oftheradius vector theorbit would begin toretrace itselfsothattheorbitisclosed. Ateach rt;such thattheinequality inEq.(3.43) issatisﬁed, itispossible to establish astable circular orbit bygiving theparticle aninitial energy andangular momentum prescribed byEqs.(3.40) and(3.41). Thequestion naturally aiises as towhat form theforce lawmust takeinorder thattheslightly perturbed orbit about anyofthese circular orbits should beclosed. Itisclear thatunder these conditions )3must notonlybearational number, itmust alsobethesame rational number at alldistances thatacircular orbit ispossible. Otherwise, since )3cantakeononly discrete values, thenumber ofoscillatory periods would change discontinuousl y with ro,andindeed theorbits could notbeclosed atthediscontinuity. With 132 everywhere constant, thedeﬁning equation for/32,Eq.(3.46), becomes ineffect adifferential equation fortheforce lawfinterms oftheindependent variable r0. Wecanindeed consider Eq.(3.46) tobewritten interms ofrifwekeepin mind thattheequation isvalidonlyovertheranges inrforwhich stable circular orbits arepossible. Aslight rearrangement ofEq.(3.46) leadstotheequation dlnf __2E-17_53, (3.47) /// \\I \ \ \ /1/ FIGURE 3.13 Orbit formotion inacentral force deviating slightly from acircular orbit for)9=5. 3.7IChapter 3TheCentral Force Problem which canbeimmediately integrated togiveaforce law: k fm--)7, (3.48) Allforce lawsofthisform, with)9arational number, leadtoclosed stable orbits forinitial conditions thatdiffer onlyslightly from conditions deﬁning acircular orbit. Included within thepossibilities allowed byEq.(3.48) aresome familiar forces suchastheinverse-square law()9El),butofcourse many other behaviors, suchasf=—kr'2/9(}3 =Q),arealsopermitted. Suppose theinitial conditions deviate more thanslightly from therequirements forcircular orbits; willthese sameforce lawsstillgivecircular orbits? Theques- tioncanbeanswered directly bykeeping anadditional term intheTaylor series expansion oftheforcelawandsolving theresultant orbitequation. I.Bertrand solved thisproblem in1873andfound thatformorethanﬁrst-order deviations from circularity, theorbits areclosed onlyforﬁz=landﬁz=4.The ﬁrstofthese values ofﬁz,byEq.(3.48), leads tothefamiliar attractive inveise- square law;thesecond isanattractive force proportional totheradial distance- Hooke’s law!These force laws, andonlythese, could possibly produce closed orbits foranyarbitrary combination oflandE(E <0),andinfactweknow from direct solution oftheorbit equation thattheydo.Hence, wehaveBei'trand’s theorem: Theonlycentral forces thatresult tnclosed orbits_f0r allbound particles aretheinverse-square lawandHooke ’slaw. Thisisaremarkable result, wellworth thetedious algebra required. Itisacom- monplace astronomical observation thatbound celestial objects move inorbits thatareinﬁrstapproximation closed. Forthemost part,thesmall deviations from aclosed orbit aretraceable toperturbations suchasthepresence ofother bodies. Theprevalence ofclosed orbits holds truewhether weconsider onlythesolarsys- tem,orlooktothemany examples oftruebinary stars thathave been observed. Now, I-looke’s lawisamost unrealistic force lawtohold atalldistances, forit implies aforce increasing indeﬁnitely toinﬁnity. Thus, theexistence ofclosed orbits forawide range ofinitial conditions byitself leads totheconclusion that thegravitational force varies astheinverse-square ofthedistance. Wecanphrase thisconclusion inaslightly different marmer, onethatisof somewhat more signiﬁcance inmodem physics. Theorbital motion inaplane canbelooked onascompounded oftwooscillatory motions, oneinrandone in6with thesame period. Thecharacter oforbits inagravitational ﬁeld ﬁxes theform oftheforce law.Later onweshall encounter other formulations ofthe relation between degeneracy andthenature ofthepotential. THE KEPLER PROBLEM: INVERSE-SQUARE LAW OFFORCE Theinverse-square lawisthemost important ofallthecentral force laws, andit deserves detailed treatment. Forthiscase, theforce andpotential canbewritten 3.7 TheKepler Problem: Inverse-Square LawofForce 93 as k kf=——2 V=——. (3.49) r r There areseveral ways tointegrate theequation fortheorbit, thesimplest being to substitute (3.49) inthedifferential equation fortheorbit (3.33). Another approach istostartwithEq.(3.39) withnsetequal to-2forthegravitational force d0=e’- I-—-“i-, (3.50) where theintegral isnowtaken asindeﬁnite. Thequantity 9'appearing in(3.50) isaconstant ofintegration determined bytheinitial conditions andwillnotnec- essarily bethesame astheinitial angle 90attimet=0.Theindeﬁnite integral is ofthestandard form, I ‘ix 1aros5+2” (351) Z4?‘ cc‘-4, . \/a+f3x+)/x3 ~/-1’ \/‘Y where q=ti’—4w- Toapply thisto(3.50), wemustset 2mE 2mk andthediscriminant qistherefore Zmk2 2511q= (1+W). (3.53) With these substitutes, Eq.(3.50) becomes '2-"-19=6’—arccos--""4-. (3.54)2512 \/1+W Finally, bysolving foru,El/r,theequation oftheorbit isfound tobe imk l2512 ,;_lT 1+1+mk2cos(6—9) . (3.55) Theconstant ofintegration 9’cannowbeidentiﬁed from Eq.(3.55) asoneofthe turning angles oftheorbit. Note thatonlythree ofthefourconstants ofintegration appear intheorbitequation; thisisalways acharacteristic property oftheorbit. In Chapter 3TheCentral Force Problem effect, thefourth constant locates theinitial position oftheparticle ontheorbit. If weareinterested solely intheorbit equation, thisinformation isclearly irrelevant andhence does notappear intheanswer. Ofcourse, themissing constant hasto besupplied ifwewish tocomplete thesolution byﬁnding rand6asfunctions oftime. Thus, ifwechoose tointegrate theconservation theorem forangular momentum, mrz d9=ldt, bymeans of(3.55), wemust additionally specify theinitial angle 90. Now, thegeneral equation ofaconic withonefocus attheorigin is 5-=C[1+ ecos(6 —6')], (3.56) where eistheeccentricity oftheconic section. Bycomparison withEq.(3.55), it follows thattheorbit isalways aconic section, withtheeccentricity l 2El2 Thenature oftheorbit depends upon themagnitude ofeaccording tothefollow- ingscheme: e>1. E>0: hyperbola, e=l, E=O: parabola, e<1, E<O: ellipse, 2mke=0, E=—i: circle.212 This classiﬁcation agrees withthequalitative discussion oftheorbits onthe energy diagram oftheequivalent one-dimensional potential V’.Thecondition for circular motion appears hereinasomewhat different form, butitcaneasily be derived asaconsequence oftheprevious conditions forcircularity. Foracircular orbit, TandVareconstant intime, andfrom thevirial theorem V VEETV=—— =—. + 2+V 2 Hence kE=-——. (3.58) Zrg ButfromEq.(3.41), thestatement ofequilibrium between thecentral force and the“effective force,” wecanwrite k12 7'3 mrg , 3.7 TheKepler Problem: lnverse-Square LawofForce 95 or I2 !‘()=——. (3.59) mk Withthisformula fortheorbital radius, Eq.(3.58) becomes k2 E=—L, 212 theabove condition forcircular motion. Inthecaseofelliptic orbits, itcanbeshown themajor axisdepends solely upon theenergy, atheorem ofconsiderable importance intheBohr theory ofthe atom. Thesemimajor axisisone-half thesumofthetwoapsidal distances r1and r2(cf.Fig.3.6).Bydeﬁnition, theradial velocity iszeroatthese points, andthe conservation ofenergy implies thattheapsidal distances aretherefore therootsof theequation (cf.Eq.(3.15)) 12 1<E—-——- —=0,Zmrz +r or 3+5--12-=0 (360)E 2mE ' ' Now, thecoefﬁcient ofthelinear terminaquadratic equation isthenegative of thesumoftheroots. Hence, thesemimajor axisisgiven by T1-I-F2 k=————— =——. 3." 2 2E (61) Notethatinthecircular limit, Eq.(3.61) agrees withEq.(3.58). Interms ofthe semimajor axis,theeccentricity oftheellipse canbewritten Z2 e=1---, (3.62)mka (arelation wewillhave useforinalaterchapter). Further, from Eq.(3.62) we havetheexpression [2 2-——=a(l—e), (3.63)mk interms ofwhich theelliptical orbit equation (3.55) canbewritten a(l-82)=---_-. 3.64r1+ecos(6 —9') () Chapter 3TheCentral Force Problem s=0 £=0.5 s=O75 s=O.9 FIGURE 3.14 Ellipses withthesame major axesandeccentricities from 0.0to0.9. From Eq.(3.64), 1tfollows thatthetwoapsidal distances (which occur when 9-9’ is0andJZ,respectively) areequal toa(l—e)anda(l+e),asistobeexpected from theproperties ofanellipse. Figure 3.14shows sketches offourelliptical orbits withthesame major axis a.andhence thesame energy, butwitheccentricities s=0.0,0.5,0.75, and0.9. Figure 3.15shows howr1andF2depend ontheeccentricity 5. Thevelocity vector v||oftheparticle along theelliptical pathcanberesolved intoaradial component Ur=i'=p,/mplusanangular component vg=rt-l= l/mr V||=Uri‘"I"U96. Theradial component withthemagnitude vr=evosin9/(l—£2)vanishes atthetwoapsidal distances, while U9attains itsmaximum value atperihelion anditsminimum ataphelion. Table 3.1listsangular velocity values attheap- sidal distances forseveral eccentiicities. Figure 3.16 presents plots ofthera- dialvelocity component v,versus theradius vector rforthehalfcycle when v,-points outward, i.e.,itispositive. During theremaining halfcycle vrisnega- 2 aphelion distance r4 1 perihelion distance 01 I l 0 8 t FIGURE 3.15 Dependence ofnormalized apsidal distances r1(lower line)andr2(upper line)ontheeccentricity e. 3.7 TheKepler Problem: Inverse-Square LawofForce 97 TABLE 3.1 Normalired angular speeds 9andv9=relatperihelion (r1)andaphelion (/-2),respectively, mKeplerian orbits ofvarious eccentricities (5).Thenormalized radial distances atperihelion andaphelion arelisted incolumns 2and3,respectively. The nonnalization iswithrespect tomotion inacircle withtheradius aandtheangular niomentunil =mavg =ma26-Q. Eccentricity Perihelion Aphelion Angular speed Linear angular speed ti/41 F2/a 91/90 92/90 voi/vo "oz/vo 1 l l 1s l—r‘ 1+s (1_£)2 0 1 l 1 0.l 0.9 l.l 1.234 0.3 0.7 1.3 2041 0.5 0.5 1.5 4000 07 0.3 1.7 11.111 0.9 0.1 l.9 100.000(1+s)2 l-8 1 0.826 0.592 0.444 0.346 0.277l 1.1ll 1.429 2.000 3.333 10.0001+s 1 0.909 0.769 0.667 0.588 0.526 tive,andtheplotofFig.3.16repeats itselfforthenegative range below Ur=0 (notshown). Figure 3.17 shows analogous plots oftheangular velocity com- ponent v9versus theangle 6.Inthese plots andinthetable thevelocities are normalized relative tothequantities vqand90obtained from theexpressions l=mr20 =mrvg =mazéq =mavq fortheconservation ofangular momentum intheelliptic orbits ofsemimajor axisa,andinthecircle ofradius a. c=05 0.6 0.4 VrV0 s=03 02 s=01 0 /\AZ"kll .. a\__._KII FIGURE 3.16 Normalized radial velocity, vr,versus rforthree values oftheeccentric- itys. 3.8IChapter 3TheCentral Force Problem 2 s=05 ‘I8 15 V” s=0.3 l_ F=01 0 I00 200 560"'HI 9 FIGURE 3.17 Normalized orbital velocity, vg,versus 6forthree values oftheeccen- Iricity a. THE MOTION INTIME INTHE KEPLER PROBLEM Theorbital equation formotion inacentral inverse-square force lawcanthusbe solved inafairly straightforward manner withresults thatcanbestated insimple closed expressions. Describing themotion oftheparticle intimeasittraverses the orbitishowever amuch more involved matter. Inprinciple, therelation between theradial distance oftheparticle randthetime (relative tosome starting point) isgiven byEq.(3.18), which heretakes ontheform t .1r=‘/gf (3.65) '°\/F-W”? Similarly, thepolar angle 9andthetimeareconnected through theconserva- tionofangular momentum, 2 dr=Kd6,l which combined withtheorbitequation (3.51) leads to r-‘B/9 d9 (366)Tmkz,0[1+€COS(9 -0511' ' Either ofthese integrals canbecarried outintenns ofelementary functions. How- ever, therelations areverycomplex, andtheir inversions togiveror6asfunc- tionsoftposefonnidable problems, especially when onewants thehighprecision needed forastronomical observations. Toillustrate some ofthese involvements, letusconsider thesituation for parabolic motion (e=l),where theintegrations canbemost simply carried out.Itiscustomary tomeasure theplane polar angle from theradius vector at 3.8 TheMotion in‘Fme intheKepler Problem 99 thepoint ofclosest approach—a point most usually designated astheperihe- l1'on.* Thisconvention corresponds tosetting 6'intheorbit equation (3.51) equal tozero. Correspondingly, timeismeasured from themoment, T,ofperihelion passage. Using thetrigonometric identity l+cos9 =2cos2%, Eq.(3.66) thenreduces forparabolic motion totheform Z3 H 46 =———- ‘—d6. I4mk,[0 sec2 Theintegration iseasily perfomied byachange ofvariable toir=tan(6/2), leading totheintegral £3 lan(9/2) Z= ‘/Q (l+X2)dX, 01' 3 t=ﬁ€5(tan%+%tan3 (3.67) Inthisequation, —rr<6<rt,where fort—>—oo theparticle starts ap- proaching from inﬁnitely faraway located at6=—rr.Thetimet=0corre- sponds to6=0,where theparticle isatperihelion. Finally t—>+00corresponds to9—>rrastheparticle moves inﬁnitely faraway. Thisisastraightforward rela- tionfortasafunction of9;inversion toobtain 6atagiven timerequires solving acubic equation fortan(6l/2), thenﬁnding thecorresponding arctan. Theradial distance atagiven timeisgiven through theorbital equation. Forelliptical motion, Eq.(3.65) ismost conveniently integrated through an auxiliary variable 1//,denoted astheeccentric an0maly,* anddeﬁned bytherela- tion r=a(l -ecosi//). (3.68) Bycomparison withtheorbit equation, (3.64), itisclear that1/1alsocovers the interval 0to21:as9goes through acomplete revolution, andthattheperihelion occurs at1/1=O(where 9=0byconvention) andtheaphelion attr=77.’=9. *L.iterally, thetermshould berestricted toorbits around theSun,while themore general terrrishould beperiap.ri.\'. However, ithasbecome customary touseperihelion nomatter where thecenter offorce isEven forspaoc craft orbiting theMoon, ofﬁcial descriptions oftheorbital parameters refer to perihelion where pericynthion would bethepedantic term *Merlieval astronomers expected theangular motion tobeconstant. Theangle calculated bymulti- plying thisaverage angular velocity (221/period) bythetime since thelastperihelion passage was called themean anomaly Fivm themean anomaly theeccentric anomaly could becalculated andthen usedtocalculate thetrueanomaly. Theangle 9iscalled thetrueanomaly _|ustasitwasinmedieval astronomy. Chapter 3TheCentral Force Problem Expressing EandZinterms ofa,e,andk,Eq.(3.65) canberewritten for elliptic motion as :=-/ﬁfraw , (3.69)2k ['0 'r_£_ __Hllgﬂz) where, bytheconvention onthestarting time, roistheperihelion distance. Substi- tution ofrinterms of1/1from Eq.(3.68) reduces thisintegral, aftersome algebra, tothesimple form 3-rr= (1—ec0S1//)a'1U. (3.70)0 First, wemaynotethatEq.(3.70) provides anexpression fortheperiod, r,of elliptical motion, iftheintegral iscarried overthefullrange in10of21:: 1=2na3/lg. (3.71) This important result canalsobeobtained directly from theproperties ofanel- lipse. From theconservation ofangular momentum, thearealvelocity isconstant andisgiven by dA12 z Theareaoftheorbit, A,istobefound byintegrating (3.72) over acomplete period r: 'dA lrL'Zi?dl—A—ﬁ. Now, theareaofanellipse is A=Tfdb, where, bythedeﬁnition ofeccentricity, thesemiminor axisbisrelated toaac- cording totheformula b=avl—e2. By(3.62), thesemiminor axiscanalsobewritten as b=a]/2 Zmk’ 3.8 TheMotion inlime intheKepler Problem 101 andtheperiod istherefore 2 /Z2 ,l1:=Tmna3/2 E=2rra3/2 %, aswasfound previously. Equation (3.71) states that,other things being equal, thesquare oftheperiod isproportional tothecube ofthemajor axis, andthis conclusion isoften referred toasthethird ofKepler’s laws.* Actually, Kepler wasconcerned withthespeciﬁc problem ofplanetary motion inthegravitational ﬁeldoftheSun.Amore precise statement ofthisthird lawwould therefore be: Thesquare oftheperiods ofthevarious planets areproportional tothecubeof their major axes. Inthisform, thelawisonlyapproximately true.Recall thatthe motion ofaplanet about theSunisatwo-body problem andmin(3.71)must be replaced bythereduced mass: (cf.Eq.(3.4)) m1m2 /1»=————.m1+mg where m1maybetaken asreferring totheplanet andmgtotheSun.Further, the gravitational lawofattraction is mimg f--0-3-. sothattheconstant kis k=Gmlmg. (3.73) Under these conditions, (3.71) becomes 3/2 3/2 r=____2"“ %_2Z“__, (3.74),/G(mi +mg) sfGm; ifweneglect themass oftheplanet compared totheSun.ltistheapproximate version ofEq.(3.74) thatisKepler’s third law,foritstates thatrisproportional toa3/2, withthesame constant ofproportionality forallplanets. However, the planetary mass mlisnotalways completely negligible compared totheSun’s; for example, Jupiter hasamass ofabout 0.1% ofthemass oftheSun.Ontheother hand, Kepler’s third lawisrigorously truefortheelectron orbits intheBohr atom, since itandkarethenthesame forallorbits inagiven atom. Toreturn tothegeneral problem oftheposition intimeforanelliptic orbit, we mayrewrite Eq.(3.70) slightly byintroducing thefrequency ofrevolution noas *Kepler‘s three lawsofplanetary motion, published around 1610, were theresult ofhispioneenng analysis ofplanetary observations andlaidthegroundwork forNewton’s great advances Thesecond law,theconservation oiareal velocity. isageneral theorem forcentral force motion, ashasbeen noted previously. However, theﬁrst-—that theplanets move inelliptical orbits about theSunatone focus—and thethirdarerestricted speciﬁcally totheinverse-square lawofforce. 3.9 IChapter 3TheCentral Force Problem 2 Itw=T” =,/m—0l:,,. (3.75) Theintegration inEq.(3.70) isofcourse easily performed, resulting intherelation wt=1/1—esin11/, (3.76) known asKepler’s equation. Thequantity totgoes through therange Oto272', along with(Irand0,inthecourse ofacomplete orbital revolution andistherefore alsodenoted asananomaly, speciﬁcally themean anomaly. Toﬁndtheposition inorbit atagiven timet,Kepler’s equation, (3.76), would ﬁrstbeinverted toobtain thecorresponding eccentric anomaly r/r.Equation (3.68) thenyields theradial distance, while thepolar angle 6canbeexpressed interms of1/1bycomparing thedeﬁning equation (3.68) withtheorbit equation (3.64): 1_ 2 1+€COS6= . Withalittlealgebraic manipulation, thiscanbesimpliﬁed, to cos9=%.isZ/C‘; Z’. (3.77) Bysuccessively adding andsubtracting bothsides ofEq.(3.77) from unity and taking theratio oftheresulting twoequations, weareledtothealternative fOI'Il'l 9 ll-l-e 1/1—= i —. 3. tanz 1_etan2 (78) Either Eq.(3.77) or(3.78) thusprovides 6,oncerhisknown. Thesolution of thetranscendental Kepler’s equation (3.76) togivethevalue ofrhcorresponding toagiven timeisaproblem thathasattracted theattention ofmany famous math- ematicians eversinceKepler posed thequestion earlyintheseventeenth century. Newton, forexample, contributed what today would becalled ananalog solution. Indeed, itcanbeclaimed thatthepractical need tosolve Kepler’s equation toac- curacies ofasecond ofarcoverthewhole range ofeccentricity fathered many ofthedevelopments innumerical mathematics intheeighteenth andnineteenth centuries. Afewofthemore than100methods ofsolution developed inthepre- computer eraareconsidered intheexercises tothischapter. THE l.APl.ACE—RUNGE-LENZ VECTOR TheKepler problem isalsodistinguished bytheexistence ofanadditional con- served vector besides theangular momentum. Forageneral central force, New- 3.9 TheLaplace-Runge-Lenz Vector 103 ton’s second lawofmotion canbewritten vectorially as p=f(r),? (3.79) Thecross product ofpwiththeconstant angular momentum vector Ltherefore canbeexpanded as pxi.=@[r><(rxr)] =E[in-1-)-ﬁr]. (3.80)I” Equation (3.80) canbefurther simpliﬁed bynoting that rr—1d(rr)—'_2dr _N (or,inlessformal terms, thecomponent ofthevelocity intheradial direction isr). AsLisconstant, Eq.(3.80) canthenberewritten, after alittlemanipulation, as a’ rrr Eu»XL)=—m.r<r)r2 (;-r-2). OT d dEn)xL)=—mf(r)r2E . (3.81) Without specifying theform off(r),wecangonofurther. ButEq.(3.81) canbe immediately integrated iff(r)isinversely proportional tor2—the Kepler prob- lem.Writing f(r)intheformprescribed byEq.(3.49), Eq.(3.81) thenbecomes d dmkr E(PXL)—Z5(T), which saysthatfortheKepler problem thereexists aconserved vector Adeﬁned by A=PXL-mic; (3.82) Therelationships between thethreevectors inEq.(3.82) andtheconservation of Aareillustrated inFig.3.18, which shows thethree vectors atdifferent positions intheorbit. lnrecent times, thevector Ahasbecome known amongst physicists astheRunge-Lenz vector, butpriority belongs toLaplace. From thedeﬁnition ofA,wecaneasily seethat A-L=0, (3.83) since Lisperpendicular topxLandrisperpendicular toL=rxp.Itfollows from thisorthogonality ofAtoLthatAmust besome ﬁxed vector intheplane of Chapter 3TheCentral Force Problem A mk P pXL "ii AZr- "4 I|-PXL mk P P PXL ink A EIGURE 3.18 ‘Thevectors p,L,andAatthree positions 1I1aKeplenan orbit. Atperihe- l10l'I(extreme lett)|p><1,|=mk(1+e) andataphelion (extreme right) |p><Ll=mk(1—e). Thevector Aalways points inthesame direction withamagnitude mke. meoﬂ51t.'1iB ‘isused todenote theanglebetween randthehxed direction oiA, thenthedotproduct ofrandAisgiven by A-r=Arcos6=r-(px L)—mkr. (3.84) Now, bypermutation oftheterms inthetriple dotproduct, wehave r-(pxL)=L-(rxp)=l2, sothatEq.(3.84) becomes ArC056 =l2—mkr. or l mk A ;="l? 1+—kC0sl) . m TheLaplace-Runge-Lenz vector thusprovides stillanother wayofderiving the orbit equation fortheKepler problem! Comparing Eq.(3.85) withtheorbit equa- tionintheform ofEq.(3.55) shows thatAisinthedirection oftheradius vector totheperihelion point ontheorbit, andhasamagnitude A=mke. (3.86) FortheKepler problem wehave thusidentiﬁed twovector constants ofthe motion LandA,andascalar E.Since avector must have allthree independent components, thiscorresponds toseven conserved quantities inall.Now, asystem such asthiswith three degrees offreedom hassixindependent constants ofthe motion, corresponding, saytothethree components ofboth theinitial position 3.9 TheLaplace-Runge-Lenz Vector 105 andtheinitial velocity oftheparticle. Further, theconstants ofthemotion we havefound areallalgebraic functions ofrandpthatdescribe theorbit asawhole (orientation inspace, eccentricity, etc.); noneofthese seven conserved quantities relate towhere theparticle islocated intheorbit attheinitial time. Since one constant ofthemotion must relate tothisinformation, sayintheform ofT.the timeoftheperihelion passage, there canbeonlyﬁveindependent constants ofthe motion describing thesize, shape, andorientation oftheorbit. Wecantherefore conclude thatnotallofthequantities making upL,A,andEcanbeindependent; there must infactbetworelations connecting these quantities. Onesuchrelation hasalready been obtained astheorthogonality ofAandL,Eq.(3.83). Theother follows from Eq.(3.86) when theeccentricity isexpressed interms ofEandl from Eq.(3.57), leading to A2=mzkz+2mEl2, (3.37) thusconﬁrming thatthere areonlyﬁveindependent constants outoftheseven. Theangular momentum vector andtheenergy alone contain only fourinde- pendent constants ofthemotion: TheLaplace—Runge—Lenz vector thusaddsone more. Itisnatural toaskwhythere should notexist foranygeneral central force lawsome conserved quantity thattogether withLandEserves todeﬁne theorbit inamanner similar totheLaplace—Runge—Lenz vector forthespecial caseofthe Kepler problem. Theanswer seems tobethatsuch conserved quantities canin factbeconstructed, butthattheyareingeneral rather peculiar functions ofthe motion. Theconstants ofthemotion relating totheorbitbetween themdeﬁne the orbit. i.e.,leadtotheorbit equation giving rasafunction of6.Wehave seen thatingeneral orbits forcentral force motion arenotclosed; thearguments of Section 3.6show thatclosed orbits imply rather stringent conditions onthefonn oftheforce law.Itisaproperty ofnonclosed orbits thatthecurve willeventually passthrough anyarbitrary (r,6)point thatliesbetween thebounds oftheturning points ofr.intuitively thiscanbeseenfrom thenonclosed nature oftheorbit; as 6goesaround afullcycle, theparticle must never retrace itsfootsteps onanypre- vious orbit. Thus, theorbit equation issuch thatrisamultivalued function of9 (modulo 27:);infact,itisaninﬁnite-valuedﬁmction of9.Thecorresponding con- served quantity additional toLandEdeﬁning theorbit must similarly involve an inﬁnite-valued function oftheparticle motion. Suppose thervariable isperiodic withangular frequency (Orandtheangular coordinate 6isperiodic withangular frequency wg.Ifthese twofrequencies havearatio(co,/<09) thatisaninteger or integer fraction, periods aresaidtobecommensurate. Commensurate orbits are closed withtheorbiting mass continually retracing itspath. When we>ai,the orbitwillspiral about theorigin asthedistance varies between theapsidal (max- imum andminimum) values, closing only ifthefrequencies arecommensurate. If,asintheKepler problem, cu,=we,theperiods aresaidtobedegenerate. If theorbits aredegenerate there exists anadditional conserved quantity thatisan algebraic function ofrandp,suchastheRunge—Lenz vector. From these arguments wewould expect asimple analog ofsuch avector to exist forthecaseofaHooke’s lawforce, where, aswehave seen. theorbits are 3.10 IChapter 3TheCentral Force Problem alsodegenerate. Thisisindeed thecase,except thatthenatural waytoformulate theconstant ofthemotion leads nottoavector buttoatensor ofthesecond rank(cf.Section 7.5). Thus, theexistence ofanadditional constant orintegral of themotion, beyond EandL,thatisasimple algebraic function ofthemotion issufﬁcient toindicate thatthemotion isdegenerate andthebounded orbits are closed SCATTERING INACENTRAL FORCE FIELD Historically, theinterest incentral forces arose outoftheastronomical problems ofplanetary motion. There isnoreason, however, whycentral force motion must bethought ofonlyinterms ofsuchproblems; mention hasalready beenmade oftheorbits intheBohr atom. Another ﬁeldthatcanbeinvestigated interms of classical mechanics isthescattering ofparticles bycentral force ﬁelds. Ofcourse, iftheparticles areontheatomic scale, itmust beexpected thatthespeciﬁc results ofaclassical treatment willoften beincorrect physically, forquantum effects areusually largeinsuchregions. Nevertheless, many classical predictions remain valid toagood approximation. More important, theprocedures fordescribing scattering phenomena arethesame whether themechanics isclassical orquan- tum; wecanlearn tospeak thelanguage equally aswellonthebasis ofclassical physics. Initsone-body formulation, thescattering problem isconcerned withthescat- tering ofparticles byacenter offorce. Weconsider auniform beam ofparticles- whether electrons, ora-particles. orplanets isirrelevant—all ofthesame mass andenergy incident uponacenter offorce. ltwillbeassumed thattheforce falls offtozeroforverylarge distances. Theincident beam ischaracterized byspeci- fyiug itsintensity I(alsocalled Iluxdensity), which gives thenumber ofparticles crossing unitareanormal tothebeam inunittime. Asaparticle approaches the center offorce, itwillbeeither attracted orrepelled, anditsorbit willdeviate from theincident straight-line trajectory. After passing thecenter offorce, the force acting ontheparticle willeventually diminish sothattheorbitonceagain approaches astraight line.Ingeneral, theﬁnaldirection ofmotion isnotthesa'ne astheincident direction. andtheparticle issaidtobescattered. Thecross section forscattering inagiven direction, a(Q), isdeﬁned by U(Q)dg:number ofparticles scattered intosolid angle dS2perunittime incident intensity ’ (3.88) where dS2isanelement ofsolid angle inthedirection Q.Often 0(Q)isalsodes- ignated asthedijferential scattering crosssection. Withcentral forces theremust becomplete symmetry around theaxisoftheincident beam; hence theelement ofsolid angle canbeWritten dS2=21:sin@d®. (3.89) 3.10 Scattering inaCentral Force Field 107 eh- Ejoii Q —i>- ds sniztiiiiiiiiiilnytQ?“ FIGURE 3.19 Scattering ofanincident beam ofparticles byacenter offorce. where 6istheangle between thescattered andincident directions, known asthe scattering angle (cf.Fig.3.19,where repulsive scattering isillustrated). Notethat thename “cross section” isdeserved inthat0(0) hasthedimensions ofanarea. Foranygiven particle theconstants oftheorbit, andhence theamount ofscat- ' 'etto terin aredetermined byitsenergy andangular momentum. Itisconveni n g. express theangular momentum interms oftheenergy andaquantity known as h ter himaitarameter, s,deﬁned astheperpendicular distance between tecen tepcp . offorce andtheincident velocity. IfU0istheincident speed oftheparticle, then ‘ (3.99) l=mvgs =S»2mE. Edrareﬁxed theangle ofscattering G-)isthendetermined uniquely.* Once an.- , Forthemoment, itwillbeassumed thatdifferent values ofscannot leadtothe 'anleTherefore thenumber ofparticles scattered intoasolid same scattering g. , anled9lying between G)and6+d®must beequal tothenumber ofthe g incident particles withimpact parameter lying between thecorresponding sand s+ds: 2rrI.r|ds| =210(6))! sin®|d(~)|. (3.91) Absolute value signs areintroduced inEq.(3.91) because numbers ofparticles '' ' ‘ .'td'rections. mustofcourse always bepositive, wlule sand®oftenvaryinopposi ei Ifsisconsidered asafunction oftheenergy andthecorresponding scattering angle, s=s(®, E), (3.92) "itisatthis ointinthef0l'l1'il.lld£l0I'l thatclassical andquantum mechanics partcompany. Indeed, P itisfundamentally characteristic ofquantum mechanics thatwecannot unequivocally predict the trajectory ofanyparticular particle. Wecanonlygiveprobabilities forscattering invarious directions. Chapter 3TheCentral Force Problem ‘Prm (1) ‘I’ _l 7 FIGURE 3.20 Relation oforbit parameters andscattering angle 111anexample ofrepul- sivescattering. thenthedependence ofthedifferential cross section on(-9isgiven by sa's Afonnal expression forthescattering angle (-9asafunction ofscanbedi- rectly obtained from theorbit equation, Eq.(3.36). Again, forsimplicity, wewill COI'lS1d6I' thecaseofpurely repulsive scattering (cf.Fig.320).Astheorbit must besymmetric about thedirection oftheperiapsis, thescattering angle isgiven by ('-)=1:—2\I/, (3.94) where IIIistheangle between thedirection oftheincoming asymptote andthe periapsis (closest approach) direction. Intum,Wcanbeobtained fromEq.(3.36) bysetting 7'9=oowhen 90=7?.’(theincoming directionl, whence 9=JZ—\I1 when r=r,,,,thedistance ofclosest approach. Atrivial rearrangement then leads to O0 d. q»=Ig’. (3.95)r,,, r2 _.zl _L12 12 ,2 Expressing linterms oftheimpact parameter s(Eq.(3.90)). theresultant expres- sionfor®(s) is °° don)=JZ-zf_-_‘L--. (3.96) r’"rlrz(1— —s3 or,changing rtoI/u um ‘d ois)=21'-2IL (3.97)0/1_ _A2u2 3.10 Scattering inaCentral Force Field 109 Equations (3.96) and(3.97) arerarely usedexcept fordirect numerical compu- tation ofthescattering angle. However, when ananalytic expression isavailable fortheorbits, therelation between (:1andscanoften beobtained almost byin- spection. Anhistorically important illustration ofsuchaprocedure istherepulsive scattering ofcharged particles byaCoulomb lield. Thescattering force ﬁeldisthat produced byaﬁxed charge —Ze acting ontheincident particles having acharge —Z’esothattheforce canbewritten as f_ZZ’e2_r2, i.e.,arepulsive inverse-square law.Theresults ofSection 3.7canbetaken over herewithnomore change thatwriting theforce constant as /<=-zz'@2. (3.98) Theenergy Eisgreater thanzeio,andtheorbit isahypetbola withtheeccentricity given by“ 61+ 2512 1+25‘2 (399‘ :: g : -i—- _ I m(ZZ’e2)2 ZZ’e ’ ' Where usehasbeen made ofEq.(3.90). If6’inEq.(3.55) ischosen tobeIr. peiiapsis corresponds to6=0andtheorbitequation becomes 122'-=la coséi-1). (3.100;r I2 This hyperbolic orbit equation hasthesame form astheelliptic orbit equa- tion(3.56) except forachange insign. Thedirection oftheincoming asymptote. \l1,isthendetermined bythecondition r—>co: 1cos\l/=-e or,byEq.(3.94), sin®—l2_e' Hence, 6)cotz5=62—1, andusing Eq.(3.99) *Toavoid confusion withtheelectron charge e,theeccentricity willtemporarily hedenoted bye. Chapter 3TheCentral Force Problem C[®_ 2Es °2_zz'e' Thedesired functional relationship between theimpact parameter andthescatter- iiigangle istherefore ZZ’e2 o=i — .l s 2E cot2, (301) sothatoncarrying through themanipulation required byEq.(3.93), weﬁndthat or(@) isgiven by izz'1 2oa(o)=Z(T") CSC45. (3.102) Equation (3.102) gives thefamous Rutherford scattering cross section, orig- inally derived byRutherford forthescattering oforparticles byatomic nuclei. Quantum mechanics inthenonrelativistic limit yields across section identical withthisclassical result. Inatomic physics, theconcept ofatotalscattering cross section or,deﬁned as 71 01-=I.o'(.Q)dS2 =2:r/ 0(6)) sin(~) d®. .42: 0 isofconsiderable importance. However, ifweattempt tocalculate thetotalcross section forCoulomb scattering bysubstituting Eq.(3.102) inthisdeﬁnition, we obtain aninﬁnite result! Thephysical reason behind thisbehavior isnotdifﬁ- culttodiscem. From itsdeﬁnition thetotalcross section isthenumber ofparti- clesscattered inalldirections perutiittimeforunitincident intensity. Now, the Coulomb ﬁeldisanexample ofa“long-range” force; itseffects extend toinﬁnity. Theverysmall deﬂections occur onlyforparticles withverylarge impact param- eters. Hence, allparticles inanincident beam ofinﬁnite lateral extent willbe scattered tosome extent andmust beincluded inthetotalscattering cross section. ltistherefore clear thattheinﬁnite value fororisnotpeculiar totheCoulomb ﬁeld; itoccurs inclassical mechanics whenever thescattering ﬁeld isdifferent from zeroatalldistances, nomatter howlarge.‘ Only iftheforce ﬁeld“cuts off,” i.e.,iszerobeyond acertain distance, willthescattering cross section beﬁnite. Physically, such acut-off occurs fortheCoulomb ﬁeldofanucleus asaresult of thepresence oftheatomic electrons. which “screen” thenucleus andeffectively cancel itscharge outside theatom. *ci1- isalsointinite fortheCoulomb held inquantum mechanics. since ithasbeen stated that Eq(3102)reiiiams valid there. However, notall“long-range” forces giverisetoinﬁnite totalcross sections inquantum mechanics. Ittums outthatallpotentials thatfallofffaster atlarger distances than1/r2produce aﬁnite qurintum-mechanical totalscattering cross section 3.10 Scattering inaCentral Force Field 111 lnRutherford scattering, thescattering angle G)isasmooth monotonic func- tionoftheimpact parameter s.From Eq.(3.l01) weseethatassdecreases from inﬁnity, E)increases monotonically from zero, reaching thevalue rtassgoes to zero. However, other types ofbehavior arepossible inclassical systems, requiring some modiﬁcation intheprescription, Eq.(3.93), fortheclassical cross section. Forexample, with arepulsive potential andparticle energy qualitatively ofthe nature shown inFig.3.2l(a), itiseasytoseephysically thatthecurve of9ver- sussmaybehave asindicated inFig.3.21ib). Thus, with veiylarge values of theimpact parameter, asnoted above, theparticle always remains atlargeradial distances fromthecenter offorce andsuffers onlyminor deﬂection. Attheother extreme, fors=0,theparticle travels inastraight lineintothecenter offorce, andiftheenergy isgreater thanthemaximum ofthepotential, itwillcontinue onthrough thecenter without being scattered atall.Hence, forboth limits ins, thescattering angle goes tozero. Forsome intermediate value ofs,thescatter- ingangle must passthrough amaximum ®,,,.When G)<®,,,,there willbetwo values ofvthatcangive risetothesame scattering angle Each will contribute tothescattering cross section atthatangle, andEq.(3.93) should accordingly be modiﬁed tothefonn ,<10(9)=Zﬁ+® , (3.103)) I where forG)75®,,,theindex itakes onthevalues 1and2.Here thesubscript i distinguishes thevarious values ofsgiving risetothesame value of(-). Ofparticular interest isthecross section atthemaximum angle ofscattering (-'),,,.Asthederivative of®withrespect tosvanishes atthisangle, itfollows from Eq.(3.93) or(3.103) thatthecross section must become inﬁnite atG)—>Om.But foralllarger angles thecross section iszero, since thescattering angle cannot exceed (':),,,.Thephenomenon oftheinﬁnite riseofthecross section followed by abrupt disappearance isveiysimilar towhat occurs inthegeometrical optics ofthe scattering ofsunlight byraindrops. Onthebasisofthissimilarity, thephenomenon iscalled rainbow scattering. 1f E l l__9'1___V (-3 ris» s—-——- (ii) (5) FIGURE 3.21 Repulsive nonsingular scattering potential anddouble-valued cuwc of scattenng angle 6:)versus impact parameter soforsufﬁciently highenergy. Chapter 3TheCentral Force Problem Sofar,theexamples havebeen forpurely repulsive scattering. Ifthescattering involves attractive forces, further complications mayarise. Theeffect ofattraction willbetopulltheparticle intoward thecenter instead oftherepulsive deﬂection outward shown inFig.3.20. Inconsequence, theangle \l/between theincoming direction andtheperiapsis direction maybegreater thanJr/2,andthescatteiing angle asgiven byEq.(3.94) isthennegative. This initself isnogreat difficulty asclearly itisthemagnitude ofG-)thatisinvolved inﬁnding thecross section. But,under circumstances (-)ascalculated byEq.(3.96) maybegreater thanZn. Thatis,theparticle undergoing scattering maycircle thecenter offorce forone ormore revolutions before going offﬁnally inthescattered direction. Toseehowthismayoccur physically, consider ascattering potential shown as thes=0ctu've inFig.3.22. Itistypical oftheintermolecular potentials assumed inmany kinetic theory problems—an attractive potential atlarge distances falling offmore rapidly than1/r2, witharapidly rising repulsive potential atsmall dis- tances. Theother curves inFig.3.22show theeffective one-dimensional potential V’(r),Eq.(3.22’),forvarious values oftheimpact parameter s(equivalently var- iousvalues ofl).Since therepulsive centrifugal barrier dominates atlargerfor allvalues ofs>0,theequivalent potential forsmall swillexhibit ahump. Now letusconsider anincoming particle withimpact parameter s1andatthe energy E|corresponding tothemaximum ofthehump. Asnoted inSection 3.3, thedifference between E1andV’(r) isproportional tothesquare oftheradial velocity atthatdistance When theincoming particle reaches r-1,thelocation of themaximum inV’,theradial velocity iszero. Indeed, recall from thediscussion V'(r) Y S3 E- ._ ___ __.2 S2 E|- ——-- 51 g l1 I’! l‘—-ik l l s=O,V'=V FIGURE 3.22 Acombined attractive andrepulsive scattering potential, andthecorre- sponding equivalent one-dimensional potential atseveral values oftheimpact parameter s. 3.10 Scattering inaCentral Force Field 113 inSection 3.6thatwehaveheretheconditions foranunstable circular orbit atthe distance i-1.Intheabsence ofanyperturbation, theincoming particle withparam- eters E1ands1,once having reached r,would circle around thecenter offorce indeﬁnitely atthatdistance without everemerging! Forthesame impact param- eterbutatanenergy Eslightly higher thanE1,notruecircular orbit would be established. However, when theparticle isintheimmediate vicinity ofr|thera- dialspeed would beverysmall, andtheparticle would spend adisproportionately large timeintheneighbourhood ofthehump Theangular velocity, 9,meanwhile would notbeaffected bytheexistence ofamaximum, being given atr,by(3.90) - l si2EQ=j=—2 i_ mrl r] m Thus, inthetimeittakes theparticle togetthrough theregion ofthehump, the angular velocity mayhave canied theparticle through angles larger than21:or evenmultiples thereof. Insuchinstances, theclassical scattering ISsaidtoexhibit orbiting orspiraling. Astheimpact parameter isincreased, thewell andhump intheequivalent potential V’tendtoﬂatten out,until atsome parameter sgthere isonly apoint ofinﬂection inV’atanenergy E2(cf.Fig3.22). Forparticle energies above E2,there willnolonger beorbiting. Butthecombined effects oftheattractive andrepulsive components oftheeffective potential canleadeveninsuchcases to zerodeﬂection forsome ﬁnite value oftheimpact parameter. Atlarge energies and small impact parameters, themajor scattering effects arecaused bythestrongly repulsive potentials atsmall distances, andthescattering qualitatively resembles thebehavior ofRutherford scattering. Wehaveseenthatthescattered particle maybedeﬂected bymore than7!when orbiting takes place. Ontheother hand, theobserved scattering angle inthelab- oratory liesbetween Oandrt.Itistherefore helpful insuch ambiguous cases to distinguish between adeﬂection angle (D,ascalculated bytheright-hand sides of Eqs.(3.96) or(3.97), andtheobserved scattering angle G).Forgiven <l>,theangle (9istobedetermined from therelation G)=i<l>-Zmrr, mapositive integer. Thesignandthevalue ofmaretobechosen sothatE)liesbetween Oandrt.The suminEq.(3.103) thencovers allvalues of<l>leading tothesame G).Figure 3.23 shows curves of6-)versus sforthepotential ofFig.3.22attwodifferent energies Theorbiting thattakes place forE=E1shows upasasingularity inthecurve at s=51.When E>E2,orbiting nolonger takes place, butthere isarainbow effect at(9=—'(although there isanonvanishing cross section athigher scattering angles). Note that6)vanishes ats=s3,which means from Eq.(3.93) thatthe cross section becomes infinite intheforward direction through thevanishing of sin9.Thecross section cansimilarly become inﬁnite inthebackward direction 4 3.11 IChapter 3TheCentral force Problem 1r <1- Sip- O|__, '4U) i_-__-1jii_-.-LY-n'l>'-——-- 15-5,£>r2 FIGURE 3.23 Curves ofdeﬂection angle <1>versus s,forthepotential ofFig.3.22attwo different energies. providing ‘do9|ds’ remains ﬁnite at(9=:r.These inﬁnities intheforward orbackward scattering angles arereferred toasglory scattering. again inanalogy tothecorresponding phenomenon inmeteorological optics.* Amore general treatment would involve quantum corrections, butinsome in- stances quantum effects aresmall, asinthescattering oflow-energy ionsincrystal lattices, andtheclassical calculations aredirectly useful. Even when quantum- mechanical corrections ateiinpoi-taiil, itoften suffices touseanapproximation method (the“semiclassical” approximation) forwhich aknowledge oftheclas- sicaltrajectoiy isrequired. Foralmost allpotentials ofpractical interest, itisim- possible toﬁndananalytic forrri fortheorbit, andEq.(3.96) (orvariant forms) is either approximated forparticular regions ofsorintegrated numerically. TRANSFORMATION OFTHE SCATTERING PROBLEM TOLABORATORY COORDINATES Intheprevious section wewere concerned with theone-body problem ofthe scattering ofaparticle byaﬁxed center offorce. Lnpractice, thescattering always involved twobodies; e.g.,inRutherford scattering wehavetheorparticle andthe atomic nucleus. Thesecond particle, mg,isnotﬁxed butrecoils from itsinitial position asaresult ofthescattering. Since ithasbeen shown thatanytwo-body *The backward glory isfami.iar toairplane travelers astheringoflight observed toencircle the shadow oftheplane projected onclouds undemeath 3.11 Transformation oftheScattering Problem 115 / / / // I / / § (2') t \ \ \\ \ \ FIGURE 3.24 Scattering oftwoparticles asviewed inthelaboratory system. central force problem canbereduced toaone-body problem, itmight beth0ught thattheonlychange istoreplace mbythereduced mass pt.However. thematter isnotquitethatsimple. Thescattering angle actually measured inthelaboratory, which weshall denote by15‘,istheangle between theﬁnalandincident directions ofthescattered particle inlaboratory coordinates.T Ontheother hand, theangle G)calculated {win theequivalent one-body pioblern istheangle between theﬁnal andinitial directions oftherelative vector between thetwoparticles inthecen- terofmass coordinates. These twoangles, 9and6*),would bethesame onlyif thesecond particle remains stationary through thescattering process. Ingeneral, however, thesecond particle, though initially atrest,isitselfsetinmotion bythe mutual force between thetwoparticles, and,asisindicated inFig.3.24, thetwo angles thenhave different values. Theequivalent one-body problem thusdoes notdirectly furnish thescattering angle asmeasured inthelaboratory coordinate system. Therelationship between thescattering angles (9and29canbedetermined byexamining howthescattering takes place inacoordinate system moving with thecenter ofmass ofbothparticles. Insuch asystem thetotallinear momentum ISzero, ofcourse, andthetwoparticles always move withequal andopposite momenta. Figure 3.25illustrates theappearance ofthescattering process toan observer inthecenter ofmass system. Before thescattering, theparticles are moving directly toward eachother: after, theyaremoving directly away from each other. Theangle between theinitial andﬁnaldirections oftherelative vector, E), must therefore bethesame asthescattering angle ofeither particle inthecenter- of-mass system. TheCOIl11CC'fi0[l between thetwoscattering angles ('1)and15‘can thusbeobtained byconsidering thetransformation between thecenter-of-mass system andthelaboratory system. Thescatterng angle :9must notbeconfused withtheangle coordinate 6'oftherelative vector, r, between thetwoparticles Chapter 3TheCentral Force Problem I I I I /e /NII I I II FIGURE 3.25 Scattering oftwoparticles asviewed inthecenter ofmass system. Itisconvenient heretousetheterminology ofSection 3.1,withslight modiﬁ- cations: riandv|aretheposition andvelocity, afterscattering. oftheincident particle, mi,inthelaboratory system, r’landv’|aretheposition andvelocity, after scattering, ofparticle m|I11the center ofmasssystem, and RandV aretheposition and(constant) velocity inthecenter ofmass inthe laboratory system. Atanyinstant, bydeﬁnition r1=R+r], andconsequently w=v+ﬂ. GMW Figure 3.26graphically portrays thisvector relation evaluated after thescattering hastaken place; atwhich timev1andV;make theangles 19andE),respectively, _I4V-‘F2-vg sI' v'1 V1 @ I19 FIGURE 3.26 Therelations between thevelocities inthecenter ofmass andlaboratory coordinates. 3.11 Transformation oftheScattering Problem 117 withthevector Vlying along theinitial direction. Since thetarget isinitially sta- tionary inthelaboratory system, theincident velocity ofparticle linthatsystem, vr),isthesame astheinitial relative velocity oftheparticles. Byconservation of total linear momentum, theconstant velocity ofthecenter ofmass istherefore given by (mi+m2)V =mi‘/0. Or v=-‘ivo, (5.105)m2 where n=m|mQ/(m.| +1112). From Fig.3.26, itisreadily seenthat visiniﬁ‘ =v]sin® and vicos19=vicosG)+V. (3.106) Theratioofthese twoequations gives arelation between 19and(1-): maria=i, (3.101)cost-3 +p where ,0isdeﬁned as pE (3.103) mgvi Analtemative relation canbeobtained byexpressing v1interms oftheother speeds through thecosine lawasapplied tothetriangle ofFig.3.26: sf=ti?+V2+zvgvcost~>. (3.109) When thisisused toeliminate vifrom Eq.(3.106) andVisexpressed interms of v()byEq.(3.105), weﬁnd .ocos19=L (3.110)(/1+2pcos(-i~)+p2 Both these relations stillinvolve aratio ofspeeds through 0.Bythedeﬁnition ofcenter ofmass, thespeed ofparticle linthecenter-of-mass system, vi,iscon- nected withtherelative speed vafterthecollision, bytheequation (cf.Eq.(3.2)), where v=Ifl: 1 I1-U] 1 -—U. "11 Chapter 3TheCentral Force Problem Hence, pcanalsobewritten as p=3%, (3.10s’)mgv where v,itshould beemphasized. istherelative speed afterthecollision. When thecollision iselastic, thetotalkinetic energy ofthetwoparticles remains unal- tered andvmust equal vosothatpissimply p=5, (elastic collision) (2111)m2 independent ofenergies orspeeds. Ifthecollision isinelastic, thetotal kinetic energy ofthetwoparticles isaltered (e.g., some ofthekinetic energy goesinto theform ofinternal excitation energy ofthetarget). Since thetotalenergy iscon- served andmomentum isconserved, theenergy change resulting from‘thecolli- sioncanbeexpressed as 2 2 #%=uT%+Q. (3.112) Theso-called Qvalue oftheinelastic collision isclearly negative inmagnitude, butthesignconvention ischosen toconform tothatused ingeneral foratomic andnuclear reactions. From Eq.(3.112) theratioofrelative speeds before and aftercollision canbewritten 1:/1,112, (3)13,‘U0 mg E 2where E=émvo istheenergy oftheincoming particle (inthelaboratory sys- tem). Thus, forinelastic scattering pbecomes p=£1-——-. (inelastic scattering) (3.114) ma/1+ Notonlyarethescattering angles 13andG)ingeneral different inmagnitude, butthevalues ofthedifferential scattering cross section depend upon which of thetwoangles isused astheargument ofcr.Theconnection between thetwo functional fonns isobtained from theobservation thatinaparticular experiment thenumber ofparticles scattered intoagiven element ofsolid angle must bethe same whether wemeasure theevent intenns of19or9.Asanequation, this statement canbewritten 2rrIcr(@) sin®|d®|=Zrrlc/(19) sinz9|d1?|, 3.11 Transformation oftheSC€1lI€|'lI‘lg Problem 119 or "G)d(-) d(cos (-9)'1?=t-1-5'1‘ ‘_l= o 3.115a()G()sini? dz? G()d(cosz9) ( ) where a’(19)isthedifferential scattering cross section expressed interms ofthe scattering angle inthelaboratory system. Thederivative caneasily beevaluated fromEq.(3.110), leading totheresult 1+2 ®+“/2aw)=a(@) . (3.116) Note that0(6)) isnotthecross section anobserver would measure inthe center-of-mass system. Both0(6)) and0’(13)arecross sections measured inthe laboratory system; theyaremerely expressed intenns ofdifferent coordinates. An observer ﬁxed inthecenter-of-mass system would seeadifferent ﬂuxdensity of incident particles from thatmeasured inthelaboratory system. andthistransfor- mation ofﬂuxdensity would have tobeincluded if(forsome reason) wewanted torelate thecross sections asmeasured inthetwodifferent systems. Thetwoscattering angles haveaparticularly simple relation forelastic scat- tering when thetwomasses ofparticles areequal. Itthenfollows thatp=1,and fromEq,(3.110) wehave cos13=‘/2&9-=cos9,2 2 or 9l9=—-, =1. 2 (p ) Thus, withequal masses, scattering angles greater than90°cannot occur inthe laboratory system; allthescattering isintheforward hemisphere. Correspond- ingly, thescattering cross section isgiven interms of(9fromEq.(3.116) as o'(z9)=4cosz?-0(6), 195%, (p=l). Even when thescattering isisotropic interms ofG),i.e.,ot(®) isconstant, in- dependent of(9,thenthecross section interms of19varies asthecosine ofthe angle! When, however, thescattering massmgisverylargecompared totheinci- dent particle mass mlandthescattering iselastic, then from Eq.(3.11 1)p%O, socr'(z7) %a(®) from Eq.(3.116). Wehave seen thateven inelastic collisions, where thetotal kinetic energy remains constant. acollision withaninitially stationary target results inatransfer ofkinetic energy tothetarget withacorresponding decrease inthekinetic energy oftheincident particle. Inother words, thecollision slows down theincident Chapter 3TheCentral Force Problem particle. Thedegree ofslowing down canbeobtained fromEq.(3.109) ifviand Vareexpressed interms ofvgbyEqs.(3.108) and(3.105). respectively: U2 M2 -fa= (1+2pcosc-)+p2) (3.117)v3 "120 Forelastic collisions p=ml/mg, andEq.(3.!l7)canbesimpliﬁed to E1 1—l-ZpC0S(':)-l-p2 , ,_—=—-—-——i, (elastic collision) (3.ll7’)E0 (1+/>)2 where E0istheinitial kinetic energy oftheincident particle inthelaboratory system andE1thecorresponding energy after scattering. When theparticles are ofequal mass. thisrelation becomes E;_l+cosE-)_c0S§ E0" 2" ' Thus, atthemaximum scattering angle (E-J=yr,19=It/2), theincident particle loses allitsenergy andiscompletely stopped inthelaboratory system. Thistransfer ofkinetic energy byscattering is,ofcourse, theprinciple behind the“moderator” inathermal neutron reactor. Fastneutrons produced byﬁssion make successive elastic scatterings untiltheirkinetic energy isreduced tothennal energies, where theyaremore liable tocause ﬁssion thantobecaptured. Clearly thebestmoderators willbethelight elements, ideally hydrogen (,0=1).Fora nuclear reactor, hydrogen ispractical onlywhen contained aspartofamixture orcompound, such aswater. Other light elements useful fortheir moderating properties include deuterium, ofmass 2,andcarbon, ofmass 12.Hydrogen, as present inparaffin, water, orplastics. isfrequently used inthelaboratory toslow down neutrons. Despite theircurrent useful applications, these calculations ofthetransfom'ia- tionfrom laboratory tocenter ofmass coordinates, andofthetransfer ofkinetic energy, arenotparticularly “modern” or“quantum” innanire. Noristheclassi- calmechanics involved particularly advanced ordifﬁcult. Allthathasbeen used, essentially, istheconservation ofmomentum andenergy. Indeed, similar calcula- tions may befound infreshman textbooks, usually interms ofelastic collisions between, say,billiard balls. Butitistheirelementary nature thatresults inthe widespread validity ofthesecalculations. Solongasmomentum isconversed (and thiswillbetrueinquantum mechanics) andtheQvalue isknown, thedetails of thescattering process areirrelevant. Ineffect, thevicinity ofthescattering par- ticle1Sa“black box,” andweareconcerned onlywithwhatgoesinandwhat comes out.Itmatters notatallwhether thephenomena occurring inside thebox are“classical” or“quantum.” Consequently, theformulae ofthissection maybe used intheexperimental analysis ofphenomena essentially quantum innature, asforexample, neutron-proton scattering, solongastheenergies arelowenough thatrelativistic effects maybeneglected. (SeeSection 7.7foradiscussion ofthe relativistic treatment ofthekinematics ofcollisions.) 3.12 I3.12 TheThree-Body Problem 121 THE THREE-BODY PROBLEM Thus far,wehavetreated integrable problems inwhich theequations ofmotion canbeintegrated togiveaclosed-form solution. Forthetwo-body case ofthe inverse-square law,wefound solutions involving motion inelliptic, parabolic, andhyperbolic orbits, thefonner ofwhich constitute closed orbits. Solutions can alsobefound forsome additional power lawsoftheform V(r) =ar".Neverthe- less,foralmost allother possible central force potentials, theequations ofmotion cannot beintegrated. When onemore mass isadded, thesituation becomes much more complex. Even forinverse-square lawiorces, thisthree-body Kepler-type problem hasnoknown general solution. Inthepresent section weshallexamine some simple examples ofwhat happens when thisthird mass isadded. TheNewtonian three-body problem involves three masses m1,mg,andm3at therespective positions r1,1'2,andr3,interacting witheachother viagravitational forces. Weassume thattheposition vectors r1,P2,andr3areexpressed inthe center ofmass system. Itiseasytowrite theequation ofmotion oftheﬁrstmass since byNewton’s second lawm1i‘1 equals thegravitational forces thattheother twomasses exert onm1: ,, 1‘—I'__ I‘—l‘_ 1»,=-cm2—‘-_°, -Gm3% (3.118)ll‘1—I'2|" ll‘:—1'3| andanalogously fortheothertwomasses. Ifwemake useoftherelative-position vectors deﬁned by s,=1‘, —ri (3.119) inFig.3.27, thenclearly S1+ S3-l-S3 =O. (3.120) "11 5'0_ S1 X1 m/J *rI S3 "2 FIGURE 3.27 Position vectors s,=rJ—rkforthethree-body problem. Adapted from Hestenes, NewFoundations forClassical Mechanics, 1999, Fig.5.1. Chapter 3TheCentral Force Problem After alittle algebra, tneequations ofmotion assume thesymmetrical form §,=-mci§+m,G (3121)S; where i=1,2,3,thequantity misthesumofthethree masses m=m;+m2+m3 (3122) andthevector Gisgiten by G=G(S-:',,+S-1+5-Z). (3123) S1 S2 S3 Thethree coupled equations inthesymmetrical form, (3.121), cannot besolved in general, buttheydoprovide solutions tothethree-body problem forsome simple cases. There isasolution duetoEuler inwhich mass mgalways liesonthestraight linebetween theother twomasses sothatr1,r2,r3,s1,s2,s3,andGareall collinear. Figure 3.28shows Euler’s negative-energy (i.e.,bound-state) solution forthemass ratiom1<m2<m3inwhich themasses move along confocal ellipses withthesame period r.During eachperiod, themasses passthrough bothaperihelion coriﬁguration, inwhich theylieclose together along theaxisof theellipses. andanaphelion conﬁguration, inwhich theyliealong thissame axis butfarapart. Theaphelion positions intheorbits areindicated inFigure 3.28. Ifthevector G=0.theequations ofmotion decouple, andEq.(3.121) reduces tothetwo-body form oftheKepler problem, .. _s,s,=-mo-3-, (3.124) sl witheachmass moving along anelliptical orbit lying inthesame plane withthe same focal point andthesame period. This decoupling occurs when thethree 71’1 #7 FIGURE 3.28 Euler’s collinear solution tothethree-body problem forthemass ra- tiom]<m2<m3.Three ofthedotsshow aphelion positions. Adapted from Hes— tenes, NewFoundatzons forClassical Mechanics, 1999, Fig.5.2 $.12 TheThree-Body Problem 123 ms m, ml FIGURE 3.29 Lagrange’s equilateral triangle solution tothethree-body problem for themass ratio m1<m2<m3.Adapted from I-Iestenes, New Foundations forClassr- calMechanics, 1999. Fig.53. masses areatthevertices ofanequilateral triangle. Asthemotion proceeds, the equations remain uncoupled sotheequilateral triangle condition continues tobe satisﬁed, butthetriangle changes insizeandorientation. Figure 3.29presents La- granges elliptic solution casewiththesamemassratioasbefore, m1<m2<m3. T'hcﬁgure shows theconﬁguration when thcmasses arcclose together, each atits respective perihelion point, andalsoindicates theanalogous aphelion arrange- ment. Various asymptotic solutions havebeenworked outforthethree-body prob- lem.Forexample, ifthetotalenergy ispositive, thenallthreemasses canmove away fromeachother, oronecanescape, carrying away mostoftheenergy, and leave theother twobehind inelliptic orbits. Iftheenergy .lSnegative, onecan escape andleave theother twoinabound state, orallthree canmove inbound orbits. Therestricted three-body problem isoneinwhich twoofthemasses arelarge andbound, andthethirdissmall andmerely perturbs themotion oftheothertwo. Examples areaspacecraft inorbitbetween Earth andtheMoon, orthepertur- bation oftheSunontheM00n‘s orbit. Inthespacecraft case, thefirstapproach istoassume thattheEarth andMoon move intheirunperturbed orbits, andthe satellite interacts withthemthrough theirrespective inverse-square gravitational forces. Weshould alsonotethatsatellites orbiting Earth ataltitudes of90miles or150kilometers havetheirorbits perturbed byEarth's nonspherical massdistri- bution. Chapter 3TheCentral Force Problem Acomplicating factor intherestricted three-body problem isthedistribution ofgravitational potential energy inthevicinity oftheEar-th—Moon system. Close toEarth, weexperience agravitational force directed toward Earth, andclose to theMoon, theforceisdirected toward theMoon. Thismeans thattheequipoten- tials, orcurves ofconstant gravitational energy, areclosed curves thatencircle theEarth, (mi) andMoon, (mg), respectively, asshown inFig.3.30. Incontrast tothis,farfromtheEarth andMoon, theequipotentials encircle theEarth—Moon pair,asshown intheﬁgure. Atsome point, called Lagrange point Lg,along the horizontal lineintheﬁgure between theEarth andMoon, theattraction tothetwo bodies isequal inmagnitude andopposite indirection sotheforceexperienced by asmall massplaced thereiszero.Inotherwords, L2isalocalpotential minimum along thisline.More precisely, thispoint isasaddle point because thepotential energy isaminimum onlyalong theEarth-Moon axis,anddecreases indirections perpendicular tothisaxis.TwootherLagrange points, L1andL3,along thissame axisbetween theEarth andMoon arelocated atthetransition points between or- bl[Sthatencircle theEarth andtheMoon individually, andorbits thatencircle the Q1-L, nwﬁ FIGURE 3.30 Contour mapofequipotential curves oftwomasses m1>mgplotted in areference system rotating withthetwomasses around each other. From Hestenes, New Foundations forClassical Mechanics. 1986, Fig.5.5. 3.12 TheThree-Body Problem 125 twotogether asapair.These arealsosaddle points. Thefourth andﬁfthLagrange points, L4andL5,which arenotcollinear withtheotherthree, correspond tolo- calmintma inthegravitational potential energy. Masses inthevicinity ofthese twopoints experience aforce ofattraction toward them, andcanﬁndthemselves instable elliptical-shaped orbits around them. Wecanverify thepreceding statements byconsidering thesolutions found inSections 3.7and3.8fortwomassive bodies inthecenter-of-mass frame and asking iftherearelocations where asmall testbodywillremain atrestrelative to thetwobodies. Byatestbodywemean onewhose massissufﬁciently small that wecanneglect itseffect onthemotions oftheother twobodies. Forsimplicity, wewilllimitourattention totherestricted casewhere thebodies undergo circular motion about thecenter ofmass. TheLagrangian forthemotion ofthetestmass, m,canbewritten, ingeneral, as L=%m(i‘2+#92)-V(r.0,1), (3.125) where V(r,6,r)isthetime-dependent potential duetothetwomassive bodies. Asaconsequence ofthecircular motion. theradius vector, r,between thetwo bodies isofconstant length androtates withaconstant frequency, co,intheinertial frame. Ifwegotoacoordinate system rotating atthefrequency, thetwomassive bodies appear tobeatrestandwecanwritetheLagrangian interms oftherotating system hyusing 9’=9+0.): asthetransformation totherotating frame Thus, the Lagrangian intherotating coordinates canbewritten interms ofthecylindrical coordinates, ,0,9=6’—wt,andz,withpbeing thedistance from thecenter ofmassand(9thecounterclockwise angle fromthelinejoining thetwomasses shown inFig.3.30. So L=ém(02+p2té'—w>2+2’)—v’</>,@, Z), (3.126) OI‘ L=%m(;32+pit)”+22)-(mwp2é’ -%mp2a>2 +v’(/>,0,z)).(3.121) Theﬁfthandsixthterms arethepotentials fortheCoriolis effect (cf.Section 4.10) andthecentrifugal effect, respectively. Theprocedure thenistoﬁndtheLagrange equations andlookforsolutions withtheconditions that,5=2=9=0.Thesolutions aretheﬁveLagrange points shown inFig.3.30. Stability canbedetermined byinvestigating theef- fects ofsmall displacements from these. positions using themethods discussed m Chapters 6and12.Only L4andL5arestable. Even though theL2point isnotstable against displacements along theline between themasses, ithasbeenuseful forstudies oftheSun.TheL2between the Earth andSunistheapproximate location inthe1990s forthesolarandhelio- spheric observatory, SOHO, which orbits theL2point inaplane perpendicular to Chapter 3TheCentral Force Problem theEarth-Sun line.Thesatellite carmot beexactly attheL2point, orwecould notreceive itstransmissions against thebright Sun.Small steering rockets correct fortheslowdrifttoward, oraway from, L2. DERIVATIONS 1.Consider asystem inwhich thetotalforces actliig ontheparticles consist ofconserva- tiveforces Ffandfrictional forces f,proportional tothevelocity. Show thatforsuch asystem thevirial theorem holds 111theform - 1 T 'r|, 1' providing themotion reaches asteady stateandisnotallowed todiedown asaresult ofthefrictional forces. 2.Byexpanding esinilrinaFourier series inwt,show thatKepler’s equation hasthe foirnal solution °°2tr=wt+ZZinnia) Sinwt, n=l where .l,,istheBessel function oforder n.l<orsmall argument, theBessel function canbeapproximated inapower series oitheargument. Accordingly, fromthisresult derive theﬁrstfewterms intheexpansion of1/rinpowers ofe. 3.Ifthedifference tp—wtisrepresented byp,Kepler's equation canbewritten p=esin(wt +p). Successive approximations topcanbeobtained byexpanding sinpinaTaylor series inp,andthenreplacing pbyitsexpression given byKepler’s equation. Show thatthe ﬁrstapproximation bypisp|,given by tan _esinwt p1— l—ecoswt’ andthatthenextapproximation isfound from sui(p2 —p1)=—e3sin(wt +p|)(l+ecoswt), anexpression thatisaccurate through terms oforder e4. 4.Show thatforrepulsive scattering, Eq.(3.96) fortheangle ofscattering asafunction oftheimpact parameter, s,canberewntten as Mp1 ®=zr—4s/ L. °,/r.%,(1—{-)-s2<1-p2) Derivations 127 S 6. 7. 8 9OI‘ l e=1:-of 2 dp e.. ° (1/(rm) —vol)+s2t1—/>2) bychanging thevariable ofintegration tosome function p(r). Show thatforare- pulsive potential theintegrand isnever singular inthelimitr-—>rm.Because of thedeﬁnite limits ofintegration, these formulations haveadvantages fornumencal calculations of®(s)andallow naturally fortheuseofGauss-Legendre quadrature schctncs. Apply theformulation ofthepreceding exercise tocompute numerically G(s)andthe differential crosssection ofcr(®) fortherepulsive potential VV=__°_ 1+1 andforatotalenergy E=l.2l/0. Itissuggested that16-point Gauss—Legendre quadrature willgiveadequate accuracy. Doesthescattering exhibit arainbow’? Ifarepulsive potential drops ofmonotonically withr,thenforenergies highcom- pared toV(rm) theangle ofscattering willbesmall. Under theseconditions showthat Eq.(397)canbemanipulated sothatthedeﬂection angle isgiven approximately by ®_if‘<v(um)-v(u))d>» "E0<1-W/2 ‘ where y,obviously, isu/um. Show further, thatifV(u)isoftheformCu",where nisapositive integer, thenin thehigh-energy limit thecross section isproportional to®'2(H'u"). (a)Show thattheangle ofrecoil ofthetarget particle relative totheincident direction ofthescattered particle issimply <l>=%(2r—9). (b)Itisobserved thatinelastic scattering thescattering cross section isisotropic in terms of6.What: arethecorresponding probability distributions forthescattered energy oftheincident particle, E1,andfortherecoil energy ofthetarget particle, E2’? Show thattheangle ofscattering inthelaboratory system, 17,isrelated totheenergy before scattering, E0,andtheenergy afterscattering E1,according totheequation cos§=(T12+m1 _E1_m2—_mi EQ+ m2Q g_ 27111 F0 2m] E1 27111-‘/E()E| Show thatthecentral forceproblem issoluble interms ofelliptic functions when the forceISapower-law function ofthedistance withthefollowing fractional exponents: n_35l5'7 '2’2’s'3’3' Chapter 3TheCentral Force Problem EXERCISES 10.Aplanet ofmass Misinanorbit ofeccentricity e=1—ozwheie oz<<1,about the Sun.Assume themotion oftheSuncanbeneglected andthatonlygravitational forces act.When theplanet isatitsgreatest distance from theSun,itisstruck byacomet of mass m.where m<<Mtraveling inatangential direction. Assuming thecollision is completely inelastic, ﬁndtheminimum kinetic energy thecomet must have tochange theneworbit toaparabola. 11.Twoparticles move about eachother incircular orbits under theinﬂuence ofgravita- tional forces, withaaeriod 1'.Their motion issuddenly stopped atagiven instant of time, andtheyatethenreleased andallowed tofallintoeach other. Prove thatthey collide afteratimer/4\/E. 12.Suppose thatthere arelong-range interactions between atoms inagasintheform of central forces derivable from apotential kU(l)— ‘F. where risthedistance between anypairofatoms andmisapositive integer. Assume further thatrelative toanygiven atom theother atoms aredistributed inspace such thattheirvolume density isgiven bytheBoltzmann factor: P0,)=%€-rm)/tr, where Nisthetotalnumber ofatoms inii.volume V.Findtheaddition tothevirialof Clausius resulting fromtheseforces between pairsotatoms, andcompute theresulting correction toBoyle’s law.Take Nsolarge thatsums maybereplaced byintegrals While closed results canbefound foranypositive m,ifdesired, themathematics can besimpliﬁed bytaking m=+1 13.(a)Show thatit‘aparticle describes acircular orbit under theinﬂuence ofanattractive central force directed toward apoint onthecircle, thentheforce varies asthe inverse-ﬁfth power ofthedistance. (b)Show thatfortheorbit described thetotalenergy ofthepaiticle iszero. (c)Findtheperiod oftheinotion. (d)Find2?,5»,andvasafunction ofangle around thecircle andshow thatallthree quantities areinﬁnite astheparticle goesthrough thecenter offorce. 14.(a)Forcircular andparabolic orbits inanattractive 1/rpotential having thesame angular momentum, show thattheperihelion distance oftheparabola isone-half theradius ofthecircle. (II)Prove thatinthesame central force asinport (ti)thespeed oftiparticle atany point inaparabolic orbit is\/5times thespeed inacircular orbitpassing through thesame point. 15.Ameteor isobserved tostrike Earth withaspced v.making anangle ¢witl"the zenith. Suppose thatfarfrom Earth the1iieteor’s speed wasv’anditwasproceeding inadirection making azenith angle ¢/,theeffect oiEarth's gravity being topullitinto Exercises 129 ahyperbolic orbitintersecting Earth’s surface. Show howv’and¢-’canbedetermined from vand¢interms ofknown constants. Prove thatinaKepler elliptic orbit with small eccentricity etheangular motion of upurticlc asvicwcd from theempty focus ofthecllipsc isuniform (theempty focus isthefocus thatisnotthecenter ofattraction) toﬁrstorder ine.Itisthistheorem thatenables thePtoleniaic picture ofplanetary motion tobeareasonably accurate approximation. Onthispicture theSunisassumed tomove uniformly onacircle whose center isshifted fromEarth byadistance called theequant. Iftheequant is taken asthedistance between thetwofociofthecorrect elliptical orbit, thenthe angular motion isthusdescribed bythePtolemaic picture accurately toﬁrstorder in e. Oneclassic theme inscience ﬁction isatwinplanet (“Planet X”)toEarth thatis identical inmass. energy, andmomentum butislocated ontheorbit 90°outofphase withEarth sothatitishidden fromtheSun.However, because oftheelliptical nature oftheorbit, itisnotalways completely hidden. Assume thistwin planet isinthe same Keplenan orbit asEarth insuch amanner thanitisinaphelion when Earth isinperihelion. Calculate toﬁrstorder intheeccentricity ethemaximum angular separation ofthetwinJI‘lCltheSunasviewed fromtheEarth. Could suchatwinbe visible from Earth" Suppose thetwinplanet isinanelliptical orbit having thesame sizeandshape asthatofEarth, butrotated 180°fromEarth’s orbit, sothatEarth and thetwinareinperihelion atthesametime.Repeat yourcalculation andcompare the visibility inthetwosituations. Atperigee ofanelliptic gravitational orbit aparticle experiences animpulse S(cf. Exercise 11,Chapter 2)intheradial direction, sending theparticle intoanother elliptic orbit.Determine thenewsemimajor axis.eccentricity, andorientation interms ofthe old Aparticle moves inaforce ﬁelddescribed by Fm=—§exp(—§). where kandaarepositive. (a)Write theequations ofmotion andreduce themtotheequivalent one-dimensional problem. Usetheeffective potential todiscuss thequalitative nature oftheorbits fordifferent values oftheenergy andtheangular momentum. (b)Show thatiftheorbit isnearly circular, theapsides willadvance approximately byirp/a perrevolution, where pistheradius ofthecircular orbit. Auniform distribution ofdustinthesolar system adds tothegravitational attraction oftheSunonaplanet anadditional force F=—vzCr. where misthemass oftheplanet, Cisaconstant proportional tothegravitational constant andthedensity ofthedust, andristheradius vector from theSuntothe planet (bothconsidered aspoints). Thisadditional forceisverysmall compared tothe direct Sun—planet gravitational force. Chapter 3TheCentral Force Problem 21 22 23. 24 25(a)Calculate theperiod foracircular orbitofradius r0oftheplanet inthiscomhiied ﬁeld. (b)Calculate theperiod ofradial oscillations forslight disturbances from thiscircular orbit. (c)Show thatnearly circular orbits canbeapproximated byaprecessing ellipse and ﬁndtheprecession frequency. lstheprecession inthesameoropposite direction totheorbital angular velocity? Show thatthemotion ofaparticle inthepotential ﬁeld V(r) =—£+-lgrr isthesameasthatofthemotion under theKepler potential alone when expressed in terms ofacoordinate system rotating orprecessing around thecenter offorce. Fornegative totalenergy, show thatiftheadditional potential teiinISverysmall compared totheKepler potential, thentheangular speed ofprecession oftheelliptical nrhit is Q=?£'.'ﬁ_Z21: Theperihelion ofMercury isobserved toprecess (after correction forknown planetary perturbations) attherateofabout 40”ofarcperCentury, Show thatthisprecession could beaccounted forclassically ifthedimensionless quantity _h n_ka (which isameasure oftheperturbing inverse-square potential relative tothegravita- tional potential) were assmall as7x10's. (The eccentricity ofMercury’s orbit is 0.206, anditsperiod is0.24year.) Theadditional terminthepotential behaving asF2inExercise 21looks verymuch likethecentrifugal barrier termintheequivalent one-dimensional potential. Why1Sit thenthattheadditional force termcauses aprecession oftheorbit, while anaddition tothebarrier, through achange inl,doesnot? Evaluate approximately theratio ofmass oftheSuntothatofEarth, using onlythe lengths oftheyear andofthelunar month (27.3 days), andthemean radii ofEarth's orbit(1.49><103km)andoftheMoon's orbit(3.2><in‘lcm). Show thatforelliptical motion inagravitational ﬁeldtheradial speed canbewritten as _ma] r=— a2e2—(r—a)2.r Introduce theeccentric anomaly variable 1,11inplace ofrandshow thattheresulting differential equation iniﬁcanbeintegrated immediately togiveKepler’s equation. Iftheeccentricity eissmall, Kepler’s equation fortheeccentric anomaly ipasafunc- tionofwt,Eq.(3.76), iseasily solved onacomputer byaniterative technique that treats theesinittermasoflower order than1/I.Denoting rm,bythenthiterative Exercises 131 26 27 28. 29‘ 30. 31.solution, theobvious iteration relation is up”=mt+esinip,,_1. Using thisiteration procedure, ﬁndtheanalytic form foranexpansion of1,11inpowers ofeatleastthrough terms ine3. Eaiih’s period between successive perihelion transits (the“anomalistic year”) is 365.2596 mean solar days, andtheeccentricity ofitsorbit is0.0167504. Assuming motion inaKeplerian elliptical orbit, howfardoes theEarth move inangle inthe orbit, starting from perihelion, inatimeequal toone—quarter oftheanomalistic year? Giveyourresult indegrees toanaccuracy ofonesecond ofarcorbetter. Anymethod maybeused, including numencal computation withacalculator orcomputer. Inhyperbolic motion ina1/rpotential, theanalogue oftheeccentric anomaly ISF deﬁned by r=a(ecoshF —1), where a(e—1)isthedistance ofclosest approach. Find theanalogue toKepler’s equation giving tfromthetimeofclosest approach asafunction ofF. Amagnetic monopole isdeﬁned (ifoneexists) byamagnetic ﬁeldsingularity ofthe formB=br/r3,where bisaconstant (ameasure ofthemagnetic charge, asitwere). Suppose aparticle ofmassmmoves intheﬁeldofamagnetic monopole andacentral forceﬁeldderived fromthepotential V(r)=—k/r. (a)Find theform oi"Newton’s equation ofmotion, using theLorentz force given by Eq.(1.60) Bylocking attheproduct rxfrshowthatwhile themechanical angular momentum isnotconserved (theﬁeldofforceisnoncentral) thereisaconserved vector 1)=L-Q5.cr (Ii)Byparalleling thesteps leading from Eq.(3.79) toEq.(3.82), show thatforsome ftr)there isaconserved vector analogous totheLaplace-Runge—Lenz vector in which Dplays thesameroleasLinthepureKepler forceproblem. ifallthemomentum vectors ofaparticle along itstrajectory aretranslated soasto startfromthecenter offorce, thentheheads ofthevectors traceouttheparticle’s hodograph, alocus curve ofconsiderable antiquity inthehistory ofmechamcs, with something ofareviva inconnection with space vehicle dynarmcs. Bytaking thecross product ofLwith theLaplace—Runge—Lenz vector A,show thatthehodograph for elliptical Kepler motion isacircle ofradius mk/lwithorigin ontheyaxisdisplaced adistance A/Ifrom thecenter offorce. What changes, ifany.would therebeinRutherford scattering iftheCoulomb force were attractive, instead ofrepulsive? Examine thescattering produced byarepulsive central force f=kr—~l. Show that thedifferential cross section isgiven by k (1—x)dx®(IQ) =—--——-ii, U() 2Ex2(2—x)2sinn'x where xistheratioof6:)/1:andEistheenergy. Chapter 3TheCentral Force Problem Acentral force potential frequently encountered innuclear physics istherectangular well, deﬁned bythepotential V=0 r>a =—VQ rﬁd. Show thatthescattering produced bysuchapotential inclassical mechanics isiden- ticalwiththerefraction oflightraysbyasphere ofradius aandrelative index of refraction n_ E-l-V0 _{T _ (Thisequivalence demonstrates whyitwaspossible toexplain refraction phenomena bothbyHuygen’s waves andbyNewton’s mechanical corpuscles.) Show alsothatthe differential cross section is "202 (ncos£53—1)(rt—cm Q _2 4°” 2(l+n2 -2ncos0(9) = What isthetotalcrosssection? Aparticle ofmass misconstrained tomove under gravity without friction onthe inside ofaparaboloid ofrevolution whose axisisvertical Findtheone-dimensional problem equivalent toitsmotion. What isthecondition ontheparticle’s initial velocity toproduce circular motion? Findtheperiod ofsmall oscillations about thiscircular motion. Consider atruncated repulsive Coulomb potential deﬁned as kV= r>0 r It=— r5a. £1 Foraparticle oftotalenergy E>k/a,obtain expressions forthescattering angle ® asafunction ofs/so,where soistheimpact parameter forwhich thepeiiapsis occurs atthepoint r=a.(Theformulas canbegiven inclosed form buttheyarenotsimple!) Make anumencal plotof®versus s/soforthespecial caseE=2k/a.What canyou deduce about theangular scattering cross section from thedependence of(9ons/so forthisparticular case? Another version ofthetruncated Coulomb potential hastheform kAV=——— r>a 7' 0 =0 r<a. Obtain closed-form expressions forthescattering angle andthediﬁ°erential scattering cross section. These aremost conveniently expressed interms ofaparameter measur- ingthedistance ofclosest approach inunitsofa.What isthetotalcrosssection? Exercises 133 36.Therestricted three-body problem consists oftwomasses mcircular orbits about each other andathirdbodyofmuch smaller masswhose effect onthetwolarger bodies canbeneglected. (a)Deﬁne aneffective potential V(r, y)forthisproblem where thexaxisistheli-1e ofthetwolarger masses Sketch thefunction V(x,0)andshowthattherearetwo “valleys” (points ofstable equilibrium) corresponding tothetwomasses. Also show thatthere arethree “hills” (three points ofunstable equilibrium). (b)Using acomputer program, calculate some orbits fortherestricted three-body problem. Many orbits willendwith6_]6CIlOn ofthesmaller mass. Startbyassum- ingaposition andavector velocity forthesmall mass. CHAPTER 4.1I 134TheKinematics of Rigid Body Motion Arigid body wasdeﬁned previously asasystem oimass points subject tothe holonomic constraints thatthedistances between allpairsofpoints remain con- stant throughout themotion. Although something ofanidealization, theconcept isquiteuseful, andthemechanics ofrigidbodymotion deserves afullexposition. Inthischapter weshalldiscuss principally thekinematics ofrigidbodies, i.e., thenature andcharacteristics oftheirmotions. Wedevote some timetodevelop- ingthemathematical techniques involved, which areofconsiderable interest in themselves, andhave many important applications toother ﬁelds ofphysics. Ofessential importance istherotational motion ofarigid body. These consid- erations leaddirectly totherelation between thetimerateofchange ofavector inaninertial frame andthetimerateofchange ofthesame vector inarotaﬁng frame. Since itisappropriate atthatpoint, weleave kinematics anddevelop the description ofthedynamics ofmotion inarotating frame. Inthenextchapter we discuss, using theLagrangian fonriulation, howthemotion ofextended objects is generated byapplied forces andtorques. THE INDEPENDENT COORDINATES OFARIGID BODY Before discussing themotion ofarigid body, wemust ﬁrstestablish howmany independent coordinates arenecessary tospecify itsconﬁguration. From experi- ence, weexpect thatthere should besixindependent coordinates. Three extemal coordinates areneeded tospecify theposition ofsomereference point inthebody andthree more tospecify howthebody isoriented withrespect totheextemal coordinates. Inthissection weshow thatthese intuitive expectations arecorrect. Arigid body with Nparticles canatmost have 3Ndegrees offreedom, but these aregreatly reduced bytheconstraints, which canbeexpressed asequations oftheform Ti]=Cij. (4.1) I-lere ruisthedistance between theithandjthparticles andthec’sareconstants. Theactual number ofdegrees offreedom cannot beobtained simply bysubtract- ingthenumber ofconstraint equations from 3N,forthere areit-N(N—1)possible equations oftheformofEq.(4.1), which isfargreater than3Nforlarge N.In truth, theEqs.(4.1) arenotallindependent. 4.1 TheIndependent COOI'dlF|€llI€S ofaRigid Body 135 I l 2 ’|3 '23 3 / FIGURE 4.1Thelocation ofapointinarigidbodybyitsdistances fromthreereference points. Toﬁxapoint intherigidbody, itisnotnecessary tospecify itsdistances to allother points inthebody; weneedonlystatethedistances toanythree other noncollinear points (cf.Fig.4.1).Thus, oncethepositions ofthree oftheparticles oftherigidbodyaredetermined, theconstraints ﬁxthepositions ofallremaining particles. Thenumber ofdegrees offreedom therefore cannot bemore thannine Butthethreereference points arethemselves notindependent; there areinfact three equations ofrigid constraint imposed onthem, 712:‘-'12, r23=(-'23’ rl3='Cl3, thatreduce thenumber ofdegrees offreedom tosix.That onlysixcoordinates areneeded canalsobeseenfrom thefollowing considerations. Toestablish the position ofoneofthereference points, three coordinates must besupplied. But oncepoint 1isﬁxed, point2canbespeciﬁed byonlytwocoordinates, since itis constrained tomove onthesurface ofasphere centered atpoint 1.With these two points determined, point 3hasonlyonedegree offreedom, foritcanonlyrotate about theaxisjoining theother twopoints. Hence, atotalofsixcoordinates is sufﬁcient. Arigidbody inspace thusneeds sixindependent generalized coordinates to specify itsconﬁguration, nomatter howmany particles itmaycontain—even in thelimitofacontinuous body. Ofcourse, theremaybeadditional constraints on thebody besides theconstraint ofrigidity. Forexample, thebody may becon- strained tomove onasurface, orwithonepoint ﬁxed. Insuchcase, theadditional constraints willfurther reduce thenumber ofdegrees offreedom, andhence the number ofindependent coordinates. Howshallthese coordinates beassigned“ Note thatthesetofconﬁguration ofarigidbodyiscompletely speciﬁed bylocating aCartesian setofcoordinates Chapter 4TheKinematics ofRigid Body Motion ifll, 7ify /./ ~tFIGURE 4.2 Unprtmed axesrepresent anexternal reference setofaxes: theprimed axes areﬁxed 111therigid body. ﬁxed intherigid body (theprimed axesshown inFig.4.2)relative tothecoor- dinate axesoftheexternal space. Clearly three ofthecoordinates areneeded to specify thecoordinates oftheorigin ofthis“body” setofaxes. Theremaining three coordinates must thenspecify theorientation oftheprimed axesrelative to acoordinate system parallel totheexternal axes,butwiththesame origin asthe primed axes. There aremany ways ofspecifying theorientation ofaCartesian setofaxes relative toanother setwithcommon origin. Onefruitful procedure istostatethe direction cosines oftheprimed axesrelative totheunprimed. Thus, thex’axis could bespeciﬁed byitsthreedirection cosines 0:1,ctg,a3,withrespect tothex, y,zaxes. If,ascustomary, i,j,karethree unitvectors along x,y,z,andi’,j’.k’ perform thesame function intheprimed system (cf.Fig.4.31.thenthese direction cosines aredeﬁned as z=x3 2'=rg k 9% Y’=It tr 0,,J’ 622 ___ 912 -l y=x2 I ell ii x’=xi /I=II FIGURE 4.3 Direction cosines ofthebody setofaxesrelative toanexternal setofaxes. 4.1 TheIndependent Coordinates ofaRigid Body 137 in\0lﬂv cos911=cos(i’-i)= =i-i' cos612=cos(i'-j)=i'-j=j-i’ cos921=cos(j'~i)—j'-i—i-j' cos6g; =cos(j'-j)=j'-j =j-j’ (4.2) andsimilarly forcos613,cos631,etc.Notethattheangle 0,1isdeﬁned sothat theﬁrstindex refers totheprimed system andthesecond index totheunprimed system. These direction cosines canalsobeusedtoexpress theunitvector inthe primed system intenns oftheunitvectors oftheunprimed system giving i’=cos911i +cos91;j+cos613k j’=cos621i+cosBggj+cos923k k’=cos631i+cosQggj+cos933k. (4.3) These setsofninedirections cosines thencompletely specify theorientation of thex’,y’,z’axesrelative tothex,y,zset.Wecanequally wellinvert theprocess, andusethedirection cosines toexpress thei,j,kunitvectors intemis oftheir components along theprimed axes.Thus, wecanwrite r—xi+yj+zk=x'i'+y'j'+z'k' (4.4) by x’=(r-i’)=cos611x +cos612y +cos013Z y'=(r-j’)=cos621x+cos922;;+cos6232 z’=(r-k’)=cos631x+cos632)‘+cos6332 (4.5) withanalogous equations fori,jandk. Thedirection cosines alsofumish directly therelations between thecoordi- nates oiagiven point inonesystem andthecoordinates intheother system. Thus, thecoordinates ofapoint inagiven reference frame arethecomponents of theposition vector, r,along theprimed andunprimed axesofthesystem, respec- tively. Theprimed coordinates arethengiven interms ofx,y,and2,asshown in Eq.(4.5). What hasbeen done hereforthecomponents ofthervector canobvi- ously bedone foranyarbitrary vector. IfGissome vector, thenthecomponent of Galong thex’axiswillberelated toitsx-,y-,z-components by Gxt=G-i’=cos611Gx +oos912G,- +cos013GZ, (4.6) andsoon.Thesetofninedirection cosines thuscompletely spells outthetrans- formation between thetwocoordinate systems. Iftheprimed axesaretaken asﬁxedinthebody, thentheninedirection cosines willbefunctions oftimeasthebodychanges itsorientation inthecourse ofthe Chapter 4TheKinematics ofRigid Body Motion motion. Inthissense, thedirection cosines canbeconsidered ascoordinates de- scribing theinstantaneous orientation ofthebody, relative toacoordinate system ﬁxedinspace butwithorigin incommon withthebodysystem. But,clearly, they arenotindependent coordinates, fortherearenineofthemandithasbeenshown thatonlythree coordinates areneeded tospecify anorientation. Theconnections between thedirection cosines arisefrom thefactthatthebasis vectors inboth coordinate systems areorthogonal toeach other andhave unit magnitude; insymbols, .n.Z'nkZkn-Z0’ and ‘J J I (4.7) j.i=j.j=](.k=1_ withsimilar relations fori’,j’,andk’.Wecanobtain theconditions satisﬁed bythe ninecoefﬁcients byforming allpossible dotproducts among thethree equations fori,j,andkinterms ofi’,j’,andk’(asinEq.(4.4)), making useoftheEqs.(4.7): 3 Zoos 61",»cos61",=0 mgém’ l=l 3 (4.8) Zcos26'1,"=l. l=l These twosetsofthreeequations eachareexactly sufﬁcient toreduce thenumber ofindependent quantities fromninetothree. Formally, thesixequations canbe combined intoonebytsing theKronecker 8-symbol 81,",deﬁned by almil linl =0 lgém. Equations (4.8)canthenbewritten as 3 Zcos61,,’cos91",=8,,,',,, (4.9) I=l Itistherefore notpossible tosetupaLagrangian andsubsequent equations ofmotion withtheninedirection cosines asgeneralized coordinates. Forthis purpose, wemust usesome setofthreeindependent ftuictions ofthedirection cosines. Anumber ofsuchsetsofindependent variables willbedescribed later, themostimportant being theEuler angles. Theuseofdirection cosines tode- scribe thecormections between twoCartesian coordinate systems nevertheless has anumber ofimportant advantages. With theiraid,many ofthetheorems about the motion ofrigid bodies canbeexpressed withgreat elegance andgenerality, andin aform naturally leading totheprocedures used inspecial relativity andquantum mechanics. Such amode ofdescription therefore merits anextended discussion here. 4.2 Orthogonal Transformations 139 42IORTHOGONAL TRANSFORMATIONS Tostudy theproperties oftheninedirection cosines withgreater ease,itiscon- venient tochange thenotation anddenote allcoordinates byx,distinguishing the axesbysubscripts: X—>x1 y—>x2 (4.10) Z—>X3 asshown inFig.4.3.Wealsochange thenotation forthedirection cosines to a,J=cos6,] (4.11) Equations (4.5) and(4.6) constitute agroup oftransfomiation equations from asetofcoordinates x1,Jig,x5toanew setxi,xé,xé.Inparticular, they form an example ofalinear orvector transformation, deﬁned bytransformation equations oftheform Xi=r1iiXi+ ai212+al3X3 xé=a21x1 +ag2x2+a23x3 (4.12) X§,=@3111 +H2212 +@3313. where thea11,a12,...,areanysetofconstant (independent ofx,x’)coefﬁ- cients.* Tosimplify theappearance ofmany oftheexpressions, wewillalsomake useofthesummation convention ﬁrstintroduced byEinstein: Whenever anindex occurs twoormore times inaterm, itisimplied, without anyfurther symbols, that theterms aretobesummed overallpossible values oftheindex. Thus, Eqs.(4.12) canbewritten most compactly inaccordance withthisconvention as x,'=a,Jx,, i=1,2,3. (4.1'2’) Therepeated appearance oftheindex jindicates thattheleft-hand sideof Eq.(4.l2’) isasumoverthedummy index jforallpossible values (here, j=1, 2,3).Some ambiguity ispossible where powers ofanindexed quantity occur, and forthatreason, anexpression suchas ZIx?l appears under thesutmnation convention as X1X;'. *Equanons (4.I2)ofcourse arenotthemostgeneral setoftransformauon equauons, cf.,forexample, those fromther’$totheq’$(1-38). Chapter 4TheKinematics ofRigid Body Motion Fortherestofthebook thesummation convention should beautomatically assumed inreading theequations unless otherwise explicitly indicated. Where convenient, ortoremove ambiguity, thesummation Signmay beoccasionally displayed explicitly, e.g.,when certain values oftheindex aretobeexcluded from thesummation. Thetransformation represented byEqs.(4.11) isonlyaspecial caseofthegen- erallinear transformation, Eqs.(4.12), since thedirection cosines arenotallinde- pendent. Theconnections between thecoefﬁcients, Eqs.(4.8)arerederived here intenns ofthenewer notation. Since bothcoordinate systems areCartesian, the magnitude ofavector isgiven interms ofthesumofsquares ofthecomponents. Further, since theactual vector remains unchanged nomatter which coordinate system isused, themagnitude ofthevector must bethesame inbothsystems In symbols, wecanstatetheinvariance ofthemagnitude as x:xl' =x,x,. (4.13) Theleft-hand sideofEq.(4.13) istherefore aljalkxjxks anditwillreduce totheright-hand sideofEq.(4.13), if,andonlyif auatk =1 =k =0 1¢k, (4.14) or,inamore compact form, if a,ja,-k =51)‘, j,k= 1,2,3. (4.l5) when theaucoethcients areexpressed interms ofthedirection cosines, thesix equations contained inEq.(4.15) become identical withtheEqs.(4.9). Anylinear transformation, Eq.(4.12), thathastheproperties required by Eq.(415)iscalled anorthogonal transformation, andEq.(4.15) itselfisknown astheorthogonality condition. Thus, thetransition from coordinates ﬁxed in space tocoordinates ﬁxed intherigid body (with common origin) isaccom- plished bymeans ofanorthogonal transformation. Thearray oftransformation quantities (thedirection cosines), written as 0|lH12013 H21422H23 , (4-16) 031I132rm iscalled thematrix oftransfomzation, andwillbedenoted byacapital letter A. Thequantities a,Jarecorrespondingly known asthematrix elements ofthetrans- formation. Tomake these formal considerations more meaningful, consider thesimple ex- ample ofmotion inaplane, sothatwearerestricted totwo-dimensional rotations, 4.2 Orthogonal Transformations 141 andthetransformation matrix reduces totheform an H12 0 H211122 © Q '-‘Q Thefourmatrix elements, av,areconnected bythree orthogonality conditions: aljalkzajka jskz I929 andtherefore onlyoneindependent parameter isneeded tospecify thetransfor- mation. Butthisconclusion isnotsurprising. Atwo-dimensional transformation from oneCartesian coordinate system toanother corresponds toarotation ofthe axesintheplane (cf.Fig.4.4), andsucharotation canbespeciﬁed completely by onlyonequantity, therotation angle ¢.Expressed interms ofthissingle parame- ter,thetransformation equations become xi=x1cosdr +x2sin¢ xé=-x1sin¢ -l-X2cos¢ X4=13. Thematrix elements aretherefore a11=cos¢ £l12=Sll1¢ a13=0 £12]=—smtﬁ an=cos45 a23=0 (4.17) 031=0 a32=0 ¢133=1, sothatthematrix Acanbewritten x x'2 2 r »='. l ¢ *1 FIGURE 4.4 Rotation ofthecoordinate axes, asequivalent totwo-dimensional orthog- onaltransformation. Chapter 4TheKinematics ofRigid Body Motion cos4;sin¢ 0 A= —sin¢ cos¢ 0 (4.17’) 0 0 1 Thethree nontrivial orthogonality conditions expand 1I1lZ0theequations allall +az1a21 =1 ¢l12r112 +0226122 =1 altar: +4121022 =0- These conditions areobviously satisﬁed bythematrix (4-17’), forinterms ofthe matrix elements (4.17) theyreduce totheidentities cos2¢+sin2¢ =1 sin2¢+cos2¢=1 cos¢sin¢ —sin¢cos¢ =0. Thetransformation matrix Acanbethought ofasanoperator that,acting ontheunprimed system, transforms itintotheprimed system. Symbolically. the process might bewritten (r)’=Ar, (4.18) which istoberead: Thematrix Aoperating onthecomponents ofavector inthe unprimed system yields thecomponents ofthevector intheprimed system. Note thatinthedevelopment ofthesubject sofar,Aactsonthecoordinate system only, thevector isunchanged, andweaskmerely foritscomponents intwodifferent coordinate frames. Parentheses havetherefore beenplaced around rontheleftin Eq.(4.18) tomake clearthatthesame vector isinvolved onbothsides ontheequa- tion.Onlythecomponents havechanged. Inthreedimensions, thetransformation ofcoordinates, asshovxn earlier, issimply arotation, andAisthenidentical with therotation operator inaplane. Despite this,notethatWithout changing theformal mathematics, Acanalsobe thought ofasanoperator acting onthevector r,changing ittoadifferent vector r’2 r’=Ar, (4.19) withbothvectors expressed inthesame coordinate system. Thus, intwodimen- sions. instead ofrotating thecoordinate system counterclockwise, wecanrotate thevector rclockwise byanangle 45toanewvector r’,asshown inFig.4.5.The components ofthenewvector willthenberelated tothecomponents oftheold bythesame Eqs. (4.12) thatdescribe thetransformation ofcoordinates. From a formal standpoint, itistherefore notnecessary touseparentheses inEq.(4.18); rather, itcanbewritten asinEq.(4.19) andinterpreted equally asanoperation on thecoordinate system oronthevector. Thealgebra remains thesame nomatter 4.2 Orthogonal Transformations 143 *2 Y ¢ " xi FIGURE 4.5 Interpretation ofanorthogonal transformation asarotation ofthevector, leaving thecoordinate system unchanged. which ofthese twopoints ofview isfollowed. Theinterpretation asanoperator acting onthecoordinates isthemore pertinent onewhen using theorthogonal transformation tospecify theorientation ofarigidbody. Ontheother hand, the notion ofanoperator changing onevector intoanother hasthemore widespread application. Inthemathematical discussion either interpretation willbefreely used, assuitstheconvenience ofthesituation. Ofcourse, notethatthenature oftheoperation represented byAwillchange according towhich interpretation isselected. Thus, ifAcorresponds toacounterclockwise rotation byanangle ¢ when applied tothecoordinate system, itwillcorrespond toaclockwise rotation when applied tothevector. Thesame duality ofroles often occurs with other types ofcoordinate transfor- mations thataremoregeneral thanorthogon altransformations. Theymayattimes belooked onasaffecting onlythecoordinate system, expressing some given quan- tityorfunction interms ofanewcoordinate system. Atother times, theymaybe considered asoperating onthequantity orﬁinctions themselves, changing themto newquantities inthesamecoordinate system. When thetransformation istaken asacting only onthecoordinate system, wespeak ofthepassive rolcofthetrans- fonnation. Intheactive sense, thetransformation islooked onaschanging the vector orother physical quantity. These altemative interpretations ofatransfor- mation willbeencountered invarious formulations ofclassical mechanics tobe considered below (cf.Chapter 9)andindeed occur inmany ﬁelds ofphysics. Todevelop further thekinematics ofrigidbodymotion about aﬁxedorigin, we shall make much useofthealgebra governing themanipulation ofthetransforma- tionmatrix. Thefollowing section istherefore abrief summary oftheelementary aspects ofmatrix algebra with speciﬁc application toorthogonal matrices. For those unacquainted withthisbranch ofmathematics, thesection should provide anintroduction adequate fortheimmediate purpose. Thematerial alsodetails the particular terminology andnotation wewillemploy. Those already thoroughly fa- 4.3 IChapter 4TheKinematics ofRigid Body Motion miliar withmatrix algebra mayhowever omitthesection andproceed directly to Section 4.4. FORMAL PROPERTIES OFTHE TRANSFORMATION MATRIX Letusconsider what happens when twosuccessive transformations aremade— corresponding totwosuccessive displacements oftherigid body. Lettheﬁrst transformation fromrtor’bedenoted byB: xi;=bqxj, (420) andthesucceeding transformation from 1"toathird coordinate setr”byA: x,"=a,kx,'c. (421) Therelation between xi’andxJcanthenbeobtained bycombining thetwoEqs. (4.20) and(4.21): X1” Z Thismayalsobewritten as X!” Z CIJXJ, where CU 1' 1 Thesuccessive application oftwoorthogonal transformations A,BISthus equivalent to21third linear transformation C.Itcanbeshown thatCisalsoan orthogonal transformation inconsequence oftheorthogonality ofAandB.The detailed proof willbeleftfortheexercises. Symbolically, theresultant operator C ca11beconsidered astheproduct ofthetwooperators AandB: C=AB, andthematrix elements cuarebydeﬁnition theelements ofthesquare matrix obtained bymultiplying thetwosquare matrices AandB. Notethatthis“matrix” oroperator multiplication isnotcommutative, BAgéAB, for.bydeﬁnition, theelements ofthetransformation D=BAare dz;=bikﬂ/Q, (4-24) 4.3 Formal Properties 0’rtheTranstormation Matrix 145 which generally donotagree withthematrix elements ofC,Eq.(4.23). Thus, the ﬁnalcoordinate system depends upon theorder ofapplication oftheoperators A andB,i.e.,whether ﬁrstAthen B,orﬁrstBandthenA.However, matrix mul- tiplication isassociative; inaproduct ofthreeormore matrices theorder ofthe multiplications isunimportant: (AB)C =A(BC). (4.25) InEq.(4.19) thejuxtaposition ofAandr,toindicate theoperation ofAon thecoordinate system (oronthevector), wassaidtobemerely symbolic. But,by extending ourconcept ofmatrices, itmay alsobetaken asindicating anactual matrix multiplication. Thus far,thematrices used have been square, i.e.,with equal number ofrows andcolumns. However, wemayalsohaveone-column matrices, suchasxandx’deﬁned by x1 xi x=X2 , x'= x§ . (4.26) x3 x-I; Theproduct Ax,bydeﬁnition, shallbetaken asaone-column matrix, withthe elements Hence, Eq.(4.19) canalsobewritten asthematrix equation x’=Ax. Theaddition oftwomatrices, while notasimportant aconcept asmultiplica- tion,isafrequently usedoperation. ThesumA+BISamatrix Cwhose elements arethesumofthecorresponding elements ofAandB: C1] ialj +b1J. Ofgreater importance isthetransformation inverse toA.theoperation that changes r’back tor.This treuisfumiatiuu willbecalled A"1 anditsmatrix ele- ments designated by41:].Wethenhave thesetofequations x,=af1x3, (4.27) which mustbeconsistent with xi=ak,x,. (4.28) Substituting x,from (4.27), Eq.(4.28) becomes xi=ak,a,5]-x’,-. (4.29) 46 Chapter 4TheKinematics ofRigid Body Motion Since thecomponents ofr’areindependent, Eq.(4.29) iscorrect onlyifthesum- mation reduces identically tox£.Thecoefficient ofx}musttherefore beIfor j=Itand0forjqék;insymbols, ak,a{J=5,, (4.30) Theleft-hand sideofEq.(4.30) iseasily recognized asthematrix element forthe product AA“, while theright-hand sideistheelement ofthematrix known as theunitmatrix 1: 1OO 1= 0l0. (431) 001 Equation (4.30) cantherefore bewritten as AA-‘ =1, (4.32) which indicates thereason forthedesignation oftheinverse matrix byA"1.The transformation corresponding to1isknown astheidentity transformation, pro- ducing nochange inthecoordinate system: x=1x. Similarly multiplying anymatrix Aby1,inanyorder, leaves Aunaffected: 1A=A1=A. Byslightly changing theorder oftheproof ofEq.(4.28), itcanbeshown thatA andA"commute. instead ofsubstituting x,inEq.(4.29) interms ofx’,wecould equally aswelldemand consistency byeliminating x’fromthetwoequations, leading inanalogous fashion to Inmatrix notation, thisreads A-1A= 1, (4.33) which proves thestatement. Nowletusconsider thedouble sum akidtiﬂfl. which canbewritten either as c1,a,'J withc1,=akiak, 4.3 Formal Properties oftheTransformation Matrix 147 01'aS akidkj With dkj=ak,a,'J. Applying theorthogonality conditions, Eq.(4.15), thesumintheﬁrstform re- duces to I (Shall =af]. Ontheother hand, thesame sumfrom thesecond point ofview, andwiththehelp oflziq.(4.50), canbewritten dkldkj =a11. Thus, theelements ofthedirect matrix Aandthereciprocal A_1arerelated by G;-I =61]]. Ingeneral, thematrix obtained from Abyinterchanging rows andcolumns is known asthetransposed matrix, indicated bythetildethusA.Equation (4.34) therefore states thatfororthogonal matrices thereciprocal matrix istobeidenti- ﬁedasthetransposed matrix; symbolically. A"=A. (4.35) Ifthisresult issubstituted inEq.(4.33), weobtain AA=1, (4.36) which isidentical withthesetoforthogonality conditions, Eq.(4.15), written in abbreviated form, ascanbeveriﬁed bydirect expansion. Similarly, analternative formoftheorthogonality conditions canbeobtained fromEq.(4.30) bysubsti- tuting (4.34): (1/“£11, =5/(J. (4.37) Insymbolic form, (4.37) canbewritten AA=1 andmaybederived directly from(4.36) bymultiplying itfromtheleftbyAand fromtherightbyA‘‘. Arectangular matrix 1Ssaidtobeofdimension m><nifithasmrows andn columns; i.e.,ifthematrix element isa;J-,thenirunsfrom Itom,andjfrom 1 ton.Clearly thetranspose ofsuch amatrix hasthedimension n><m.Ifavector column matrix isconsidered asarectangular matrix ofdimension m><l,the transpose ofavector isofdimension lxrn,i.e.,aone-row matrix. Theproduct Chapter 4TheKinematics ofRigid Body Motion ABoftworectangular matrices exists onlyifthenumber ofcolumns ofAisthe sameasthenumber ofrowsofB.Thisisanobvious consequence ofthedeﬁnition ofthemultiplication operation leading toamatrix element‘ Crj=atkb/(_]' From thisviewpoint, theproduct ofavector colunm matrix withasquare matrix doesnotexist. Theonlyproduct between these quantities thatcanbeformed is thatofasquare matrix with asingle column matrix. Butnotethatasingle row matrix, i.e.,avector transpose, canindeed pre-multiply asquare matrix. Fora vector, however, thedistinction between thecolumn matrix anditstranspose is often ofnoconsequence. Thesymbol xmaytherefore beused todenote either acolumn orarowniatiix, asthesituation warrants!‘ Thus intheexpression Ax, where Aisasquare matrix, thesymbol Xstands foracoluinti matrix, whereas in theexpression XAitrepresents thesame elements arranged inasingle row.Note thattheithcomponent ofAxcanbewritten as AUX] =¥J(A)J,'. Hence, wehaveauseful commutation property oftheproduct ofavector anda square matrix that AX =Xi. Asquare matrix thatisthesame asitstranspose, 4.,=4).. <438> issaid(forobvious reasons) tobesymmetric. When thetranspose isthenegative oftheoriginal matrix, AU Z TA”, thematrix isantisymmetric orskew symmetric. Clearly inanantisymmetric ma- trix,thediagonal elements arealways zero. Thetwointerpretations ofanoperator astransforming thevector, oraltema- tively thecoordinate system, arebothinvolved ifweﬁndthetransformation oi"an operator under achange ofcoordinates. LetAbeconsidered anoperator acting upon avector F(orasingle-column matrix F)toproduce avector G: G=AF. Ifthecooitliiiate system istransformed byamatrix B,thecomponents orthe vector Ginthenewsystem willbegiven by so=BAF, "The trans oscsinonvector matrices willoccasional] beretained where itisuseful toemhasize P 3 Y thedistinction between column androwmatrices 4.3 Fo'mal Properties oftheTransformation Matrix 149 which canalsobewritten ac=BAB-‘BF. (4.40) Equation (4.40) canbestated astheoperator BAB'1 acting ‘.]p0l1 thevector F, expressed inthenewsystem, produces thevector G,likewise expressed inthe newcoordinates. Wemaytherefore consider BAB" tobetheformtaken bythe operator Awhen transformed toanewsetofaxes: A’=BAB-1. (4.41) Anytransformation ofamatrix having theform ofEq.(4.41) isknown asasimi- larity transformation. Itisappropriate atthispoint toconsider theproperties ofthedeterminant formed fromtheelements ofasquare matrix. Asiscustomary, weshalldenote such adeterminant byvertical bars, thus: IA|.Note thatthedeﬁnition ofmatrix multiplication rsidentical withthatforthemultiplication ofdeterminants mm=|Al-lB|. (4.41') Since thedeterminant oftheunitmatrix rsl,thedeterminantal form oftheor- thogonality conditions, Eq.(4.36), canbewritten nit-tAt=1- Further, asthevalue ofadeterminant isunaffected byinterchanging rows and columns, wecanwrite |A|2=1, (4.42) which implies thatthedeterminant ofanorthogonal matrix canonlybe+1or—1. (Thegeometrical signiﬁcance ofthese twovalues willbeconsidered inthenext section.) When thematrix isnotorthogonal, thedeterminant doesnothavethese simple values, ofcourse. Itcanbeshown however thatthevalue ofthedeterminant is invariant under asimilarity transfonnation. Multiplying Eq.(4.41) forthetrans- formed matrix from therightbyB.weobtain therelation A’B=BA, orindetenninantal form IA’!-IBI=IBI~IAI- Since thedeterminant ofBlSmerely anumber, andnotzero,* wecandivide by *Ifitwere zero. there could benoinverse operator B_'(byCramer‘s rule), Nh.lCh isrequired for Eq.(44|)tomake sense. 4.4 IChapter 4TheKinematics ofRigid Body Motion |B|onbothsidestoobtain thedesired result: IA’!=IAI- Indiscussing rigidbody motion later. allthese properties ofmatrix transfor- mations, especially oforthogonal matrices, willbeemployed. Inaddition, other properties areneeded, andtheywillbederived astheoccasion requires. THE EULER ANGLES Wehavenoted (cf.p.I37)thatthenineelements ab,arenotsuitable asgeneralized coordinates because theyarenotindependent quantities. Thesixrelations that express theorthogonality conditions, Eqs.(4.9) orEqs. (415),ofcourse reduce thenumber ofindependent elements tothree. Butinorder tocharacterize the motion ofarigid body, there isanadditional requirement thematrix elements mustsatisfy, beyond those implied byorthogonality. Intheprevious section we pointed outthatthedeterminant ofarealorthogonal matrix could havethevalue +1or-1.Thefollowing argument shows however thatanorthogonal matrix whose determinant is-1cannot represent aphysical displacement ofarigidbody. Consider thesimplest 3><3matrix withthedeterminant -1: -1 O 0 §= 0-1 O=—1. 0 O-1 Thetransformation Shastheeffect ofchanging thesignofeachofthecomponents orcoordinate axes (cf.Fig. 4.6). Such anoperation transforms aright-handed coordinate system intoaleft-handed oneandisknown asaninversion ofthe coordinate axes. Onemethod ofperforming aninversion istorotate about acoordinate axisby 180°andthenreﬂect inthatcoordinate axisdirection. Forthez-direction, this gives rotate reﬂect by180° inthe =inversion. about 2 xyplane Z i S.y—'~y X I Z FIGURE 4.6Inversion ofthecoordinate axes.xi 4.4 TheEuler Angles 151 Inmatrix notation, thishastheform —1 00 10 O -1 O0 0-1O 010= 0-1 0. 0 01 00 1 00 1 where the180°rotation isobtained bysetting ¢=180°inEq.(4.17). From thenature ofthisoperation, itisclearthataninversion ofaright-handed system intoaleft-handed onecannot beaccomplished byanyrigidchange inthe orientation ofthecoordinate axes Aninversion therefore never corresponds toa physical displacement ofarigidbody. What istruefortheinversion Sisequally valid foranymatrix whose determinant is-1,foranysuchmatrix canbewrit- tenastheproduct ofSwithamatrix whose determinant is+1,andthusincludes theinversion operation. Consequently, itcannot describe arigid change inon- entation Therefore, thetransformations representing rigidbodymotion mustbe restricted tomatrices having thedeterminant +1.Another method ofreaching this conclusion starts fromthefactthatthematrix oftransformation mustevolve con- tinuously fromtheunitmatrix, which ofcourse hasthedeterminant +1.Itwould beincompatible withthecontinuity ofthemotion tohavethematrix determinant suddenly change fromitsinitial value +1to-1atsome given time. Orthogonal transformations withdeterminant +1aresaidtobeproper, andthose withthe determinant -1arecalled improper. Inorder todescribe themotion ofrigidbodies intheLagrangian formulation ofmechanics, itwilltherefore benecessary toseekthreeindependent parameters thatspecify theorientation ofarigidbody insuchamanner thatthecorrespond- ingorthogonal matrix oftransformation hasthedeterminant +1.Onlywhen such generalized coordinates havebeenfound canwewrite aLagrangian forthesys- temnndobtain theLagrangian cquations ofmotion. Anumber ofsuch setsof parameters have been described intheliterature, butthemost common anduseful aretheEuler orEulerian angles. Weshalltherefore deﬁne these angles atthis point, andshowhowtheelements oftheorthogonal transformation matrix canbe expressed interms ofthem. Wecancarry outthetransformation fromagiven Cartesian coordinate sys- temtoanother bymeans ofthree successive rotations performed inaspeciﬁc sequence. TheEuler angles arethendeﬁned asthethree successive angles ofrota- tion.Within limits, thechoice ofrotation angles isarbitrary. Themainconvention thatwillbefollowed hereisusedwidely incelestial mechanics, applied mechan- ics,andfrequently inmolecular andsolid-state physics. Other conventions will bedescribed below andinAppendix A. Thesequence employed hereisstarted byrotating theinitial system ofaxes, xyz,byanangle ¢counterclockwise about the1axis,andtheresultant coordinate system islabeled the$172;axes. Inthesecond stage, theintermediate axes, $175, arerotated about the.5axiscounterclockwise byanangle 6toproduce another in- termediate set,the5'11’§’axes. TheE’axisisattheintersection ofthexyandE'17’ planes andisknown asthelineofnodes. Finally, the£,=’17’2;’ axesarerotated coun- Chapter 4TheKinematics ofRigid Body Motion FIGURE 4.7Therotations deﬁning theEulerian angles. terclockwise byanangle ipabout the4"axistoproduce thedesired x'y'z' system ofaxes. Figure 4.7illustrates thevarious stages ofthesequence. TheEuler angles 0,¢,and1/1thuscompletely specify theorientation ofthex'y'z' system relative tothexyzandcantherefore actasthethreeneeded generalized coordinates. * Theelements ofthecomplete transformation Acanbeobtained bywriting the matrix asthetriple product oftheseparate rotations, eachofWl'II(‘h hasarelatively simple matrix form. Thus, theinitial rotation about zcanbedescribed byamatrix D: §=DX, where §andxstand forcolumn matrices. Similarly, thetransformation from £174‘ to$’r7'§’ canbedescribed byamatrix C, *Anumber ofminor variations willbefound intheliterature cwcnwithin thisconvention Thediffer- ences arenotverygreat, buttheyareoften sufﬁcient tofrustrate easycomparison ofthecndformulae. suchasthematrix elements. Greatest confusion. perhaps, arises fromtheoccasional useofleft-handed coordinate systems 4.4 TheEuler Angles 153 €’=cs. andthelastrotation tox’y'z’byamatrix B, X’=B§'. Hence, thematrix ofthecomplete transformation, x’=Ax, istheproduct ofthesuccessive matrices, A=BCD. Now theDtransformation isarotation about z,andhence hasamatrix ofthe form(cf.Eq.(4.17)) cos¢ sin¢0 D=—sin¢ cos¢0. (4.43! 0 0 l TheCtransformation corresponds toarotation about 5,withthematrix l 0 0 C=0cos6sin0, (4.44-l 0—sin6cos9 andﬁnally Bisarotation about §’andtherefore hasthesameformasD: cos1/1 sin1/!0 B=—sin1,0cos1,110. (4.45) O O 1 Theproduct matrix A=BCD thenfollows as A=[—§tfl¢COQ¢—COS9S|l‘l¢C05tl! —sint/1s.n¢+cos9cos¢cosilr cos¢sin9 .costhcosip —cos9sin¢s1n1!/ costlrsm¢+cos9cos¢ sintli sint//sm9 sin9sin¢ —sin6cos¢ cos6 (4.46) Theinverse transformation from body coordinates tospace axes x=A'1x’ isthengiven immediately bythetransposed matrix A: A-1= W |:COS'lPCOS¢—COS9Sln¢Slll\l! —sintI/cos¢-cos9sin¢cos1// sin6sin¢ A= c0stlrsin¢+cos6cos¢sin1,0 —sin1ﬁ's1n¢+C0s6lcos¢cos1,h —srn9cos¢ . sin9sinili sin9cos1,0 cos9 (4.47) 4.5 IChapter 4TheKinematics ofRigid Body Motion Veriﬁcation ofthemultiplication, anddemonstration thatArepresents aproper, orthogonal matrix willbelefttotheexercises. Notethatthesequence ofrotations usedtodeﬁne theﬁnalorientation ofthe coordinate system istosome extent arbitrary. Theinitial rotation could betaken about anyofthethree Cartesian axes. Inthesubsequent tworotations, theonly limitation isthatnotwosuccessive rotations canbeabout thesame axis. Atotal of12conventions istherefore possible indeﬁning theEuler angles (inaright- handed coordinate system). Thetwomost frequently usedconventions differ only inthechoice ofaxisforthesecond rotation. IntheEuler’s angle deﬁnitions de- scribed above, andused throughout thebook. thesecond rotation isabout the intermediate xaxis.Wewillrefertothischoice asthex-convention. Inquan- tummechanics, nuclear physics, andparticle physics, weoften takethesecond deﬁning rotation about theintermediate yaxis; thisform willbedenoted asthe y—c0nventi0n. Athird convention iscommonly usedinengineering applications relating to theorientation ofmoving vehicles snch asaircraft andsatellites Roth the1r-and y—conventions havethedrawback thatwhen theprimed coordinate system isonly slightly different fromtheunprimed system, theangles ¢and1/Ibecome indistin- guishable, astheirrespective axesofrotation, zandz’arethennearly coincident. Togetaround thisproblem, allthree rotations aretaken around different axes. Theﬁrstrotation isabout thevertical axisandgives theheading oryawangle. Thesecond isaround aperpendicular axisﬁxed inthevehicle andnormal tothe ﬁgure axis;itismeasured bythepitch orattitude angle. Finally, thethirdangle isoneofrotation about theﬁgure axisofthevehicle andistherollorbank an- gle.Because allthree axesareinvolved intherotations, itwillbedesignated as thexyz-convention (although theorder ofaxeschosen mayactually bedifferent). Thislastconvention issometimes referred toastheTair—Bryan angles. While only thex-convention willbeused inthetext, forreference purposes Appendix Alistsformulae involving Euler’s angles, suchasrotation matrices, in boththey-andxyz-conventions. THE CAYLEY-KLEIN PARAMETERS AND RELATED QUANTITIES Wehaveseenthatonlythreeindependent quantities areneeded tospecify theori- entation ofarigidbody. Nonetheless, thereareoccasions when itisdesirable to usesetsofvariables containing more thantheminimum number ofquantities to describe arotation, eventhough theyarenotsuitable asgeneralized coordinates. Thus, Felix Klein introduced thesetoffour parameters bearing hisname tofa- cilitate theintegration ofcomplicated gyroscopic problems TheEuler angles are difﬁcult touseinnumerical computation because ofthelarge number oftrigono- metric functions involved, andthefour-parameter representations aremuch better adapted foruseoncomputers. Ftuther, thefour-parameter setsareofgreat the- oretical interest inbranches ofphysics beyond thescope ofthisbook, wherever 4.6I4.6 Euler's Theorem ontheMotion ofaRigid Body 155 rotations orrotational symmetry areinvolved. lttherefore seems worthwhile to brieﬂy describe these parameters, leaving thedetails toAppendix A. ThefourCayley-Klein parameters arecomplex numbers denoted byoz,B,y, and8with theconstraints thatB=y*and6=a*.Interms ofthese numbers, thetransformation matrix ofarotated body isgiven by |\)Iv-|\)1-1|--|[\)s---toﬁ-1/2+62-to -oz—a’+@2 -an 1/5-at A=-(<12+Y2-at-8’)§(¢¥2+1/2+#2+62)—t<¢a+ya‘ ﬂd—ozy i(<xy +135) 0z8+ﬁ)/ Thematrix Aisrealinspiteofits appearance, aswecanseebywriting 0t=e0+ie3 /3=@2+i-<21, where thefourrealquantities eq,e1,82,ande3areoftenreferred toastheCayley- Klein parameters butshould becalled theEuler parameters tobecorrect. They satisfy therelation %+J+%+§=L Abitofalgebraic manipulation thenshows thatthematrix Acanbewritten in terms ofthefourrealparameters intheform 8%+6%—(2%—eg 2(e1e1 +6063) 2(e1e3 —eoeg) A= 2(¢t¢z —elm) 8%—ei+v;—9,? 2(@2¢s +@0@i) -(4-47') 2(¢1Pz +@062) 2(€2@3 -@081) 66—8%-8%+6% Thereality ofthematrix elements isnowmanifest. Itcanalsobeeasily demon- strated thatthematrix Aintenns ofthese parameters cannot beputintheform of theinversion transformation S.Anexamination oftheoff-diagonal elements and theirlransposes shows thattheyallvanish onlyifatleastthreeoftheparameters arezero. Wecannot thenchoose theremaining nonzero parameter such thatall three ofthediagonal elements (oronlyoneofthem) are——1. EULER'S THEOREM ONTHEMOTION OFARIGID BODY Thediscussions oftheprevious sections provide acomplete mathematical tech- nique fordescribing themotions ofarigid body. Atanyinstant, theorientation of thebodycanbespeciﬁed byanorthogonal transformation. theelements ofwhich maybeexpressed interms ofsome suitable setofparameters. Astimeprogresses, theorientation willchange, andhence thematrix oftransformation willbeafunc- Chapter 4TheKinematics ofRigid Body Motion tionoftimeandmaybewritten A(t). Ifthebody axesarechosen coincident with thespace axesatthetimet=O,thenthetransformation isinitially simply the identity transformation: A(O) =1. Atanylatertime,A(r)willingeneral differ fromtheidentity transformation, but sincethephysical motion mustbecontinuous, A(t)mustbeacontinuous function oftime. Thetransformation maythusbesaidtoevolve continuously from the identity transformation. With thismethod ofdescribing themotion, andusing onlythemathematical apparatus already introduced, wearenowinaposition toobtain theimportant characteristics ofrigid body motion. Ofbasic importance is: Euler’s Theorem" Thegeneral displacement ofarigid body withone poinlﬁxed isarotation about some axis. Thetheorem means thatforevery suchrotation itisalways possible toﬁndan axisthrough theﬁxedpoint oriented atparticular polar angles 6and¢suchthat arotation bytheparticular angle 1/1about thisaxisduplicates thegeneral rota- tion.Thus, three parameters (angles) characterize thegeneral rotation. Itisalso possible toﬁndthreeEuler angles toproduce thesamerotation. Iftheﬁxedpoint (notnecessarily atthecenter otmassoitheobject) IStaken astheorigin ofthebody setofaxes, thenthedisplacement oftherigidbody involves notranslation ofthebody axes; theonlychange isinorientation. The theorem thenstates thatthebodysetofaxesatanytimetcanalways beobtained byasingle rotation oftheinitial setofaxes(taken ascoincident withthespace set).Inother words, theoperation implied inthematrix Adescribing thephysical motion oftherigidbodyisarotation. Nowitischaracteristic ofarotation thatone direction, namely, theaxisofrotation, isleftunaffected bytheoperation. Thus. anyvector lying along theaxisofrotation musthavethesamecomponents inboth theinitial andﬁnalaxes. Theothernecessary condition forarotation, thatthemagnitude ofthevectors beunaffected, isautomatically provided bytheorthogonality conditions. Hence. Euler’s theorem willbeproven ifitcanbeshown thatthere exists avector Rhav- mgthesamecomponents inbothsystems. Using matrix notation forthevector, R’=AR=R. (4.48) Equation (4.48) constitutes aspecial caseofthemore general equation: R’=AR=AR, (4.49) where Aissome constant, which maybecomplex. Thevalues ofAforwhich Eq.(4.49) issoluble areknown asthecharacteristic values, oreigenvalues,* of *Th|s term1sdenved from theGerman Ezgenwerre literally “proper values " 4.6 Euler's Theorem ontheMotion ofaRigid Body 157 thematrix. Since equations oftheformof(4.49) areofgeneral interest andwillbe usedinChapter 6,weshallexamine Eq.(4.49) andthenspecialize thediscussion toEq.(4.48). Thepioblem ofﬁnding vectors thatsatisfy Eq.(4.49) istherefore called the eigenvalue problem forthegiven matrix, andEq.(4.49) itself isreferred toasthe eigenvalue equation. Correspondingly, thevector solutions aretheeigenvectors ofA.Euler’s theorem cannowberestated inthefollowing language: Therealorthogonal matrix specifying thephysical motion ofarigid body withonepoint ﬁxed always hastheeigenvalue +1. Theeigenvalue equations (4.49) maybewritten (A—).1)R =0, (4.50) or,inexpanded form, (ﬂ1|— K)X+41127’ —4132 =0 Cl21X +(Q22—}.)Y—a23Z =0 (4.51) a31X +a32Y +((133—}.)Z=0. Equations (4.51) comprise asetofthreehomogeneous simultaneous equations for thecomponents X,Y,Zoftheeigenvector R.Assuch, theycannever furnish def- initevalues forthethree components, butonlyratios ofcomponents. Physically, thiscorresponds tothecircumstance thatonlythedirection oftheeigenvector can beﬁxed; themagnitude remains undetermined. Theproduct ofaconstant withan eigenvector isalsoaneigenvector. Inanycase,being homogeneous, Eqs.(4.51) canhaveanontrivial solution onlywhen thedeterminant ofthecoefﬁcients van- ishes. 6111—K 012 013 IA—)L1| = (121 (Z22 -K G23 =O. (4.52) ﬂat 432 Q33—7» Equation (4.52) isknown asthecharacteristic orsecular equation ofthematrix, andthevalues ofAforwhich theequation issatisﬁed arethedesired eigenvalues. Euler’s theorem reduces tothestatement that,fortherealorthogonal matrices under consideration, thesecular equation must have therootJL=+1. Ingeneral, thesecular equation willhave three roots withthree corresponding eigenvectors. Forconvenience, thenotation X1,X2,X3willoften beusedinstead ofX,Y.Z.Insuch anotation, thecomponents oftheeigenvectors might be labeled asX,k,theﬁrstsubscript indicating theparticular component, thesecond denoting which ofthethreeeigenvectors ininvolved. Atypical member ofthe group ofEqs.(4.51) would thenbewritten (withexplicit summation) as Zazjxjk =7-kxrk ] Chapter 4TheKinematics ofRigid Body Motion Or,altematively. as Za,,X,k =ZX,]5Jk7tk. (4.53) J 1' BothsidesofEq.(4.53) thenhavetheformofamatrix product element; theleft sideastheproduct ofAwithamatrix Xhaving theelements XJk,theright side astheproduct ofXwithamatrix whose jkthelement isSjkitk. Thelastmatrix is diagonal, anditsdiagonal elements aretheeigenvalues ofA.Weshall therefore designate thematrix byA: L1 0 0 A= Olg 0 . (4.54) 0 0X3 Equation (4.53) thusimplies thematrix equation AX=XA, or,multiplying fromtheleftbyX'1, x-1Ax =x. (4.55) Now, theleftsideisintheform ofasimilarity transformation operating onA.(We haveonlytodenote X‘1bythesymbol Ytoreduce ittothefonn Eq.(4.4l).) Thus, Eq.(4.55) provides thefollowing altemative approach totheeigenvalue problem: Weseektodiagonalize Abyasimilarity transformation. Eachcolumn ofthema- trixusedtocarry outthesimilarity transformation consists ofthecomponents of aneigenveclut. Theelements oflliediagonaliaed form ofAarethecorresponding eigenvalues. _Euler’s theorem canbeproven directly byusing theorthogonality property of A.Consider theexpression (A—1)A=1—A. Ifwetakethedetenninant ofthematrices formi.ng bothsides (cf.Eq.(4.41')), we canwritetheequality |A-1||;i| =|1-A|. (4.56) Todescribe themotion ofa.rigid body, thematrix A(t) must correspond toa. proper rotation; therefore thedeterminant ofA,andofitstranspose, mustbe+1. Further, since ingeneral thedeterminant ofthetranspose ofamatrix isthesame asthatofthematrix, thetranspose signs inEq.(4.56) canberemoved: IA—1|=I1—AI. (4.57) 4.6 Euler's Theorem ontheMotion ofaRigid Body 159 Equation (4.57) saysthatthedeterminant ofaparticular matrix isthesame asthe determinant ofthenegative ofthematrix. Suppose Bissome n><nmatrix. Then itisawell-known property ofdeterminants that 1-Bi=(—1)"iB|- Since weareworking inathree-dimensional space (n=3),itisclear that Eq.(4.57) canholdforanyarbitrary proper rotation onlyif IA—1|=O. (4.58) Comparing Eq.(4.58) withthesecular equation (4.52), wecanseethatoneofthe eigenvalues satisfying Eq.(4.52) mustalways be7.=+1,which isthedesired result ofEuler’s theorem. Notehowtheproof ofEulerie theorem emphasizes theimportance ofthenum- berofdimensions inthespace considered. Inspaces withanevennumber of dimensions, Eq.(4.57) isanidentity forallmatrices andEuler’s theorem doesn’t hold.Thus, fortwodimensions thereisnovector inthespace thatisleftunaltered byarotation—the axisofrotation isperpendicular totheplane andtherefore out ofthespace. itisnowasimple matter todetermine thepI'OpOI‘tieS oftheothereigenvalues inthreedimensions. Designate the+1eigenvalue asA3.Thedeterminant ofany matrix isunaffected byasimilarity transformation (cf.Section 4.3).Hence, by Eqs.(4.54) and(4.55) andtheproperties ofAasaproper rotation, [AI=Mkgks =M12 =1. (4.59) Further, since Aisarealmatrix, then ifAlSasolution ofthesecular equa- tion(4.52), thecomplex con_1ugate }t*must alsobeasolution. Ifagiven eigenvalue A,iscomplex, thenthecorresponding eigenvector, R,-, thatsatisﬁes Eq.(4.59) willingeneral alsobecomplex. Wehavenotpreviously dealtwiththeproperties ofcomplex vectors under (real) orthogonal transforma- tions, andtherearesome modiﬁcations toprevious deﬁnitions. Thesquare ofthe length ormagnitude ofacomplex vector RisR-R*,orinmatrix notation §R*, where thetranspose signontheleft-hand vector indicates itisrepresented bya rowmatrix Under arealorthogonal transformation, thesquare ofthemagnitude isinvariant ﬁ'R'*=(A‘iz)AR* =RAAR* =iuz*. Suppose nowthatRisacomplex eigenvector corresponding toacomplex eigen- value )t.Hence, byEq.(4.49), wehave ﬁ'R!* =l\.h.*|iR*, Chapter 4TheKinematics ofRigidBodyMotion which leads totheconclusion thatalleigenvalues have unitmagnitude: MU=1. (1-.60) From thcsc properties itmay beconcludcd thatthcrc arethrcc possible CilSlII’i— butions ofeigenvalues. ifalloftheeigenvalues arereal,thenonlytwosituations arepossible: 1.Alleigenvalues are+1.Thetransformation matrix isthenjust 1,acasewe mayjustly calltrivial. 2.Oneeigenvalue is+1andtheother twoareboth-1.Such atransfonnation maybecharacterized asaninversion intwocoordinate axeswiththethird unchanged. Equally itisarotation through theangle Jrabout thedirection oftheunchanged axis. Ifnotalloftheeigenvalues arereal,thereisonlyoneadditional possibility: 3.Oneeigenvalue IS+1,andtheother twoarecomplex conjugates ofeach other oftheforme‘°ande"°. Amore complete statement ofEuler’s theorem thusisthatanynontrivial real orthogonal matrix hasone,andonlyone,eigenvalue +1. Thedirection cosines oftheaxisofrotation canthenbeobtained bysetting A=I1l‘ltheeigenvalue equations (4.51) andsolving forX,Y,andZ.*The angle ofrotation canlikewise beobtained without difﬁculty. Bymeans ofsome similarity transformation, itisalways possible totransfoim thematrix Atoa system ofcoordinates where thezaxisliesalong theaxisofrotation. Insucha system ofcoordinates. A’represents arotation about thezaxisthrough anangle <l>,andtherefore hastheform cossin<1)0 A'= —sin cos 0. 0 0 l Thetrace ofA’issimply 1+2cos<l>. Since thetrace isalways invariant under asimilarity transformation, thetrace of Awithrespect toanyinitial coordinate system must havethesame form, TrA=an=1+2cos, (4.61) *lfthere aremultiple roots tothesecular equation, thenthecorresponding etgenvectors cannot be found assimply (ctSections 54and6.2)Indeed, itisnotalways possible tocompletely dl£1g0ﬂE1l1LB ageneral matrix iftheeigenvalues arenotalldistinct These exceptions areotnoimportance forthe present considerations. asEuler’s theorem shows thattorallnontrivial orthogonal matrices +1isa single root 4.7I4.7 Finite Rotations 161 which gives thevalue of<1)interms ofthematrix elements. Therotation angle <l> istobeidentiﬁed alsowiththephase angle ofthecomplex eigenvalues A,asthe sumoftheeigenvalues isjustthetrace ofAinitsdiagonal form, Eq.(4.54). By Euler’s theorem andtheproperties oftheeigenvalues, thissumis TrA=Z)t, =l+e‘¢+e"° =1+2cos<l>. I Weseethatthesituations inwhich theeigenvalues areallrealareactually special cases ofAhaving complex eigenvalues. Allthe7t,=+1corresponds toarotation angle £1»=0(theidentity transformation), while thecasewithadouble eigenvalue —lcorresponds to<l>=rt,aspreviously noted. Theprescriptions forthedirection oftherotation axisandfortherotation angle arenotunambiguous. Clearly ifRisaneigenvector, sois—R;hence thesense of thedirection oftherotation axisisnotspeciﬁed. Further, —<l>satisﬁes Eq.(4.61) ifdoes. indeed, itisclear thattheeigenvalue solution does notuniquely ﬁx theorthogonal transformation matrix A.From thedeterrmnantal secular equa- tion(4.52), itfollows thattheinverse matrix A“!=Ahasthesame eigenvalues andeigenvectois asA.However, theambiguities canatleast beameliorated by assigning <l>toAand—<l>toA",andﬁxing thesense oftheaxesofrotation by theright-hand screw rule. Finally, noteshould bemade ofanimmediate corollary ofEuler’s theorem, sometimes called Chasles' Theorem: Themostgeneral displacement ofarigid body is atranslation plusarotation. Detailed proof ishardly iiecessary. Simply stated, removing theconstraint ofmo- tionwith onepoint ﬁxed introduces three translatory degrees offreedom forthe origin ofthebody system ofaxes.* FINITE ROTATIONS Therelative orientation oftwoCartesian coordinate systems with common ori- ginhasbeen described byvarious representations, including thethree successive Euler angles ofrotation thattransform onecoordinate system totheother. inthe previous section itwasshown thatthecoordinate transformation canbecarried through byasingle rotation about asuitable direction. Itistherefore natural to seek arepresentation ofthecoordinate transformation intenns oftheparame- *MChaslcs (1793-1881) alsoproved astronger form ofthetheorem, namely. thatitispossible to choose theorigin ofthebody setofcoordinates sothatthetranslation isinthesame direction asthe axisofrotation. Such acombination ottranslation androtation iscalled ascrew motion Thisl'oi-rnalism hassome useincrystallograpluc studies ofcrystals withascrew axisorsymmetry. Such syininctry produces strange optical properties. Aside from thatapplication, there seems tobe little present useforthisversion ofChasles’ theorem, norfortheelaborate mathematics ofscrew motions developed inthenineteenth century. Chapter 4TheKinematics ofRigid Body Motion tersoftherotation-the angle ofrotation andthedirection cosines oftheaxisof rotation. With thehelpofsome simple vector algebra, wecanderive such arepresen- tation Forthispurpose, itisconvenient totreat thetransformation initsactive sense, i.e.,asonethatrotates thevector inaﬁxed coordinate system (cf.Sec- tion4.2).Recall thatacounterclockwise rotation ofthecoordinate system then appears asaclockwise rotation ofthevector. InFig.4.8(a) theinitial position of thevector risdenoted by$9andtheﬁnalposition r’by5Q, while theunit vector along theaxisofrotation isdenoted byn.Thedistance between 0andN hasthemagnitude n-r,sothatthevector WV canbeWI'lﬂ'.CI1 asn(n-r).Fig- ure4.8(b) sketches thevectors intheplane normal totheaxisofrotation. The vector IW5canbedescribed alsoasr—n(n-r),butitsmagnitude isthesame as thatofthevectors ITQandrxn.Toobtain thedesired relation between r’andr, weconstruct r’asthesumofthree vectors: r’=OW+W+@ or r’=n(n~r)+[r—n(n-r)]cos<l> +(rxn)sin<l>. Aslight rearrangement oftenns leads totheﬁnalresult: r’=rcos+n(n-r)(1—cos<l>)+(rxn)sin<l>. (4.62) Equation (4.62) willbereferred toastherotation formula. Note thatEq.(4.62) holds foranyrotation, nomatter what itsmagnitude, andthusisaﬁnite-rotation version (inaclockwise sense) ofthedescription given inSection 2.6,forthe change ofavector under inﬁnitesimal rotation (cfalsoSection 48) N \- VQ. n(n-r) r YIP Q 0 (b) Theplane normal to (ﬁ)0\'Bl’==1|1Vl6\‘1 theaxisofrotation FIGURE 4.8 Vector diagrams fordenvanon oftherotation formula. 4.8 I4.8lnﬁnitesimal Rotations 163 Itisstraightforward toexpress therotation angle, <1),interms oftheEuler an- gles.Equation (4.61) gives thetrace oftherotation matrix intheplane perpendic- ulartotheaxisofrotation. Since thetrace ofamatrix isinvariant, thisexpression must equal thetrace ofAasgiven inEq(4.46) Ifweusethisequality, addone(1) tobothsides, andusetrigonometric identities, wegetanequation whose square rootis <l> 6cos—2—=cos2%” cos (4.63) where thesignofthesquare ruulisﬁxed bylliephysical requireuient that.Q—>O as¢.rlr,and9 —>0. INFINITESIMAI. ROTATIONS Intheprevious sections various matrices havebeen associated withthedescrip- tionoftherigid body orientation. However, thenumber ofmatrix elements has always been larger thanthenumber ofindependent variables, andvarious sub- sidiary conditions havehadtobetagged on.Nowthatwehaveestablished that anygiven orientation canbeobtained byasingle rotation about some axis, itis tempting totrytoassociate avector, characterized bythree independent quanti- ties,withtheﬁmte displacement orarigidbodyabout aﬁxedp01I1L Certainly a direction suggests itself obviously—that oftheaxisofrotation—and anyfunction oftherotation angle would seem suitable asthemagnitude. Butitsoon becomes evident thatsuch acorrespondence cannot bemade successfully. Suppose Aand Baretwosuch“vectors” associated withtransformations AandB.Then toqualify asvectors theymust becommutative inaddition: A+B=B+A. Buttheaddition oftworotations, i.e.,onerotation performed afteranother, ithas been seen, corresponds totheproduct ABofthetwomatrices. However, matrix multiplication isnotcommutative, ABgéBA,andhence A,Barenotcommuta- tiveinaddition andcannot beaccepted asvectors. Thisconclusion, thatthesum ofﬁnite rotations depends upon theorder oftherotations, isstrikingly demon- strated byasimple experiment. Thus, Fig.4.9illustrates thesequence ofevents inrotating ablock ﬁrstthrough 90°about thez’axisﬁxed intheblock, andthen 90°about they’axis,while Fig.4.10presents thesame rotations inreverse order. Theﬁnalposition ismarkedly different inthetwosequences. While aﬁnite rotation thuscannot berepresented byasmgle vector, thesame objections donotholdifonlyinﬁnitesimal rotations areconsidered. Aninﬁnites- imal rotation isanorthogonal transformation ofcoordinate axes inwhich the components ofavector arealmost thesame inbothsetsofaxes—the change isinﬁnitesimal. Thus, thexicomponent ofsomevector r(onthepassive interpre- tation ofthetransformation) would bepractically thesame asx1,thedifference Chapter 4TheKinematics ofRigid Body Motion (a)Vertical positior (b)Rotated 90°about 2' (c)Rotated 90°about intermediate y FIGURE 4.9 Theeffect oftworotations performed inagiven order. >1! / yl / xi 1’ Z’ xi Z’ (a)v6l'l.‘lC=ll position (b)Romled 90°about y’ (c)Rotated 90°about intermediate z’ FIGURE 4.10 Thetworotations shown inFig.4.9,butperfonned inreverse order. being extremely small: xi=x1+ e11x1+ 6121!; +613263. (4.64) Thematrix elements e11,e12,etc.,aretobeconsidered asinﬁnitesimals, sothatin subsequent calculations onlytheﬁrstnonvanishing order in6,,needberetained. Foranygeneral component x",theequations ofinﬁnitesimal transformation can bewn'tten as xf=x,+e,_,xJ or xi’=(5,,+e,J)xJ. (4.65) Thequantity 8,,willberecognized astheelement oftheunitmatrix, and Eq.(4.65) appears inmatrix notation as x’=(1+e)x. (4.66) 4.8 infinitesimal Rotations 165 Equation (4.66) states thatthetypical formforthematrix ofaninﬁnitesimal trans- formation is1+e;i.e.,itisalmost theidentity transformation, differing atmost byaninﬁnitesimal operator. Itcannowbeseenthatthesequence ofoperations isunimportant forinﬁnites- imaltransformations; inother words, theycommute. If1+6|and1+E2aretwo inﬁnitesimal transformations, thenoneofthepossible products is (1+61)(1+ 62)=12 +611 +162 +6162 =1+€1+62, (4.67) neglecting higher-order inﬁnitesimals. Theproduct inreverse order merely inter- changes eiand62;thishasnoeffect ontheresult, asmatrix addition isalways commutative. Thecommutative property ofinﬁnitesimal transformations over- comes theobjection totheirrepresentation byvectors. Forexample, therotation matrix (4.46) forinﬁnitesimal Euler rotation angles isgiven by l (d¢+dtb) 0 A=—(d¢ +dt/I) 1 d6 0 —d6 1 and d$'Z=id6 +k(d¢ +di//), where iandkaretheunitvectors inthex-andz-directions, respectively. Theinverse matrix foraninﬁnitesimal transformation isreadily obtained. Ii A=1+eisthematrix ofthetransformation, thentheinverse is A-1=I-6. (4.68) Asproof, notethattheproduct AA_l reduces totheunitmatrix, AA_]=(1+e)(i—e)=1, inagreement with thedeﬁnition_f0r theinverse matrix, Eq(432) Further, the orthogonality ofAimplies thatAE(1+E)must beequal toA4asgiven by Eq.(4.68). Hence, theinﬁnitesimal matrix isantisymmetric* (cf.Eq.(4.39)): €=—e. Since thediagonal elements ofanantisymmetric matrix arenecessarily zero, there canbeonlythree distinct elements inany3><3antisymmetric matrix. Hence, *Inthissection wehave assumed implicitly thatanniinitesimal orthogonal transformation corre- sponds toarotation. Inasense thisassumption isobvious; an“inﬁnitesimal inversion” isacontradii:- tionintenns. Formally. thestatement follows from theantisymmetry ofeAllthediagonal elements of1+earethenunity, andtoﬁrstorder insmall quantities, thedeterminant ofthetransformation is always +.which isthemarkofaproper rotation. Chapter 4TheKinematics ofRigid Body Motion thereisnolossofgenerality inwriting einthefoim O £15.23 —dQg G= —!1Q3 0 IIQ1 (4.69) (IQ; —dQ1 0 Thethree quantities (191, (K22, d§Z3 areclearly tobeidentiﬁed with thethree independent parameters specifying therotation. Wewillnowshow thatthese three quantities alsoform thecomponents ofaparticular kindofvector. ByEq.(4.66) thechange inthecomponents ofavector tinder theinﬁnitesimal transformation ofthecoordinate system canbeexpressed bythematrix equation r’—rEdr'=er, (4.70) which inexpanded form, withisgiven by(4.69), becomes dX1=X2£lQ3—X3(lQ2 dx; =X36191 —Xi]£193 (4.71) dX3 =X1d§Zg—x2dQ1. Theright-hand sideofeachofEqs.(4.71) isintheformofacomponent ofthe cross product oftwovectors, namely, thecross product ofrwithavector d§2hav- ingcomponents* (191,dS22,dQ3.Wecantherefore writeEq.(4.71) equivalently as dr=rxdﬂ. (4.72) Thevector rtransforms under anorthogonal matrix Baccording totherelations (cf.Eq.(4.20)) x,’=bi,-xj. (4.73) Ifdﬂistobeavector inthesame sense asr,itmust transform under Binthe same way.Asweshall see,dﬂpasses most ofthistestforavector, although in onerespect itfailstomake thegrade. Onewayofexamining thetransfonnation properties ofdﬂistoﬁndhowthematrix etransforms under acoordinate trans- formation. Aswasshown inSection 4.3,thetransformed matrix e’isobtained by asimilarity transformation: e’=BeB_1. *ltcannot beemphasized toostrongly thatdﬂISnotthediﬁerential ofavector. Thecombmation dﬂ stands foradifferential vector, thatis,avector ofdifferential inagiiitude. Unfortunately, notational convention results inhaving thevector characteristic applied onlytoQ,butitshould beclear tothe reader there isnovector ofwhich dﬂrepresents adifferential. Aswehave seen, aﬁnite rotation cannot berepresented byasingle vector 4.8 lnfinitesimal Rotations 167 Astheantisymmetry property ofamatrix ispreserved under anorthogonal simi- larity transformation (seeDerivation 3),e’canalsobeputintheform ofEq.(4.69) withnonvanishing elements d.Q.'.Adetailed study ofthese elements shows that 6transforms under thesiniilaiity/I transformation suchthat Thetransformation ofdQisthusalmost thesame asforr,butdiffers bythefactor |B|,thedeterminant ofthetransformation matrix. There ishowever asimpler waytouncover thevector characteristics ofdﬂ, andindeed toverify itstransformation properties asgiven byl:.q.(4.74). Lnthe previous section avector formula wasderived forthechange inthecomponents ofrunder aﬁnite rotation ofthecoordinate system. Byletting gotothe limitofaninﬁnitesimal angle d,thecorresponding formula foraninﬁnitesimal rotation canbeobtained. Inthislimit, cos<l>inEq.(4.62) approaches unity, and sin<bgoesto(D;theresultant expression fortheinﬁnitesimal change inristhen r’—rzdr=rxnd. (4.75) Comparison withEq.(4.72) indicates thatd.O.isindeed avector andisdetermined by dﬂ=ndd). (4.76) Equation (4.75) canofcourse bederived directly without recourse totheﬁnite rotation formula. Considered initsactive sense, theinﬁnitesimal coordinate trans- formation corresponds toarotation ofavector rclockwise through anangle d<l> about theaxisofrotation, asituation thatisdepicted inFig.4.11.* Themagnitude ofdr,toﬁrstorder ind<Dis,from theﬁgure, dr=rsin6d<l>, andthedirection dris,inthislimit, perpendicular toboth randdﬂ=nd. Finally, thesense ofdrisinthedirection aright-hand screw advances asris turned intodﬂ.Figure 4.11thusshows thatinmagnitude. direction, andsense dr isthesameasthatpredicted byEq.(4.75). Thetransformation properties ofd.Q.,asdeﬁned byEq.(4.76), arestilltobe discussed. Asiswellknown from elementary vector algebra, there aretwokinds ofvectors inregard totransfomiation properties under aninversion. Vectors that transform according toEq.(4.72) areknown aspolar vectors. Under athree- diinensional inversion, -1 O O S= 0—l O 0 0——1 *Figure 4.11istheclockwise-rotation version ofFig.2.8. Chapter 4TheKinematics ofRigid Body Motion A nd¢=dQ d‘r k I r I 9 FIGURE 4.11 Change inavector produced byaninﬁnitesimal clockwise rntntinn ofthe VCCEOII whose components are SI]=-5:1: allcomponents ofapolarvector change sign. Ontheother hand, thecomponents ofaxial vectors orpseudovecrors donot change signunder inversion. Thesimplest example ofanaxialvector isacross product oftwopolar vectors, V*=D><F, where thecomponents ofthecross product aregiven, ascustomary, bythedeﬁni- tions: v;*_D117,-F,D,.. i,j,kincyclicorder. <4."/7) Thecomponents ofDandFchange signunder inversion; hence those ofCdonot. Many familiar physical quantities areaxial vectors, such astheangular momen- tumL=rxp,andthemagnetic ﬁeldintensity. Thetransformation lawforan axialvector isoftheformofEq.(4.74). Forproper orthogonal transformations, axial andpolar vectors areindistinguishable, butforimproper transformations, i.e.,involving inversion, thedeterminant |V*|is—l.andthetwotypes ofvectors behave differently. Another waytoexplain thisproperty istodeﬁne aparity operator P.Theoper- atorPperforms theinversion x—>—x,y—>—y,z—>—-z.Then ifSisscalar. V apolarvector, andV*anaxialvector, 4.8 nfinitesimal Rotations 169 PS=S PV=—V PV*=V*, and,obviously, P(V-V*)=—(V-V*). Thus, V-V*isapseudoscalar S*withtheproperty PS*=—S*andofcourse P(SS*) =—.S'.S'*, P(.S'V} =-SV, P(SV*) =.§‘V* Onthepassive interpretation ofthetransformation, itiseasytoseewhypo- larvectors behave astheydounder inversion. Thevector remains unaffected by thetransformation, butthecoordinate axes,andtherefore thecomponents, change sign. What thenisdifferent foranaxial vector? Itappears thatanaxial vector al- ways carries withita“handedness” convention, asimplied, e.g.,bythedeﬁnition, Eq.(4.77), ofacross product. Under inversion aright-handed coordinate system changes toaleft-handed system, andthecyclic order requirement ofEq.(4.77) implies asimilar change from theright-hand screw convention toaleft-hand con- vention. Hence, even onthepassive interpretation, there isanactual change inthe direction ofthecross product upon inversion. Itisclearnowwhya'.Qtransforms asanaxialvector according toEq.(4.74). Algebraically, weseethatsince both randdrinEq.(4.75) arepolar vectors, then n,andtherefore dﬂ,must beaxial vectors. Geometrically, theinversion ofthe coordinates corresponds totheswitch from aright-hand screw lawtoaleft-hand screw todeﬁne thesense ofn. Thediscussion ofthecross product provides anopportunity tointroduce a notation thatwillbemost useful onfuture occasions. Thepermutation symbol orLevi—Civita density‘ 6,11,isdeﬁned tobezero ifanytwooftheindices ijk areequal, andotherwise either +1or—laccording asijkisaneven orodd permutation of1,2,3.Thus, interms ofthepermutation symbol, Eq.(4.77) for thecomponents ofacross product canbewritten C,=€UkDJ Fk, (4.'i'7') where theusual summation convention hasbeenemployed. Thedescriptions ofrotation presented sofarinthischapter have been devel— oped sothatwecanrepresent theorientation ofarigid body. Note thatthetrans- formations primarily involve rotation ofthecoordinate system (cf.Fig.4.l2a). Thecorresponding “active” interpretation ofrotation ofavector inaﬁxed co- ordinate system therefore implies arotation intheopposite direction, i.e,ina clockwise sense. Buttherearemany areasofmechanics, orofphysics ingeneral forthatmatter, where weareconcemed withtheeffects ofrotating thephysical system andassociated vectors (cf.Fig.4.12b). Theconnection between invariance ofthesystem under rotation andconservation ofangular momentum hasalready *Also known interchangeably asthealternating tensor orisotropic tentorofrank3. 7 Chapter 4theKinematics oiRigid Body Motion Z Z I z z’ (DIDyl Y x’/ ) , xX Y xi (11) (b) FIGURE 4.12 (a)Transformation from thecoordinate system (x,y,z)toanewcoor- dinate system (x',y’,z’).Byconvention, thistransformation isconsidered positive inthe clockwise sense. Werefer tothisasapassive transformation. (b)Therotation ofabody through anangle <l>’.Byconvention, therotation ispositive inacounterclockwise sense. Before therotation, thecoordinates ofpoints ofthebodyweregivenby(x,y,z);afterthe rotation, theyaregiven by(x’,y’,z’).This iscalled anactive transformation because the physical body moves. been pointed out(cf.Section 2.6).Insuchapplications itisnecessary toconsider theconsequences ofrotation ofvectors intheusual counterclockwise sense. For reference purposes. anumber ofrotation formulae given above willhelisted here, butforcounterclockwise rotation ofvectors. Allequations andstatements from heretotheendofthissection apply onlyforsuchcounterclockwise rotations. Therotation formula, Eq.(4.62), becomes r’=rcos +n(n-r)(1—cos)+(nxr)sin, (4.62’) andthecorresponding innmtesimal rotation, Eq.(4.75), appears as dr'=d.Q xr=(nxr)d<l> =—(r><n)d<l>. (4.75') Theantisymmetric matrix oftheinﬁnitesimal rotation, Eq.(4.69), becomes O -619’; dQ7 0 —n3 rig 6== £19’; 0 —dQ1 = 713 O -711 d<l>, (4-.69’) —clQ2 dﬂq O -712 ft| 0 where n,arethecomponents oftheunitvector iialong theaxisofrotation. Letting drstand fortheinﬁnitesimal change r’—r,Eq.(4.66) canthentaketheformof amatrix differential equation withrespect totherotation angle: dri =—N . 4.7 dCb r (3) where Nisthetranspose ofthematrix onrightinEq.(4.69') withelements NU= Guknk. 4.9 I4.9RateofChange ofaVector 171 Another useful representation istowrite einEq.(4.69’) as eZ 711M, d¢ where M,arethethree matrices: O0 O OO1 O—l O M1=O0-1 ,M2= 0OO,M3=l OO. Ol 0 —l OO O O0 (4.79) Thematrices M,areknown astheinﬁnitesimal rotation generators andhavethe property thattheir products are M,M,-MJM,E[M,,M,-]=e,,kM;,. (4.80) Thedifference between thetwomatrix products. orcommutator. isalsocalled the Liebracket orM,,andEq.(4.80) deﬁnes theLiealgebra oftherotation group parametrized interms oftherotation angle. Togofurther intothegroup theory of rotation would takeustoofaraﬁeld, butweshall have occasion torefer tothese properties oftherotation operation. (cf.Section 9.5andAppendix B) RATE OFCHANGE OFAVECTOR Theconcept ofaninﬁnitesimal rotation provides apowerful toolfordescribing themotion ofarigidbodyintime.Letusconsider some arbitrary vector orpseu- dovector Ginvolved inthemechanical problem, such astheposition vector ofa point inthebody, orthetotal angular momentum. Usually such avector willvary intimeasthebody moves, butthechange willoften depend upon thecoordinate system towhich theobservations arereferred. Forexample, ifthevector happens tobetheradius vector from theorigin ofthebody setofaxestoapoint intherigid body, thenclearly suchavector appears constant when measured bythebody set ofaxes. However, toanobserver ﬁxed inthespace setofaxes, thecomponents ofthcvector (asmeasured onthespace axes) willvary intime ifthebody isin motion. Thechange 11'1atimedtofthecomponents ofageneral vector Gasseenbyan observer inthebody system ofaxeswilldiffer from thecorresponding change as seenbyanobserver inthespace system. Arelation between thetwodifferential changes inGcanbedenved onthebasis ofphysical arguments. Wecanwrite that theonly difference between thetwoistheeffect ofrotation ofthebody axes: (dG)spacc =(dG)body +(dG)rot- Nowconsider avector ﬁxedintherigidbody. Asthebodyrotates, thereisof course nochange inthecomponents ofthisvector asseenbythebody observer, Chapter 4TheKinematics ofRigid Body Motion i.e.,relative tobodyaxes.Theonlycontribution to(dG)§pa¢g isthentheeffect of therotation ofthebody. Butsince thevector isﬁxed inthebody system, itrotates withitcounterclockwise, andthechange inthevector asobserved inspace isthatIgiven byEq.(4.75 ),andhence (dG)m, isgiven by (dG)rot =d9XG- Foranarbitrary vector, thechange relative tothespace axesisthesumofthetwo effects: (dG)§pa¢e Z XG. Thetime rateofchange ofthevector Gasseenbythetwoobservers isthen obtained bydividing theterms inEq.(4.8!) bythedifferential timeelement dt under consideration; = +(.0xG. (4.82) ‘It space dt body Here toistheinstantaneous angular velocity ofthebody deﬁned bytherelation* wdt=dﬂ. (4.83) Thevector toliesalong theaxisoftheinﬁnitesimal rotation occurring between t andz+dt,adirection known astheinstantaneous axisqfrntation. Inmagnitude, tomeasures theinstantaneous rateofrotaiionofthebody. Amore formal derivation ofthebasic Eq.(4.82) canbegiven interms ofthe orthogonal matrix oftransformation between thespace andbody coordinates. The component ofGalong theithspace axisisrelated tothecomponents along the body axes: _~ 1_ I Asthebodymoves intime,thecomponents G’willchange aswilltheelements a,-,4ofthetransformation matrix. Hence, thechange inG,inadifferential time element dzis dG,=a,,dG3+daj,G3. (4.84) Itisnolossofgenerality totakethespace andbodyaxesasinstantaneously coincident atthetime t.Components inthetwosystems willthenbethesame instantaneously, butdifferentials willnotbethesame, since thetwosystems are moving relative toeach other. Thus, G’=GJbuta,,dG; =dG§, theprime emphasizing thedifferential ismeasured inthebody axissystem. Thechange in thematrix Ainthetimedtisthusachange fromtheunitmatrix andtherefore ‘Note thattoisnotthedernraiive ofanyvector. 4.9 RateofChange ofaVector 173 corresponds tothematrix eoftheinfinitesimal rotation. Hence, dajl =(EL, =_5i]i using theantisymmetry property ofe.Interms ofthepermutation symbol euk, theelements ofearesuchthat(cf.Eq.(4.69)) —6ij =—€,jkdQk =6,/qdﬁk. Equation (4.84) cannowbewritten dG, =dG: +6,k]dQkG_,. Thelastterm ontheright willberecogmzed astheexpression fortheithcom- ponent ofacross product, sothattheﬁnalexpression fortherelation between differentials inthetwosystems is dG,=dc;+(doX0),, <4.ss) which isthesame astheithcomponent ofEq.(4.81). Equation (4.81) isnotsomuch anequation about aparticular vector Gasitisa statement oithetransformation ofthetimederivative between thetwocoordinate systems. Thearbitrary nature ofthevector Gmade useofinthederivation canbe emphasized bywriting Eq.(4.82) asanoperator equation acting onsome given vector: d d-=- . 4.8((11). (di),+‘°" (6) Here thesubscripts sandrindicate thetime derivatives observed inthespace andbody (rotating) system ofaxes, respectively. Theresultant vector equation canthenofcourse beresolved along anydesired setofaxes, ﬁxed ormoving. But again notethatthetimerateofchange isonlyrelative tothespeciﬁed coordinate system. When atimederivative ofavector iswithrespect toonecoordinate sys- tem, components may betaken along another setofcoordinate axes only after the differentiation hasbeen carried out. Itisoften convenient toexpress theangular velocity vector interms oftheEu- lerangles andtheirtimederivatives. Thegeneral inﬁnitesimal rotation associated withancanbeconsidered asconsisting ofthreesuccessive inﬁnitesimal rotations withangular velocities 10¢=qi,(09=9,we= Inconsequence ofthevector property ofinﬁnitesimal rotations. thevector cocanbeobtained asthesumofthe three separate angular velocity vectors. Unfortunately, thedirections m¢,cog,and 10,),arenotsymmetrically placed: cod,isalong thespace zaxis, (.09isalong the line0:"nodes, while coy,alone isalong thebody z’axis.However, theorthogonal transformations B,C,DofSection 4.4maybeusedtofurnish thecomponents of these vectors along anydesired setofaxes. 74 4.10 IChapter 4TheKinematics ofRigid Body Motion Thebody setofaxesproves most useful fordiscussing theequations ofmotion, andweshalltherefore obtain thecomponents oftoforsuchacoordinate system. Since 00¢isparallel tothespace zaxis, itscomponents along thebody axesare given byapplying thecomplete orthogonal transformation A=BCD, Eq.(4.46): (co¢,),,/ =sin!)sin1/1, (w¢)y/ =sin6cos1/1, (co¢)z» =cos6. Notethat hastheprojection sin8inthex’,y’plane, anditisperpendicular to thelineofnodes. Thelineofnodes, which isthedirection of(.09,coincides withthe5’axis,so thatthecomponents ofwewithrespect tothebodyaxesareflJITl1Sl‘1CCl byapplying onlytheﬁnalorthogonal transformation B,Eq.(4.45): (cu9),,» =Qcos10, (¢o9)y» =-9sin'¢, (co9)z/ =0. N0transformation isnecessary forthecomponents ofrow,which liesalong thez’ axis.Adding these components oftheseparate angular velocities, thecomponents oftowithrespect tothebody axesare 0),,=q§sin6sin¢ +dcostl/ coy’=q§sin9cos1// —ésintﬁ oz.=¢l»¢0s0 +ti. (4.27) Similar techniques maybeusedtoexpress thecomponents oftoalong thespace setofaxesinterms oftheEuler angles. THE CORIOLIS EFFECT Equation (4.86) isthebasickinematical lawuponwhich thedynamical equations ofmotion forarigidbodyarefounded. Butitsvalidity isnotrestricted solely to rigidbodymotion. Itmaybeusedwhenever wewishtodiscuss themotion ofa particle, orsystem ofparticles, relative toarotating coordinate system. Aparticularly important problem inthislatter category isthedescription of particle motion relative tocoordinate axesrotating withEarth. Recall thatinSec- tion1.1aninertial system wasdeﬁned asoneinwhich Newton’s lawsofmotion arevalid. Formany purposes, asystem ofcoordinates ﬁxedintherotating Earth isasuﬂicient approximation toaninertial system. However, thesystem ofcoordi- nates inwhich thelocalstarsareﬁxedcomes stillcloser totheidealinertial sys- tem. Detailed examination shows there areobservable effects arising from Earth’s rotation relative tothisnearly inertial system. Equation (4.86) provides theneeded modiﬁcations oftheequations ofmotion relative tothenoninertial system ﬁxed intherotating Earth. Theinitial stepistoapply Eq.(4.86) totheradius vector, r,from theorigin of theterrestrial system tothegiven particle: 4.10 TheCoriolis Effect 175 v_,=v,+toxr, (4.88) where vsandv,arethevelocities oftheparticle relative tothespace androtating setofaxes, respectively, andtoisthe(constant) angular velocity ofBarth relative totheinertial system. Inthesecond step,bq.(4.86) ISusedtoobtain thetimerate ofchange ofv,: —a +coxvdz,_s_ aft, S =a,+2(¢oxv,)+wx(wxr), (4.89) where v,hasbeensubstituted fromEq.(4.88), andwhere a,anda,aretheaccel- erations oftheparticle inthetwosystems. Finally, theequation ofmotion, which intheinertial system issimply F=mas, expands, when expressed intherotating coordinates, intotheequation F—2m(w xv,)—mmx(coxr)=ma,. (4.90) Toanobserver intherotating system, ittherefore appears asiftheparticle is moving under theinﬂuence ofaneffective forceFeffl Fe“=F—2m(w xv,)—mwx(cox1'). (4.91) Letusexamine thenature oftheterms appearing inEq.(4.91).Thelasttermis avector normal totoandpointing outward. Further, itsmagnitude ismmzr sin0. Itwill therefore berecognized thatthistcrm provides thefamiliar centrifugal force. When theparticle isstationary inthemoving system, thecentrifugal force istheonlyadded term intheeffective force. However, when theparticle ismov- ing,themiddle tennknown astheCoriolis eﬁ’ect* comes intoplay.Theorder ofmagnitude ofbothofthese quantities mayeasily becalculated foraparticle onEaith’s surface. Earth rotates counterclockwise about thenorth polewithan angular velocity relative totheﬁxed stats. 27$ _ _5 _l °’_(24><3600) (365.5) ‘7292X10S' Heretheﬁrstsetofparentheses gives theangular velocity relative totheradius vector totheSun. Thequantity inthesecond parentheses, theratio ofthenumber ofsidereal daysinayeartothecorresponding number ofsolardays,isthecorrec- tionfactor togivetheangular velocity relative totheﬁxedstars. Withthisvalue *The termCoriolis eﬁect isusedinstead oftheolder term, Coriolis force, toremind usthatthiseffect exists because weareusing anoninerual frame. Inaproper inertial frame, theeffect doesnotexist Youcanalways visualize theCoriolis effect byasking what ishappemng inaninertial frame. 7 Chapter 4TheKinematics ofRigid Body Motion forw,andwithrequal toEarth’s equatorial radius, themaximum centripetal acceleration is wzr.-3.32;cm/S2, orabout 0.3% oftheacceleration ofgravity. While small, thisacceleration is bynomeans negligible. However, themeasured effects ofgravity represent the combination ofthegravitational ﬁeld ofthemass distribution ofEarth andthe effects ofcentripetal acceleration. Ithasbecome customary tospeak ofthesum ofthetwoasEarth’s gravity held, asdistinguished from itsgI‘(lVlI£ll'l()l’l(ll held. Thesituation isfurther complicated bytheeffect ofthecentripetal acceleration inﬂattening therotating Earth. IfEarth were completely ﬂuid, theeffect ofrota- tionwould betodeform itintotheshape ofanellipsoid whose surface would be anequipotential surface ofthecombined gravity ﬁeld. Themean level ofEarth’s seasconforms veryclosely tothisequilibrium ellipsoid (except forlocalvaria- tions ofwind andtide)anddehnes what iscalled thegeoid. Except foreffects oflocalperturbations, theforce ofgravity willbeperpen- dicular totheequipotennal surface ofthegeoid. Accordingly, thelocal vertical is deﬁned asthedirection perpendicular tothegeoid atthegiven point onthesur- face. Forphenomena thatoccur inthevicinity ofaparticular spotonEarth, the centripetal acceleration tenns inEq.(4.91) canbeconsidered asswallowed upin thegravitational acceleration g,which willbeoriented inthelocalvertical direc- tion.Themagnitude ofgofcourse varies withthelatitude onEarth. Theeffects ofcentripetal acceleration andtheﬂattening ofEarth combine tomake gabout 0.53% lessattheequator thanatthepoles. Incidentally, thecentrifugal force onaparticle arising from Earth’s revolution around theSunisappreciable compared togravity, butitisalmost exactly bal- anced bythegravitational attraction totheSun.Ifweanalyze themotion ofthe Sun-Earth system from aframe rotating withEarth, itisofcourse justthebal- ance between thecentrifugal effect andthegravitational attraction thatkeeps the Earth (andallthatareonit)andSunseparated. Ananalysis inaNewtonian iner- tialframe gives adifferent picture. Aswasdescribed inSection 3.3,theangular momentum contributes totheeffective potential energy tokeep theEarth inorbit. TheCoriolis effect onamoving particle isperpendicular toboth toandv.* Inthenorthem hemisphere. where topoints outoftheground, theCoriolis effect 2m(vxco)tends todeflect aprojective shotalong Earth’s surface, totheright ofitsdirection oftravel (cf.Fig.4.13). TheCoriolis deflection reverses direction inthesouthem hemisphere andiszeroattheequator, where coishorizontal. The magnitude oftheCoriolis acceleration isalways lessthan 2cm)21.5x10-41;, *From hereon.thesubscript rwillbedropped fromvasallvelocities Willbetaken withrespect to therotating coordinate axesonly 4.10 TheCoriolis Effect 177 “'2 "0 VXQ Horizontal traiectory FIGURE 4.13 Direction ofCoriolis deﬂection inthenorthem hemisphere. which foravelocity of105cm/s (roughly 2000 mi/h) is15cm/s2, orabout 0.0l5g. Nonnally, such anacceleration isextremely small, butthere areinstances when itbecomes important. Totakeanartiﬁcial illustration, suppose aprojectile were ﬁredhorizontally atthenorthpole.TheCoriolis acceleration would thenhavethe magnitude 2wv, sothatthelinear deﬂection after atime tiswvtz. Theangular deﬂection would bethelinear deﬂection divided bythedistance oftravel: 2 0=%=cut, (4.92)U!‘ which istheangle Earth rotates inthetime t.Physically, thisresult means that aprojectile shotoffatthenorth polehasnoinitial rotational motion andhence itstrajectory intheinertial space isastraight line, theapparent deﬂection be- ingduetoEarth rotating beneath it.Some ideaofthemagnitude oftheeffect canbeobtained bysubstituting atimeofﬂight of100s—not unusual forlarge projectiles--—in Eq.(4.92). Theangular deﬂection isthenoftheorder of7><l0'3 radians, about 0.4°. which isnotinconsiderable. Clearly theeffect iseven more important forlong-range missiles, which haveamuch longer timeofﬂight. TheCoriolis effect alsoplays asigniﬁcant roleinmany oceanographic and meteorological phenomena involving displacements ofmasses ofmatter overlong distances, suchasthecirculation pattem ofthetrade winds andthecourse of theGulf stream. Afulldescription ofthese phenomena requires thesolution of complex hydrodynamic problems inwhich theCoriolis acceleration isonlyone among many terms involved. Itispossible however togivesome indication ofthe contribution ofCoriolis effects byconsidering ahighly simpliﬁed picture ofone particular meteorological problem—the large-scale horizontal wind circulation. Masses ofairtendtomove, other things being equal, from regions ofhighpressure toregions oflowpressure—the so-called pressure-gradient ﬂow. inthevertical direction thepressure gradient isroughly balanced bygravitational forces sothat 7 Chap er4TheKinematics ofRigid Body Motion I lsobars _/ //1/ /__i_ LO“ I High FIGURE 4.14 Deﬂection ofwind from thedirection ofthepressure gradient bythe Coriolis effect (shown forthenorthem hemisphere). itisonlyinthehorizontal plane thattherearepersistent long-range motions of airmasses—which weperceive aswinds. Thepressure gradient forces arequite modest, andcomparable inmagnitude tntheCoriolis effects acting onairmasses moving atusual speeds. Intheabsence ofCoriolis effects, thewind directions would ideally beperpendicular totheisobars, asshown inFig.4.14.However. the Coriolis effects deﬂect thewind totherightofthisdirection inthesense indicated intheﬁgure. Thedeﬂection totheright continues untilthewind vector isparallel totheisobars andtheCoriolis effect isintheopposite direction to,andideally justbalances, thepressure-gradient force. Thewind thencontinues parallel tothe isobars, circulating inthenorthem hemisphere inacounterclockwise direction about acenter oflowpressure. Inthesouthem hemisphere, theCoriolis effect actsintheopposite direction, andthecyclonic direction (i.e., theﬂow around alow-pressure center) isclockwise. (Such awind ﬂow, deﬂected parallel tothe isobars, isknown asageostrophic wind.) Inthissimpliﬁed picture, theeffect of friction hasbeen neglected. Atatmospheric altitudes below several kilometers. thefnction effects ofeddy viscosity become important, andtheequilibrium wind direction never becomes quiteparallel totheisobars, asindicated inFig.4.15. Another classical instance where Coriolis effect produces ameasurable effect isinthedeﬂection fromthevertical ofafreely falling particle. Since theparti- clevelocity isalmost vertical andcoliesinthenorth-south vertical plane, the .7L/_(a)ldealized (b)Actual FIGURE 4.15 Cyclone pattem inthenorthem hemisphere. 4.10 lhe(.oriolis lzftect 179 deﬂecting force 2m(vxm)isintheeast-west direction. Thus, inthenorthern hemisphere, abody falling freely willbedeﬂected totheEast. Calculation ofthe deﬂection isgreatly simpliﬁed bychoosing thezaxisoftheterrestrial coordinate system tobealong thedirection oftheupward vertical aspreviously deﬁned. If theyaxisistaken aspointing North, andthefrictional effect oftheatmosphere is neglected, thentheequation ofmotion inthex(East) direction is dzxmg-I-5 =—2m(co xv)x =—2m<ov3 sin0, (4.93) where 6istheco-latitude Theeffect oftheCoriolis effect onU2would constitute asmall correction tothedeﬂection, which itself isverysmall. Hence, thevertical velocity appearing in(4.93) maybecomputed asifCoriolis effects wereabsent. U3 Z _gt. /2zI= —. 8 With these values, Eq.(4.93) maybeeasily integrated togivethedeﬂection* asTheintegral ofthisis x=%gt3sin0 wl(2z)3 .x=—ism0. 3s Anorder ofmagnitude ofthedeﬂection canbeobtained byassuming 6=Jr/'2 (corresponding totheequator) andz=100m.Thedeﬂection isthen, roughly,O1‘ x1'2.2cm. Thcactual experiment isdifﬁcult toperform, asthesmall deﬂection may often be masked bytheeffects ofwind currents, viscosity, orother disturbing inﬂuencesl More easily observable isthewell-known experiment oftheFoucault pendu- lum.Ifapendulum issetswinging atthenorth poleinagiven plane inspace, thenitslinear momentum perpendicular totheplane iszero,anditwillcontinue toswing inthisinvariable plane while Earth rotates beneath it.Toanobserver onEarth, theplane ofoscillation appears torotate once aday.Atother latitudes theresult ismore complicated, butthephenomenon isqualitatively thesame and detailed calculation willbeleftasanexercise. *Again, weneglect thefrictional eﬁects oftheatmosphere litiseasytoshow, using Eq.(4.93), thataparticle projected upward willfallback totheground westward oftheoriginal launching spot. 80 Chapter 4theKinematics oiRigidBodyMotion Effects duetotheCoriolis terms alsoappear inatomic physics. Thus, twotypes ofmotion mayoccur simultaneously inpolyatomic molecules: Themolecule ro- tates asarigid whole, andtheatoms vibrate about theirequilibrium positions As aresult ofthevibrations, theatoms areinmotion relative totherotating coordi- natesystem ofthemolecule. TheCoriolis term willthenbedifferent from zero andwillcause theatoms tomove inadirection perpendicular totheoriginal os- cillations. Perturbations inmolecular spectra duetoCoriolis effects thusappear asinteractions between therotational andvibrational motions ofthemolecule. DERIVATIONS 1.Prove thatmatrix multiplication isassociative. Show thattheproduct oftwoorthogo- nalmatrices isalsoorthogonal. 2.Prove thefollowing properties ofthetransposed andadjoint matnces: rsv -~ as=BA. (Ami=BW. 3.Show thatthetraceofamatrix 1Sinvariant under anysimilarity transformation. Show alsothattheantisymnieiry property ofamatrix ispreserved under anorthogonal sim- ilarity liaiisfuriiialiuii 4.(a)Byexamining theeigenvalues ofanantisymmetric 3x3realmatrix A,showthat 1iAisnonsingular. (li)Show thenthatunder thesameconditions thematrix s=(1+A)(l—A)" isorthogonal. 5.Ocitain thematrix elements ofthegeneral rotation matrix inl6l‘lTlS oftheEuler angles, Eq.(4.46), byperforming themultiplications ofthesuccessive component rotation matrices. Verify directly thatthematrix elements obey theorthogonality conditions. 6.Thebody setofaxescanberelated tothespace setinterms ofE.uler’s angles bythe following setofrotations: (a)Rotation about thexaxisbyanangle 6 (b)Rotation about thez’axisbyanangle 1//. (c)Rotation about theoldzaxisbyanangle ¢. Show thatthissequence leads tothesame elements ofthematrix oftransformation as thesequence ofrotations given inthebook. [Hint ltisnotnecessary tocarry outthe explicit multiplication oftherotation matrices] 7.lfAisthematrix ofarotation through 180°about anyaxis, show thatit Pi=go1A), Derivations 181 8. 9. 10. 11. 12. 13. 14.thenPi=Pi.Obtain theelements ofPiinanysuitable system, andﬁndageometric interpretation oftheoperation P+andP_onanyvector F. (a)Show thattherotation matrix inthefonn ofEq.(4.4-7') cannot beputinthefonn afthematnx oftheinversion transformation S. (b)Venfy bydirect multiplication thatthematrix inEq.(4.4'7’) isorthogonal. Show thatanyrotation canberepresented bysuccessive reﬂection intwoplanes, both passing through theaxisofrotation withtheplanar angle <l>/2between them. IfBisasquare matrix andAistheexponential ofB,deﬁned bytheinﬁnite series expansion oftheexponential, 1 B"AEeB=i+B+-B2+---+—+---,2 n! thenprove thefollowing properties‘ (a)egec =e8+c, providing BandCcommute (b)A"1=8"“ (C) eCBC"' =cAc—l (d)Aisorthogonal ifBisantisymmetric. Verify therelation I-5! =(-1)"|Bl forthedeterminant ofannxnmatrix B. Inasetofaxeswhere tl-ezaxisistheaxisofrotation ofaﬁnite rotation, therotation matrix isgiven byEq.(4.43) with0replaced bytheangle ofﬁnite rotation <l>.Derive therotation rormula. bq.(4.62), bytransforming toanarbitrary coordinate system, expressing theorthogonal matrix oftransformation interms ofthedirection cosines oftheaxisoftheﬁnite rotation. (a)Suppose twosuccessive coordinate rotations through angles <b|and<l>2arecar- riedout.equivalent toasingle rotation through anangle <l>.Show that1,(D2,and IDcanbeconsidered asthesides ofaspherical tnangle withtheangle opposite to rbgiven bytheangle between thetwoaxesofrotation. (b)Show thatarotation about anygiven axiscanbeobtained astheproduct oftwo successive rotations. eachthrough 180° (a)Verify thatthepermutation symbol satisﬁes thefollowing identity interms of Kronecker delta symbols: Gtjpérmp =8r*8_1m _8IP7’£8]l"' (h)Show that 6,-J-peuk =23!,/C. Chapter 4TheKinematics ofRigid Body Motion 15. 16 17 18 190 20Show thatthecomponents oftheangular velocity along thespace setofaxesaregiven interms oftheEuler angles by wx=écosqﬁ +tisinésingb, my=9sin¢ —ilrsin6cos¢, mz=cos9+ Show thattheEuler parameter e0hastheequation ofmotion —2éQ =e|wxi +82(0),! +e3wz/, Where theprime denotes thebody setofaxes. Findthecorresponding equations forthe other three Euler parameters andforthecomplex Cayley-Klein parameters aand,6. Venfy directly thatthematrix generators ofinﬁnitesimal rotation. M,,asgiven by Eq.(4.79) obey thecommutation relations [MinM1]=El]/(M/0 (a)Find thevector equation describing thereﬂection ofrinaplane whose unitnor- malisn. (b)Show thatifl,,i=1,2,3,arethedirection cosines ofii,then thematrix of transformation hastheelements AU =8;] — andverify thatAisanimproper orthogonal matrix. Figures 4.9and4.10show thattheorder ofﬁnite rotations leadstodifferent results. Usethenotation thatA(o:, 1,,)where Aisarotation inthedirection of1,,through an angle onLetn1andn2betwoorthogonal directions. (a)Ifxistheposition vector ofapoint onarigid body, which isthenrotated byan angle 9aruuiid theorigin, sliuw thatthenew value ofxis it’=(1,,-x)1,,+[X-i,,(1,,-X)]cOS9 -.,,><xsin6. From this,obtain thefomiula forAtrr/2, l,,)andderive thetworotations inthe ﬁgures. (b)Discuss these tworotations. [Hint: Theanswer willinvolve arotation bytheangle §zinadirection(1/~/§)(1,1, 1).] Express the“rolling” constraint ofasphere onaplane surface interms oftheEuler angles. Show thattheconditions arenonintegrable andthattheconstraint istherefore nonholonomic. EXERCISES 21.Aparticle isthrown upvertically withinitial speed vo,reaches amaximum height andfallsback toground. Show thattheCoriolis deﬂection when itagain reaches the ground isopposite indirection, andfourtimes greater inmagnitude, thantheCoriolis deﬂection when itisdropped atrestfromthesamemaximum height. Exercises 183 Aprojectile isfiredhorizontally along Earth’s surface. Show thattoaﬁrstapproxima- tiontheangular deviation from thedirection ofﬁreresulting from theCoriolis effect varies linearly withtimeatarate woos 6, where mistheangular frequency ofEarth’s rotation and0istheco-latitude, thedi- rection ofdeviation being totheright inthenorthem hemisphere. TheFoucault pendulum experiment consists insetting alongpendulum inmotion at apoint onthesurface oftherotating Earth withitsmomentum originally inthever- tical plaiie containing thependulum bubandthepoint ofsuspeiisiuii. Show thatthe pendulum’s subsequent motion maybedescribed bysaying thattheplane ofoscilla- tionrotates uniformly Zncos0radians perday,where 6istheco-latitude. What isthe direction ofrotation? Theapproximation ofsmall oscillations maybeused, ifdesired. Awagon wheel withspokes ismounted onavertical axissoit1Sfreetorotate inthe horizontal plane. Thewheel isrotating withanangular speed ofw=3.0radianls. A bugcrawls outononeofthespokes ofthewheel withavelocity of0.5cmls holding ontothespoke withacoefﬁcient offriction /J.=0.30. How farcanthebugcrawl along thespoke before itstairs toslip? Acarousel (counter-clockwise merry-go-round) starts fromrestandaccelerates ata constant angular accleration of0.02revolutions/s2. Agirlsitting onabench onthe platform 7.0mfrom thecenter isholding a3.0kgball.Calculate themagnitude and direction oftheforce shemustexerttoholdtheball(1.0safterthecarousel starts to move. Givethedirection withrespect tothelinefromthecenter ofrotation tothegirl. CHAPTER 5.1I 184TheRigid Body Equations ofMotion Chapter 4presents allthekinematical tools needed inthcdiscussion ofrigid body motion. IntheEuler angles wehaveasetofthree coordinates, deﬁned rather unsymmetrically itistrue,yetsuitable foruseasthegeneralized coordinates de- scribing theorientation oftherigidbody. Inaddition, themethod oforthogonal transformations, andtheassociated matrix algebra, furnish apowerful andele- ganttechnique forinvestigating thecharacteristics ofrigid body motion. Wehave already hadoneapplication ofthetechnique inderiving Eq.(4.86), therelation between thestates ofchange ofavector asviewed inthespace system andin thebody system. These tools willnowbeapplied toobtain theEuler dynamical equations ofmotion oftherigidbodyintheirmostconvenient form. Withthehelp oftheequations ofmotion, some simple buthighly important problems ofrigid bodymotion canbediscussed. ANGULAR MOMENTUM AND KINETIC ENERGY OFMOTION ABOUT APOINT Chasles’ theorem states thatanygeneral displacement ofarigidbody canberep- resented byattranslation plus arotation. The theorem suggests thatitOught to bepossible tosplittheproblem ofrigidbodymotion intotwoseparate phases, oneconcemed solely withthetranslational motion ofthebody, theother, withits rotational motion. Ofcourse, ifonepoint ofthebody isﬁxed, theseparation is obvious, forthenthere isonlyarotational motion about theﬁxed point, without anytranslation. Buteven forageneral typeofmotion suchaseparation isoften possible. Thesixcoordinates needed todescribe themotion havealready been formed intotwosetsinaccordance withsuchadivision: thethree Cartesian coor- dinates ofapoint ﬁxedintherigidbodytodescribe thetranslational motion and. say,thethreeEuler angles forthemotion about thepoint. If,further, theorigin of thebody system ischosen tobethecenter ofmass, thenbyEq.(1.28) thetotal angular momentum divides naturally intocontributions fromthetranslation ofthe center ofmass andfrom therotation about thecenter otmass. '1hetormer term willinvolve onlytheCartesian coordinates ofthecenter ofmass, thelatter only theangle coordinates. ByEq.(1.31), asimilar division holds forthetotalkinetic energy T,which canbewritten intheform T=%Mv’+T'<¢.e. in. 5.1 Angular Momentum andKinetic Energy ofMotion abou- aPoint 185 asthesumofthekinetic energy oftheentire bodyasifconcentrated atthecenter ofmass, plusthekinetic energy ofmotion about thecenter ofmass. Often thepotential energy canbesimilarly divided, eachterminvolving only oneofthecoordinate sets, either thetranslational orrotational Thus, thepoten- tialenergy inauniform gravitational ﬁeldwilldepend onlyupontheCartesian vertical coordinate ofthecenter ofgravity/.* Oriftheforce onabody isdueto auniform magnetic ﬁeld,B,acting onitsmagnetic dipole moment, M,thenthe potential isproportional toM-B,which involves onlytheonentation ofthebody. Certainly, almost allproblems soluble inpractice willallow forsuchaseparation. Insuchacase, theentire mechanical problem doesindeed splitintotwo.TheLa grangian, L=T—V,divides intotwoparts, oneinvolving onlythetranslational coordinates, theotheronlytheangle coordinates. These twogroups ofcoordinates willthenbecompletely separated, andthetranslational androtational problems canbesolved independently ofeach other. Itisofobvious importance therefore toobtain expressions fortheangular mo- mentum andkinetic energy ofthemotion about some point ﬁxed inthebody. To doso,wewillmake abundant useofEq.(4.86) linking derivatives relative toa coordinate system ﬁxedatsome point intherigidbody. Itisintuitively obvious thattherotation angle ofarigidbodydisplacement, asalsotheinstantaneous an- gular telocity vector, isindependent ofthechoice oforigin ofthebodysystem ofaxes.Theessence oftherigidbodyconstraint isthatallparticles ofthebody move androtate together. However, aformal proof iseasily constructed. LetR1andR2betheposition vectors, relative toaﬁxedsetofcoordinates, of theorigins oftwosetsofbodycoordinates (cf.Fig.5.1).Thedifference vector is denoted byR: R2=Rr+R. R2 RI X FIGURE 5.1 Vectorial relation between setsofrigid body coordinates with different origins.Z H‘T K J’ *The center ofgravity ofcourse corncrdes withthecenter ofmass rnal1[lllOITﬂ gravitational ﬁeld. Chapter 5TheRigid Body Equations ofMotion Iftheorigin ofthesecond setofaxesisconsidered asapoint deﬁned relative to theﬁrst,thenthetimederivative ofR2relative tothespace axesisgiven by <e>ei+<s><s>»~»-R dr,_ dz, dz,— dz, 1' Thelaststepfollows from Eq.(4.86), recalling thatthederivatives ofRrelative toanyrigid body axesmust vanish, andwith to]asbeing theangular velocity vector appropriate totheﬁrstcoordinate system. Altematively, theorigin ofthe ﬁrstcoordinate system canbeconsidered asﬁxed inthesecond system withthe position vector —R.Inthesame manner, then, thederivative oftheposition vector R1tothisorigin relative totheﬁxed-space axescanbewritten as (“RU(‘“”)("“)(ml Z =i -—— =—— —w;xR.dz3 dzS dz5 dz5 Acomparison ofthese twoexpressions shows (ml—(1)2)xR=0.Anydiﬁ°er- enceintheangular velocity vectors attwoarbitrary points must liealong theline joining thetwopoints. Assuming thetovector ﬁeldiscontinuous, theonlypossi- blesolution forallpairsofpoints isthatthetwoangular velocity vectors mustbe equal: ml—¢og.* Theangular velocity vector isthesame forallcoordinate systems ﬁxed inthe rigidbody. When arigid body moves withonepoint stationary, thetotalangular momen- tumabout thatpointis L=mr(rr xvi)a (5-I) (employing thesummation convention) where 1',andv,aretheradius vector and velocity, respectively, oftheithparticle relative tothegiven point. Since r,1Sa ﬁxed vector relative tothebody, thevelocity v,withrespect tothespace setof axes arises solely from therotational motion oftherigid body about theﬁxed point. From Eq.(4.86), v,isthen v,=toxr,. (5.2) Hence, Eq.(5.1)canbewritten as T.=m,[r,x(mxr,)], or,expanding thetnple crossproduct, L=m,[turf —r,(r, -m):|. (5.3) *See alsoNA.Lemos, Am..h:Phys ,68(7) 2000, pp.668-669. 5.1Angular Momentum andKinetic Energy ofMotion abouta Point 187 Again expanding, thex-component oftheangular momentum becomes Lx=f-0:mt(7',2 —35,2)—wym-ix: Y1_¢9zmzx:Zz, (5-4) with similar equations fortheother components ofL.Thus, each component of theangular momentum isalinear function ofallthecomponents oftheangular velocity. Theangular momentum vector isrelated totheangular velocity bya linear transformation. "lbemphasize thesimilarity of(5.4) withtheequations of alinear transformation, (4.12), wemaywrite L,as L): Z IXXGJX + [x)(l)): + IXZCUZ. Analogously, forLyandLZwehave Ly ='Iyxwx +1y)1(D)- +Iyzwz, L2 1- IZXCUX + Izytoy + IZZLDZ. Theninecoefﬁcients I“,Ix)»,etc.,arethenineelements ofthetransfonnation matrix. Thediagonal elements areknown asmoment ofinertia coeﬁicients, and have thefollowing form 1,,=m,(r,2-16,2), (5.6) while theoff-diagonal elements aredesignated asproducts ofinertia, atypical onebeing Ixy=—mzx1)’z- (5-7) InEqs.(5.6)and(5.7), thematrix elements appear intheformsuitable ifthe rigidbodyiscomposed ofdiscrete particles. Forcontinuous bodies thesumma- tionisreplaced byavolume integration, with theparticle mass becoming amass density. Thus, thediagonal element Ixxappears as [xx=/Vp<r><r2—x2>dv. <16’) With aslight change innotation, anexpression forallmatrix elements canbe stated forcontinuous bodies. Ifthecoordinate axesaredenoted byx,,j=1,2,3, thenthematrix element IJkcanbewritten 1,:=[Vp(r)(r25_,k—xJxk)dV. (5.8) 5.2 IChapter 5TheRigid Body Equations ofMotion Thusfar,thecoordinate system usedinresolving thecomponents ofLhasnot beenspeciﬁed. From nowon,wewilltakeittobeasystem ﬁxedinthebody.* Thevarious distances x,,3;,z,arethenconstant intime, sothatthematrix el- ements arelikewise constants, peculiar tnthehotly involved, anddependent on theorigin andorientation oftheparticular body setofaxesinwhich theyare expressed. Equations (5.5) relating thecomponents ofLandmcanbesummarized bya single operator equation, L=lcu, (5.9) where thesymbol Istands fortheoperator whose matrix elements arethein- ertiacoefﬁcients appearing in(5.5), and0:andLarecolunm matrices. Ofthe twointerpretations thathavebeengiven totheoperator ofalinear transformation (cf.Section 4.2),itisclearthathereImustbethought ofasacting uponthevector nu.andnotupon thecoordinate system. Thevectors Landtoaretwophysically different vectors, having different dimensions, andarenotmerely thesame vector expressed intwodiffeient coordinate systems. Unlike theoperator ofrotation, I willhavedimensions—mass times length squared—and itisnotrestricted byany orthogonality conditions. Equation (5.9)istobereadastheoperator Iacting upon thevector toresults inthephysically newvector L. While fullusewillbemade ofthematrix algebra techniques developed in thediscussion oftherotation operator, more attention must bepaidheretothe nature andphysical character oftheoperator perse.However, acertain amount ofpreliminary mathematical formalism needs ﬁrsttobediscussed. Those already familiar withtensors canproceed immediately toSection 53. TENSO RS Thequantity Imaybeconsidered asdeﬁmng thequotient ofLandwfortheprod- uctofIandtogives L.Now, thequotient oftwoquantities isoftennotamember ofthesame class asthedividing factors, butmaybelong toamore complicated class. Thus, thequotient oftwointegers isingeneral notaninteger butrather a rational number. Similarly, thequotient oftwovectors, as1Swellknown, cannot bedeﬁned consistently within theclass ofvectors. Itisnotsurprising, therefore toﬁndthatIisanewtypeofquantity, atensor ofthesecond rank. InaCartesian three-dimensional space, atensor ToftheNthrankmaybede- ﬁnedforourpurposes asaquantity having 3”components T}1;,(with Nindices) thattransform under anorthogonal transformation ofcoordinates, A,according to *l.nChapter 4,such asystem wasdenoted byprimes. Ascomponents along spatial axesarerarely usedhere, thisconvention willhedropped from nowontosimplify thenotation. Unless otherwise speciﬁed, allcoordinates usedfortherestofthechapter refertosystems ﬁxed intherigid body. 5.2 Tensors 189 thefollowing scheme:* T531 =ailaymakn ---Tlmlt (X)- (5-10) Bythisdeﬁnition, ateiisoi ofthenew ldl1l\ hasonecouiponenl, which isiiivariaiil under anorthogonal transformation. Hence, ascalar isatensor ofzerorank. A tensor oftheﬁrstrankhasthree components transforming as I 71=al]TJ' Comparison with thetransformation equations foravector, (4.l2'), shows that atensor oftheﬁrst rankiscompletely equivalent toavet-t0r.l Finally, thenine components ofatensor ofthesecond ranktransform as Tl;=61,ktlJ1T/<1. (5.11) Rigorously speaking, wemust distinguish between asecond-rank tensor Tand thesquare matrix formed from itscomponents. Atensor isdeﬁned onlyinterms of itstransformation properties under orthogonal coordinate transformations. Onthe other hand, amatrix isinnowayrestricted inthetypes oftnmsfomiations itmay undergo andindeed maybeconsidered entirely independently ofitsproperties under someparticular classoftransformations. Nevertheless, thedistinction must notbestressed unduly. Within therestricted domain oforthogonal transforma- tions, there isapractical identity. Thetensor components andthematrix elements aremanipulated inthesame fashion; forevery tensor equation there willbea corresponding matrix equation. andviceversa. ByEq.(4.41), thecomponents of asquare matrix Ttransform under alinear change ofcoordinates deﬁned bythe matrix Aaccording toasimilarity transfomiationz T’=ATA-'. Foranorthogonal transformation. wetherefore have T’=Ar/K (5.12) “tinaCartesian space (that is,with orthogonal straight-luie axes) there isnodistinction between “co- variant“ and“contravariant” ll1(.llC8\, andtheterminology willnotbeneeded Indeed, strictly speaking thetensors deﬁned hereshould bedenoted as“Cartesian tensors." Asthisistheonlytypeoftensor thatWl.l.lbeusedinthisbook (except inChapters 7and13),the3Clj6ClIlVC willbeomitted |.l'1subsequent discussions. IApsciidotensor inthree dimensions transfomis asatensor except under inversion. Ingeneral, the transforrnation equation forapseudotensor T*oftheNthrankis(cf.Eq.(4.74)) =lAlalla_]makﬂ T13,“ , andthenarity operation Pgives PTIIK =(_])N-l-l-I-4 Asrigidbody motion ll‘|V0lVC< onlyproper rotations nofurther usewillhemade hereotthegeneral pseudotensor Chapter 5TheRigid Body Equations ofMotion Of 713=¢1ik7iz¢1;i- (5-I3) Comparison withEq.(5.11) thusshows thatthematrix components transform identically, under anorthogonal transformation, withthecomponents ofatensor ofthesecond rank. Alltheterminology andoperations ofmatrix algebra, such as“transpose” and“antisymrnetrrical” canbeapplied totensors without change. Theequivalence between thetensor andthematrix isnotrestricted totensors of thesecond rank. Forexample, wealready know thatthecomponents ofavec- tor,which isatensor oftheﬁrstrank. form acolumn orrowmatrix andvector manipulation maybetreated completely interms ofthese associated matrices. Twovectors canbeusedtoconstruct asecond-rank tensor, T.LetAandBbe vectors withcomponents A,andB;andconstruct thetensor T,by Tu-A,B]. (5.14) Forexample, ifAandBaretwo-dimensional vectors,* T=(1..Ts)=(AXBX AxByTyx A)rBx Since eachindividual vector transforms asavector under aCartesian transforma- tion,eachcomponent ofTwilltransform asrequired byEq.(5.10). Forexample, 3 3 I I I Txy = Z 61x;61y_,A;B] 1 aa|A;a)vjBj Z AXBW i=1j=l soTisatensor. Thetypes ofoperations performed withvectors canbecombined withtensors inanobvious way.There isaunittensor, 1,whose components are 1,’;=5,] (5.15) where 8,-jisthedeltafunction (alsocalled theKronecker delta), 6,,=Iifi=j, andzerootherwise. Thedotproduct ontherightofatensor Twithavector Cis deﬁned asthevector Dby 3 D=T-C where1>,=Zi",,c,=r,,c,, 1=l *Todistinguish between matnces which aretransfonnanons andtensors whch arephysical quantities weuse[Iformatnces and()fortensors. 5.3I5.3 Tl'eInertia Tensor andtheMoment ofInertia 191 andthedotproduct ontheleftwithavector Fisdeﬁned asthevector Eby 3 E=F-T whereE, =ZF,r,, =F,r,,.j=1 Ascalar Scanbeconstructed byadouble dotproduct 33 s=F-r-cwhere s=ZZF,i",,c, =F,r,,c,.1-l_;—l These processes aretermed contraction. Ifthetensor Tisconstructed oftwovec- torsAandBasinEq.(5.14), then T-C=A(B-C)=(B-C)A. and F-T=(F-A)B=(A-F)B. THE INERTIA TENSOR AND THE MOMENT OFINERTIA Considered asalinear operator thattransforms tointoL,thematrix lhaselements thatbehave astheelements ofasecond-rank tensor. Thequantity Iistherefore identiﬁed asasecond-rank tensor andisusually called themoment ofinertia tensor orbrieﬂy theinertia tensor. Thekinetic energy ofmotion about apointis T 1 -%m,U,2, where v,isthevelocity oftheithparticle relative totheﬁxedpoint asmeasured inthespace axes.ByEq.(5.2), Tmayalsobewritten as T=%m,v, -(tox1-,), which, upon permuting thevectors inthetriple dotproduct, becomes T: %-m,(r, xv,). Thequantity summed overiwillberecognized astheangular momentum ofthe body about theorigin, andinconsequence thekinetic energy canbewritten inthe form to-L (D-|-0)T= 2= 2 . (5.16) Letnbeaunitvector inthedirection ofatsothatto=wn.Thenanaltemative formforthekinetic energy is Chapter 5TheRigid Body Equations ofMotion (02 1T=—- -|-=-I 2, 5.17 2n n2w () where Iisascalar, deﬁned by I=Il-|~l1=m,[T‘2—(I','I1)2:|, (5.1s) andknown asthemoment ofinertia about theaxisofrotation. Intheusual elementary discussions, themoment ofinertia about anaxisis deﬁned asthesum, overtheparticles ofthebody, oftheproduct oftheparticle mass andthesquare oftheperpendicular distance from theaxis.Itmust beshown thatthisdeﬁnition isinaccord withtheexpression given inEq.(5.18). Theper- pendicular distance isequal tothemagnitude ofthevector r,xn(cf.Fig.5.2). Therefore, thecustomary deﬁnition ofImaybewritten as 1=m,(r, xn)-(r,xn). (5.19) Multiplying anddividing bycoz,thisdeﬁnition ofImayalsobewritten as I=§%(wxr,)-(mxr,). Buteachvector inthedotproduct isexactly therelative velocity v,asmeasured inthespace system ofaxes.Hence, Isodeﬁned isrelated tothekinetic energy by I2T_602, which isthesame asEq(517),andtherefore Imust beidentical with thescalar deﬁned byEq.(5.19). Thevalue ofthemoment ofinertia depends upon thedirection oftheaxisof rotation. Asmusually changes itsdirection withrespect tothebody inthecourse FIGURE 5.2 Thedeﬁnition ofthemoment ofinertia. 5.3 TheInertia Tensor andtheMoment ofInertia 193 ofmass 4 b FIGURE 5.3 Thevectors involved intherelation between moments ofinertia about parallel axes. oftime,themoment ofinertia mustalsobeconsidered afunction oftime.When thebodyisconstrained soastorotate onlyabout aﬁxed axis,thenthemoment ofinertia isaconstant. Insuchacase, thekinetic energy (5.16) isalmost inthe form required tofashion theLagrangian andtheequations ofmotion. Theone further stepneeded istoexpress toasthetimederivative ofsome angle. which canusually bedonewithout difﬁculty. Along withtheinertia tensor, themoment ofinertia alsodepends upon the choice oforigin ofthebody setofaxes. However, themoment ofinertia about some given axisisrelated simply tothemoment about aparallel axisthrough the center ofmass. Letthevector from thegiven origin Otothecenter ofmass be R.andlettheradii vectors from 0andthecenter ofmass totheithparticle be r,andrf.respectively. Thethreevectors sodeﬁned areconnected bytherelation (cf.Fig.5.3) r,=R+rf. (5.20) Themoment ofinertia about theaxisaistherefore Ia=m,(r, Xn)2=m,[(r: +R)xn]2 O1’ la=M(R xn)2+m,(1"§ xn)-2+2m,(Rxn)-(r:xn), where Misthetotal mass ofthebody. Thelastterm inthisexpression canbe rearranged as —2(R xn)-(nxm,r§). Chapter 5TheRigid Body Equations ofMotion Bythedeﬁnition ofcenter ofmass, thesuimnation m,r: vanishes. Hence, Iacan beexpressed interms ofthemoment about theparallel axisbas 1,,=1,,+M(RX“)2 (5.21) =1,,+MR2sin2t9. Themagnitude ofR><ii,which hasthevalue Rsin6,where 9istheangle between Randn,istheperpendicular distance ofthecenter ofmass from theaxispassing through O.Consequently, themoment ofinertia about agiven axisisequal tothe moment ofinertia about aparallel axisthrough thecenter ofmass plusthemoment ofinertia ofthebody, asifconcentrated atthecenter ofmass, withrespect tothe original axis. Theinertia tensor isdeﬁned ingeneral fromthekinetic energy ofrotation about anaxis,andiswritten as Z 2 Trotation =%ml((o Xrt) =%¢9awﬁmi(5u;3Tl "'7‘iu7'i;?)» where Greek letters indicate thecomponents ofcoandr,.Inaninertial frame, the sumisovertheparticles inthebody, andrmistheathcomponent oftheposition oftheithparticle. Because 1Z}.,t,,,,,,,, isabilinear fomiinthecomponents ofco,it canbewritten as 7rotation =‘éIaﬁmamﬂ , where 1.;=mi<8”?-rm) <522> isthemoment ofinertia tensor. Togetthemoment ofinertia about anaxisthrough thecenter ofmass, choose therotation about thisaxis Forabody with acontin- uous distiibution ofdensity p(r), thesums inthecomponents ofthemoment of inertia tensor inEq.(5.22) reduce to i,,,,=IVp(r)(5,,,;r2 -rarﬂ)dV. (523) Asanexample, letusconsider ahomogeneous cube ofdensity p,mass M, andsidea.Choose theorigin tobeatonecorner andthethree edges adjacent tothatcorner tolieonthe+x,+y.and+zaxes. Ifwedeﬁne b=Maz, then straightforward integration ofEq.(5.23) gives 2 1 1 I=—§b %b-51; . —§b -)1» gt; Thus, boththemoment ofinertia andtheinertia tensor possess atypeofrevolu- tion,relative tothecenter ofmass, verysimilar tothatfound forthelinear and angular momentum andthekinetic energy inSection (1.2). 5.4I5.4 TheEigenvalues oftheInertia Tensor 195 THE EIGENVALUES OFTHE iNERTlA TENSOR AND THE PRINCIPAL AXIS TRANSFORMATION Thepreceding discussion emphasizes theimportant roletheinertia tensor playsin thediscussion ofthemotion ofrigid bodies. Anexamination, atthispoint, ofthe properties ofthistensor anditsassociated matrix willtherefore prove ofconsid- erable interest. From thedeﬁning equation, (5.7), itisseenthatthecomponents ofthetensor aresymmetrical; thatis Ixy 1 Iyx. This means that.while theinertia tensor willingeneral have ninecomponents, onlysixofthem willbeindependent—-the three along thediagonal plusthree of theoff-diagonal elements. Theinertia coefﬁcients depend bothuponthelocation oftheorigin ofthebody setofaxesandupon theorientation ofthese axeswithrespect tothebody. This symmetry suggests thatthere exists asetofcoordinates inwhich thetensor is diagonal withthethreeprincipal values I1,I2,andI3.Inthissystem, thecompo- nents ofLwould involve onlythecorresponding component ofoi,thus* /-i=Iiwi. L2=l2w2. Ls=I3w3~ (5-Z5) Asimilar simpliﬁcation would alsooccur intheformofthekinetic energy: to-I-m 1'>l lT=? =51105;+Elgwg+513o§. (5.26) Wecanshow thatitisalways possible toﬁndsuchaxes, andtheproof isbased essentially onthesyiiuiieuiu nature oftheinertia tensor. There areseveral ways tounderstand vectors andtensors. Forexample, avector isaquantity deﬁned byitstransformation properties. hianysetofcoordinates, a vector isspeciﬁed byitsthree components. e.g., V Z Vxi + + Vzk, orbyitsmagnitude anddirection. Inanyframe, themagnitude isgiven by /V}+V}+Viz.andthedirection isgiven bythepolar angles 6and¢.An alternative istousetheﬁrsttwoEuler angles tospecify anewzaxischosen such thatthevector’s direction isalong thataxis.Since thevector liesalong thatzaxis, thethirdEuler angle isnotneeded. Anapproach similar tothislatter method canbeused forthesymmetric mo- ment ofinertia tensor. Consider themoment ofinertia ofabody about anaxis passing through thecenter ofmass ofthebody. Asimilarity transfomiation per- *W"ith aneyetofuture applications, components relative tothese axeswillbedenoted bysubscripts 1,2,3. 6 Chapter 5TheRigid Body Equations ofMotion formed byarotation matrix Rcanbechosen suchthat :0=Rni. (5.23) Thisrotation canbeexpressed interms oftheEuler angles ¢,6,and10asshown inEqs.(4.46) and(4.47). Aproper choice ofthese angles willtransform Iintoits diagonal form 11 O 0 |D= O I2 0 (5.29) 0 0 I3 where I1,I2,andI3,which aretheeigenvalues ofI,arereferred toasthecom- ponents oftheprincipal moment ofinertia tensor. Thedirections ofx’,y’,and Z’deﬁned bytherotation matrix inEq.(5.28) arecalled theprincipal axes, or eigenvectors oftheinertia tensor. These eigenvectors liealong thedirections x’, y’,andz’. Once theprincipal moments andtheirdirections relative tothesurface ofa body areknown, theinertia tensor relative toanyother setofaxisthrough the center ofmasscanbefound byasimilarity transformation deﬁned bytheEuler angles relating thetwocoordinate systems. IfSisthattransformation, then |=s|,,S, (5.30) gives themoment ofinertia inthatframe. Equation (5.21) canthenbeusedto transfonn therotation center toanydesired location. Theprincipal values ofIcan bedetermined bythemethods ofmatrix algebra. Thethreeprincipal values ofthemoment ofinertia tensor inEq.(5.29) canbe found bysolving thecubic equation forIthatarises from thedetenmnant Ixx i I Ix): Izx lzx Iyz Izz_I where thesymmetry ofIhasbeen displayed explicitly. Equation (5.31) isthesec- ularequation, whose three roots arethedesired principal moments. Foreach of theseroots, Eqs.(5.28) canbesolved toobtain thedirection ofthecorresponding principal axis.Inmostoftheeasily soluble problems inrigiddynamics, theprin- cipal axescanbedetermined byinspection. Forexample, weoften have todeal withrigidbodies thataresolids ofrevolution about some axis,withtheorigin of thebody system onthesymmetry axis. Alldirections perpendicular totheaxisof symmetry arethenalike, which isthemark ofadouble roottothesecular equa- tion.Theprincipal axesarethenthesymmetry axisandanytwoperpendicular axesintheplane normal tothesymmetry axis. Theprincipal moments ofinertia cannot benegative, because asthediagonal elements intheprincipal axessystem theyhavetheformofsumsofsquares. Thus. 5.4 TheEigenvalues oftheInertia Tensor I97 Inisgiven by(cf.Eq.(5.6)) 1..=mm?+2%). Foroneoftheprincipal moments tovanish, allpoints ofthebody must besuch thattwocoordinates ofeach particle arezero. Clearly thiscanhappen onlyifall pOinlS ofthebodyarecollinear withtheprincipal axiscorresponding tothezero principal moment. Anytwoaxesperpendicular tothelineofthebody willthen betheother principal axes. lndeed, thisisclearly alimiting caseofabody with anaxisofsymmetry passing through theorigin. Wecanalsounderstand theconcept ofprincipal axesthrough some geometri- calconsiderations thathistorically formed theﬁrstapproach tothesubject. The moment ofinertia about agiven axishasbeen deﬁned asI=n-I~n.Letthe direction cosines oftheaxisbeoz,13.andysothat n=ai+}3j+yk; 1thencanbewritten as 1=1,,,,a2+1_,._,a2+12,;/2+z1,,.a;3 +21,,ay +211;}/0!, (5.32) using thesymmetry ofIexplicitly. Itisconvenient todeﬁne avector pbythe equation np=—. (5.33) ~/7 Themagnitude ofpisthusrelated tothemoment ofinertia about theaxiswhose direction isgiven byn.Interms ofthecomponents ofthisnewvector, Eq.(5.32) takes ontheform 1=1xxP%+ Twp; +1zz.<>§+ 21xyPlP2 +2])-zP2P3 _21zxP3Pl- (5-34) Considered asafunction ofthethree variables pl,pg,p3,Eq.(5.34) isthe equation ofsome surface inpspace. Inparticular, Eq.(5.34) istheequation ofan ellipsoid designated astheinertial ellipsoid. Wecanalways transform toasetof Cartesian axesinwhich theequation ofanellipsoid takes onitsnormal form: 1=1tp'i+120%+npﬂi. (5.35) withtheprincipal axesoftheellipsoid along thenewcoordinate axes.But(5.35) issimply theformEq.(5.34) hasinasystem ofcoordinates inwhich theinertia tensor Iisdiagonal. Hence, thecoordinate transformation thatputs theequation ofellipsoid intoitsnonnal formisexactly theprincipal axistransformation pre- viously discussed. Theprincipal moments ofinertia detennine thelengths ofthe axesoftheinertia ellipsoid. Iftwooftheroots ofthesecular equation areequal, theinertia ellipsoid thushastwoequal axesandisanellipsoid ofrevolution. Ifall three principal moments areequal, theinertia ellipsoid isasphere. 5.5IChapter 5TheRigid Body Equations ofMotion Aquantity closely related tothemoment ofinertia istheradius ofgyration, Rn,deﬁned bytheequation 1=MR3. (5.36) Interms oftheradius ofgyration, thevector pcanbewritten as n P=Z-R0./M Theradius vector toapoint ontheinertia ellipsoid isthusinversely proportional totheradius ofgyration about thedirection ofthevector. Itisworth reemphasizing thattheinertia tensor Iandallthequantities associ- atedwithit—pn'ncipal axes, principal moments, inertia ellipsoid, etc.-—are only relative tosome particular pointﬁxedinthebody. Ifthepointisshifted elsewhere inthebody, allthequantities willingeneral bechanged. Thus, Eq.(5.21) gives theeffect ofmoving thereference point from thecenter ofmass tosome other point. Theprincipal axistransformation thatdiagonalizes I’atthecenter ofmass willnotnecessarily diagonalize Iabout another axis, andhence isnotingeneral theprincipal axistransfonnation fortheshifted tensor I.Onlyiftheshiftvector Risalong oneofthepnncipal axesrelative tothecenter ofmasswillthediffer- encetensor bediagonal inthatsystem. Thenewinertia tensor Iwillinthatspecial casehavethesame principal axesasatthecenter ofmass. However, theprincipal moments ofinertia arechanged, except forthatcorresponding totheshiftaxis, where thediagonal element ofthedifference tensor isclearly zero.The“paral- lelaxis” theorem forthediagonalized formoftheinertia tensor thushasarather specialized andrestricted form. SOLVING RIGID BODY PROBLEMS AND THE EULER EQUATIONS OFMOTION Practically allthetoolsnecessary forsetting upandsolving problems inrigid body dynamics havebynowbeenassembled. Ifnonholonomic constraints are present, then special means must betaken toinclude theeffects ofthese con- straints intheequations ofmotion. Forexample, ifthere are“rolling constraints," thesemustbeintroduced intotheequations ofmotion bythemethod ofLagrange undetermined multipliers, asinSection 2.4.Asdiscussed inSection 5.1,weusu- allyseekaparticular reference point inthebody suchthattheproblem canbe splitintotwoseparate parts, onepurely translational andtheother purely rota- tional about thereference pomt. Ofcourse, ifonepoint ofthengrclbody1sﬁxed inaninertial system, thenthatistheobvious reference point. Allthathastobe considered thenistherotational problem about theﬁxed point. Forbodies without aﬁxed point, themostuseful reference point isalmost always thecenter ofmass. Wehavealready seenthatthetotalkinetic energy and angular momentum thensplitneatly intoonetennrelating tothetranslational 5.5 Solving Rigid Body Problems andtheEuler Equations ofMotion 199 motion ofthecenter ofmassandanother involving rotation about thecenter of mass. Thus, Eq.(l.3l)cannowbewritten T=%Mv2 +3131012. Formany problems (certainly allthose thatwillbeconsidered here), asimilar sortofdivision canbemade forthepotential energy. Wecanthensolve individu- allyforthetranslational motion ofthecenter ofmass andfortherotational motion about thecenter ofmass. Forexample, theNewtonian equations ofmotion canbe useddirectly: Eq.(1.22) forthemotion ofthecenter ofmassandEq.(1.26) for themotion about thatpoint. Withholonomic conservative systems, theLagrangian formulation isavailable, withtheLagrangian taking theform L':q= =L¢(qCs qt‘)+Lb(qb= éb)' I-Iere LCisthatpartoftheLagrangian involving thegeneralized coordinates q¢ (andvelocitieS éc)ofthecenter ofmass, andL1,thepartrelating totheorienta- tionofthebody about thecenter ofmass, asdescribed byqb,db.Ineffect then, there aretwodistinct problems, onewithLagrangian LCandtheother withLa- grangian Lb. InboththeNewtonian andLagrangian formulations, itisconvenient towork intenns oftheprincipal axessystem ofthepoint ofreference, sothatthekinetic energy ofrotation takes thesimple form given inEq.(5.26). Sofar,theonly suitable generalized coordinates wehave fortherotational motion oftherigid body aretheEuler angles. Ofcourse, themotion isoften effectively conﬁned to twodimensions, asinthemotion ofarigidlamina inaplane. Theaxisofrotation isthen ﬁxed inthedirection perpendicular totheplane; only oneangle ofrotation isnecessary andwemaydispense withthecumbersome machinery oftheEuler angles. Fortherotational motion about aﬁxedpoint orthecenter ofmass, thedirect Newtonian approach leads toasetofequations known asEuler’s equations of motion. Weconsider either aninertial frame whose origin isattheﬁxed point of therigid body, orasystem ofspace axes with origin atthecenter ofmass. Inthese twosituations, Eq.(1.26) holds, which hereappears simply as dL(5).? =N. Thesubscript sisused because thetime derivative iswith respect toaxes thatdo notshare therotation ofthebody. However, Eq.(4.86) canbeusedtoobtain the derivatives withrespect toaxesﬁxedinthebody: +wxLdz,_ dr_b ‘ 5.6IChapter 5TheRigid Body Equations ofMotion or,bydropping the“body” subscript: dLE+(0xL=N. (5.37) Equation (5.37) isthustheappropriate formoftheNewtonian equation ofmotion relative tobodyaxes.Theithcomponent ofEq.(5.37) canbewritten dL| I +€;JkCUJLk =N,. Ifnow tliebody axes aietakeii asthepiiiiuipal axes relative totheiefeieiice point, thentheangular momentum components areL;=l,w,. ByEq.(5.25), Eq.(538)takes thefomi (nosummation oni*) .11,-£1”-ii+6,,-,,w,w,.i,. =N,, (5.39) since theprincipal moments ofinertia areofcourse timeindependent. Inexpanded form, thethree equations making upEq.(5.39) looklike 1irbi- w2w3(T2 —13)=Ni 12032—w3¢v1(Is —11)=N2 (5-39') 116513-wiw2(li -T2)=N3- Equations (5.39) or(5.39') areEuler’s equations ofmotion forarigid body withonepoint ﬁxed. They canalsobederived from Lagrange’s equations in theform ofEq.(1.53) where thegeneralized forces Qjarethetorques. NJ, corresponding totheEuler angles ofrotation. However, onlyoneoftheEuler angles hasitsassociated torque along oneofthebody axes, andtheremaining twoEuler’s equations must beobtained bycyclic permutation (cf.Derivation 4). Consider thecasewhere I1=I2qéI3.Atorque withcomponents N|orN2 willcause bothwiand0.73tochange without affecting (03.Weshallretum toa discussion ofthisinSection 5.7when weconsider theheavy symmetric topwith onepoint ﬁxed. Letusﬁrstconsider thetorque-free motion ofarigid body. TORQUE-FREE MOTION OFARIGID BODY Oneproblem inrigiddynamics where Euler’s equations areapplicable isinthe motion ofarigidbodynotsubject toanynetforces ortorques. Thecenter ofmass isthcn either atrestofmoving unifonrily. anditdoes notdecrease thegenerality ofthesolution todiscuss therotational motion inareference frame inwhich the center ofmass isstationary. Insuchacase,theangular momentum arises only from rotation about thecenter ofmass, andEuler’s equations aretheequations of *Itshould beobvious thatliq(5.39), astheithcomponent ofavector equation, doesnotinvolve a summation overz,although summation isimplied overtherepeated indices jandk. 5.6Torque-free Motion ofaRigid Body 201 motion forthecomplete system. Intheabsence ofanynettorques, theyreduce to Iiﬁbi=w2¢v3(12 —13) 12032 =w3wi(T3 —71) (5-40) 116113=wiwz(11 —I2)- Thesame equations. ofcourse. willalsodescribe themotion ofarigidbody when onepoint isﬁxed andthere arenonetapplied torques. Weknow twoim- mediate integrals ofthemotion, forboththekinetic energy andthetotalangular momentum vector must beconstant intime. With these twointegrals it1Spossible tointegrate (5.40) completely interms ofelliptic ﬁinctions, butsuchatreatment is notveryilluminating. However, itisalsopossible toderive anelegant geometri- caldesciiption ofthemotion, known asPoinsot’s construction, without requiring acomplete solution totheproblem. Letusconsider acoordinate system oriented along theprincipal axes ofthe bodybutwhose axesmeasure thecomponents ofavector palong theinstanta- neous axisofrotation asdeﬁned byEq.(5.33). Forourpurposes, itisconvenient tomake useofEq.(5.17) forthekinetic energy (here constant) andwrite the deﬁnition ofpintheform (.0 (Ii) Inthispspace, wedeﬁne afunction nm=p-Lp=ﬁm mu) where thesurfaces ofconstant Fareellipsoids, theparticular surface F=1being theinertia elhpsoid. Asthedirection ottheaxisofrotation changes intime. the parallel vector pmoves accordingly, itstipalways deﬁning apoint ontheinertia ellipsoid. Thegradient ofF,evaluated atthispoint, furnishes thedirection of thecorresponding normal totheinertia ellipsoid. From Eq.(5.42) forF(p).the gradient ofFwithrespect tophastheform 2|-toVpF=2I'P=-\Ti?, %F=J;L mo) Thus. thetovector willalways move suchthatthecorresponding nomial tothe inertia ellipsoid isinthedirection oftheangular momentum. Intheparticular case under discussion, thedirection ofLisﬁxedinspace, anditistheinertia ellipsoid (ﬁxed withrespect tothebody) thatmustmove inspace inorder topreserve this connection between toandL(cf.Fig.5.4).OI‘ Chapter 5TheRigid Body Equations ofMotion Inertia ellipsoid \ Invanable plane Herpolh ode \\ l.FIGURE 5.4Themotion oftheinertia ellipsoid relative totheinvariable plane. ltcanalsobeshown thatthedistance between theorigin oftheellipsoid andthe plane tangent toitatthepointpmustsimilarly beconstant intime.Thisdistance isequal totheprojection ofponLandisgiven by p'L Q) ‘I1 L=L\/2T OI.‘ -L~/2T PT=T’ (544)where usehasbeen made ofEq.(5.16). Both T,thekinetic energy, andL,the angular momentum, areconstants ofthemotion, andthetangent plane istherefore always aﬁxed distance fromtheorigin oftheellipsoid. Siuce thenormal tothe plane, being along L,alsohasaﬁxed direction, thetangent plane isknown as theinvariable plane. Wecanpicture theforce-free motion oftherigidbody as being suchthattheinertia ellipsoid rolls,without slipping, ontheinvariable plane, with thecenter oftheellipsoid aconstant height above theplane. The rolling occurs without slipping because thepoint ofcontact isdeﬁned bytheposition of p,which, being along theinstantaneous axisofrotation, istheonedirection in thebodymomentarily atrest.Thecurve traced outbythepoint ofcontact onthe inertia ellipsoid isknown asthepolhode, while thesimilar curve ontheinvariable plane iscalled theherp0lh0de.* Poinsot’s geometrical discussion isquite adequate todescribe completely the force-free motion ofthebody. Thedirection oftheinvariable plane andtheheight oftheinertia ellipsoid above itaredetermined bythevalues ofTandL,which areamong theinitial conditions oftheproblem. ltisthenamatter ofgeometry to *I'I8l1CB.tI10_|2lIDb0l'WOCI(liln-S0|JIldlI1g statement: thepolhode rollswithout shppmg ontheherpolhode lying i.ntheinvanable plane. 5.6Torque-free Motion ofaRigid Body 203 traceoutthepolhode andtheherpolhode.* Thedirection oftheangular velocity inspace isgiven bythedirection ofp,while theinstantaneous orientation ofthe bodyisprovided bytheorientation oftheinertia ellipsoid, which isﬁxedinthe body. Many elaborate descriptions offorce-free motion obtained inthisfashion canbefound intheliterature. Inthespecial caseofasymmetrical body, theinertia ellipsoid isanellipsoid ofrevolution, sothatthepolhode ontheellipsoid isclearly acircle about the symmetry axis. Theherpolhode ontheinvariable plane islikewise acircle. An observer ﬁxedinthebodyseestheangular velocity vector wmove onthesurface ofacone—-called thebodycone--whose intersection withtheinertia ellipsoid is thepolhode. Correspondingly, anobserver ﬁxed inthespace axesseestomove onthesurface ofaspace conewhose intersection withtheinvariable plane isthe herpolhode. Thus, thefreemotion ofthesymmetrical rigid body issometimes described astherolling ofthebody cone onthespace cone. Ifthemoment of inertia about thesymmetry axisislessthan thatabout theother twoprincipal axes. thenfrom Eq.(5.35) theinertia ellipsoid isprolate, i.e.,football shaped- somewhat asisshown inFig.5.4.Inthatcase, thebody cone isoutside thespace cone. When themoment ofinertia about thesymmetry axisisthegreater, the ellipsoid isoblate andthebody conerollsaround theinside ofthespace cone. Ineither case, thephysical description ofthemotion isthatthedirection ofto precesses intimeabout theaxisofsymmetry ofthebody. ThePoinsot construction shows howtomoves, butgivesnoinformation asto howtheLvector appears tomove inthebody system ofaxes. Another geomet- ricaldescription isavailable however todescribe thepathoftheLvector asseen byanobserver intheprincipal axessystem. Equations (5.25) and(5.26) imply thatinthissystem thekinetic energy isrelated tothecomponents oftheangular momentum bytheequation L2L2L2T=—" —y —‘. ‘.45211+212+213 (D ) Since Tisconstant, thisrelation deﬁnes anellipsoid, referred toastheBinet ellipsoid, alsoﬁxedinthebodyaxesbutnotthesame astheinertia ellipsoid Ifweadopt theconvention 13512511, andwrite theequations fortheellipsoid inthestandard form LEL?»L? -- -—— _=1 5.45’2TI| +ZTI; +ZTI3 ( ) thenweseethattheellipsoid sketched onFig.5.5ahassemimajor axes,inorder ofdecreasing size,oft/2T1], ,/2TI2, and./ZTI3. Theconservation ofthetotal *The herpolhode ISalways concave totheorigin, belying itsname, which means “snakelike.” Chapter 5TheRigid Body Equations ofMotion angular momentum, L,gives us L}+L1;+Lgiui =1, (5.46) theequation forasphere inLxL)ILz space. Thevector Lmoves insuchawaythat itdescribes apathonboththeellipsoid ofEq.(5.45) andthesphere ofEq.(5.46). Inother words, thepathofListheintersection oftheellipsoid andthesphere. Thecomponents Lsatisfy theequation 14+ Lg+1.§={.Z+t§.+1.§_ 2T1, 2112 2TI3 L1’ ltiseasytoshow thatthese twosurfaces willintersect forvalues ofLlarger thantheellipsoid semiminor axisandlessthanthesemimajor axis,thatis, ,/2:r1_, <L<,/2111. Thesphere isoutside theellipsoid ontheLzaxisandinside theellipsoid along Lx.Figure 5.5depicts curves where thesphere intersects theellipsoid forvarious values ofL.Fig.5.5ashows aperspective view andFig.5.5bshows theview asseenfrom theL>.axis.Thecurves thatappear asstraight linesonFig5.5b correspond tothecasewhere L=,/2Tl2. With thehelpofthisgeometrical construction, something canbesaidabout the possible motions ofafreeasymmetric body. Itiseasytoseethatasteady rotation LI (a) (b) FIGURE 5.5 (a)Thekinetic energy, orBinet, ellipsoid ﬁxed inthebody axes, andsome possible paths oftheLvector initssurface. (b)Sideview ofBinet ellipsoid. 5.6Torque-free Motion ofaRigid Body 205 ofsuch abody ispossible only about oneoftheprincipal axes. From theEuler equations (5.40), allthecomponents ofcocanbeconstant onlyif wiw2(11 12)=wzv->s(12 —13)=wswi (13—11)—0. which requires thatatleasttwoofthecomponents w,bezero; i.e.,toisalong onlyoneoftheprincipal axes. However, notallofthese possible motions are stable—that is,notmoving farfromtheprincipal axisunder small perturbation. Forexample, steady motion about theLZaxiswilloccur when L2=2Tlg.When there areslight deviations from thiscondition, theradius oftheangular momen- tumsphere isjustslightly smaller thanthisvalue, andtheintersection withthe kinetic energy ellipsoid isasmall circle about theLzaxis. Themotion isthus stable, theLvector never being farfrom theaxis. Similarly, attheother extreme, when themotion about theaxisofsmallest Iis perturbed, theradius oftheangular momentum sphere isjustslightly larger than thesmallest semimajor axis.Theintersection isagain asmall closed ﬁgure around theprincipal axis, andthemotion isstable. However, themotion about theinter- mediate axisISunstable. Thisisclearly shown inFig.5.5.Fortheintermediate (L))axis,thekinetic energy hastwoorbits thatencircle theellipsoid andcross eachother where the:l:Ly passthrough theellipsoid. Hence, there aretwodiffer- entorbits withvalues slightly lessthan,/2T1; andtwoother distinctly different orbits withvalues slightly exceeding \/ZTIZ, allfourofwhich havequite long paths onthesurface. Thisbehavior canbebestunderstood byrecognizing thatattheintermediate axistheradius ofcurvature oftheellipsoid inonedirection isgreater thanthat ofthecontact sphere, andlessintheperpendicular direction. Attheother two extremes, theradiiofcurvature areeither greater orsmaller thanthesphere radius inalldirections. These conclusions onthestability offree-body motion havebeen known foralongtime. butapplications, e.g.,tothestability ofspinning space- craft, have brought them outoftheobscurity ofoldmonographs onrigid body dynamics." Forasymmetrical rigidbody, theanalytical solution fortheforce-free mo- tionisnotdifﬁciilt tonhtain, andwecandirectly conﬁmi theprecessing motion predicted bythePoinsot construction. Letthesymmetry axisbetaken astheL? principal axissothat1|=I2.Euler’s equations (5.40) reduce thento *lfthere aredissipative mechanisms present, these stability arguments havetobemodiﬁed. Itiseasy toseethatforabody withconstant L,butslowly decreasing T,theonlystable rotation ISabout the principal axiswiththelargest moment ofinertia Thekinetic energy ofrotation about theill]pillicipal axisforgiven LisT=L2/21, ,which isleastfortheaxiswiththelargest I,.Ifabody issetspinning about anyother pnncipal axis,theettcct ofaslowly decreasing kinetic energy istocause theangular velocity vector toshiftuntilthespinning isabout theaxisrequinng theleastvalue ofTforthegiven LSuch dissipative effects arepresent inspacecraft because oftheﬂexing ofvarious members inthe course ofthemotion, especially ofthelongbooms camed bymany ofthem These factswere leamed thehardwaybytheearly designers ofspacecraft‘ 2 Chapter 5TheRigid Body Equations ofMotion 11651=(11—1s)w3w2 11032=(Ii-73)w3w1 (547) l3£b3 =O. Thelastofthese equations states that(03isaconstant, anditcantherefore be treated asoneoftheknown initial conditions oftheproblem. Theremaining two equations cannowbewritten ab]=—S'Zw2, ab;=Qwi, (5.48) where Qisanangular frequency Q= (5.49)l Elimination of(02bctwccn Eqs.(5.48) leads tothestandard diffeiential equation forsimple harmonic motion (B1 =-9260], withthetypical solution 0)]=AcosQt. Thecorresponding solution for0);canbefound bysubstituting thisexpression forco],back intheﬁrstofEqs.(5.48): (03=AsinS2t. Thesolutions foranand602show thatthevector a)1i+wgjhasaconstant magni- tudeandrotates uniformly about thezaxisofthebody withtheangular frequency S2(cf.Fig.5.6).Hence, Z116totalangular velocity onisalsoconstant inmagnitude andprecesses about thezaxiswiththesame frequency, exactly aspredicted by thePoinsot construction. *Recall thattheprecession described here isrelative to thebodyaxes, which arethemselves rotating inspace withthelarger frequency w.From Eq.(5.49), itisseenthatthecloser I1isto10,.theslower willbethe precession frequency Qcompared totherotation frequency cu.Theconstants A (theamplitude oftheprecession) and(1)3canbeevaluated interms ofthemore usual constants ofthemotion, namely, thekinetic energy andthemagnitude of theangular momentum Roth TandL2canbewritten asfunctions ofAandm3: *The precession canbedeinorstrated inanother fashion bydetinin gavector Itlying along thezaxis withmagnitude given by(549)Equations (547)arethenessentially equivalent tothevector equarion di=coxQ, which immediately reveals theprecession of0)withthefrequency S2. 5.6Torque-free Motion ofaRigid Body 207 Q Z “‘=‘ FIGURE 5.6Precession oftheangular velocity about theaxisofsymmetry intheforce- freemotion ofasymmetrical rigid body. T=%I1A2 +%I30)§, L2=1,2,4?+150%, andthese relations intummaybesolved forAand(1)3interms ofTandL. Wewould expect thatEarth’s axisofrotation should exhibit thisprecession, for theexternal torques acting onEarth aresoweak thattherotational motion maybe considered asthatofafreebody. Earth isapproximately symmetrical about the polaraxisandslightly ﬂattened atthepolessothatI1islessthanI3.Numerically, theratioofthemoments issuchthat ﬁil =000327I1 i ’ andthemagnitude oftheprecession angular frequency should therefore be 033 503s2=-_-~_.305.8lO39 306 Since w3ispractically thesame asthemagnitude ofcu,thisresult predicts aperiod ofprecession ofapproximately 306days orabout 10months. lfsome circumstance disturbed theaxisofrotation from theﬁgure axisofEarth, wewould therefore expect theaxisofrotation toprocess around theﬁgure axis(i.e.,around thenorth pole) onceevery 10months. Practically, suchamotion should showup 5.7 IChapter 5TheRigid Body Equations ofMotion asaperiodic change intheapparent latitude ofpoints onEarth’s surface. Careful measurements oflatitude atanetwork oflocations around theworld, carried out nowforabout acentury, show thattherotation axisisindeed moving about the polewithanamplitude oftheorder ofafewtenths ofasecond oflatitude (about 10m).Butthesituation isfarmore complicated (andinteresting) thantheabove simple analysis would suggest. Thedeviations between theﬁgure androtation axesareveryirregular sothat it’smore a“wobble” thanaprecession. Careful frequency analysis shows the existence ofanannual period inthemotion, thought toarisefrom theannual cycle ofseasons andthecorresponding mean displacement ofatmospheric masses about theglobe. Additionally, astrong frequency component iscentered about a period of420days, known astheChandler wobble. Thepresent belief isthatthis motion represents thefree-body precession derived above. Itisthought thatthe difference inperiod arises from thefactthatEarth isnotarigid body butisto some degree elastic. Ineffect, some partofEarth follows along withtheshiftin therotation axis. which hastheeffect ofreducing thedifference intheprincipal moments ofinertia andtherefore increasing theperiod. (If,forexample, Earth were completely ﬂuid, thentheﬁgure axiswould instantaneously adjust tothe rotation axisandtherecould benoprecession.) There arestillother obscure features totheobserved wobble. Thefrequency analysis indicates strong damping effects arepresent, believed toarisefrom either tidal friction ordissipative effects inthecoupling between themantle andthecore. Thedamping period ought tobeontheorder ofl0-20 years. Butnosuchdecay oftheamplitude oftheChandler wobble hasbeenobserved; some sortofran- domexcitation mustbepresent tokeepthewobble going. Various sources ofthe excitation havebeensuggested. Present speculation points todeepearthquakes, orthemantle phenomena underlying them, aspossibly producing discontinuous changes intheinertia tensor large enough tokeep exciting thefree-body preces- sion.* THE HEAVY SYMMETRICAL TOP WITH ONE POINT FIXED Asafurther andmore complicated example oftheapplication ofthemethods ofrigiddynamics, letusconsider themotion ofasymmetrical body inauni- form gravitational ﬁeldwhen onepoint onthesymmetry axisisﬁxed inspace. A widevariety ofphysical systems, ranging fromachild’s toptocomplicated gyro- scopic navigational instruments, areapproximated bysuchaheavy symmetrical top.Both foritspractical applications andasanillustration ofmany ofthetech- *Thc treeprecession ofEarth‘s axisisnottobeconfused withitsslowprecession about thenormal totheecliptic Thisu.i-imnmniral precession oftheequinoxes isduetothegravitational torques of theSunandMoon, which wereconsidered negligible mtheabove discussicn. Thattheassumption is justiﬁed isshown bythelongperiod oftheprecession oftheequinoxes (26,000 years) compared to aperiod ofroughly oneyearfortheforce-free precession Theastronomical precession isdiscussed further below 57TheHeavy Symmetrical Topwith One PomtFixed 209 Verlical Z 0 i " avxty %~.>.:‘1;,;¢.\_=;»¢:.:o1>» .,—:<1;*r‘ '~>>-,2-511:" "M i2..,,__. -*_*‘-_ X .¢ 4’I Line ofnudes FIGURE 5.7Euler's angles specifying theorientation ofasymmetrical top. niques previously developed. themotion oftheheavy symmetrical topdeserves a detailed exposition. Thesymmetry axisisofcourse oneoftheprincipal axesandwillbechosen as thezaxisofthecoordinate system ﬁxed inthebody.* Since onepoint isstationary, theconﬁguration ofthetopiscompletely speciﬁed bythethree Euler angles: 9 gives theinclination ofthezaxisfrom thevertical, qﬁmeasures theazimuth ofthe topabout thevertical, while 11/istherotation angle ofthetopabout itsown2axis (cf.Fig.5.7).Thedistance ofthecenter ofgravity (located onthesymmetry axis) fromtheﬁxedpoint willbedenoted byI Therateofchange ofthese three angles givethecharacteristic motions ofthe topas 1/}=rotation ofthetopabout itsownﬁgure axis,z J2=precession orrotation oftheﬁgure axiszabout thevertical axisZ’ ' ' theverti- fi=nutation orbobbing upanddown ofthezﬁgure 8.X1Srelative to calspace axisZ’. Formany cases ofinterest sucli asthetopandthegyioscope, wehave >>ti>> ¢.Since 1|=I2géI3,Euler’s equations (5.39') become h itwilltherefore beconvenient todesign ate * thebody axes need speciﬁc identiﬁcation ere, ' ti‘thesaceaxes, which willOnly them inthissection asthexyzaxes, without fearofconfusing them wi p bedesignated bythex'y'z' axes 0 Lhapter 5TheRigid Body Equations ofMotion 1i@31+ w2w3(I3 —12)=Ni, 12032+wiw3(11 —T3)=N2, and I3d)3 =N3. Letusconsider thecasewhere initially N3=0=N2,N1-7‘:O,andwl= C02=0,(0375O,then0);;willbeconstant. Thetorque N1willcause antochange since 011;éO.Since cu]isnolonger zero, thesecond equation requires thatcog begin tochange also. What thismeans interms ofanobservation isnotobvious. Weobserve thechanges intheEuler angles 1,5, 9andtheir associated angles inthex’,y’,z’laboratory frame rather thanthecbi,032,digandtheirassociated angles intheprincipal axissystem. This suggests thattheEuler equations may notprovide themostuseful description ofthemotion. TheLagrangian procedure, rather thanEuler’s equations, willbeusedtoobtain asolution forthemotion ofthetop.Since thebody issymmenical, thekinetic energy canbewritten as T=%Ii(a)% +01%) +%I3a)%, or,interms ofEuler’s angles, andusing Eqs.(4.87), as T=L2](é2 +Q52sinz9)+ +cosl9l2, (5.50) where the 9cross terms inco?andmgcancel. Itisawell-known elementary theorem thatinaconstant gravitational ﬁeldthe potential energy isthesame asifthebody wereconcentrated atthecenter ofmass. Wewillhowever giveabnef fonnal proot here. Thepotential energy ofthebody isthesumoveralltheparticles: V=—mi-r.-g, where gistheconstant vector fortheacceleration ofgravity. ByEq.(1.21), deﬁn- ingthecenter ofmass, thisisequivalent to V=-—MR -g, (5.51) which proves thetheorem. Interms oftheEuler angles, V=Mglcost-3, (5.Sl’) sothattheLagrangian is 1:=l5‘(é2+q‘>%-.1139) +£23-(ti,+<iicosl9)2 -Mglcos9. (5.52) 5.7TheHeavy Symmetrical TopwithOnePoint Fixed 211 Notethatif)and‘l/Idonotappear explicitly intheLagrangian; theyaretherefore cyclic coordinates, indicating thatthecorresponding generalized momenta are constant intime.Now, wehaveseenthatthemomentum conjugate toarotation angle isthecomponent ofthetotal angular momentum along theaxisofrotation, which forqt)isthevertical axis, andfor(0,thezaxisinthebody. Wecaninfact show from elementary principles thatthese components oftheangular momentum must beconstant intime. Since thetorque ofgravity isalong thelineofnodes, there isnocomponent ofthetorque along either thevertical orthebody 2axis, forbydeﬁnition bothofthese axesareperpendicular tothelineofnodes. Hence, thecomponents oftheangular momentum along these twoaxes must beconstant intime. Wetherefore havetwoimmediate ﬁrstintegrals ofthemotion: at ..pip= =I3(i// +450059) =l3a)3 =Ila (5.53) and at ..P,=Q=(11sin2l9+13cos26)¢ +13¢cost?=1,1». (554) Herethetwoconstants ofthemotion areexpressed interms ofnewconstants a andb.There isonefurther ﬁrstintegral available; sincethesystem isconservative, thetotalenergy Eisconstant intime: 1.. 1E=T+v=5&0’+¢2$11120)+55*-mg+Mglcosﬁ. (5.55) Only three additional quadratures areneeded tosolve theproblem, andtheyare easily obtained from these three ﬁrstintegrals without directly using theLagrange equations. From Eq.(5.53), 'l/Iisgiven interms ofq5by 13¢=1,“-13¢cos0, (5.56) andthisresult canbesubstituted in(5.54) toeliminate ilr: I1¢isinz9+Ilacos 9=Ilb, OI.‘ .b—acos9 =-i. 5.57¢ sinz9 () Thus, if9wereknown asafunction oftime, Eq.(5.57) could beintegrated to furnish thedependence of4)ontime. Substituting Eq.(5.57) backinEq.(5.56) results inacorresponding expression for1//: Chapter STheRigid Body Equations ofMotion .I1a b—acos6ilr I3 cos9 sinz9. (558) which furnishes (11if9isknown. Finally, Eqs.(5.57) and(5.58) canbeusedto eliminate (1.3and1/}fromtheenergy equation, resulting inadiiferential equation involving 9alone. First notice thatEq.(5.53) says603isconstant intimeandequal to(I1/I3)a. Therefore, E—I;w§/2 isaconstant ofthemotion, which weshall designate asE’. Making useofEq.(5.57), theenergy equation canthusbewritten as 11¢?’ It(b—acos6)2E’=—— ——i—- l.9. 5.5 2+2 sinze +Mg cos (9) Equation (5.59) hastheform ofanequivalent one-dimensional problem inthe variable i9,withtheeffective potential V’(9)given by , 11»- e1v(0)=Mglcosfi +-é . (5.60) Thus, wehave fourconstants associated with themotion, thetwoangular mo- menta p,;,and13¢,theenergy temiE—%I3w§_. andthepotential energy term Mgl.Itiscommon todeﬁne fournormalized constants ofthemotion as 25-1,05%(1= It 2M1p=73- (5.61) _Pla_Il i>="_¢Ii Interms ofthese constants, theenergy equation (5.55) canbewritten as a=6i2+ i_—il§°S—f9)—% +)6’cos6. (5.62)sin6' Wewillusethisone-dimensional problem todiscuss themotion in9,very similarly towhat wasdone inSection 3.3indescribing theradial motion forthe central force problem. Itismore convenient t0Change variables aswedidforthe central force problem. Using thevariable u=cos9,rewrite Eq.(5.62) as 1:42=(i-u2)(o:-ﬂu)-(b-dlt)2, (5.62') which canbereduced immediately toaquadrature: 5.7TheHeavy Symmetrical TopwithOnePoint Fixed 213 “(') dut=f --_ . (5.63) u(0) \/(1 —M2)(t1 —ﬂu)—(b—au)2 Wiili thisresult, andEqs.(5.57) and(5.58), ¢andilrcanalsobereduced to quadratures. However, thepolynomial intheradical isacubic sothatwehave to dealwithelliptic integrals. These solutions canbegenerated oncurrent desk-top computers. Inthecaseoftheforce-free motion, thephysics tends tobeobscured intheprofusion ofmathematics. Fortunately, thegeneral nature ofthemotion can bediscovered without actually performing theintegrations. Before proceeding withthestudy ofthepossible solutions ofEq.(5.63), afew comments ontheconstants deﬁned inEqs.(5.61) willbeuseful. Figure 5.7shows thecasewhere theﬁxedpoint isnotatthecenter ofmass. Ifthetopisspinning on ahorizontal surface, bothoiand/3aregreater thanzero.Ifthe topissupported by astand thatallows ittodipbelow horizontal, /3isstilllarger thanzero, butorcould bepositive ornegative. Another common application isthegyroscope where the center oimass istheﬁxed point. Lnterms ofFig.5.7,ozistheenergy inthesystem excluding thex3angular kinetic energy. Forthegyroscope, B=0andoz30. Weshallrestrict ourattention tosituations inwhich therotational kinetic energy about thex3axisismuch larger thanthekinetic energy about theother twoaxes. Itisconvenient todesignate theright-hand sideofEq.(5.62’) asafunction f(u)anddiscuss thebehavior ofthecubic equation f(u)=ﬂl43-((1+a2)u2+(2ab-is)“+lo:-62). Forthegyroscope, f(u)isonlyaquadratic equation since ,3=0,while forthetop thefullcubic equation must beconsidered. Since many oftheapplications ofthe gyroscope usetorque-free mountings, piecession andiiulations aresuppressed so thegyroscope motions aretrivial. Tounderstand thegeneral motions ofaspinning body. wewillconsider onlycases where ,8>0. Therootsofthecubic polynomial furnish theangles atwhich 9changes Sign, thatis,the“turning angles” in6?.Knowing these angles willgivequalitative in- fomiation about themotion. There arethree roots toacubic equation andthree possible combinations ofsolutions. There canbeonerealrootandacomplex conjugate pairofroots; there canbethree realroots, twoofwhich areequal; and therecanbethreerealandunequal roots. These possibilities depend upontherel- ative signs andmagnitudes ofthefourconstants inEqs.(5.61). There isalsothe physical constraint thatthesolution umust satisfy -1<u51.Wewilldraw all ﬁgures asifu>0,which would bethecaseifthetopissupported byahorizontal surface. Recall thatapoint support could allow thesmallest roottobelessthan zero. Forularge, thedominant temi inf(u)is,Bu3. Since /3(cf.Eqs. (5.61)) is always apositive constant f(u)ispositive forlarge positive uandnegative for large negative u.Atpoints u=:l:1,f(u)becomes equa, to—(bIFa)2andis therefore always negative, except fortheunusual casewhere u=:l:lisaroot 24 Chapter 5TheRigid Body Equations ofMotion flu) u=—1 u=+1 F 1» ~24-»-1 “A "3 FIGURE 5.8 Illustrating thelocation oftheturning angles of9inthemotion ofaheavy symmetric topsupported onahorizontal plane. Apoint support could allow oneofthe roots tobenegative. (corresponding toavertical top). Hence, atleast onerootmust lieintheregion u>l.aregion thatdoesnotcorrespond torealangles. Indeed, physical motion ofthetopcanoccur onlywhen uzispositive somewhere intheinterval between u=-1andu=+1,thatis,6between 0and+:rr.Wemustconclude theretore thatforanyactual topf(u)willhavetworoots, u1andM2,between -1and+1 (cf.Fig.5.8),andthatthetopmoves suchthatcos9always remains between these tworoots. Thelocation ofthese roots, andthebehavior ofand forvalues of0 between them, provide much qualitative information about themotion ofthetop. Itiscustomary todepict themotion ofthetopbytracing thecurve ofthein- tersection oftheﬁgure axisonasphere orunitradius about theﬁxedpoint. This curve isknown asthelocus oftheﬁgure axis.Thepolar coordinates ofapointon thelocus areidentical withtheEuler angles 6,¢forthebody system. From the discussion inthepreceding paragraph, wecanseethatthelocus liesbetween the twobounding circles ofcolatitude 6|=anccos u|and62=arccos u2,withd van- ishing atbothcircles. Theshape ofthelocuscurve isinlargemeasure determined bythevalue oftherootoib—au,which wedenote byu’: u’= (5.64)a Suppose, forexample, theinitial conditions aresuch thatu’islarger thanM2. Then, byFlt](557), willalways have thesame sign fortheallowed inclination angles between 01and6;.Hence, thelocus oftheﬁgure axismustbetangent to thebounding circles insuchamanner that isinthesame direction atboth61 and0;,asisshown inFig.5.9(a). Since ¢therefore increases secularly inone direction ortheother, theaxisofthetopmaybesaidtoprecess about thevertical axis. Butitisnottheregular precession encountered inforce-free motion, foras theﬁgure axisgoes around, itnods upanddown between thebounding angles 0| and01-the topnutates during theprecession. Should b/abesuchthatu’liesbetween u;andug,thedirection ofthepreces- sionwillbedifferent atthetwobounding circles, andthelocus oftheﬁgure axis exhibits loops, asshown inFig.5.9(b). Theaverage ofwillnotvanish how- eversothatthere isalways anetprecession inonedirection ortheother. Itcan 5,7 TheHeavy Symmetrical TopwithOnePoint Fixed Z15 "1, ° ‘*1 at1'l!121~.\‘M 01 (3) (b) (C) FIGURE 5.9 Thepossible shapes forthelocus oftheﬁgure axisontheunitsphere. alsohappen thatu’coincides withoneoftheroots off(u).Atthecorresponding bounding circles, both 63and must then vanish, which requires thatthelocus have cusps touching thecircle, asshown inFig.5.9(c). This lastcaseisnotasexceptional asitsounds; itcorresponds infacttothe initial conditions usually stipulated inelementary discussions oftops:Weassume thatinitially thesymmetrical topisspinning about itsﬁgure axis,which isﬁxed insome direction 60.Attimet=0,theﬁgure axisisreleased andtheproblem is todescribe thesubsequent motion. Explicitly, these initial conditions arethatat t=0,9=00and9==O.Thequantity uo=cos90must therefore beoneof therootsoff(u);infact,itcorresponds totheupper circle: M9=U2=u’=E. (5.65) Forproof, notethatwiththese initial conditions E’isequal toMglcos90,and thattheterms inE’derived from thetop’s kinetic energy cannever benegative. Hence. as(9and begin todiffer from their initial zerovalues, energy canbe conserved onlybyadecrease inMglcos6,i.e.,byanincrease in9.Theinitial 90 istherefore thesame as(92,theminimum value 6canhave. When released inthis manner, thetopalways starts tofall,andcontinues tofalluntiltheother bounding angle 61isreached, precessing themeanwhile. Theﬁgure axisthenbegins torise again to02,thecomplete motion being asshown inFig.5.9(c). Some quantitative predictions canbemade about themotion ofthetopun- derthese initial conditions ofvanishing éand(ii,provided thattheinitial kinetic energy ofrotation about thez-axis isassumed large compared tothemaximum change inpotential energy: §13w§>>2Mgl. (5.66) Theeffects ofthegravitational torques, namely, theprecession andaccompanying nutation, willthenbeonlysmall perturbations onthedominant rotation ofthetop about itsﬁgure axis.Inthissituation, wespeak ofthetopasbeing a“fasttop.” Chapter 5TheRigid Body Equations ofMotion Withthisassumption wecanobtain expressions fortheextent ofthenutation, the nutation frequency, andtheaverage frequency ofprecession. Theextent ofthenutation under these given initial conditions isgiven by ui—uo,where uiistheother physical rootoff(u).Theinitial conditions E’=Mglcos60isequivalent totheequality a=,BuQ. With thisrelation, andtheconditions ofEq.(5.65), f(u)canberewritten more simply as fa)=(uo-in[ﬂu-uh—@2010—u)]- om) Theroots of_f(u) other thannoaregiven bytheroots ofthequadratic expression inthebrackets, andthedesired rootu1therefore satisﬁes theequation 2 (1-iii)-%(u(,-U1)=0. (5.62) Denoting U0—ubyxanduo—uibyx1,Eq.(5.68) canberewritten as if+pxl-q=0, (5.69) where a2 _2p=F—2cos09. q=s1n 00. Thecondition fora“fast” top,Eq.(5.66), implies thatpismuch larger thanq. This canbeseen bywriting thcratio a2//3 as a2_(I3) Igwg [3 I12Mgl' Except inthecasethatI3<<11(which would correspond toatopintheunusual shape ofacigar), theratio ismuch greater than unity, andp>8q.Toﬁrstorder inthesmall quantity q/p,theonlyphysically realizable rootofEq.(5.68) isthen QX1=—. P Neglecting 2cos90compared toa3/fl,thisresult canbewritten 5'20 I2M1_ _Jr]=3%‘! = S1112 99. (5./0) (1 I3I3w.4 Thus, theextent ofthenutation, asmeasured byx1=no—M],goes down as 1/w§. Thefaster thetop1Sspun, thelessisthenutation. 5.7 TheHeavy Symmetrtcal TopwithOnePoint Fixed 217 Thefrequency ofnutation likewise caneasily befound forthe“fast” top.Since theamount ofnutation issmall, theterm (1—uz)inEq.(5.67) canbereplaced by itsinitial value, sinz69.Equation (5.67)thenreads, withthehelpofEq.(5.70), fut)=xi=azx(x1 -x). Ifweshifttheorigin ofxtothemidpoint ofitsrange, bychanging variable to >'=x—ﬂ,2 thenthedifferential equation becomes ’) -.2=2 ﬁ_2,,(,y),which ondifferentiation again reduces tothefamiliar equation forsimple har- monic motion ¥=—a2y- Inviewoftheinitial condition x=0att=0,thecomplete solution is x=£21-(1-cosat), (5.71) where x1isgiven by(5.70). Theangular frequency ofnutation oftheﬁgure axis between 60and91istherefore a=13013, (5.72) It which increases thefaster thetopisspun initially. Finally, theangular velocity ofprecession, from(5.57), isgiven by .a(ug —u) ax ¢=.i~..s1n29 s1n26Q or,substituting Eqs.(5.72) and(5.70), d:=%(1 —cosat). (5.73) Therateofprecession istherefore notunifonn butvaries harmonically withtime, withthesame frequency asthenutation. Theaverage precession frequency how- everis =—=———, 5.74¢ 2a [3603 ( ) Lhapter 5TheRigid Body Equations ofMotion which indicates thattherateofprecession decreases astheinitial rotational ve- locity ofthetopisincreased. Wearenowinaposition topresent acomplete picture ofthemotion ofthefast topwhen theﬁgure axisinitially haszerovelocity. Immediately after theﬁgure axisisreleased, theinitial motion ofthetopisalways tofallunder theinﬂuence of gravity. ButasitFalls, theresultant torque around theaxisoffallcauses thetopto pickupaprecession velocity, directly proportional totheextent ofitsfall,which starts theﬁgure axismoving sideways about thevertical. Theinitial fallresults inaperiodic nutation oftheﬁgure axisinaddition totheprecession. Asthetop isspunfaster andfaster, theextent ofthenutation decreases rapidly, although thefrequency ofnutation increases, while atthesame timetheprecession about thevertical becomes slower. Inpractice, forasufficiently fasttopthenutation is damped outbythefriction atthepivot andbecomes unobservable. Thetopthen appears toprecess uniformly about thevertical axis. Because theprecession is regular onlyinappearance, Klein andSommerfeld havedubbed itapseudoregular precession. Inmost oftheelementary discussions ofprecession, thephenomenon ofnutation isneglected Asaconsequence, suchderivations seem toleadtothe paradoxical conclusion thatupon release thetopimmediately begins toprecess unifonnly, amotion thatisnormal totheforces ofgravity thataretheultimate cause oftheprecession. Ourdiscussion ofpseudoregular precession serves to resolve theparadox; theprecession builds upcontinuously from restwithout any inﬁnite accelerations, andtheinitial tendency ofthetopistomove inthedirection oftheforces ofgravity. Itisofinterest todetermine exactly whatinitial conditions willresult inatrue regular precession. insuchacase, theangle 6remains constant atitsinitial value 60,which means that01=02=69.Inother words. f(u)must haveadouble root atug(cf.Fig.5.10), or . dff(u)=u2=0, I-=0; u=ug.ll Theﬁrstofthese conditions, from Eq.(5.62’) withii=0,implies b_ 2 ta—ﬂan= (5.15)1uo f(u) u=-1 u=+1 l i I Ho u_’ FIGURE 5.10 Appearance ofj(u)foraregular precession. 5.7TheHeavy Symmetrical TopwithOnePoint Fixed 219 thesecond corresponds to +9a(l?—W0) (I1—.3110)—= — . 5.762 l—14% uo 1—14% ( ) Substitution ofEq.(5.75) inEq.(5.76) leads, inview ofEq.(5.57) for toa quadratic equation for¢: 2=aqi—<52cos60. (5.76') With thedeﬁnitions ofBanda,Eq.(5.61), thiscanbewritten intwoalternative forms. depending onwhether aisexpressed interms ofL03orthe(constant) 1,0 and¢ Mgl=q5(I3w3 -11¢»cosao), (5.77) OI‘ Mal=<15(I31l/—<11—In¢¢ose@>. (SW) Theinitial conditions fortheproblem oftheheavy toprequire thespeciﬁcation of9,¢,1/r,9,(iv,and,say,either or603atthetimet=0.Because theyarecyclic, theinitial values of¢and1/1arelargely irrelevant, andingeneral wecanchoose anydesired value foreachofthefourothers. Butifinaddition werequire thatthe motion oftheﬁgure axisbeoneofunifomt precession without nutation, thenour choice ofthese fourinitial values isnolonger completely unrestricted. lnstead, theymust satisfy either ofEqs. (5.77). For9=0,wemaystillchoose initial values of9and603,almost arbitrarily, butthevalue ot1sthendeterrmned. The phrase “almost arbitrarily” isusedbecause Eqs.(5.77) arequadratic, andfor to bereal,thediscriminant ofEq.(5.77) mustbepositive: 1§w§>4Mgl11 cos00. (5.18) For99>H/Z(atopmounted soitscenter ofmass 1Sbelow theﬁxed point), then anyvalue of(03canleadtouniform precession. Butfor90<rr/2, m3must be chosen tobeabove aminimum value cog, , 2weon>(03=T3‘/Mglll cos99 (5.79) toachieve thesame situation. Similar conditions canbeobtained fromEq.(5.77’) fortheallowable values oftn.Asaresult ofthequadratic nature ofEq.(5.77), there willingeneral betwosolutions forqi,known asthe“fast” and“slow” pre- cession. Also notethat(5.77) cannever besatisﬁed by =0forﬁnite orm3; toobtain uniform precession, wemust always givethetopashove tostartitonits Chapter 5TheRigid Body Equations ofMotion way.Without thiscorrect initial precessional velocity, wecanobtain atbestonly apseudoregular precession. lfthe precession isslow, sothat cos90maybeneglected compared toa,then anapproximate solution for is ¢zg=yfii (Slow). which agrees withtheaverage rateofpseudoregular precession forafasttop.This result istobeexpected ofcourse; iftherateofprecession isslow, there islittle difference between starting thegytoscope offwith alittle shove orwith noshove atall.Note thatwith thisvalue of theneglect ofqicos 60compared toais equivalent torequiring thatco;bemuch greater thantheminimum allowed value. Forsuchlarge values ofm3,the“fast” precession 1Sobtained when issolarge thatMglissmall compared totheother terms inEq.(5.771: . I30); ¢_I1COS99 (fast). Thefastprecession isindependent ofthegravitational torques andcaninfactbe related totheprecession ofafreebody(seeDerivation 6aintheExercises). Onefurther casedeserves some attention, namely, when u=lcorresponds tooneoftheroots of_f(u).* Suppose, forinstance, atopissetspinning withits ﬁgure axisinitially vertical. Clearly thenb=a,forIlbandIlaaretheconstant components oftheangular momentum about thevertical axisandtheﬁgure axis respectively, andthese axes areinitially coincident. Since theinitial angular ve- locity isonly about theﬁgure axis, theenergy equation (5.59) evaluated attime t=C-states that E’=E—%13w§ =Mgl. Bythedeﬁnitions ofozand[3(Eq.(5.61)|, itfollows thatoz=B. Theenergy equation atanyangle maytherefore bewritten as 1.22=(1-u2);3(1— Ll)-a2(1— “)2 O1’ 112=(1-u)2[,s(1+ u)-:12]. Theformoftheequation indicates thatlt=lisalways adouble root,withthe third rootgiven by 2 u3=aF—l. *Note thatthismustbetreated asaspecial case.smceIIItheprevious discussions factors ofsmz9 were repeatedly divided outoftheexpressions. 5.7 TheHeavy Symmetrical TopwithOnePoint Fixed 221 f(1¢) f(u) = u_>u+1 H-; M1 u+1 u (a)w3>nu’ (b)03<w’ FIGURE 5.11 Plotoff(a)when theﬁgure axisisinitially vertical. lfa2/,6 >2(which corresponds tothecondition fora“fast” top), a3islarger thanlandtheonlypossible motion isforu=1;thetopmerely continues tospin about thevertical. Forthisstateofaffairs, theplotoff(u)appears asshown in Fig.5.11-fa). Ontheother hand, ifa2/,6 <2,thethirdrootu;isthenlessthan 1,f(u)takesontheformshown inFig.5.1l(b),andthetopwillnutate between 6=0and9=93.There isthusacritical angular velocity, co’,above which only vertical motion ispossible, whose value isgiven by a_(Ig)1-1_<*)”_,l\> isT112Mgl— 01’ M11a/2=4%, (5.80)3 which isidentical withEq.(5.79) fortheminimum frequency foruniform preces- sionwith00=O. Inpractice, ifatopisstarted spinning withitsaxisvertical andwithco3greater thanthecritical angular velocity, itwillcontinue tospinquietly forawhile about thevertical (hence thedesignation asa“sleeping” lop). However, friction grad- ually reduces thefrequency ofrotation below thecritical value, andthetopthen begins towobble ineverlarger amounts asitslows down. Theeffects offriction (which ofcourse cannot bedirectly included intheLa- grangian framework) cangiverisetounexpected phenomena inthebehavior of tops.Anotable example isthe“tippie-top,” which consists basically ofsomewhat more thanhalfasphere with4stemadded ontheﬂatsurface. When setrotating withthespherical surface downwards onahardsurface, itproceeds toskidand nutate until iteventually turns upside down. pivoting onthestem, where itthen behaves asanormal “sleeping” top.Thecomplete reversal oftheangular mo- mentum vector istheresult offrictional torque occurring asthetopskids onits spherical surface. Chapter 5TheRigid Body Equations ofMotion Alarge andinﬂuential technology isbased ontheapplications ofrapidly spin- ningrigid bodies, particularly through theuseofwhat arecalled “gyroscopes.” Basically, athree-frame gyroscope isasymmetrical toprotated veryrapidly by external means about theﬁgure axisandmounted ingimbals sothatthemotion of theﬁgure axisisunrestricted about three perpendicular spatial axeswhile thecen- terofgravity remains stationary. Theﬁgure axismaintains thesame direction in space nomatter howthemounting isreoriented, aphenomenon called gyroscopic inertia. Such aninstrument canindicate theroll,pitch, andattitude directions of anairplane ﬂying “blind” byusing thexyzEuler angle convention described in Section 4.4andAppendix A. Ifexternal torques aresuitably exerted onthegyroscope, itwillundergo the precession andnutation motions described earlier fortheheavy top.However, thecondition forthe“fast” topisabundantly satisﬁed, sothattheextent ofthe nutation isalways verysmall, andmoreover isdeliberately damped outbythe method ofmounting. Theonlygyroscopic phenomenon thenobserved ispreces- sion,andthemathematical treatment required todescribe thisprecession canbe greatly simpliﬁed. Wecanseehowtodothisbygeneralization from thecaseof theheavy symmetrical top. IfRistheradius vector along theﬁgure axisfrom theﬁxed point tothecenter ofgravity, thenthegravitational torque exerted onthetopis N=Rx Mg, (5.81) where gisthedownward vector oftheacceleration ofgravity. IfL3isthevec- toralong theﬁgure axis,describing theangular momentum ofrotation about the ﬁgure axis, andwp,known astheprecession vector, isaligned along thevertical withmagnitude equal tothemean precession angular velocity ()5,Eq.(5.74). then thesense andmagnitude ofthe(pseudoregular) precession isgiven by w,,xL3=N. (5.82) Since anytorque about theﬁxedpoint orcenter ofmass canbeputintheform RxF,similar toEq.(5.81), theresulting average precession ratefor:1“fast” top canalways bederived from Eq.(5.82), withthedirection oftheforce Fdeﬁning theprecession axis. Almost allengineering applications ofgyroscopes involve theequilibrium behavior (i.e., neglecting transients) which canbederived from Eq.(5.82). Freefrom anytorques, agyroscope spinaxiswillalways preserve itsoriginal direction relative toaninertial system. Gyros cantherefore beused toindicate ormaintain speciﬁc directions, e.g.,provide stabilized platforms. Asindicated by Eq.(5.82), through theprecession phenomena theycansense andmeasure angular rotation rates andapplied torques. Note from Eq.(5.82) thattheprecession rate isproportional tothetorque. whereas inanonspinning body itistheangular acceleration thatisgiven bythetorque. Once thetorque isremoved, anonspirming 5.8I5.8 Precession oftheEquinoxes andofSatellite Orbits 223 body willcontinue tomove; under similar conditions agyrosimply continues spinning without precessing. Thegyrocompass involves more complicated considerations because herewe aredealing withthebehavior ofagyroscope ﬁxedinanoninertial system, while Earth rotates underneath it.Inagyrocompass, anadditional precession isauto- matically applied byanextemal torque ataratejustenough tobalance Earth’s rotation rate.Once setinthedirection ofEarths rotation, i.e.,thenorth direction, thegyrocompass thenpreserves thisdirection, atleastinslowly moving vehicles. What hasbeenpresented hereisadmittedly anoversimpliﬁed, highly compressed viewofthefascinating technological usesoftastspinning bodies. Tocontinue further inthisdirection would regrettably leadustoofaraﬁeld. There arehowever twoexamples ofprecession phenomena innature forwhich asomewhat fuller discussion would bevaluable, bothforthegreat interest inthe phenomena themselves andasexamples ofthetechniques derived inthischapter. Theﬁrstconcerns thetypes ofprecession thatarisefromthetorques induced by Earth’s eqnatoiial “bulge,” andthesecond istheprecession ofmoving charges in amagnetic ﬁeld.Thenexttwosections areconcerned withtheseexamples. PRECESSION OFTHE EQUINOXES AND OFSATELLITE ORBITS Ithasbeenmentioned previously thatEarth isatopwhose ﬁgure axisisprecess- ingabout thenormal totheecliptic, theplane ofEarth’s orbit, amotion known astronomically astheprecession oftheequinoxes. Were Earth completely spher- ical,noneoftheother members ofthesolarsystem could exeitagravitational torque onit.But,ashasbeenpointed out,Earth deviates slightly fromasphere, being closely approximated byanoblate spheruid ofrevolution. Itis_|ustthenet torque ontheresultant equatorial “bulge” arising from gravitational attraction, chieﬂy oftheSunandMoon, thatsetsEarth’s axisprecessing inspace. Tocalculate therateofthisprecession, aslight excursion intopotential theory isneeded toﬁndthemutual gravitational potential ofamasspoint (representing thesunorthemoon) andanonspherical distribution ofmatter. Wewillﬁndthe properties oftheinertia tensor asobtained above very useful inthederivation of thisPotential. Consider adistribution ofmass points forming onebody, andasingle mass point, massM,representing theother (cf.Fig.5.12). Ifr,isthedistance between theithpointinthedistribution andthemasspointM,thenthemutual gravitational potential between thetwobodies is* v=-GM“ =- GMm' . (5.s3ir, 2I, rl+()—27‘COSIII, *Itmaybeworth areminder thatsummation isimplied overrepeated subscripts~i|._‘l_ Chapter 5TheRigid Body Equations ofMotion O O Ogbm .OO". O .0.. r M I‘. .0 FIGURE 5.12 Geometry involved ingravitational potential between amextended body andamass point. Inthislastexpression theterminology ofFig.5.121Sused: rfistheradius vector totheithparticle fromaparticular point, which willlaterbetaken tobethecenter ofmassoftheﬁrstbooy. risthecorresponding radius vector tothemass point M,and11/,istheangle between thetwovectors. Itiswellknown thatasimple expansion intemis ofLegendre polynomials canbegiven forEq.(5.83); infact, thereciprocal ofthesquare rootinEq.(5.83) isknown asthegenerating function forLegendre polynomials, sothat §/'\-.“ ;,/=GMv=-— Z -P,(cos111,), (5.254)T "=0 I’ rovidin r,thedistance fromtheoriintoM,ismuch reater thananr’.We P 8 8 8 Y, shallmake useofonlytheﬁrstthreeLegendre polynomials that,forreference. are Poo)=1,P1(x)=x. P2(x)=%(3x2-1). (sss) Foracontinuous spherical body, withonly aradial variation ofdensity, all terms except theﬁrstinEq.(5.84) caneasily beshown tovanish. Thus, thenth terminside thesummation, forabody withspherical symmetry andmass density p(r’), canbewritten ,rn. dV'p(r') P,,(cos11/). Using spherical polar coordinates, withthepolar axisalong r,thisbecomes 2 7.1P1+1 fr’ dr'p(r') /1 r1(cosi/r)P,,(cos ilr). From theorthonormal properties ofP,,withrespect toP0,theintegral overcos1,11 vanishes except forn=0,which proves thestatement. Ifthebodydeviates onlyslightly fromspherical symmetry, asisthecasewith Earth, wewould expect theterms inEq.(5.84) beyond n=0todecrease rapidly 5.8 Precession oftheEquinoxes andofSatellite Orbits 225 withincreasing n.Itwilltherefore besufﬁcient toretain onlytheﬁrstnonvanish- ingcorrection terminEq.(5.48)tothepotential forasphere. Now, thechoice of thecenter ofmass asorigin causes then=1termtovanish identically, sinceit canbewritten GM , GM ,—7m,rl COS]!/‘l ='-71‘-m,|',, which iszero, bydeﬁnition ofthecenter ofmass. Thenextterm, forn=2,can bewritten GM 2-é7m|T: —3C052 ll/i). Simple tensor manipulation gives thecomplete second-order approximation tothe nonspherical potential as GM GMv=-__’”- +_3(31, -Tn),r Zr where misthemassoftheﬁrstbody(Earth), I,isthemoment ofinertia about the direction ofr,andIisthemoment ofinertia tensor intheprincipal axissystem. From thediagonal representation oftheinertia tensor intheprincipal axissystem, itstraceisjustthesun1oftheprincipal moments ofinertia, sothatVcanbe written as v=-£191 +gut, -(1,+I2+13)]. (5.86)r 2r Equation (5.86) issometimes known asMacCullagh’s formula. Sofar,noas- sumption ofrotational syrnmetry hasbeen made. Letusnowtaketheaxisof symmetry tobealong thethirdprincipal axis,sothatI1=I2.Ifoz,ﬂ,yarethe direction cosines ofrrelative totheprincipal axes,thenthemoment ofinertia I, canbeexpressed as 1,=Ito’+51)+av’=I1+<13-my’. <5-81> Withthisformfor1,,thepotential, Eq.(5.86), becomes GM GM I—I v=--—’"+-‘T3-,;—‘)<$#y2 -1>,7' 4!‘ 01' GM GM I—I v=-T"’+-—(r5‘3—‘-)-P2<y>- <5-88> Thegeneral fonnofEq.(5.88) could havebeenforetold fromthestart,forthe potential fromamassdistribution obeys Poisson’s equation. Thesolution appro- priate tothesymmetry ofthebody, asiswellknown. isanexpansion ofterms Chapter 5TheRigid Body Equations ofMotion ofthefonn P,,(y)/r"+1, ofwhich Eq.(5.88) shows theﬁrsttwononvanishing terms. However, thisapproach doesnotgivethecoefﬁcients oftheterms any more simply thanthederivation employed here. Itshould alsoberemarked that theexpansion ofVisthegravitational analog ofthemultipole expansion of,say, theelectrostatic potential ofanarbitrary charged body. Then=1tennis absent herebecause there isonlyonesignofgraxitational “charge” andthere canbeno gravitational dipole moment. Further, theinertia tensor isdeﬁned analogously to thequadrupole moment tensor. Therefore, themechanical effects weareseek- ingcanbesaidtoarisefromthegravitational quadrupole moment oftheoblate Eaith.* Oftheterms inEq.(5.88) forthepotential, theonlyonethatdepends onthe orientation ofthebody, andthuscould giverisetotorques, is V2=G”%_"lP2o>- (5.89) Fortheexample ofEarth’s precession, itshould beremembered thatyisthedi- rection cosine between theﬁgure axisofEarth andtheradius vector fromEarth’s center totheSunorMoon. Asthesebodies goaround theirapparent orbits, ywill change. Therelation ofytothemorecustomary astronomical angles canbeseen fromFig.5.13where theorbitoftheSunorMoon istaken asbeing inthexy plane, andtheﬁgure axisofthebody inthexzplane Theangle 6between the ﬁgure axisandthezdirection istheobliquity oftheﬁgure axis.Thedotproduct ofaunitvector along theﬁgure axiswiththeradius vector tothecelestial body involves onlytheproducts oftheirx-components, sothat y=sin6cos17. Hence, V;canbewritten GM(I —I) _V2= (351I126COS27] —1). z 6 cos-17 Y 17 x FIGURE 5.13 Figure axisofEarth relative toorbitofmasspoint. *Note thatsofarnothing intheargument restricts thepotential ofEq(5.88) tortgrd bodies. The constraint ofrigidity enters on_ywhen werequire fromhereonthattheprincipal axesbeﬁxed inthe body andtheassociated moments ofinertia beconstant intime. 5.8 Precession oftheEquinoxes andofSatellite Orbits 227 Asweshallsee,theorbital motion isveryrapid compared totheprecessional motion, andforthepurpose ofobtaining themean precession rate,itwillbead- equate toaverage V;overacomplete orbital period ofthecelestial body consid- ered. Since theapparent orbits oftheSunandMoon ltave loweccentrlcities, rcan beassumed constant andtheonlyvariation isincos17.Theaverage ofcosz17over acomplete period is%,andtheaveraged potential isthen _ GMI—I 3. GMI—I l3V2= (5s1n26—l)= (5—icos29), or,ﬁnally, v2=- P2(cos9). (5.90)r Thetorque derived fromEq.(5.90) isperpendicular toboththeﬁgure axisand thenormal totheorbit(which plays thesameroleasthevertical axisfortheheavy top). Hence, theprecession isabout thedirection oftheorbit normal vector. The magnitude oftheprecession ratecanbeobtained fromEq.(5.82), butbecause the potential differs informfromthatfortheheavy top,itmaybemore satisfying to obtain amore formal derivation. Foranysymmetric body inwhich thepotential isafunction ofcos9only, theLagrangian canbewritten, following Eq.(5.52), as 1.=%(e'2+<;152sin20) +1230i+¢E¢Qse)2 -V(cos9). (5.91) Ifwearetoassume onlyuniform precession andarenotconcemed about the necessary initial conditions, wecansimply take6and9tobezerointheequations ofmotion. TheLagrange equation corresponding to9isthen dL -_ ._.. 8V_—=I1¢2s1n9cos9 —I3¢s1n9(1,'/ +¢cos6) ———=0d6 36 or 1 . . dV I3co3¢-1,¢2cost)=56559-), (5.92) which istheanalog ofEq.(5.76') foramore general potential. Forslow pre- cession, which means basically that¢<<03,the(62terms inEq.(5.92) canbe neglected, andtherateofuniform precession isgiven by - 1 8V=-——?-. 5.3¢ 13053 8(cos6) (9) From Eq.(5.51’) weseethatfortheheavy topEq.(5.93) agrees withtheaverage result ofEq.(5.74).Withthepotential ofEq.(5.90), theprecession rateis Chapter 5TheRigid Body Equations ofMotion . 3GM I3—I1 ¢=— -TCOS9. Forthecaseoftheprecession duetotheSun,thisformula canbeputina simpler form, bytaking rasthesemimajor axisofEarth’s orbitandusing Kepler’s law,Eq.(3.71), intheform 2_21;2_GML00 — T — Theprecession rate,relative totheorbital angular velocity, tog,isthen 3=-5951 cos0. (5.95)wt) Zws I3 With thevalue of(I3—I1)/13 asgiven inSection 5.6,and9=23°27’, Eq.(5.95) saysthatthesolar-induced precession would besuchastocause acomplete tota- tionoftheﬁgure axisabout thenormal totheecliptic (plane ofEarth’s orbit) in about 81,000 years. TheMoon isfarlessmassive thantheSun,butitisalsomuch closer; thenetre- sultisthatthelunar-induced precession rateisovertwice thatcaused bytheSun. Since thelunar orbit isclose totheecliptic andhasthesame sense astheapparent solarorbit, thetwoprecessions nearly addtogether arittnnetically, andthecom- bined lunisolar precession rateis50.25”/year, oronecomplete rotation inabout 26,000 years. Note thatthisrateofprecession issoslow thattheapproximation ofneglecting compared to<03isabundantly satisﬁed. Because theSun.Moon, andEarth areinconstant relative motion, andtheMoon’s orbitisinclined about 5°tutheecliptic, theprecession exhibits irregularities designated asastronomical nutation. Theextent ofthese periodic irregularities isnotla.rge—about 9”ofarc in9andabout 18"in¢.Even so,theyarefarlarger thanthetruenutation that,as Klein andSommerfeld haveshown, ismanifested bytheChandler wobble whose amplitude isnever more thanafewtenths ofanarcsecond. Onefurther application canbemade ofthepotential, Eq.(5.88), andassoci- ateduniform precession rate,Eq.(5.93). Ithasbeenstressed thatthepotential represents amutual gravitational interaction; ifitresults intorques acting onthe spinning Earth, italsogives riseto(noncentral) forces acting onthemass point M. Theeffect ofthese small forces appears asaprecession oftheplane oftheorbit ofthemass point, relative toaninertial frame. Itispossible toobtain anapprox- itnate formula forthisprecession byanargument again based onthebehavior of spinning ngid bodies. Since theprecession rates aresmall compared totheorbital angular velocity. wecartagain average overtheorbit. Theaveraging ineﬁect replaces thepar- ticlebyarigidringofmass Mwiththesame radius asthe(assumed circular) orbit, spinning about theﬁgure axisoftheringwiththeorbital frequency. Equa- tion(5.90) gives thepotential ﬁeldinwhich thisringislocated, with6theangle 5.8 Precession oftheEquinoxes andofSatellite Orbits 229 between theﬁgure axesoftheringandEarth. Theaverage precession rateisstill given byEq.(5.93), butnowI3and(03refer tothespinning ringandnotEarth. Itwould therefore bebetter torewrite Eq.(5.93) forthisapplication as . 1' 3V , ¢- <5-93) andEq.(5.94) appears as ._ 133G(13 —I1) I Equat1on (5.94’) could beused, forexample, toﬁndtheprecession oftheorbit of theMoon duetoEarth’s oblateness. Amore current application would betothe precession ofnearly circular orbits ofartiﬁcial satellites revolving about Earth. Thefraction ofacomplete pI'CCCSS10l1 rotation inoneperiod ofthesatellite is ¢3r_ 1Z3G(I3—I|)ETCOS6. Anapplication ofKepler’s law,thistimefortheperiod ofthesatellite. reduces thisresult to ¢ 31Ig=—E%cos6, (5.96) where misEarth’s mass. IfEarth wereauniform sphere, thentheprincipal mo- ments ofinertia would be [3~I1=%mR2, withREarth’s radius. Because thecoreismuch moredense thantheouter layers, themoment ofinertia issmaller, suchthatinfact* I3=O.33lmRZ ~§mR2. Theapproximate precession isthusgiven by at 113-11 R2—=——— — . 5. 2” 2I3 (r) cost) (97) Fora“close” satellite where risveryclosetoR,andtheinclination ofthesatellite orbittotheequator is,say,30°,Eq.(5.97) saysthattheplane oftheorbitprecesses completely around 221inabout 700orbits ofthesatellite. Since theperiod ofa close satellite isabout 1%hours, complete rotation oftheorbital plane occurs inalittleoversixweeks time. Clearly theeffect isquite signiﬁcant. Weshall rederive theprecession ofthesatellite orbitlateron,when wediscuss theSL1lJ_]6C[ ofperturbation theory (cf.Section 12.3). *The bestvalues ofI;arenowobtained from observation ofjustsucheffects onsatellite O1‘bllS. 5.9IChapter 5TheRigid Body Equations ofMotion PRECESSION OFSYSTEMS OFCHARGES INAMAGNETIC FIELD Themotion ofsystems ofcharged particles inmagnetic ﬁelds does notnormally involve rigid body motion. Inanumber ofparticular instances, themotion ishow- evermostelegantly discussed using thetechniques developed hereforrigidbody motion. Forthisreason. andbecause oftheir importance inatomic andnuclear physics, afewexamples willbegiven here. Themagnetic moment ofasystem ofmoving charges (relative toaparticular origin) isdeﬁned as IM=5q,(r, xv,)—>%fdVpe(r)(r xv). (5.98) Here theﬁrstexpression isasumoverdiscrete particles withcharge qr:while the second isthecorresponding generalization toacontinuous distribution ofcharge density pg(r).Theangular momentum ofthesystem under corresponding con- ventions is L=m,(r, xv,)—>fdVp,,,(r)(r xv). Boththemagnetic moment andtheangular momentum haveasimilar form. Weshallrestrict thediscussion tosituations inwhich Misdirectly proportional toL: M=yL, (5.99) most naturally byhaving auniform q/mratio forallparticles oratallpoints in thecontinuous system. Insuch cases, thegyromagnetic ratio yisgiven by y=Hi, (5.100)Am but,withaneyetomodels ofparticle andatomic spin,ywilloften beleftunspec- iﬁed. Theforces andtorques onamagnetic dipole maybeconsidered asderived fromapotential V=—(M-B). (5.101) Itisimplied along withEq.(5.101) thatthemagnetic ﬁeldissubstantially constant overthesystem. hideed, thepicture applies besttoapointlike magnetic moment whose magnitude isnotaffected bythemotion itundergoes—a picture appropriate topermanent magnets orsystems onanatomic orsmall scale. With uniform B,thepotential depends onlyontheorientation ofMrelative toB;no forces areexerted onthemagnetic moment, butthere isatorque N=MxB. (5.102) 5.9 Precession ofSystems ofCharges inaMagnetic Field 231 (Compare withEq.(5.81).) Thetimerateofchange ofthetotalangular momen- tumisequal tothistorque, sothatinview ofEq.(5.99) wecanwrite dLE=1/LxB. (5.103) Butthisisexactly theequation ofmotion foravector ofconstant magnitude rotating inspace about thedirection ofBwith anangular velocity to=-1/B. Theeffect ofauniform magnetic ﬁeldonapermanent magnetic dipole istocause theangular momentum vector (andthemagnetic moment) toprecess uniformly. Fortheclassical gyromagnetic ratio, Eq.(5.100), theprecession angular veloc- ityis (.0; Z _"' known astheLarmor frequency. Forelectrons qisnegative, andtheLamior pre- cession iscounterclockwise around thedirection ofB. Asasecond example, consider acollection ofmoving charged particles, with- outrestrictions onthenature oftheirmotion, butassumed toallhavethesame q/mratio, andtobeinaregion ofuniform constant magnetic ﬁeld. Itwillalso beassumed thatanyinteraction potential between particles depends onlyonthe scalar distance between theparticles. TheLagrangian forthesystem canbewrit- ten(cf.F.q.(1.63)) IL=—m,vl2+im,v, -A,-(r,)+v(|i-,-1-,|), (5.105)2 m where theconstant magnetic ﬁeldBisgenerated byavector potential A: A=tnXr. (5.106) Interms ofB,theLagrangian hastheform (permuting dotandcross products) lL=—rn,v,2+2-(r, xm,-v,)+V(|r, —rJ-I). (5.107)2 2m Theinteraction term withthemagnetic ﬁeld canbevariously written (cf. Eqs.(5.101) and(5104)) B-L5'-27;=M-B=-00,.(r,Xm,v,). (5.10s) Suppose nowweexpress theLagrangian interms ofcoordinates relative to “primed” axeshaving acommon origin withtheoriginal set,butrotating uni- formly about thedirection ofBwithangular velocity col.Distance vectors from theorigin areunchanged asofcourse arescalar distances suchasIr,—rJI.How- ever, velocities relative tothenewaxesdiffer from theoriginal velocities bythe relation v;=vf+w1xr,. Chapter 5TheRigid Body Equations ofMotion Thetwoterms intheLagrangian affected bythetransfonnation are '* 2mv" mv’ m f=42> +miv1-(wiX ri)+7'(wi Xrt)-(wiXrt), -0);-r,-xm,v, =-001-(r, xm,v;) -co;-(r,xm,(c0; xr,-)). Bypermuting dotandcross product, wecanseethatthetenns linear inanand vjarejustequal andopposite andtherefore cancel intheLagrangian. Asimilar permutation intheterms quadratic incolshowthattheyareofthesame formand arerelated tothemoment ofinertia ofthesystem about theaxisdeﬁned bymi(cf Section 5.3).Thequadratic termintheLagrangian caninfactbewritten as l 1-5"ii(m, Xr,)-((1)1xr,)=-500;-I-ts,=-Eimf, (5.109) where I;denotes themoment ofinertia about theaxisof(01.I11terms ofcoordi- nates intherotating system, theLagrangian thus hasthesimple form L=§m,v;2 +V(]r,-r,-|)-%11w,2. (5.110) fromwhich alllinear tenns inthemagnetic ﬁeldhavedisappeared. Wecangetanideaoftherelative magnitude ofthequadratic termbycon- sidering asituation inwhich themotion ofthesystem consists ofarotation with some frequency cu,e.g.,anelectron revolving around theatomic nucleus. Then for systems nottoofarfromspherical symmetry, thekinetic energy isapproximately -é-Iwz (without subscripts onthemoment ofinertia) andthelinear terminw;is ontheorder ofco;~LwIcolco. Hence, thequadratic terminEq.(5.110) ison theorder of(co;/w)2 compared tothekinetic energy, andontheorder of(cu;/cu) relative tothelinear term. Inmost systems ontheatomic orsmaller scale, thenatural frequencies are much larger thantheLarmor frequency. Compare, forexample, thefrequency of aspectral line(which isadifference ofnatural frequencies) tothefrequency shift inthesimple Zeeman effect, whichisproportional totheLarmor frequency. Thus, forsuchsystems themotion intherotating system isthesameasinthelaboratory system when there isnomagnetic ﬁeld What wehave isLarmor’s theorem, which states thattoﬁrstorder inB,theeffect ofaconstant magnetic ﬁeldonaclassical system istosuperimpose onitsnormal motion aunifomi precession withangular frequency (oi. DERIVATIONS 1.IfR,isanantisymmetric matrix associated withthecoordinates oftheithmass point ofasystem, withelements R,,,,, =e,,,,,;xi show thatthematrix oftheinertia tensor canbewritten as |=—m,(R,)2. Derivations 233 2.Show directly byvector manipulation thatthedeﬁnition ofthemoment ofinertia as 3. 4. 5. 6.I=m;(r,xn)-(r, xn) “cduces toEq(S18) Prove thatforageneral rigid body motion about aﬁxed point, thetimevariation of "hekinetic energy Tisgiven by (IT —= -N. at"’ Derive Euler’s equations ofmotion, Eq(5.39'), from theLagrange equation ofmo- tion,intheform ofEq.(1.53), forthegeneralized coordinate 1/r. Equation (5.38) holds forthemotions ofsystems thatarenotrigid, relative toachosen rotating setofcoordinates. Forgeneral nonrigid motion, iftherotating axesarechosen tocoincide withthe(instantaneous) principal axesofthecontinuous system, show that Eqs.(5.39) aretobereplaced by d(1w) dl _—-(2% +e,JkcuJw/¢Ik —w,T' =N,, z=1,2.3, where 1,=/dVp(r)€,jkx,-v;'c withp(r)themassdensity atpoint r,andv’thevelocity ofthesystem point atr relative totherotating axes. These equations aresometimes known astheLiouville equations andhave applications fordiscussing almost-rigid motion, such asthatof Earth including theatmosphere andoceans. (a)Show thattheangular momentum ofthetorque-free symmetrical toprotates in thebody coordinates about thesymmetry axiswithanangular frequency S2.Show alsothatthesyrmnetry axisrotates inspace about theﬁxed direction oftheangular momentum withtheangular frequency - I1;60'; ¢ ,I1c0s9 where ¢istheEuler angle ofthelineofnodes withrespect totheangular mo- mentum asthespace zaxis. [b)Using theresults ofExercise 15,Chapter 4,show thattorotates inspace about theangular momentum withthesame frequency ¢,butthattheangle 6’between anandLisgiven by sin9'= sin9”, where 9”istheinclination oftotothesymmetry axis. Using thedatagiven in Section 5.6,show therefore thatEarth’s rotation axisandtheaxisofangular mo- mentum are1lC\6I‘ more thanI.5cmapart onEarth’s surface. Chapter 5TheRigid Body Equations ofMotion (c)Show from parts (a)and(b)thatthemotion oftheforce-free symmetrical top canbedescribed interms oftherotation ofacone ﬁxed inthebody whose axis isthesymmetry axis, rolling onaﬁxed cone inspace whose axisisalong the angular momentum. Theangular velocity vector isalong thelineofcontact ofthe twocones. Show thatthesame description follows immediately from thePoinsot construction intemis oftheinertia ellipsoid. Forthegeneral asymmetrical ngidbody, verify analytically thestability theorem shown geometrically above onp.204byexaiinning thesolution ofEuler’s equations forsmall deviations from rotation about each oftheprincipal axes. Thedirection of toisassumed todiffer soslightly from aprincipal axisthatthecomponent oftoalong theaxiscanbel2ll(C‘l asconstant, while theproduct ofcomponents perpendicular to theaxiscanbeneglected. Discuss theboundedness oftheresultant motion foreachof thethree principal axes. When therigid body isnotsymmetrical, ananalytic solution toEuler’s equation for thetorque-free motion cannot begiven interms ofelementary functions. Show, how- ever. thattheconservation ofenergy andangular momentum canbeused toobtain expressions forthebody components ofnointemis ofelliptic integrals. Apply Euler's equauons totheproblem oftheheavy symmetrical top.expressing 0), interms oftheEuler angles. Show thatthetwointegrals ofmotion, Eqs. (5.53) and i5.54), canbeobtained directly from Euler’s equations inthisform. Obtain from Eulcr’s equations ofmotion thecondition (5.77) fortheuniform preces- sionofasymmetrical topinagravitational ﬁeld,byimposing therequirement thatthe motion beauniform: precession without nutation Show thatthemagnitude oftheangular momentum foraheavy symmetrical topcan beexpressed asafunction of6andtheconstants ofthemotion only. Prove thatasa result theangular momentum vector precesses uniformly onlywhen there isuniform precession ofthesymmetry axis (a)Consider apnmed setofaxescoincident inorigin withaninertial setofaxes butrotating withrespect totheinertial frame withﬁxed angular velocity mo.Ifa system ofmass points issubject toforces derived from aconservative potential Vdepending onlyonthedistance totheorigin, show thattheLagrangian forthe system interms ofcoordinates relative totheprimed setcanbeWritten as L=T'+w0-L'+%mQ-I’-w0—V, where primes indicate thequantities evaluated relative totheprimed setofaxes. What isthephysical signiﬁcance ofeachofthetwoadditional temis? (b)suppose thatmgisinthexéxéplane, andthatasymmetric topisconstrained to move with itsﬁgure axisinthexéxl plane. sothatonly twoEuler angles are needed todescribe itsorientation. I.fthebody ismounted sothatthecenter of mass isﬁxed attheorigin andV=0,show thattheﬁgure axisofthebody oscillates about thexgaxisaccording totheplane-pendulum equation ofmotion andﬁndthefrequency ofsmall oscillations. Thisillustrates theprinciple ofthe gyrocompass. Exercises 235 EXERCISES 131 14. 15 16. 17 18. 19.Twothinrodseachofmass mandlength Iareconnected toanideal (nofriction) hinge andahorizontal thread. Thesystem restsonasmooth surface asshown intheﬁgure. Attime t-O,thethread iscut.Neglecting thernnss ofthehinge andthethread, and considering onlymotion inthexyplane (a)Find thespeed atwhich thehinge hitstheﬂoor. [b)Findthetimeittakes forthehinge tohitthefloor. Y thread 30° 30°:- X What istheheight-to-diameter ratio ofatight cylinder suc:ithattheinertia ellipsoid atthecenter ofthecylinder isasphere? l"lll(ltheprincipal moments ofinertia about thecenter ofmass ofaﬂatrigid body in theshape ofa45°righttriangle withuniform mass density. What aretheprincipal axes" Three equal mass points arelocated at(a.0,0),(O.a,Za),(0,2a,a).Findtheprinci- palmoments ofinertia about theorigin andasetofpnncipal axes. Auniform right circular cone ofheight h,half-angle oz,anddensity prollsonits sidewithout slipping onaunifomi horizontal plane insuchamanner thatitreturns toitsoriginal position inatime 1:.Find expressions forthekinetic energy andthe components oftheangular momentum ofthecone. (a)Abarofnegligible weight andlength lhasequal mass points matthetwoends. Thebarismade torotate uniformly about anaxispassing through thecenter ofthebarandmaking anangle 6withthebar.From Euler’s equations ﬁndthe components along theprincipal axesofthebarofthetorque driving thebar. (b)From thefundamental torque equation (l26)ﬁndthecomponents ofthetorque along axesﬁxed inspace. Show thatthese components areconsistent withthose found inpart(st). Auniform barofmass Mandlength 2lissuspended from oneendbyaspring of force constant k.Thebarcanswing freely onlyinonevertical plane, andthespring is constrained tomove onlyinthevertical direction. Setuptheequations ofmotion in theLagrangian formulation. Chapter 5TheRigidBodyEquations ofMotion *1-suspension point attachment \h_ point Aplane pendulum consists ofauniform rodoflength landnegligible thickness with mass m,suspended inavertical plane byoneend.Attheother endauniform diskof radius aandmass M'sattached soitcanrotate freely initsownplane, which isthe vertical plane. Setuptheequations ofmotion intheLagrangian formulation. Acompound pendulum consists ofangidbodyintheshape ofalamina suspended inthevertical plane atapoint other than thecenter ofgravity. Compute theperiod forsmall oscillations intemis oftheradius ofgyration about thecenter ofgravity andtheseparation ofthepoint ofsuspension from thecenter ofgravity. Show thatif thependulum hasthesame period fortwopoints ofsuspension atunequal distances from thecenter ofgravity, thenthesumofthese distances isequal tothelength ofthe equivalent simple pendulum. Auniform rodslides withitsendsinside asmooth vertical circle Iftherodsubtends anangle of120° atthecenter ofthecircle. show thattheequivalent simple pendulum hasalength equal totheradius ofthecircle. Anautomobile isstarted from restwith oneofitsdoors initially atright angles. If thehinges ofthedoor aretoward thefront ofthecar,thedoor willslam shutasthe automobile picks upspeed. Obtain aformula forthetimeneeded forthedoortoclose iftheacceleration fisconstant, theradius ofgyration ofthedoor about theaxisof rotation isrg,andthecenter ofmass isatadistance afrom thehinges. Show that iffis0.3m/s2 andthedoor isauniform rectangle 1.2mwide, thetime willbe approximately 3.04s. Awheel rollsdown aﬁatinclined surface thatmakes anangle ozwiththehorizontal. Thewheel isconstrained sothatitsplane isalways perpendicular totheinclined plane, butitmayrotate about theaxisnormal tothesurface. Obtain thesolution for thetwo-dimensional motion ofthewheel, Jblng Lagrange‘s equations andthemethod ofundetermined multipliers. (a)Express interms ofEuler’s angles theconstraint conditions forauniform sphere rolling without slipping onaﬂathorizontal surface. Show thattheyarenonholo- nomic. (b)SetuptheLagrangian equations forthisproblem bythemethod ofLagrange multipliers. Show thattheU'ﬂI1SlﬂlIOI'l'¢l androtational parts ofthekinetic energy areseparately conserved. Arethere anyother constants ofmotion? Fortheaxially symmetnc body precessing uniformly intheabsence oftorques. find analytical solutions fortheEuler angles asafunction oftime. Exercises 237 9 InSection 5.6,theprecession ofEarth s:ixisofrotation about thepolewascalculated onthebasis thatthere were notorques acting onEarth. Section 5.8,ontheother hand, showed thatEarth isundergoing aforced precession duetothetorques oftheSun andMoon. Actually bothresults arevalid: Themotion oftheaxisofrotation about thesyrmnetry axisappears asthenutation oftheEarth inthecourse ofitstorced precession. Toprove thisstatement, calculate 0and asafunction oftime fora heavy symmetrical topthatisgiven aninitial velocity Q50,which islarge compared withthenetprecession velocity /3/2a, butwhich issmall compared withm3.Under these conditions, thebounding circles fortheﬁgure axisstilllieclose together, butthe orbit oftheﬁgure axisappears asinFig.5.9(b), thatis,shows large loops thatmove only slowly around thevertical Show forthiscase that(571)remains valid butnow 2. xi=((1% —$0) Sinz90. From these values or6andqi,obtain co;and(1)2,andshow thatfor'8/2a small com- pared with¢(),thevector cuprecesscs around theﬁgure axiswithanangular velocity 9:10”Ii inagreement withEq.(5.49). Verify fromthenumbers given inSection 5.6that corresponds toaperiod ofabout 1600years, sothat($0iscertainly small compared withthedaily rotation andissufficiently large compared withﬁ/2a, which corre- sponds totheprecession period of26,000 years. Suppose thatinasymmetrical topeachelement ofmasshasaproportionate charge associated withit,sothatthee/mratioisconstant—the so-called charged symmetric top.Ifsuch abody rotates inauniform magnetic ﬁeldtheLagrangian, from (5.108), is L=T—(ii);-L. Show thatTisaconstant (which isamanifestation oftheproperty oftheLorentz force thatamagnetic ﬁeld does nowork onamoving charge) andﬁndtheother constants ofmotion Under theassumption thatco;ismuch smaller thantheinitial rotational velocity about theiigure axis, obtain expressions forthefrequencies and amplitudes ofnutation andprecession. From where dothekinetic energies ofnutation andprecession come? Ahomogeneous cubeofsides Iisinitially atrestinunstable equilibrium withoneedge mcontact withahorizontal plane. Thecubeisgiven asmall angular displacement and allowed tofall.What istheangular velocity ofthecube when onefacecontacts the plane if: (a)theedge uicontact withtheplane cannot slide? (b)theplane isfnctionless sotheedge canslide‘? Adoor isconstructed ofathinhomogeneous material. Ithasaheight of2manda width of0.9m.Ifthedoorisopened by90°andreleased from rest,itisobserved that thedoorcloses itself in3s.Assuming thatthehinges arefrictionless, what angle do these hinges make withthevertical? CHAPTER 6.1I 233Oscillations Aclass ofmechanical motions thatcanbest bctrcatcd intheLagrangian for- mulation isthatoftheoscillations ofasystem about positions ofequilibrium. Thetheory ofsmall oscillations ﬁndswidespread physical applications inacous- tics,molecular spectra, vibrations ofmechanisms, andcoupled electrical cir- cuits. Ifthedeviations ofthesystem from stable equilibrium conditions are small enough, themotion cangenerally bedescribed asthatofasystem of coupled linear harmonic oscillators. Itwillbeassumed thereader isfamiliar with theproperties ofasimple harmonic oscillator ofonedegree offreedom, bothin freeandforced oscillation, withandwithout damping. Heretheemphasis willbe onmethods appropriate todiscrete systems withmore thanonedegree offree- dom. Aswillbeseen, themathematical techniques required turnouttobevery similar tothose employed instudying rigidbodymotion, although themechanical systems considered need notinvolve rigid bodies atall.Analogous treatments of oscillations about stable motions canalsobedeveloped, butthese aremost easily doneinthe1-lamiltonian formulation presented inChapter 8. FORMULATION OFTHE PROBLEM Weconsider conservative systems inwhich thepotential energy isafunction of position only. Itwillbeassumed thatthetransformation equations deﬁning the generalized coordinates ofthesystem, q1,..,q,,,donotinvolve thetimeexplic- itly.Thus, time-dependent constraints aretobeexcluded. Thesystem issaidtobe inequilibrium when thegeneralized forces acting onthesystem vanish: Qt=" =0- 51-0 Thepotential energy therefore hasanextremum attheequilibrium conﬁguration ofthesystem, qol,(102, ..,qQ,,.Iftheconﬁguration isinitially attheequilib- rium position, with zero initial velocities q",,,then thesystem will continue in equilibrium indeﬁnitely. Examples oftheequilibrium ofmechanical systems are legion—a pendulum atrest,asuspension galvanometer atitszeroposition, anegg standing onend. Anequilibrium position isclassiﬁed asstable ifasmall disturbance ofthe system fromequilibrium results onlyinsmall bounded motion about therestpo- 6.1 Formulation oftheProblem 239 sition. Theequilibrium isunstable ifaninﬁnitesimal disturbance eventually pro- duces unbounded motion. Apendulum atrestisinstable equilibrium, butthe eggstanding onendisanobvious illustration ofunstable equilibrium. Itcanbe readily seenthatwhen theextremum ofVisaminimum theequilibrium must bestable. Suppose thesystem isdisturbed from theequilibrium byanincrease in energy dEabove theequilibrium energy. IfVisaminimum atequilibrium, any deviation fromthisposition willproduce anincrease inV.Bytheconservation of energy, thevelocities mustthendecrease andeventually come tozero,indicating bound motion. Ontheother hand, ifVdecreases astheresult ofsome departure fromequilibrium, thekinetic energy andthevelocities increase indeﬁnitely, corre- sponding tounstable motion. Thesame conclusion maybearrived atgraphically byexamining theshape ofthepotential energy curve, asshown symbolically in Fig.6.1.Amore rigorous mathematical proof thatstable equilibrium requires a minimum inVwillbegiven inthecourse ofthediscussion. Weshall beinterested inthemotion ofthesystem within theimmediate neigh- borhood ofaconﬁguration ofstable equilibrium. Since thedepartures from equi- librium aretoosmall, allfunctions maybeexpanded inaTaylor series about the equilibrium, retaining onlythelowest-order terms. Thedeviations ofthegeneral- izedcoordinates fromequilibrium willbedenoted by17,: <11=qo+tn. (6-2) andthese maybetaken asthenewgeneralized coordinates ofthemotion. Ex- panding thepotential energy about qot,weobtain 2av 1aVV(q1»----qr.) =V(qo1.....qo~)+ (a—(1i)0n. +5(%T%)0n.nj +~-. res) to+dEol----—— ____-__---- ro________ __ _.--_- E .r5o+a151__..__ ---_ ___-_ E so------ -- -----—- q,—- q,—-(a)Stable (b)Unstable FIGURE 6.1 Shape ofthepotential energy curve atequilibrium. Chapter 6Oscillations where thesummation convention hasbeen invoked, asusual. Theterms linear in 1),vanish automatically inconsequence oftheequilibrium conditions (6.1). The ﬁrsttermintheseries isthepotential energy oftheequilibrium position, andby shifting thearbitrary zeroofpotential tocoincide withtheequilibrium potential, thistermmayalsobemade tovanish. Wearetherefore leftwiththequadratic terms astheﬁrstapproximation toV: 2l 3V 1 v=_ T =— 1rI r 2(aqlaqJ)0/lillj ZVJP7 77] (64) where thesecond derivatives ofVhavebeendesignated bytheconstants V,Jde- pending only upon theequilibrium values oftheq,’s. Itisobvious from their deﬁnition thattheV,-1-‘saresymmetrical, thatis,thatV”=V1,.TheV,Jcoefﬁ- cients canvanish under avariety ofcircumstances. Thus, thepotential cansimply beindependent ofaparticular coordinate, sothatequilibrium occurs atanyar- bitrary value ofthatcoordinate. Wespeak ofsuchcases asneutral orindifferent equilibrium. Itmayalsohappen, forexample, thatthepotential behaves likea quadratic atthatpoint, again causing oneormore oftheV,-J’stovanish. Either situation callsforspecial treatment inthemathematical discussion thatfollows. Asimilar series expansion canbeobtained forthekinetic energy Since the generalized coordinates donotinvolve thetimeexplicitly, thekinetic energy 1Sa homogeneous quadratic function ofthevelocities (cf.Eq.(1.71)): T=émtjéléj =%mr_]7ll7lj- (6-5) Thecoefﬁcients m,,areingeneral functions ofthecoordinates qk,buttheymay beexpanded inaTaylor series about theequilibrium conﬁguration: 8 . mq(<1ri---iqn)=m;j(q01----J10")+ 2," 77k+"'- 34k 0 AsEq(6.5)isalready quadratic inthe1),’s,thelowest nonvanishing approxima- tiontoTisobtained bydropping allbuttheﬁrstterm intheexpansions ofm,-1. Denoting theconstant values ofthemufunctions atequilibrium byTU,wecan therefore writethekinetic energy as T=%Ti-,-tin'1,~- (6-6) ltisagain obvious thattheconstants T,Jmustbesymmetric, sincetheindivid- ualtenns inEq.(6.6)areunaffected byaninterchange ofindices. From Eqs.(6.4) and(66),theLagrangian isgiven by L= _Vij77i7lJ)- (6-7) Taking theifsasthegeneral coordinates, theLagrangian ofEq.(6.7) leads tothe following nequations ofmotion: Tzjﬁrj +V1177] =0- 6.2I6.2 TheEigenvalue Equation andthePrincipal AxisTransformation 241 where explicit usehasbeenmade ofthesymmetry property oftheVUandT,J coefﬁcients. Each ofEqs.(6.8) willinvolve, ingeneral, allofthecoordinates 17,, andit"sthissetofsimultaneous differential equations thatmust besolved to obtain themotion neartheequilibrium. Inalmost allcases ofinterest, thekinetic energy tenncanbeeasily written so astohavenocross [6l’[I1S.* Thiscorresponds totheLagrangian L=trim?-v.,n.n,-i. (6.9) which generates thefollowing equations ofmotion T,ﬁ,+l/U17] =0. (nosumoveri] (6.10) THE EIGENVALUE EQUATION AND THE PRINCIPAL AXIS TRANSFORMATION Theequations ofmotion (6.8)arelinear differential equations withconstant oo- efﬁcients, ofaformfamiliar fromelectrical circuit theory. Wearetherefore ledto tryanoscillatory solution oftheform 17,=Ca,e “”' (6.1I) Here Ca,gives thecomplex amplitude oftheoscillation foreach coordinate 17,, thefactor Cbeing introduced forconvenience asascale factor, thesame forall coordinates. Itisunderstood ofcourse thatitistherealpartofEq.(6.9) thatis tocorrespond totheactual motion. Substitution ofthetrialsolution (6.9) into theequations ofmotion leads tothefollowing equations fortheamplitude fac- tors: (v,-,-a,-wzr,-;a,) =0. (6.12) Equations (6.12) constitute nlinear homogeneous equations forthea,’s,and consequently canhave anontrivial solution onlyifthedeterminant ofthecoeffi- cients vanishes: *'Malhematically, wewould goevenfunher when thecoordinates areCartesian andmaking theT,:J-= 6,]byrescaling thecoordinates. Snch coordinates arecalled mass-weighted coordinates since they aregenerated bydividing thecoordinates bythesquare rootofthemass. Thistransforms thekinetic energy lotheform _7l|7IIT- T . Thisreduces theproblem totheeigenvalue problem ofChapters 4and5,onlyinndimensions insttaid otthree, however, themathematical simpliﬁcation canobscure thephysics, since eachcoordinate can haveadifferent characteristic scale 2-1-2 Chaptei 6Oscillations V“—a)2T11 V12—-c02T12 .. V21—w2Tz1 V22—w2T22V3‘_(02%! =0. (6.13) v This determinantal condition isineffect analgebraic equation ofthenthde- greefor(02,andtherootsofthedeterminant provide thefrequencies forwhich Eq.(6.11) represents acorrect solution totheequations ofmotion. Foreach of these values ofwz,Eqs.(6.12) maybesolved fortheamplitudes ofa,,ormore precisely, forn—1oftheamplitudes interms oftheremaining a,-. Equations (6.12) represent atypeofeigenvalue equation, forwriting TUasan element ofthematrix T,theequations maybewritten Va=ATa. (6.14) Here theeffect ofVontheeigenvector aisnotmerely toreproduce thevector times thefactoi A,asintheordinary eigenvalue problem. Instead, theeigenvector issuchthatVacting onaproduces amultiple oftheresult ofTacting ona.We shallshowthattheeigenvalues Aforwhich Eq.(6.14) canbesatisﬁed areallreal inconsequence ofthesymmetric andreality properties ofTandV,and,infact, mustbepositive. Itwillalsobeshown thattheeigenvectors aareorthogonal—in asense. Inaddition, thematrix oftheeigenvectors, A,diagonalizes bothTandV, thefomier totheunitmatrix 1andthelattertoamatrix whose diagonal elements aretheeigenvalues it.Most importantly itisnecessary toshow thataanditare real. Proceeding asinSection 5.4,letakbeacolumn matrix representing thelcth eigenvector, satisfying theeigenvalue equation* Vak =}\.rTa]. . (6.15) Assume nowthattheonlysolution toEq.(6.15) involves complex A.andak.The adjoint equation, i.e.,thetransposed complex conjugate equation, forA1hasthe form afv=ifa,lT. (6I6) Hereallstands fortheadjoint vector——the complex conjugate rowmatrix—and explicit usehasbeenmade ofthefactthattheVandTmatrices arerealand symmetric. Multiply Eq.(6.16) from theﬁght byakandsubtract theresult of thesimilar product ofEq.(6.15) fromtheleftwithaél.Theleft-hand sideofthe difference equation vanishes, leaving only 0=(ik-i_t)a}“Ta,.. (6.17) *Ithardly need beadded thatthere isnosummation overkinEq(615).Indeed, inthischapter the summation convention willapply onlytothecomponents ofmatrices ortensors (ofanyrank) andmi: tothematrices andtensors themselves. 6.2 TheEigenvalue Equation andthePrincipal Axis Transformation 243 Whenl =k,Eq.(6.17) becomes (kk-).Z)aZTa), =0. (6.1s) Thatthematrix product inEq.(6.18) isrealcanbeshown immediately bytaking itscomplex conjugate andusing thesymmetry property ofT.However, wewant toprove thatthematrix product isnotonlyrealbutispositive deﬁnite. Forthis purpose, separate akintoitsrealandimaginary components. at=at+iB1<. Thematrix product canthenbewritten as aZTak =ii/CT0tk -l-ﬁkTBk +i(&kTBk —i}kT(.!k). (6.19) Theimaginary term vanishes byvirtue ofthesymmetry ofTandtherefore, as noted earlier, thematrix product isreal.Further, thel(.lIl6[1C energy uiI-sq.(6.6) canberewritten intenns ofacolunm matrix 1')as r=gins). (6.20) Hence. theﬁrsttwoterms inEq.(6.18) aretwice thekinetic energies when the velocity matrix 'i|khasthevalues atandBk,respectively. Now, akinetic energy byitsphysical nature mustbepositive definite forrealvelocities, andtherefore thematrix product inEq.(6.18) cannot bezero.Itfollows thattheeigenvalues A), mustbereal. Since theeigenvalues arereal, theratios oftheeigenvector components ajk determined byEqs.(6.15) mustallbereal.There isstillsome indeterminateness ofcourse sincethevalue ofaparticular oneoftheaJk'scanstillbechosen atwill without violating Eqs.(6.15). Wecanrequire however thatthiscomponent shall bereal,andthereality ofit),thenensures thereality ofalltheother components. (Any complex phase factor intheamplitude oftheoscillation willbethrown into thefactor C,Eq.(6.1l).)Multiply nowEq.(6.15) by5;,from theleftandsolve forA1,: ltk= (6.21)akTak Thedenominator ofthisexpression isequal totwice thekinetic energy forveloc- ities(1,),andsince theeigenvectors areallreal,thesummustbepositive deﬁnite. Similarly, thenumerator isthepotential energy forcoordinates am,andthecon- dition thatVbeaminimum atequilibrium requires thatthesummust bepositive orzero. Neither numerator nordenominator canbenegative, andthedenominator cannot bezero, hence Aisalways ﬁnite andpositive. (Itmayhowever bezero.) Recall thatAstands for:02,sothatpositive A.corresponds torealfrequencies of oscillation. Were thepotential notalocalminimum, thenumerator inEq.(6.21) Chapter 6Oscillations might benegative, giving risetoimaginary frequencies thatwould produce anun- bounded exponential increase ofthe17,-withtime. Such motion would obviously beunstable, andwehave herethepromised mathematical proof thataminimum ofthepotential isrequired forstable motion. Letusretum forthemoment toEq.(6.17) which, inview ofthereality ofthe eigenvalues andeigenvectors, canbewn'tten (Wk—X1)§1Tak =0. (6.17) Ifalltheroots ofthesecular equation aredistinct, thenEq.(6.17’)canholdonly ifthematrix product vanishes forlnotequal tok: §i1Tak =0, lgék. (6.22a) Ithasbeen remarked several times thatthevalues oftheajk’sarenotcompletely ﬁxed bytheeigenvalue equations (6.12). Wecanremove thisindeterminacy by requiring further that 5/,Tak =1. (6.22b) There arensuchequations (6.22), andtheyuniquely ﬁxtheonearbitrary compo- nentofeach oftheneigenvectors ak.*Ifwefonn alltheeigenvectors akinto asquare matrix Awith components ajk(cf.Section 4.6), then thetwoequa- tions (6.22a andb)canbecombined intoonematrix equation: ATA=1. (6.23) When twoormore oftheroots arerepeated, theargument leading toEq.(6.22a) fallsthrough forA1=Ak.Weshall reserve adiscussion ofthisexceptional case ofdegeneracy foralater time. Forthepresent, sufﬁce ittostate thatasetof ajkcoefﬁcients canalways befound thatsatisﬁes boththeeigenvalue conditions Eqs.(6.10), andEq.(6.22a), sothatEq.(6.23) always holds. InChapter 4,thesimilarity transformation ofamatrix Cbyamatrix Bwas deﬁned bytheequation (cf.Eq.(4.41): c’=scar‘. *Equation (6.22b) maybeputinaform thatexplicitly shows thatitsufﬁces toremove theindetermi- nacy intheajk’s. Suppose itisthemagnitude ofalkthatistobeevaluated; theratioofalltheother ajk’stoalkisobtained from Eqs.(6.12). Then Eq.(6.22b) canbewritten as 21.“;/<2 _L l_} — 2' alk alk alk Theleft-hand sideiscompletely determined from theeigenvalue equations andmaybeevaluated directly toprovide a1k. 6.2 TheEigenvalue Equation andthePrincipal AxisTransformation 245 Wenowintroduce therelated concept ofthecongruence transformation ofCby Aaccording totherelation c’=ACA. (6.24) IfAisorthogonal, sothatA=A_1, there isnoessential difference between thetwotypes oftransformation (asmaybeseenbydenoting A_1bythematrix B).Equation (6.23) cantherefore bereadasthestatement thatAtransforms T byacongiuence transfonnation intoadiagonal matrix, inparticular intotheunit matrix. Ifadiagonal matrix Awithelements Mk=Ak8,),isintroduced, theeigenvalue equations (6.15) maybewritten Vijajk =Yijajilzk, which becomes inmatrix notation VA=TAIL. (6.25) Multiplying byAfrom theleft,Eq.(6.25) takes theform AVA=ATAA, which byEq.(6.23) reduces to AVA=A. (6.26) Ourﬁnal equation (6.26) states thatacongruence transformation ofVby Achanges itintoadiagonal matrix whose elements aretheeigenvalues Ak. Eq.(6.26) hassolutions |V—)t1|= O. (6.26’) Insummary wecanusenonnalized Cartesian coordinates sothatT,~j=8;1-which reduces thephysics tosolving AA=1(4.36) and AvA=vd,,g,n,1 (6.26), orwemay choose more general goordinates where T,-j758,-J-,even allowing T,-j=Tji;éOfori 75j,anduse ATA=1(6.23) and AVA=vdiaggnal (6.26), tosolve thegeneral problem. Asanexample, weconsider aparticle ofmass mwithtwodegrees offreedom (x1,X2)thatobeys theLagrangian (cf.Eq.(6.9)) -2 -2L=%m(x1+x2)—%V)]-x,-xj 24- Chapter 6Oscillations where theV1]areconstants. Thecongruence transformation (6.26) hassolutions onlywhen Eq.(6.26’) issatisﬁed, so V11—K V12 _0 V21 V22—9»_ [Q-—Thisequation hastwosolutions: M=(V11+ V22+\/(V11—V22)2 +4V12V21) K2=-(V11-F V22-'\/(V11- V22)2 +4V12V21)- Associated withtheeigenvalues A,aretheeigenvectors a1,thatsatisfy(QI- ail-(V,-J —A1611) =0and 11,21+a,22 =1 (nosumon1') Weconsider twolimiting cases. Theﬁrstcaseassumes V11>V22>0and 076V21=V12<<(V11— V22). Wewrite thesmall quantityé =[V12/(l/11- V7_g)] then, toﬁrstorder in8,theeigenvalues are A=V+V8 1 11 12 (6.27) K2-—V22-V125 whose eigenvectors are,tolowest order in5, ___6’ a:[all azl]_[1 5‘l’'2'] (628) 5 2 t~>‘?.1~35LWIN 1112(122___ __ These correspond totherelations (111=1122 and (112=-1121- Theother limiting caseassumes V12>V22>0and(V11-V22)<<V12=1'31. Wenowwrite s=(V11—V22)/8V12,which isasmall quantity. Toﬁrstorder in etheeigenvalues are 1\1=§(V11+ V22)+V12+(V11—V22)81 (6.29)K2=5(V11+ V22)-V12-(V11—V22)~'J whose eigenvectors are,tolowest order in6‘, l l —(1+2) --—(1—2) a=[all‘"11= 8*5 8. (6.30) ~/7:1112(122 -(1- 28) ﬁ(1+2@) V112 V221 _l_ I 2 1 2 3 1VIZ6.2 TheEigenvalue Equation andthePrincipal AxisTransformation 247 Therelations among thecomponents oftheeigenvectors aredifferent thaninthe previous example. Here a12=—a21 isslightly lessthan1/1/2 while a11=1122is slightly greater than1/\/2. Thepreceding approximations looked atthebehavior oftheeigenvalues and eigenvectors inlimiting cases. Thequalitative changes inthese quantities asa function ofV12/(V11—V22)from zerotothree areshown inFig.6.2.Weshall retum tothisexample afterconsidering thegeneral problem ofmultiple roots of theeigenvalue equation (6.26’). l \ “11-“22 l\)|—I§|’“‘Z __________________: - —- “12 I __ 11 I avg I *2 tiI""t\-I-IIIIIIIII"“' “21 '1 2 3 i iV11‘ V22 V11’ V22 (3) (b) FIGURE 6.2Behavior ofthe(a)eigenvalues and(b)eigenvector components asthe energy ratioWvf-'7 changes from0to3. Itremains only toconsider thecase ofmultiple roots tothesecular equation, asituation thatismoreannoying inthemathematical theory thanitisinpractice. Ifoneormore oftherootsisrepeated, it1Sfound thatthenumber ofindependent equations among theeigenvalues isinsufﬁcient todetermine even theratioofthe eigenvector COIIlp0l'l8l'llI>. Thus, iftheeigenvalue Aisadouble root.anytwoofthe components a,maybechosen arbitrarily, therestbeing ﬁxed bytheeigenvalue equations. Ingeneral, anypairofeigenvectors randomly chosen outoftheinﬁnite setof allowed vectors willnotbeorthogonal. Nevertheless, itisalways possible tocon- struct apairofallowed vectors thatareorthogonal, andthese canbeusedtoform theorthogonal matrix A.Consider forsimplicity theprocedure tobefollowed foradouble root.Letaiandafbeanytwoallowable eignenvectors foragiven Chapter 6Oscillations double rootA,which havebeennormalized soastosatisfy Eq.(6.22b). Anylinear combination ofajcandafwillalsobeaneigenvector fortherootA.Wetherefore seektoconstruct avector a1, a1=c1a;(+c2a§, (631) where c1andc2areconstants such thata1isorthogonal toai.Theorthogonality condition, Eq.(6.22a), thenrequires that ~ /__ ~//_a1Tak _C1+(‘g2IT8k —0, where usehasbeen made ofthenormalization ofa2.Ittherefore follows thatthe ratioofc1toc2must begiven by 6-‘=-a;Ta;, E-1,. (6.32)C2 Wecanillustrate these ideas byagain considering ourtwo-dimensional example given byEqs (6.27) through (6.30). Thetwolimiting cases ofthe off-diagonal potential term V12,being much lessthanandmuch greater than thedifference factor (V11—V22), provide anexcellent example oftheproblems introduced bydegeneracy. When V11=V22=V0, V12=0- thetwoeigenvalues become thesame, A1=A2=V0. Ifthelimit istaken byletting V12—>0ﬁrstandthen taking thelimit (V11->V22),theeigenvectors inEqs.(6.28) become 3]= and 3,2= . Ifthelimitistaken inthereverse order, Eqs.(6.30) give L _; b1= and 112=(‘/5), (6.34)E _ where bisusedfortheeigenvectors inEqs.(6.34) toavoid confusion withthe eigenvectors inEqs.(6.33). Eachoftheeigenvectors in(6.33) and(6.34) arelinear combinations oftheothersetofeigenvectors. Forexample,ﬁlm 1 1 bi=E011 +82), and b2=3&2 ~81). soeither setofeigenvectors isalinear combination oftheother, aswasdiscussed inthissection. These results obviously generalize totheinﬁnite set —ba1=(Z) and a2=(a), 6.2TheEigenvalue Equation andthePrhcipal AxisTransformation 249 where aandbareanypairsofnumbers thatsatisfy G2-l-b2=1. This shows thatthere isanmﬁmte setofpossible eigenvectors inthecaseof degeneracy. There isanother waytoconsider thesigniﬁcance ofthese results. Theapprox- imate eigenvectors inEqs.(6.28) areforthecasewhere themain potential energy terms areV11andV22,which areatdiagonal positions, andtheV12areintheoff- diagonal positions. ifwetaketheeigenvectors ofEq.(6.30) inthelimit s—>0 andlettheeigenvectors ofEqs.(6.30) transform VasV’=AVA, weobtain the transformed potential energy tensor V,_2(V1i +V22)+V12 2(V11— V22) %(Vll—V22) 2(V11+ V22)—V12 inwhich thedifference term(V11—V22)1Soff-diagonal. Thus, thesetofeigenvec- torsgiven byEqs.(6.30) areforthephysical situation inwhich thesmall energy tenn (V11—V22)isoff-diagonal. Returning tothemaindiscussion, therequirement thata1ofEq.(6.32) benor- malized provides another condition onthetwocoefﬁcients, which interms ofr1 deﬁned byEq.(6.32) takes thefonn 51Ta1 =l=c%+6%+2c1c2r1. Together thetwoequations ﬁxthecoefﬁcients c1andc2,andtherefore thevector a1.Botha1andakEa§,areautomatically orthogonal totheeigenvectors ofthe other distinct eigenvalues, forthentheargument based onEq.(6.l7’) remains valid Hence, wehave asetofneigenvectors ajwhose components form the matrix Asatisfying Eq.(6.23). Asimilar procedure isfollowed forarootofhigher multiplicity. IfAisan m-fold root, then orthogonal nomialized eigenvectors areformed outoflinear combinations ofanyofthemcorresponding eigenvectors ai,...,ain.Theﬁrstof the“orthonormal” eigeivectors a1isthenchosen asamultiple ofaaga2istaken asalinear combination of3'1anda§;andsoon.Inthismanner, thenumber of constants tobedetermined isequal tothesumoftheﬁrstmintegers, or%m(m—1). Thenormalization requirements provide mconditions, while thereare%m(m —l) orthogonality conditions, andtogether these arejustenough toﬁxtheconstants uniquely. Thisprocesses ofconstructing orthogonalized eigenvectors inthecaseofmul- tipleroots iscompletely analogous totheGram-Schmidt method ofconstructing asequence oforthogonal functions outofanyarbitrary setoffunctions. Phrased ingeometrical language, itisalsoseentobeidentical withtheprocedure followed inChapter 5formultiple eigenvalues oftheinertia tensor. Forexample. theadded indeterminacy intheeigenvector components foradouble rootmeans thatallof thevectors inaplane areeigenvectors. Wemerely choose anytwoperpendicular 6.3 IChapter 6Oscillations directions intheplane asbeing thenewprincipal axes, withtheeigenvectors inA asunitvectors along these axes. FREQUENCIES OFFREE VIBRATION, AND NORMAL COORDINATES Thesomewhat lengthy arguments ofthepreceding section demonstrate thatthe equations ofmotion willbesatisﬁed byanoscillatory solution oftheform(6.11), notmerely foronefrequency butingeneral forasetofnfrequencies wk.Acom- plete snlnrion oftheequations ofmotion therefore involves asuperposition of oscillations withalltheallowed frequencies. Thus, ifthesystem isdisplaced slightly fromequilibrium andthenreleased, thesystem performs small oscilla- tions about theequilibrium withthefrequencies m1,...,canThesolutions ofthe secular equation aretherefore oftendesignated asthefrequencies offreevibration orastheresonant frequencies ofthesystem. Thegeneral solution oftheequations ofmotion may now bewritten asasum- mation overanindex k: 171=C/<¢1ike'“"*'. (6-35) there being acomplex scale factor C1,foreachresonant frequency. Itmight be objected thatforeach solution A1,ofthesecular equation there aretworesonant frequencies +011,and—:o1,. Theeigenvector a1,would bethesame forthetwo frequencies, butthescalefactors CfandCI}could conceivably bedifferent. On thisbasis, thegeneral solution should appear as ,1,=11,,.(c,;"e+'"’*' +cge-W). (6.35') Recall however thattheactual motion istherealpartofthecomplex solution, and therealpartofeither (6.35) or(6.35’) canbewritten intheform 17,=f1,a,k costwkt +511), (6.36) where theamplitude fkandthephase 61,aredetermined form theinitial condi- tions. Either ofthesolutions ((6.35) and(6.36)) willtherefore represent theactual motion, andtheformer ofcourse isthemore convenient. Theorthogonality properties ofAgreatly facilitate thedetermination ofthe scale factors Ckinterms oftheinitial conditions. Att=0,therealpartof Eq.(6.35) reduces to T7z(0) =R6Ckllrk. (6-37) where Restands for“real partof.”Similarly, theinitial value ofthevelocities is obtained as 01(0) =ImCkarkwk, (6-38) 6.3 Frequencies ofFreeVibration, andNormal Coordinates 251 where TmCkdenotes theimaginary partofC1,.From these 2nequations, thereal andimaginary partsofthenconstants Ckmaybeevaluated. Tosolve Eq.(6.37), forexample, letusﬁrstwrite itinterms ofcolumn matrices 11(0) andC: 11(0) =AReC. (6.37’) Ifwemultiply byATfromtheleftanduseEq.(6.23), weimmediately obtain a solution forReC: ReC=AT1)(0l. or,taking thelthcomponent, Rec,=aJ1TJk77k (0). (6.39) Asiimlar procedure leadstotheimaginary panofthescalefactors as* 1Imc,=J1Ea,-,r,m(0). (6.40) ],k Equations (6.39) and(6.40) thuspemiit thedirect computation ofthecomplex factors C1(andtherefore theamplitudes andphases) interms oftheinitial condi- tions andtheiiialiices TandA. Thesolution foreac'icoordinate, Eq.(6.35), isingeneral asumofsimple harmonic oscillations inallofthefrequencies wksatisfying thesecular equation. Unless ithappens thatallofthefrequencies arecommensurable, thatis,rational fractions ofeachother, r;,-never repeats itsinitial value andistherefore notitself a periodic function oftime. However, itispossible totransform from the17,-toanew setofgeneralized coordinates thatareallsimple penodic functions ott1me—a set ofvariables known asthenormal coordinates. Wedeﬁne anewsetofcoordinates 4‘, or,interms ofsingle column matrices 1|and§, 1|=A§. (6.4l’) Thepotential energy, Eq.(6.4), iswritten inmatrix notation as v=gave. (6.42) Now, thesingle-row transpose matrix iiisrelated toZ‘bytheequation a=K2=ZR,*TheSllml‘l'1dI10I1 overjandkisshown explicitly because thereisnosunmation overtherepeated subscript! Chapter 6Oscillations sothatthepotential energy canbewritten alsoas v=§ZAvAg. ButAdiagonalizes Vbyacongruence transformation (cf.Eq.(6.26)). andthe potential energy therefore reduces simply to v=gilt;=%w,7;?,‘,3. (6.43) Thekinetic energy hasaneven simpler foirn inthenewcoordinates. Since the velocities transform asthecoordinates, Tasgiven inEq.(6.20) transforms to T_%§ATA§ which byvirtue ofEq.(6.23) reduces to T=;;§=%&n. 646 Equations (6.43) and(6.44) statethatinthenewcoordinates boththepotential andkinetic energies aresums ofsquares only, without anycross terms. Ofcourse, thisresult issimply another wayofsaying thatAproduces aprincipal axistrans- formation. Recall thattheprincipal axistransformation oftheinertia tensor was speciﬁcally designed toreduce themoment ofinertia toasumofsquares; thenew axesbeing thepnnci palaxesoftheinertia ellipsoid. Herethekinetic andpotential energies arealsoquadratic forms (aswasthemoment ofinertia) andbotharedi- agonalized byA.Forthisreason, theprincipal axistransfomiation employed here isaparticular example ofthewell-known algebraic process ofthesimultaneous diagonalization oftwoquadratic forms. Theequations ofmotion share inthesimpliﬁcation resulting from theiruse. ThenewLagrangian is L=aaa-an» me) sothattheLagrange equations for§kare it+win=0. (6.46) Equations (6.47) have theimmediate solutions rk=cke"“"'=‘, (6.47) which could havebeenseenofcourse directly fromEqs.(6.35) and(6.41). Each ofthenewcoordinates isthusasimply periodic function involving onlyoneof theresonant frequencies. Asmentioned earlier, itistherefore customary tocall theQ‘‘sthenormal coordinates ofthesystem. Each normal coordinate corresponds toavibration ofthesystem withonlyone frequency, andthese component oscillations arespoken ofasthenormal modes ofvibration. Alloftheparticles ineach mode vibrate with thesame frequency andwiththesame phase;* therelative amplitudes being determined bythematrix ’*PaI'llClC‘i maybeexactly outofphase itthea‘shaveopposite sign 6.4 I6.4 FreeVibrations ofaLinear Triatomic Molecule 253 elements ajk.Thecomplete motion isthenbuiltupoutofthesumofthenormal modes weighted with appropriate amplitude andphase factors contained inthe Ck’S. Harmonics ofthefundamental frequencies areabsent inthecomplete motion essentially because ofthestipulation thattheamplitude ofoscillation besmall. Wearethenallowed torepresent thepotential asaquadratic fonn, which ischar- acteristic ofsimple harmonic motion. Thenormal coordinate transformation em- phasizes thispoint. fortheLagrangian inthenormal coordinates (6.45) isseen tobethesumoftheLagrangians forharmonic oscillators offrequencies wk.We canthusconsider thecomplete motion forsmall oscillations asbeing obtained by exciting thevarious harmonic oscillators withdifferent intensities andphases.* FREE VIBRATIONS OFALINEAR TRIATOMIC MOLECULE Toillustrate thetechnique forobtaining theresonant frequencies andnormal modes, weshall consider indetail amodel based onalinear symmetrical tri- atomic molecule. Intheequilibrium conﬁguration ofthemolecule, twoatoms ofmass maresymmetrically located oneach sideofanatom ofmass M(cf. Fig.6.3). Allthree atoms areononestraight line,theequilibrium distances apart being denoted byb.Forsimplicity, weshallﬁrstconsider onlyvibrations along thelineofthemolecule, andtheactual complicated interatomic potential willbe approximated bytwosprings offorce constant kjoining thethree atoms. There arethree obvious coordinates marking theposition ofthethree atoms ontheline. Inthese coordinates, thepotential energy is k kv=56¢)-X1-b)2+56:3-it)-b)2. (6.4s) Wenowintroduce coordinates relative totheequilibrium positions: 771=-xi_7501, where X02—X01=b=X03—X02- m M m X‘ 17 X2 I7 X3 FIGURE 6.3 Model ofalinear symmetrical ti-iatomic molecule. *Note forfuture reference thatthesame sortofpicture appears inthequantization oftheelectromag- netic ﬁeld. Thefrequencies oftheharrnomc oscillators areidentiﬁed withthephoton frequencies, and theamplitudes ofexcitation become thediscrete quantized “occupation numbers”-—the number of photons ofeachfrequency. Chapter 6Oscillations Thepotential energy thenreduces to V=§(172 —1702+§(173 —112)2, or v=§<11%+211%+11%—2111111-2111111). (6.49) Hence, theVtensor hastheform F???‘—k 0 V= - 2k—k . (6.50) O—k k Thekinetic energy hasanevensimpler form: ..M.T=§<11%+11%)+711%, (6.51) sothattheTtensor isdiagonal: m0O r=0M0. (6.52) O0m Combining these twotensors, thesecular equation appears as k—cozm —k 0 |v-w2T|=—k 2k-1111M —k =0. (6.53) 0 —k k—wzm Direct evaluation ofthedeterminant leads tothecubic equation inm2: 111201-w2m)(k(M +2111)-QFM111) =0, (6.54) withtheobvious solutions w1=0, (0z=\/Z-7, a)3= (6.55) Theﬁrsteigenvalue, an=O,mayappear somewhat surprising andevenalann- i.ngatﬁrstsight. Such asolution doesnotcorrespond toanoscillatory motion at all,fortheequation ofmotion forthecorresponding normal coordinate is Z1=0, which produces auniform translational motion. Butthisisprecisely thekeyto thedifﬁculty. Thevanishing frequency arises from thefactthatthemolecule 6.4 FreeVibrations ofaLinear Triatomic Molecule 255 maybetranslated rigidly along itsaxiswithout anychange inthepotential en- ergy, anexample ofneutral equilibrium mentioned previously. Since therestoring force against such motion iszero, theeffective “frequency” must alsovanish. Wehave made theassumption thatthemolecule hasthree degrees offreedom forvibrational motion, whereas inreality oneofthem isarigid body degree of freedom. Anumber ofinteresting points canbediscussed inconnection withavanishing resonant frequency. ItisseenfromEq.(6.21) thatazerovalue oftocanoccur only when thepotential energy ispositive butisnotpositive deﬁnite; thatis,it canvanish even when notallthe17,’sarezero. Anexamination ofV,Eq.(6.49), shows thatitisnotpositive deﬁnite andthatVdoesinfactvanish when allthe n’sareequal (uniform translation). Si.nce thezerofrequency found hereisofnoconsequence forthevibration frequencies ofinterest, itisoften desirable tophrase theproblem sothattheroot iseliminated from theoutset. Wecandothisheremostsimply byimposing the condition orconstraint thatthecenter ofmass remain stationary attheorigin: m(x1 +x3) +Mxg =0. (6.56) Equation (6.56) canthenbeused toeliminate oneofthecoordinates from Vand T,reducing theproblem tooneoftwodegrees offreedom (cf.Derivation 1,this chapter). Therestriction ofthemotion tobealong themolecular axisallows onlyone possible typeofuniform rigidbodymotion. However, ifthemoregeneral problem ofvibrations inallthree directions isconsidered, thenumber ofrigid body degrees offreedom willbeincreased tosix.Themolecule maythentranslate uniformly along thethree axesorperform uniform rotations about theaxes.Hence, inany general system ofndegrees offreedom, there willbesixvanishing frequencies andonlyn—6truevibration frequencies. Again, thereduction inthenumber of degrees offreedom canbeperformed beforehand byimposing theconservation oflinear andangular momentum upon thecoordinates. Inaddition torigid body motion, ithasbeen pointed outthatzeroresonant frequencies mayalsoarise when thepotential issuchthatboththeﬁrstandsecond derivatives ofVvanish atequilibrium. Small oscillations maystillbepossible in thiscase ifthefourth derivatives donotalsovanish (thethird derivatives must vanish forastable equilibrium), butthevibrations willnotbesimple hannonic. Such asituation therefore constitutes abreakdown ofthecustomary method of small oscillations, butfortunately itisnotoffrequent occurrence. Returning nowtotheexamination oftheresonant frequencies, anwillberec- ognized asthewell-known frequency ofoscillation foramassmsuspended bya spring offorce constant k.Wearetherefore ledtoexpect thatonlytheendatoms partake inthisvibration; thecenter molecule remains stationary. Itisonlyinthe third mode ofvibration, m3,thatthemass Mcanparticipate intheoscillatory mo- tion.These predictions areveriﬁed byexamining theeigenvectors forthethree normal modes. 6 Chapter 6Oscillations Thecomponents a1jaredetermined foreachfrequency bytheequations (k—co§m)a1J —lca2_, =0 -km,+(2/<-wfM)11,, -M13,=0 (6.57a) —/C612] +(k—a>§m)a3] =O, along withthenormalization condition: 111(11fJ+11%,)+Mag]=1. (6.576) For(01=0,itfollows immediately fromtheﬁrstandthirdofEqs.(6.57a) thatall three coefﬁcients areequal: an=(Z21=(131.Thisofcourse isexactly what was expected form thetranslational nature ofthemotion (cf.Fig.6.4a). Thenormal- ization condition thenﬁxesthevalue ofa1Jsothat 1 l I =i, =i, =i. 6.58““0% ‘*1’,/am “*3,/2% (“’ Thefactors (k—w%m) vanish forthesecond mode, andEqs.(6.57a) show imme- diately thatan=O(aspredicted) andan=—a32. Thenumerical value ofthese quantities isthendetermined byEq.(6.S7b): I 1=—, =0, =——. 6.586 4:2 ‘/E 4122 432 \/2? ( ) Inthismode thecenter atomisatrest,while thetwoouter onesvibrate exactly outofphase (astheymust inorder toconserve linear momentum) (cf.Fig.6.4b). Finally, when w=603,itcanbeseenfromtheﬁrstandthirdofEqs.(6.S’7a) that (113and4133must beequal. Therestofthecalculation forthismode isnotquite as simple asfortheothers, anditwillbesufﬁcient tostatetheﬁnalresult: 1 2 1H13=it 423= H33=mm- l2111(1+%) \/2M(Z+;) /2111(1+2§) (6.586) 1tr’ II‘ I-I1 (<1) —~§i V,’ I 311 (b) :1-1 III @-I (v) FIGURE 6.4Longitudinal normal modes ofthelinear symmetric triatormc molecule 6.4 FreeVibrations ofaLinear Triatomic Molecule 257 I-lerethetwoouter atoms vibrate withthesame amplitude, while theinner one oscillates outofphase withthemandhasadifferent amplitude, (cf.Fig.6.4c.) Thenormal coordinates maybefound byinverting Eq.(6.41) as 1 4'1= (¢Eﬂ1 +x/E772 +~/;"_7l3) §2=(/5(111—173) (6-59) 4'3=\ Ii‘!-@971 +?73) _M02]- These normal modes describe eachofthebehaviors shown onFig.6.4.Anygen- erallongitudinal vibration ofthemolecule thatdoesnotinvolve arigidtranslation willbesome linear combination ofthenormal modes cogand(03.Theamplitudes ofthenormal modes, andtheirphases relative toeachother, willofcourse be determined bytheinitial conditions (cf.Exercise 5). Wehavespoken sofaronlyofvibrations along theaxis; intheactual molecule there willalsobenormal modes ofvibration perpendicular totheaxis. Thecom- plete setofnormal modes isnaturally more difﬁcult todetermine thanmerely the longitudinal modes, forthegeneral motion inalldirections corresponds tonine degrees offreedom. While theprocedure isstraightforward, thealgebra rapidly becomes quite complicated, anditisnotfeasible topresent thedetailed calcula- tionhere. However, itispossible togiveaqualitative discussion onthebasis of general principles, andmostoftheconclusions ofthecomplete solution canbe predicted beforehand. Thegeneral problem willhaveanumber ofzeroresonant frequencies cor- responding tothepossibility ofrigidbody motion. Foramolecule withnatoms there are3ndegrees offreedom. Subtracting thethree translational andthree rigid rotational degrees offreedom, there willbeingeneral 3n—6vibrational modes. Forthelinear molecule, there willbethree degrees offreedom forrigidtrans- lation, butrigid rotation canaccount foronly twodegrees offreedom. Rotation about theaxisofthemolecule isobviously meaningless andwillnotappear asa mode ofrigid body motion. Wearetherefore leftwithfourtruemodes ofvibra- tion.Twoofthese arethelongitudinal modes, which have already been examined sothatthere canonlybetwomodes ofvibration perpendicular totheaxis. How- ever, thesymmetry ofthemolecule about itsaxisshows thatthese twomodes ofperpendicular vibration mustbedegenerate. There isnothing todistinguish a vibration intheydirection fromavibration inthezdirection, andthetwofre- quencies mustbeequal. Theadditional indeterminacy oftheeigenvectors ofadegenerate mode appears here, inthatalldirections perpendicular tothemolecular axisarealike. Anytwo orthogonal axesintheplane normal tothemolecule maybechosen asthedirec- tions ofthedegenerate modes ofvibration. Thecomplete motion oftheatoms 8 Chapter 6Oscillations normal tothemolecular axiswilldepend upontheamplitudes andrelative phases ofthetwodegenerate modes. Ifboth areexcited, andtheyareexactly inphase, thentheatoms willmove onastraight linepassing through theequilibrium con- ﬁguration. Butiftheyareoutofphase, thecomposite motion isanelliptical Lis- sajous ﬁgure, exactly asinatwo-dimensional isotropic oscillator. Thetwomodes thenrepresent arotation, rather thanavibration. ltisobvious from thesymmetry ofthemolecules thattheamplitudes oftheend atoms must beidentical inmagnitude. Thecomplete calculation shows thatthe endatoms alsotravel inthesame direction along theLissajous ﬁgure. Hence, the center atom must revolve intheopposite direction, inorder toconserve angular momentum. Figure 6.5illustrates themotion forthetwodegenerate modes when theyare90°outofphase. Asthecomplexity ofthemolecule increases, thesizeofthesecular deter- minant becomes verylarge, andﬁnding thenormal frequencies andamplitudes becomes aproblem ofconsiderable magnitude. Wehave seenhowever thateven inasituation assimple asthelinear triatomic molecule, astudy ofthesymmetries tobeexpected inthevibrations greatly simpliﬁes thecalculations. Considerable mathematical ingenuity hasbeen devoted toexploiting thesymmetries inherent incomplex molecules toreduce thelabor involved inﬁnding their vibration fre- quencies. Thetheory ofsymmetry groups hasbeenapplied withgreat success in factoring thelargesecular determinant intosmaller blocks thatmaybediagonal- izedseparately. Ithasbeenpointed outhowever thatsuchelaborate mathematical manipulation wasmore appropriate inatimewhen numerical computations were difficult andtedious. Considering thespeed andmemory capacity ofpresent-day computers, astraightforward approach maybeeasier andmore accurate inthe longrun.Fastandaccurate routines forsolvi.ng theeigenvalue problems oflarge matrices arethestock-in—trade today ofscientiﬁc computers ofeven moderate size. There hastherefore been atrend toward amore brute-force approach in which mass-weighted Cartesian coordinates (seep.241) areused toformulate theproblem. Thekinetic energy ellipsoid forthemolecular vibrations isthen already asphere, andﬁnding thenonnal modes reduces todiagonalizing thepo- tential energy. These approaches areextensively applied ininfrared andRaman spectroscopy. FIGURE 6.5 Degenerate modes ofthesymmetrical tnatomic molecule. 6.5I6.5 Forced Vibrations andtheEffect ofDtsstpattve Forces 259 FORCED VIBRATIONS AND THE EFFECT OFDISSIPATIVE FORCES Freevibrations occur when thesystem isdisplaced initially from itsequilibrium conﬁguration andisthenallowed tooscillate byitself. Very often, however, the system issetintooscillation byanexternal driving force thatcontinues toacton thesystem after t=O.Thefrequency ofsuch aforced oscillation isthendeter- mined bythefrequency ofthedriving force andnotbytheresonant frequencies. Nevertheless. thenonnal modes areofgreat importance inobtaining theampli- tudes oftheforced vibration, andtheproblem isgreatly simpliﬁed byuseofthe normal coordinates obtained fromthefreemodes. IfFjisthegeneralized force corresponding tothecoordinate 1;J,then by Eq.(1.49) thegeneralized force Q,forthenormal coordinate §,-is Q1 Z Cl]; F]. Theequations ofmotion when expressed innormal coordinates nowbecome +mfg,=Q,. (6.61) Equations (6.61) areasetofninhomogeneous differential equations thatcanbe solved only when Weknow thedependence ofQ,ontime. While thesolution willnotbeassimple asinthefreecase,notethatthenormal coordinates preserve theiradvantage ofseparating thevariables, andeachequation involves onlya single coordinate. Frequently. thedriving force varies sinusoidally withtime.Inanacoustic prob- lem,forexample, thedriving forcemight arisefromthepressure ofasound wave impinging onthesystem, andQ;thenhasthesame frequency asthesound wave. Or,ifthesystem isapolyatomic molecule, asinusoidal driving force ispresent ifthemolecule isilluminated byamonochromatic lightbeam. Each atom inthe molecule isthensubject toanelectromagnetic force whose frequency isthatof theincident light. Even where thedriving force isnotsinusoidal withasingle fre- quency, itcanoften beconsidered asbuiltupasasuperposition ofsuchsinusoidal terms. Thus, ifthedriving force isperiodic, itcanberepresented byaFourier se- ries;other times, aFourier integral representation issuitable. Since Eqs. (6.61) arelinear equations, itssolutions forparticular frequencies canbesuperposed to ﬁndthecomplete solution forgiven Q,. Itistherefore ofgeneral interest tostudy thenature oftheoscillations when theforce Q,canbewritten as Qt=Q0:605(0)! +5,). (6-62) where wistheangular frequency ofanexternal force. Theequations ofmotion nowappear as +mfg,=Q0,oos(wt+5,). (6.63) Chapter 6Oscillations Acomplete solution ofEq.(6.63) consists ofthegeneral solution tothehomo- geneous equation (that is,thefreemodes ofvibration) plusaparticular solution totheinhomogeneous equation. Byaproper choice ofinitial conditions, thesu- perimposed freevibrations canbemade tovanish,* centering ourinterest onthe particular solution ofEqs.(6.63) thatwillobviously havetheform Q‘,=B,cos(a>t +5,). (6.64) Here theamplitudes B,aredetermined bysubstituting thesolution inEqs.(6.63): _ Q01B,_T_(D2. (6.65) Thecomplete motion isthen aQcos(a>t +8) T71=ant, = (6-66)cu]—a: Thus, thevibration ofeachparticle isagain composed oflinear combinations of thenormal modes, butnoweachnormal oscillation occurs atthefrequency ofthe driving force. Twofactors determine theextent towhich eachnormal mode isexcited. One istheamplitude ofthegeneralized driving force, Q0,-.Ifthe force oneachparticle hasnocomponent inthedirection ofvibration ofsome particular normal mode, thenobviously thegeneralized force corresponding tothemode willvanish and Q0,willbezero. Anextemal force canexcite anormal mode onlyifittends to move theparticles inthesome direction asinthegiven mode. Thesecond factor is thecloseness ofthedriving frequency tothefreefrequency ofthemode. Asacon- sequence ofthedenominators inEq.(6.66), thecloser wapproaches toanyto,,the stronger willthatmode beexcited relative totheother modes. indeed, Eq.(6.66) apparently predicts inﬁnite amplitude when thedriving frequency agrees exactly withoneofthew,’s--thefamiliar phenomenon ofresonance. Actually, ofcourse, thetheory behind Eq.(6.66) presumes only small oscillations about equilibrium positions; when theamplitude predicted bytheformula becomes large, thisas- sumption breaks down andEq.(6.66) isthennolonger valid. Note thattheos- cillations areinphase withthedriving force when thefrequency islessthanthe resonant frequency, butthatthere isaphase change ofrtingoing through the resonance. Ourdiscussion hasbeen unrealistic inthattheabsence ofdissipative orfric- tional forces hasbeen assumed. lnmany physical systems, these forces, when present, areproportional totheparticle velocities andcantherefore bederived *The freevibrations areessentially thetransients generated bytheapplication ofthednvmg forces IfweCOIlSldﬂ‘ thesystem tobeuiitially inanequilibrium conﬁguration, andthenslowly build up thednvmg forces from zero, these transients willnotappear. Alternatively, dissipative forces canbe assumed present (seepages following) thatwilldamp outthefreevibrations 6.5 Forced Vibrations andtheEffect ofDissipative Forces 261 fromadissipation function f(cf.Section 1.5).Letusﬁrstconsider theeffects of frictional forces onthefreemodes ofvibration. From itsdeﬁnition. .7-‘mustbeahomogeneous quadratic function oftheveloc- ities: 1-‘=ix,-r,,a,. (6.67) Thecoefﬁcients .7-"Uareclearly symmetric, F”=F1-,,andingeneral willbe functions ofthecoordinates. Since weareconcerned withonly small vibrations about equilibrium, itissufﬁcient toexpand thecoefﬁcients about equilibrium and retain only theﬁrst, constant term, exactly aswasdone forthekinetic energy. Infuture applications ofEq.(6.67), weshall take7",]asdenoting these constant factors. Recall that2.7-'istherateofenergy dissipation duetothefrictional forces (cf.Eq.(2.60)). Thedissipation function .7-'therefore cannever benegative. The complete setofLagrange equations ofmotion nowbecome (cf.Section 1.5) T1157} +7:11'71+Vii'71=0- (6-68) Clearly inorder toﬁndnormal coordinates forwhich theequations ofmotion would bedecoupled, itisnecessary toﬁndaprincipal axistransformation that simultaneously diagonalizes thethree quadratic forms T,V,andF.Aswasshown above, thisisnotingeneral possible; normal modes cannot usually befound for anyarbitrary dissipation function. There arehowever some exceptional cases when simultaneous diagonalization ispossible. Forexample, ifthefrictional force isproportional bothtotheparticle’s velocity anditsmass, then.7willbediagonal whenever Tis.When suchsimul- taneous diagonalization isfeasible, thentheequations ofmotion aredecoupled in thenormal coordinates withthefonn E,+.7-",5,+042;,=O.(nosummation) (6.69) Herethe.7-",’sarethenonnegative coefﬁcients inthediagonalized formof.7’when expressed intemis of§,.Being asetoflinear differential equations withconstant coefﬁcients, Eqs.(6.69) maybesolved byfunctions oftheform Cl Z Cl e—lm:! 9 where 00:satisﬁes thequadratic equation w,'2+iw,T.7-', —co?=0.(nosummation) (6.70) Equation (6.70) hasthetwosolutions $2 iii;=;l:‘/a>‘.2—T'—i%. (6.71) Chapter 6Oscillations Themotion istherefore notapureoscillation, forw’iscomplex. Itisseenfrom Eq.(6.71) thattheimaginary partofco:results i.nafactor exp(—F,t/2), andby reason ofthenonnegative nature ofofthe.F,’s, thisisalways anexponentially decreasing function oftime.* Thepresence ofadamping factor duetothefriction ishardly unexpected. Astheparticles vibrate, theydowork against thefrictional forces, andtheenergy ofthesystem (andhence thevibration amplitudes) must decrease withtime. TherealpartofEq.(6.71) corresponds totheoscillatory factor inthemotion; notethatthepresence offriction alsoaffects thefrequency ofthe vibration. However, ifthedissipation issmall, thesquared term inF,maybe neglected, andthefrequency ofoscillation reduces tothefriction-free value. The complete motion isthen simply anexponential damping ofthefreemodes of vibration: q,=c,@-*7’/2¢"'""’. (6.72) lfthedissipation function cannot bediagonalized along with TandV,the solution ismuch more difﬁcult toobtain. Thegeneral nature ofthesolution re- mains pretty much thesame, however: anexponential damping factor times an oscillatory exponential function. Suppose weseekasolution toEqs.(6.68) ofthe form 17]=CaJe_""' =CaJe_'”e_2”"”. (6.73) With thissolution, Eqs.(6.68) become asetofsimultaneous linear equations v,,a,-l'CU_FU'6l_] -w17",,a, =0. (6.74) Itisconvenient towritewasiy,sothat y=—icu=—/c—Zrriv, (6.75) andthus—Kistherealpartofy.Interms ofthesquare tensors ofV,T,and.7-I thesetofequations (6.74) become acolumn matrix equation involving y: Va+7/Fa+3/2Ta=0. (6.76) Thesetofhomogeneous equations (6.74) or(6.76) canbesolved fortheti,only forcertain values ofwory. Without actually evaluating thecorresponding secular equation, wecanshow thatKmust always benonnegative. Convert thematrix equation (6.76) intoa scalar equation forybymultiplying from theleftwithal: aiva+yaiFa+y2a1'Ta =0. (6.77) *Some (butnotall)IF,’smaybezero, which simply means there arenofrictional effects inthecorre- sponding nonnal modes. Theimportant point isthatthe.7-',’scannot benegative 6.5 Forced Vibrations andtheEffect ofDissipative Forces 263 Equation (6.77) isaquadratic equation forywithcoefﬁcients thatarematrix products ofthesame general typeasthose encountered inEq.(6.19). Byvirtue ofthesymmetry ofV,F,andT,thematrix products areallreal,ascanbeseenby expanding aasoz+iB(cf.Eq.(6.19)). Hence, ifyisasolution ofthequadratic equation, itscomplex conjugate y*must alsobeasolution. Now, thesumofthe tworoots ofaquadratic equation isthenegative ofthecoefﬁcient ofthelinear termdivided bythecoefﬁcient ofthesquare terrn lFa*_____i_._ y+y_2K_awa. (6.78) Hence, Kcanbeexpressed interms oftherealandimaginary parts ofa1-as _1-7:1](aza] -l-ﬂlﬁj) K_2T1<z(<1/<41: +5/J31) ' (6.79) Thedissipation function Fmust always bepositive, andTispositive deﬁnite; hence rccarmot benegative. Theoscillations ofthesystem maydecrease exponen- tially with time, buttheycannever increase withtime. Note thatif.7-7ispositive deﬁnite, 7cmust bedifferent from zero(andpositive), andallmodes willhave an exponential damping factor. Thefrequencies ofoscillation. given bytherealpan ofw,willofcourse beaffected bythedissipative forces, butthechange willbe small ifthedamping isnotverylarge during aperiod ofoscillation. Finally, wemayconsider forced sinusoidal oscillations inthepresence ofdis- sipative forces. Representing thevariation ofthedriving force withtimeby F]=F()]e""", where F0,maybecomplex, theequations ofmotion are v,,1;,-+F,-,-i7,+:r,,;;', =F(),e"°". (6.80) Ifweseekaparticular solution tothese equations ofthefomi '71=A)¢'“”'. weobtain thefollowing setofinhomogeneous linear equations fortheamplitudes AJ: (v,,-iw.7-",1-w2:r,,-)A, -F0,=0. (6.31) Thesolution tothese equations* mayeasily beobtained from Cramer’s rule: _91(0))A,_mm). (6.82) *They areofcourse merely theinhomogeneous version ofEqs.(674) Chapter 6Oscillations where D(w) isthedeterminant ofthecoefﬁcients ofA1inEq.(6.81) and DJ(w)isthemodiﬁcation inD(w) resulting when thejthcolunm isreplaced byFm...F0".Itisthedenominator D(w) thatisofpiincipal interest toushere, fortheresonances arise essentially outofthealgebraic fomi ofthedenominator. Now. Disthedeterminant appearing inthesecular equation corresponding tothe homogeneous equations (6.74); itsroots arethecomplex frequencies ofthefree modes ofvibration. Therequirement thatbothyandy*areroots ofEq.(6.77) means, onthebasisofEq.(6.75), thatifw,isarootofD(w), then—w,*isaroot. Forasystem ofndegrees offreedom, itistherefore possible torepresent D(w) as D(w) =G(a>—w1)(w —601)...(w—w,,)(w +wf)(w +0);)...(cu+(oz), where Gissome constant. Using product notation, anddenoting wby21:v,this representation canbewritten as ll 0(0))=Gn(2n(v —i),)+iK,)(2JT(‘U+‘U,) +i7c,). (6.33) i=1 When werationalize Eq.(6.83) toseparate A,intoitsrealandimaginary parts, thedenominator willbe D*(w)D(w) =00*l£[(4rr2(v -v,)2+lc,2)(47r2(v +v,)2+K2). (6.84)[=1 Theamplitudes oftheforced oscillation thusexhibit typical resonance behav- iorintheneighborhood ofthefrequencies offreeoscillations :l:v,. Asaresult of thepresence ofthedamping constants 1c,,theresonance denominators nolonger vanish atthefreemode frequencies, andtheamplitudes remain ﬁnite. Thedriving frequency atwhich theamplitude peaks isnolonger exactly atthefreefrequencies because offrequency dependence ofterms inA1-other thantheparticular reso- nance denominator. However, solongasthedamping issmall enough topreserve arecognizable resonant peak, theshiftintheresonance frequencies isusually small. Wehavediscussed theproperties ofsmall oscillations solely interms ofme- chanical systems. Thereader however hasundoubtedly noticed thesimilarity with thetheory oftheoscillations ofelectrical networks. Theequations ofmo- tion(6.68) become thecircuit equations forncoupled circuits ifwereadtheV,J coefﬁcients asreciprocal capacitances, the.7-',-_;’s asresistances, andtheT,]’s as inductarices. Driving forces arereplaced bygenerators offrequency wapplied to oneormore ofthecircuits, andtheequations offorced vibration (6.80) reduce to theelectrical circuit equations (2.42) mentioned inChapter 2. Wehavepresented hereonlyafraction ofthetechniques thathavebeendevised forhandling small oscillations, andofthegeneral theorems about themotion. For example, space does notpermit adiscussion ofthepowerful Laplace transform teclmiques tostudy theresponse ofalinearly oscillating system todriving forces 6.6I6.6 TheDamped Driven Pendulum andtheJosephson Junction 265 witharbitrary timedependencies. Norisitappropriate heretofullyconsider the extensive subject ofnonlinear oscillations, where thepotential energy contains tenns beyond thequadratic, andthemotion isnolonger simple harmonic. (Some relevant portions ofthisﬁeld willbeintroduced later when wetreat chaos and perturbation theory). Asmentioned earlier, aformal development ofthetheory ofsmall oscillations about steady motion willbegiven later inconnection with theHamiltonian version ofmechanics. Another generalization thatwilldeserve ourattention relates totheoscillation ofsystems withcontinuously inﬁnite num- bersofdegrees offreedom. Thequestion ishowwecanconstruct awayofhan- dling continuous systems thatisanalogous totheclassical mechanics ofdiscrete systems. Weshall postpone such considerations ofcontinuous systems toChap- terl3-—after wehave developed thecanonical formulation ofdiscrete mechanics, andafter wehave seenhowthestructure ofNewtonian mechanics must bemodi- ﬁedinthespecial theory ofrelativity. BEYOND SMALL OSCILLATIONS: THE DAMPED DRIVEN PENDULUM AND THEJOSEPHSON IUNCTION Asanexample offorced vibrations withdissipative forces, weconsider themo- tionofthependulum sketched inFig.6.6,which issubjected toanapplied torque N,andispermitted torotate through itsfullrange ofmotion -1:5¢5rr.In addition, thependulum issubject todamping bytheviscosity r;ofthemedium in which itrotates. Forsimplicity, wewillassume thattherodismassless, andthat allofthependulum mass isconcentrated attheendoftherod. Letusbegin byrecalling thedynamics ofasimple pendulum oflength Rand mass m.The angular acceleration ofthependulum isproduced bytherestoring TC -——; /X _-___i‘1v=0 N=;IfmgR N=mgR=Nc ¢=0 <15=30° ¢=90° (=1) (bl (tr) FIGURE 6.6 Pendulum (a)with noapplied torque, N=0,(b)withthetorque N= %mgR, and(c)withthecritical torque applied, NC=mgR. Figures 6.6,6.8,6.10, and6.11 areadapted from C.P.Poole, J1..H.A.Farach andR.J.Creswick, “Superconductivity,” Wiley, NY.1995. 66 Chapter 6Oscillations gravitational torque mgRsin¢corresponding totheequation ofmotion 2d2¢ -mR dtg+mgRs1n¢ =0, (6.85) where I=mR2 isthemoment ofinertia. Forsmall angular displacements, the approximation sin¢ k¢lineaﬁzes theproblem bymaking thetorque propor- tional tothedisplacement, andthemotion issimple harmonic, ¢=¢()sinwtwith thecharacteristic frequency mo mg=(%)'/2 (6.86) Ifatorque Nisapplied toastationary pendulum, itwillswing outthrough an angle 45.Theforce ofgravity acting onthemass mprovides therestoring torque mgR sin¢,aswenoted above, andthependulum assumes anequilibrium position attheangle ¢given by N=mgR sin¢ =0), (6.87) asindicated inFig.6.6b. Thegreater thetorque, thelarger theangle ¢.There is acritical torque N‘indicated onFig.6.6(c) forwhich theangle <1:assumes the values J1.’/2: N,=mgR. (6.22) IfNexceeds thiscritical value, thentheapplied torque becomes larger thanthe restoring torque, N>mgR sin¢,forallangles ¢.Asaresult, thependulum wﬂl begin torotate beyond qb=rr/2,anditwillcontinue torotate aslongasthetorque N>NCisapplied. Themotion willtakeplace atavariable angular speed w _Qw-dl, (6.89) anditcanpersist ifthetorque islaterremoved. With these factsinmind, letusproceed toexamine thecaseofthedamped pendulum assuming thatthedamping force Fdamp =nw1sproportional tothe angular velocity cu.Towrite thedifferential equation ofitsmotion, weaddthe restoring anddamping torques mgRsin¢and17d¢/dz‘, respectively, toEq.(6.85): N=mR2d ‘Z5+ndi+mgRsin¢. (6.90)dr° dz lfwedeﬁne acritical frequency wtcorresponding totheangular speed atwhich thedamping torque moequals thecritical torque mgR, no‘=@=E, (6.91)nn 6.6 TheDamped Driven Pendulum andtheJosephson Junction 267 thenwecanwrite thependulum equation (6.90) inthenormalized fonn 1v 1d2¢ 1d¢—=—— —— '. .2 Thesolutions ofthisequation exhibit complex timevariations oftheangular po- sition ¢(t). When aconstant torque isapplied tothependulum atrest,there willbeainitial transient behavior thateventually settles down toadynamic steady state afterthe transients dieout.Weshall examine several cases ofthisdynamic steady state. 1.Forlowapplied torques, N5N,,thereisastatic steady state N=NCsin¢, (6.93) inwhich alltimederivatives vanish aftertheinitial oscillations havedied out.This isillustrated inFig.6.6b with thependulum stationary atthe angle ¢. 2.Forundamped motion (:7=0)withaconstant applied torque, N,Eq.(6.90) assumes theform . dztorque =N—mgR s1n¢ =mR2T§. (6.94) soweseethattheacting torque isangularly dependent. Thistorque has special values atfourparticular angles: torque =N ¢=O (6.95a) torque =N—NC ¢=Tl’/2 (6.95b) torque =N ¢=JT (6.95c) torque =N+N,~ ¢=3rr/2 (6.95d) Iftheapplied torque Nexceeds thecritical torque NC,themotion willbe continuously accelerated rotation, andthependulum increases itsenergy as timegoes on.Theangular speed alsoincreases withtime, butwith ﬂuctu- ations thatrepeat every cycle, asindicated inFig.6.7.Note thatFig.6.7 isdrawn forthecase where damping ispresent. Theaverage over these oscillations provides theaverage angular speed 11¢=— 6.96 (w) <dt () which continually increases linearly withthetime. 3.When damping ispresent withwc<<wt;andN>NC,theangular speed wcontinues toincrease until thedamping temi 17d¢/dt approaches the 6 Chapter 6Oscillations l.(l-9);,-__ ..- } ? w I / / / / / ~/ / timei> FIGURE 6.7 Dependence oftheangular velocity w=d¢/dt onthetimeforanapplied torque N>NC.Theaverage value (w)increases linearly with time intheabsence of damping (linear region), andtheoverall curve applies tothecasewe<<coowithdamping. value oftheapplied torque. When thisoccurs. theaverage angular speed (w) approaches alimiting value (co)L,asshown inFig.6.7,andtheacceleration ﬂuctuates around anaverage thatiszero: (d¢2/dtz) =O.Thependulum undergoes what iscalled quasi-static motion, rotating withanangular speed wthatundergoes periodic variations butalways remains close totheaverage (wlL- Toobtain more insight intothisquasi-static behavior, weneglect theac— celeration termintheequation ofmotion (6.92), andwrite N1d¢,E — +S1I1¢, which isanequation thatcanbesolved analytically withthesolutions (co)=0 forN<NC (6.98a) <0»=w,[(N/iv,)2 -1]‘/2 forN>iv, (6.98b) (co)=(N/N¢)a>c forN>>NC, (6.98c) which areplotted inFig.6.8.Theactual cyclic variations inwforpoints AandBonthisplotarepresented inFig.6.9.Atpoint A,theapplied torque hasthevalue N=1.2Nr, sofrom Eqs.(6.95) thenettorque varies between O.2N_. and2.2N,, around thecycle, andtheangular speed isfastat thebottom andslowatthetop,withthevariations shown atthelower partof Fig.6.9.Forpoint B,wehave N=ZNCsothenettorque varies between N, and3N0, producing themore regular variations inangular speed presented 6.6 TheDamped Driven Pendulum andtheJosephson Junction 269 l l ’ ZN_ wc<<a>, B / /" ///N A A//// NC / -///// / / / / I I at 21.0L‘ I‘ (w) FIGURE 6.8 Relationship between theapplied torque Nandtheaverage angular veloc- ity(co)forwc<<wn.Weseethat(w)=0forN<NLand(co)increases withincreasing N>NC. 6_ mg) 2r:/Q (w)+or=S— (co)—a=3 2.. A1_ I l I I l l I l50 100 150 200 250 300 350 400 timer i> FIGURE 6.9 Oscillations atpoints A(N=1.2N,,) andB(2N,-) forwc<<C00indicated onFig.68forthedamped harmonic oscillator. Adapted from A.Barone andG.Paterno, “Physics andApplications oftheJosephson Effect,” Wiley, NY,1982. 70 Chapter 6Oscillations atthetopofFig.6.9.Inthelimit N>>NC,meaning (w)>>wc,theangular speed begins toapproximate asinusoidal variation withtime w(t) »'¥(co)+orsinQt, (6.99) which approximates point BinFig.6.8. Forthenegligible damping case (17—>Oandwc>>coo),thesteady-state solution (6.98a) canstilloccur forN<NCwiththependulum heldﬁxed attheangle ¢deﬁned byEq.(6.93), which means thatw=(w)=0. Inaddition, thesolution, (6.98c), inwhich thetorque balances thetime averaged damping force, nowapplies forallvalues ofN,bothlessthanand greater thanNC,andsowehave w=0 forN5NC (6.l00a) (w)=(N/N¢)wC for05N (6.l00b) These solutions areplotted inFig.6.10. Note from theﬁgure thatthesystem exhibits hysteresis, meaning thatthebehavior differs forincreasing andde- creasing torques. When thetorque isincreased forN<NC,thependulum is stabilized attheangle ¢satisfying therelation N=NCsin¢ofEq.(6.87). soto=0viaEq.(6.l00a). When Nreaches thecritical torque NC,the angular speed jumps tothevalue wc,andthenriseslinearly withfurther increases inN,asshown intheﬁgure. Fordecreasing torques, Eq.(6.l00b) applies, and(cu)remains proportional toNallthewaytotheorigin, as shown. Figure 6.8shows theresponse for(0,;<<wt),Fig.6.10 presents itfor 0),;>>con,andthequestion arises astowhat isthebehavior foraninter- mediate condition suchaswc~£00?Thisrequires solving thegeneral I I ZN‘_ wt>>wo _ /7 I lw‘ 2(oC (w) FIGURE 6.10 Relationship between theapplied torque Nandtheaverage angular \e- locity (at)forw,;>>coo.There ishysteresis forthebehavior when (cu)<wc. 6.6 TheDamped Driven Pendulum andtheJosephson Junction 271 ‘i le i 2Nc_ co‘.=2w0 _ /,‘/N NF-——r-—— - , /NC // li’//' ,1 i i w‘ 2028 (w) FIGURE 6.11 Relationship between theaverage angular velocity ofthependulum (co) andtheapplied torque N.Forlowapplied torques, thependulum oscillates andtheav- erage velocity lSzero, whereas athightorques, N>NC,motion iscontinuous with (w) proportional toN.Notethehysteresis forincreasing anddecreasing torques. equation (6.92) since noapproximations canbemade. TheNversus (w) characteristic fortheparticular caseav,=2w0isplotted inFig.6.11. We seefrom theﬁgure thatforincreasing torques there istheusual initial risein Natzerofrequency untilthecritical value NCisreached, atwhich pointthe average angular speed jumps tococ,asinthesoc>>concaseofFig.6.10.For decreasing torques, thereishysteresis withzeroaverage frequency reached atatorque Né,which islessthanNa. Thedamped-driven pendulum equation (6.92) hasaparticularly important ap- plication insolid-state physics. When twosuperconductors areinclose proximity with athinlayer ofinsulating material between them, thearrangement consti- tutes aJosephson junction, which hastheproperty thatelectric current Icanﬂow across thejunction withzeroapplied voltage, uptoacertain critical value Ic.Cur- rentexceeding thisvalue isaccompanied bythepresence ofavoltage, andplots ofcurrent Iversus voltage Vforthejunction exhibit hysteresis. TheJosephson junction satisﬁes thesame differential equation (6.93) astheclamped oscillator withthecurrent playing theroleofthetorque, thevoltage playing theroleofthe average angular speed, thecapacitance acting likeamoment ofinertia, andthe electrical conductance serving astheviscosity. Thevariable. which istheangle ¢fortheoscillator, becomes thephase difference 1/1across theJosephson junc- tion.Many physicists ﬁndithelpful toobtain anintuitive understanding ofthe operation oftheJosephson junction bystudying properties ofthedamped driven pendulum thatmimics itsbehavior. Chapter 6Oscillations DERIVATIONS 1 2Theproblem ofthelinear iriatomic molecule canbereduced tooneoftwodegrees of freedom byintroducing coordinates yi=x2—x1,yz=x3—x2, andeliminating x2by requiring thatthecenter ofmass remain atrest.Obtain thefrequencies ofthenonnal modes inthese coordinates andshow thattheyagree withtheresults ofSection 6.4. Thedistances between theatoms, y]and)/'2,areknown asintemal coordinates. Obtain thefrequencies oflongitudinal vibration ofthemolecule discussed 111Sec- l.l0l'l6.4,except thatnowthecenter atom istobeconsidered bound totheorigin bya spring offorce constant k.Show thatthetranslational mode disappears EXERCISES 3. 4. 5. 6.Abead ofmass misconstmined tomove onahoop ofradius R.Thehoop rotates withconstant angular velocity toaround adiameter ofthehoop. which isavertical axis(linealong which gravity acts). (a)setuptheLagrangian andobtain theequations ofmotion ofthebead. (b)Findthecritical angular velocity 5'2below which thebottom ofthehoop provides astable equilibrium forthebead. (c)Findthestable equilibrium position forw>Q. Obtain thenormal modes ofvibration forthedouble pendulum shown inFig.1.4, assuming equal lengths, butnotequal masses. Show thatwhen thelower mass is small compared totheupper one,thetworesonant frequencies arealmost equal. Ifthe pendula aresetinmotion bypulling theupper mass slightly away from thevertical andthenreleasing it,show thatsubsequent motion issuchthatatregular intervals one pendulum isatrestwhile theother hasitsmaximum amplitude Thisisthefamiliar phenomenon of“beats.” (a)Inthelinear triatomjc molecule, suppose theinitial condition isthatthecenter atom isatrestbutdisplaced byanamount £10ﬁ'om equilibrium, theother two being attheir equilibrium points. Find theamplitudes ofthelongitudinal small oscillations about thecenter ofmass. Give theamplitudes ofthenonnal modes (b)Repeat part(a)butwith thecenter atom initially atitsequilibrium position but withaninitial speed vo. (a)Aﬁve-atom linear molecule issimulated byaconﬁguration ofmasses andideal springs thatlooks likethefollowing diagram- m M m M m l b 2 b 3 b 4 b S Allforce constants areequal. Find theeigenfrequencies andnormal modes for longitudinal vibrations. [Hm1: Transform thecoordinates 1),to5,deﬁned by 123-=63, 7ll=§L|_'E§'~ vs=ni§~/5 ~/5 Exercises 273 7 8 9. 10. ll.with symmetrical expressions for172andr74.Thesecular determinant willthen factor intodeterinmants oflower rank.] (b)Solve thisproblem using computer techniques. Inthelinear triatormc molecule, suppose thatmotion intheyandzdirections is govemed bythepotentials IQ7e'l~J?§~7‘?k V;=-(Y2-)’l)2+503-y2)2$ V1=—(Z2—102+5&3—zz)2- Find theeigenfrequencies forsmall vibrations inthree dimensions anddescribe the normal modes. What symmetries dothezerofrequencies represent’? Youmaywant to usethekindofintermediate coordinates suggested inExercise 6. Theequilibrium conﬁguration ofamolecule isrepresented bythree atoms ofequal mass atthevertices ofa45°nghttriangle cormected bysprings ofequal force con- stant. Obtain thesecular deteriiunant forthemodes ofvibration intheplane andshow byrearrangement ofthecolumns thatthesecular equation hasatriple rootw=0. Reduce thedeteniiinant tooneofthird rankandobtain thenonvanishing frequencies offreevibration. Show directly thattheequations ofmotion ofthepreceding problem aresatisﬁed by (ii)auniform translation ofallatoms along thexaxis, (b)aunifonn translation along theyaxis. and(c)auniform rotation about thezaxis (a)Three equal mass points haveequilibrium positions atthevertices ofanequi- lateral triangle. They areconnected byequal springs thatliealong thearcsof thecircle circumscribing thetriangle. Mass points andsprings areconstrained to move onlyonthecircle. sothat,forexample, thepotential energy ofaspring is determined bythearclength covered Determine theeigenfrequencies andnormal modes ofsmall oscillations intheplane. Identify physically anyzerofrequencies. (b)Suppose oneofthesprings hasachange inforce constant 8/c,theothers remaimng unchanged. Toﬁrstorder in8k,what arethechanges intheeigenfrequencies and normal modes‘? (c)Suppose what ischanged isthemass ofoneoftheparticles byanamount Sm. Now howdothenormal eigenfrequencies andnormal modes change? Auniform baroflength Iandmass missuspended bytwoequal springs ofequilibrium length handforce constant k,asshown inthediagram. T.__.__._oiwe 5#:- Q:QL_____. I Findthenormal modes ofsmall oscillation intheplane. 274 Chapter 6Oscillations I2.Twoparticles move inonedimension attheJunction ofthree springs, asshown inthe ﬁgure. Thesprings allhaveunstretched lengths equal toa.andtheforce constants and masses areshown 7 k 3k k a m G m a A Findtheeigenfrequencies andnormal modes ofthesystem. 13.Twomass points ofequal mass mareconnected toeach other andtoﬁxed points by three equal springs offorce constant k,asshown inthediagram. 7 / k ,,, k m It a +4 t1 +4 a /, Theequilibrium length ofeachspring isa.Eachmasspointhasapositive charge +q, andtheyrepeleachotheraccording totheCoulomb law.Setupthesecular equation forthecigenfrequencies. 14.Findexpressions fortheeigcnfrequencies ofthefollowing electrical coupled circuit. éla]!écsei % 15.Ifthegeneralized driving forces Q,arenotsinusoidal, show thattheforced vibrations ofthenomial coordinates intheabsence ofdamping aregiven by l +00 G1 ZLIJI=Z _i -,1, “ml... ...3_.,1‘ “’ where G,(co)istheFourier transform ofQ,deﬁned by (E Qi(I)= Gi(w)@"‘"‘dw- Ifthedissipation function issimultaneously diagonalized along with TandV,show thattheforced vibrations aregiven by 21:—oo (H1?—w2)2+w2.7"l2 Exercises 275 which hasthetypical resonance denominator fonn. These results aresimple illus- trations ofthepowerful teclmique oftheoperational calculus forhandling transient vibrations. Amass particle moves inaconstant vertical gravitational ﬁeldalong theCUW6 deﬁned byy=ax4,where yisthevertical direction. Findtheequation ofmotion forsmall oscillations about theposition ofequilibrium. Aplane triatomic molecule consists ofequal masses matvertices ofanequilateral tnangle ofsidesa.Assume themolecule isheldtogether byforces thatarehannonic forsmall oscillations andthattheforce constants areidentical andequal tok.Allow motion onlyintheplane ofthemolecule. (a)Without writing theequations ofmotion, Justify yourreasoning onthenumber of normal modes ofthesystem andhowmany ofthese modes havezerofrequency. (b)Oneofthenormal modes corresponds toasymmetrical stretching ofallthree vertices ofthemolecule. Find thefrequency ofthismode. Aparticle inanisotropic three-dimensional harmonic oscillator potential hasanatural frequency ofcoo.Assume theparticle ischarged andthatcrossed static electric and magnetic ﬁelds areapplied. Findthevibration frequencies withthese electromagnetic ﬁelds present. Discuss theresults forthelimits ofstrong andweak ﬁelds. Show forthecaseV11>V22>0andV12=V21=0inEq.(6.27) thatthere aretwo normal modes withfrequencies 401=(V1|)l/2 andcu;=(V22)1/2. Reintroduce the mass factor manddescribe aphysical system thatwould show thisbehavior forsmall oscillations. Write theLagrangian forthecaseV11=V22=0andV12=V21>0fortheexample discussed inEqs.(6.27) to(6.30). Show there isonenormal mode ofsimple harmonic motion withthefrequency ml=(V12)l/2, andasecond mode inwhich theparticle isunbound, receding exponentially toinﬁnity forlongtimer>tinaccordance with theexpression e"‘/T, where theparameter tisgiven byr=(V|2)_l/2. Forthis unbounded mode, howdoesthedistance depend upon timewhen t<1:‘!What isthe nature ofthepomt x1=X2=0'?Restate your results with themass parameter m included explicitly. Write theLagrangian discussed inEqs.(6.27) to(6.30) inpolar coordinates forthe caseV11=V2;>0andV12=V21=0,Show thatthere isaradial normal mode r= r0cos(wr) withfrequency w=(V11)l/2 when theangular momentum iszero. Show thatinthecaseofnonzero angular momentum, theangular momentum isconserved andtheparticle cartnolonger reach r=0.Write theﬁctitious potential energy V’(r) (Chapter 3)fornonzero angular momentum. When ﬁnished, reintroduce themass parameter, m,intoallequations. Repeat Exercise 21withtheconditions V11>V22<0andV12=V21=0and discuss yourresults mterms oftheeffective potential energy ofChapter 3. Make afullanalysis oftheexample discussed inEqs.(6.27) to(6.30). CHAPTER 276TheClassical Mechanics ofthe Special Theory ofRelativity Attheendofthenineteenth century, thephysics community hadtwoincom- patible descriptions ofphenomena, Newtonian mechanics andMaxwellian elec- tromagnetic theory. Newtonian mechanics assumed thatallinertial frames were equivalent, while Maxwell’s wave equations gave auniversal speed oflight that wasthesame inallinertial frames. Albert Einstein developed thespecial theory ofrelativity toreplace Newtonian mechanics with atheory thatwasconsistent withelectromagnetic theory. After abrief historical survey, weshall review the assumptions ofthespecial theory andtheconsequences ofthese assumptions. We shall thenexamine theformalism ofthegeometric picture ofspacetime thatre- sults. Lastly, wedevelop aLagrangian formalism andstudy attempts toexpress theresults inaproper relativistic fonn. lnNewtonian mechanics, asetofwell-veriﬁed lawsapplies inaninertial frame ofreference deﬁned bytheﬁrstlaw.Anyframe moving atconstant velocity with respect toaninertial frame isalsoaninertial frame. Consider twoframes denoted bySandS’with(t,x,y,z)and(t',x’,y’.z’)thecoordinates inSandS’,respec- tively. Without lossofgenerality, weassume thecoordinate axesarealigned, x along x’,andsoon.LetS’bemoving relative toSinthe+x-direction ataspeed v,asshown inFigure 7.l. Newtonian mechanics assumes thespacetime coordinates inSarerelated to those inS’bythesimple expressions r’=r x'=x—vt y'=y (7.1) z'=z. Transformations ofthistypearecalled Galilean transformations. Under thisas- sumption, itfollows thatNewton’s second law, dF=—, dzp relating theapplied force, F,andthemomentum, p,remains invariant, and F=F’, t=t’, and p=p’. (7.2) 7.1I7.1 Basic Postulates oftheSpecial Theoiy 277 z S J x Z! S’ ,Y V ——->- xi FIGURE 7.1Galilean transformation fromStoS’byavelocity vinthe+x-direction. ThetimeinboththeSandS’frames isassumed tobe(I=r’).TheNewto- nianworld viewisthattheuniverse consists ofthree spatial directions andone timedirection. Allobservers agree onthetimedirection uptoapossible choice ofunits. Under these assumptions, there arenouniversal velocities. Ifuandu’ arethevelocities ofaparticle asmeasured intwoframes moving with relative velocity vasdeﬁned byFigure 7.1.then u’=u—v. (7.3) Maxwell’s electromagnetic equations, ontheother hand, haveauniversal con- stant(denoted byc),which isinterpreted asthespeed oflight.Since thisisincon- sistent withNewtonian mechanics, either Newtonian orMaxwellian mechanics would havetobemodiﬁed. After carefully thinking about howtheuniverse would appear toanobserver traveling atthespeed oflight, Albert Einstein decided that Maxwell’s equations arecorrect toallinertial observers andtheassumed trans- formations forNewtonian mechanics areincorrect. Thecorrect transformations make thespeed oflightthesame toallinertial observers. BASIC POSTULATES OFTHE SPECIAL THEORY Einstein usedtwopostulates todevelop what became known asthespecial theory: 1.Thelawsofphysics arethesame toallinertial Observers. 2.Thespeed oflightisthesametoallinertial observers. Afomiulation ofphysics thatexplicitly incorporates these twopostulates is saidtobecovariant. Since thespeed oflight, c,isthesame inallcoordinate systems, itisreasonable toconsider thenumerical value ofcasaconversion factor between theunitsusedinmeasuring space andtheunitsusedinmeasuring time. So,cdtisthetime interval measured inthesame units used tomeasure space units. IntheSIsystem ofunits, cdthasdimensions ofmeters. Many books Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity andarticles onrelativity setc=1andmeasure timeandspace inmeters. Inthe material thatfollows, weshall show theexplicit dependence upon c. Tosatisfy thetwopostulates, thespace andtimeofthespecial theory consist ofasingle entity thatwerefertoasspacetime. Thisspacetime isthegeometric framework within which weperform physics. Wecannot assume thatallobservers make thesame division intotime andspace inthesame way. Theseparation is unique toeach inertial frame. Thesquare ofthedistance inthatspacetime, As-2, between twopoints AandBisgiven by (As)2 =c2(time interval)2 —(space interval)2, (7.4) where theinterval isbetween thetwopoints AandB.Iftheseparation ofthe interval isassumed tobeinﬁnitesimal, theAisreplaced bythedifferential symbol d.Since apoint inspacetime consists ofaspeciﬁcation ofthreespatial coordinate values andonetimevalue, theusual convention istorefer toapoint inspacetime asanevent. Thetermevent isusedbecause suchapoint hasadeﬁnite location andadeﬁnite timeinanyframe. Thechoice ofopposite signs forthetimeandspace intervals isintrinsic to thetheory; however, thechoice ofapositive signfor(cdt)2isarbitrary. Some authors deﬁne a(ds)2, which isthenegative ofthechoice given inEq.(7.4). All signchoices makes (ds)2 =0according tothedeﬁnition inEq.(7.4)forlight, since thespace interval is:l:(cxtimeinterval). Thechoice made hereforthe relative signs usedforspace andtimeissuchthatrealbodies moving atavelocity lessthanlighthave(ds)2 >0.Thismakes dsrealforbodies moving slower than lightspeed. If(ds)2 >0,theinterval iscalled rimelike. If(ds)2 <0,theinterval iscalledspacelike. Intervals forwhich(ml=0arecalledlightlike ornull. Since, toallinertial observers, objects thattravel ontimelike paths move less thanthespeed oflight, theyarecalled tardyons. Hypothetical bodies thatalways move faster thanlight arecalled tachyons, butsuch bodies willnotconcern us here.Objects moving atthespeed oflightarecalled nullorlightlilce. Inthelimit ofsmall displacements (differential displacements), Eq.(7.4) be- comes, inaCartesian coordinate system, (mi=(cdz)2-(dxz+dyz+(112). 0.4’) Thefour-dimensional space withaninterval deﬁned byEqs.(7.4) or('7.4’), 1s often called Minkowski space todistinguish itfrom afour—dimensional Euclidean space forwhich there would benominus signinEqs. (7.4) or(7.4'). Theidea ofusing ictforthetime coordinate tomake thespace Euclidean isnolonger useful since itobscures thenon-Euclidean nature ofspacetime andmakes the generalization tononinertial frames more difﬁcult. Since theinterval between twoevents ofspacetime isageometric quantity. allinertial observers measure coordinates thatpreserve thevalue oftheinterval squared, (ds)2. IfSandS’aretwodifferent inertial frames, then ds'2=ari. (7.5) 7.1 Basic Postulates oftheSpecial Theory 279 Thus, (ds)2 iscalled thesquare oftheinvariant spacetime interval. Forthisto bepossible, thetransformations between thecoordinates inS’andthose inS, mustinvolve therelative velocity between theframes inboththespace andthe titne parts; thatis,thetime coordinate cannolonger stand independent ofthe transformation. Thismeans therelative splitting ofspacetime intospace andtime willbedifferent fordifferent inertial observers. Since thetimemeasured inalab- oratory frame isdifferent fromthatmeasured byanobserver atrestwithrespect tothebodyunder study, wemustdistinguish thesetimes. Wedistinguish themby calling thetimemeasured byclocks atrestwithrespect toabodytheproper time, while theother inertial observer usesatimethatisoften called laboratory time. Asaspecial caseofEq.(7.4), consider therelation between theproper time,r, measured byanobserver atrestwithrespect toanobject inframe S’withcoordi- nates (r.x’,y’,z’).which ismoving atavelocity, v,withrespect toalaboratory frame Swith coordinates (r,x,y,z).Intherestframe oftheobject, there isno motion, soEqs.(7.4') and(7.5)give c2(dr)2 =c2(dr)2 -v2(dt)2 =t-2(¢z¢)2 01' dz=_“’_2_ (7.6) t/1'5 Since Eq.(7.6)makes dr<dt,thiseffect ondtiscalled “time dilation": moving clocks appear torunslower. Theinvariance oftheinterval expressed inEq.(7.5), naturally divides space- timeintofourregions, sketched inFig.7.2relative toanyevent Aattimet_A(A islocated atx=y=t=0inFigure 7.2).Ifanevent Battime:3issuchthat (ds_AB)2 >0,thenallinertial observers willagree onthetimeorder oftheevents AandZ5’.Itiseven possible tochoose aninertial frame where Bhasthesarne space coordinates asA.lf15islessthantAinoneinertial frame, thenI5isless thanQ4inallinertial frames. Wecallthisregion thepast. Likewise, there isa region called thefuture where forevent C(shown inFigure 7.2),tcisgreater than t_,4forallinertial observers. Both thepastandthefuture could becausally related totheevent A.Foranyevent inside thelight cone, there exists aframe inwhich thatevent andtheorigin have thesame x,y,zcoordinates. If(dS_A_'[))2 <O,thenthere exist asetofinertial frames inwhich therelative order oft,4andt1;canbereversed orevenmade equal. Thisregion hassometimes been referred toastheelsewhere, orastheelsewhen. Intheregion inwhich event Dislocated, there exists aninertial frame S’withitsorigin atevent theAin Wll1Ch Disatthesame timeasA(butsomewhere else). There alsoexist frames inwhich thetimeofDoccurs before Aandframes inwhich thetimeofDisafter event A.Separating thepast-future andtheelsewhere isthenullorlight cone, where dsz=0.Thenullcone isthesetofspacetime points from which emitted 0 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity 7.2 IA(‘Z C 1) > nullorlzghrcone J D- / *elsewhen FIGURE 7.2 Thethree dimensions (ct,x,andy)ofthelight cone. Thethird spatial dimension hasbeen suppressed. Theevent Areferenced I11thetextislocated atx=y ct=0.Thelight cone isthesetof(ct,x,y)traced outbylight emitted from ct=x= y=0orbylightthatreaches x=y=0attimeer=0.Thepastandfuture lieinside the lightcone. Thisﬁgure isofnecessity misleading because allpoints onthelight cone have zeroseparation inspacetime. lightcould reach event A,andthose points from which light emitted from event Acould reach. Anyinterval between theorigin andapoint inside thelightcone istimelike, andanyinterval between theorigin toapoint outside thelightconeis spacelike. Understanding theimplication ofthedivision ofspacetime bythelight coneisusually allthatisneeded toresolve theapparent paradoxes ofthespecial theory. LORENTZ TRANSFORMATIONS Thesimplest setoftransformations thatpreserve theinvariance oftheinterval, ds2,arecalled theLorentz lrcmsforrnations. These transformations aresimplest in thesense thattheyarelinear inthecoordinates andastherelative velocity goesto zero,thetransformations become identity transformations. Ifweconsider parallel Cartesian coordinate systems, SandS’,whose origins coincide att=t’=O,and whose relative velocity isvalong thexaxisasmeasured byS,anddeﬁne v l/3=Z. and ]/= , thenthefollowing fourequations relate thetwosetsofcoordinates ct’=Q =]/(CI —ﬁx) (7.8al ,/1-52 x’= =3/(x—ﬂat) (7.8bl ,/l—,63 7.2 Lorentz Transformations 281 >"=Y (7-36) 1’=z. (7.sd) Here weareonlyinterested intransformations forwhich t’->tandx’—>xas ﬁ—>0.Asmatrices, these transformations appear as <11’ 1/-1/fl x Z (7‘8I) z’ 0 InthelimitofB<<1,Eqs.(7.8)reduce totheGalilean transformations asex- pected. Thegeneralization toarbitrary orientation ofthevelocity relative totheaxes isstraightforward. Since weareconsidering spacetime afour-dimensional en- tity,wewould expect todealwithfour-dimensional vectors. Using thenotation (ct,x,y,z)=(ct,r)allows thewriting ofthegeneralization ofEqs.(7.8') tothe casewhere visnotparallel toanaxis,as'-'1 O QQY O'-‘CO |—OOO|________ITilNW.H9,l___€.___I ct'=y(ct—B-1‘) 1"=r+ —B)/ct, (7.9) provided thetwosetsofaxesarealigned. Another waytoexpress thisarbitrary velocity istoconsider theLorentz transformation between twoinertial coordi- natesystems with aligned axes, asamatrix transformation relating thetwo4- quantities, x=(ct,r)andx’=(ct’,r’),where x'=Lx (7.10) Wetreatx’andxascolumn matrices andLasthesymmetric matrix 1/ "l/ﬁx _ "V/3y —1//3? -yrs1+o»—1>% 0/—1>% <y—1>%;-lieL= , 3 ...(7.11)-we o~n%% 1+o-0% o—n%%2 "7/ﬁz tr—1>% (V—1>?;,% 1+0/—1% Thisreduces totheresults given inEqs.(7.8’) when ﬁx=5,ﬂy=ﬁz=0. These transformations maptheorigin ofSandtheorigin ofS’to(0,0,0,O). Hence thecoordinates ofbothorigins correspond tothesame location inspace- time.lfthisisnotdesired, thereisamore general transformation oftheform x’=l.x+a (1.12) 7.3IChapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity where Lisaspacetime rotation (boost) andaisaspacetime translation. Thisisthe Poincare’ tranjormation ortheinhomogeneous Lorentz l‘rans_f0rmali0n. Weshall consider onlyhomogeneous transformations forwhich aofEq.(7.12) iszero. VELOCITY ADDITION AND THOMAS PRECESSION Themostgeneral homogeneous Lorentz transformation willinvolve bothaveloc- itychange andarotation ofthecoordinates. Thevelocity transformation istermed aboost andhastheformofEq.(7.11).Anyhomogeneous Lorentz transformation, L,canbewritten as L=RL0=i;,R' (7.13) where Risarotation matrix asdiscussed inChapter 4,andL0,which iscalled arestricted orproper Lorentz transformation, corresponds toapureboost. The restricted Lorentz transformations form arepresentation oftheLorentz group.* Since Risnotsymmetric andL0issymmetric, Lwill,ingeneral, havenosym- metry. Also, since LgandRarematrices, RL0;éLQR.There willexisttwoother transformations L6andR’suchthatRlg=l.{,R'. ForanyLorentz transfomiation, L,thereisaninverse transformation, L“1.such that ii-1=L-‘i=1, (7.14) where 1isthediagonal unit4x4matrix withelements 50,5.Theexistence of aninverse places fourconstraints onthediagonal element andsixontheoff- diagonal elements foratotal oftenconstraints ontheLorentz transformation. There arethenonly sixindependent components. Three ofthese correspond to thecomponents oftherelative velocity vector andthree correspond totheEuler angles oftherotation (seeSection 4.4). Consider threeinertial systems, S1,S2,andS3,withxaxesaligned. LetSgbe moving atavelocity valong thecommon x-direction withrespect toS1andlet S;bemoving atvelocity v’along thecormnon x-direction withrespect toS2.The Lorentz transformation fromSitoS3isgiven by H,_ Vi/'(1+l5l3') -1/1/’(ﬁ+ﬁ’) =-1/1/’(g+l‘3') VJ/'(1dH9»5')cc'<_‘<\‘*1 0oo‘<\ 0--co <3»—*ooooo~<‘Os o»—ooOO‘<~< >-Ooc‘omo»-co l-ocol-1-3= *Group concepts arediscussed inAppendix B. 7.3 Velocity Addition andThomas Precession 283 where Eq.(7.7)deﬁnes [3andyforvandI3’andy’forv’.Let,3”bethespeed of S3relative toS1andy”theassociated factor, thensince L1_3canbewritten asa single Lorentz transformation withavelocity ,5"withitsassociated y”as VII _//‘Bu _y//ﬂu 0 O oo"<=‘< 0»-oo i-ooo|-i-3= and,since thesetwoforms ofL1_3mustbethesame, wehave I!B+5’13_1+ﬁﬂ, (7.15) Thisistherelativistic addition ofvelocity formula forparallel velocities. Theproduct ofanytwotransformations, L1andL2isitself aLorentz trans- formation, L3.Such aLorentz transformation will,ingeneral, involve notonlya boost, butmayalsoinclude arotation ofcoordinate axes. ifbothL1andL2are pureboosts buttheirtwovelocities arenotparallel, L3willinvolve arotation in addition toaboost. Thisrotation iscalled theThomas precession rotation. The usual form fortheThomas precession assumes thesecond boost, Lghasave- locity small compared totheﬁrstboost, L1andalsothatitissmall compared to thespeed oflight. Forexample, theThomas precession canbeobserved fora gyroscope orbiting theEarth orforelectrons inatoms. Consider three inertial frames S1,SQ,andS3,withS;moving atavelocity [3 withrespect toSiandS3moving atavelocity ofB’withrespect toSg.Without lossofgenerality, wecanarrange theaxesofS1sothatBisalong thexaxisof S1andB’liesinthex’y’plane ofS2;thatis,/3,£5’deﬁne thex’y’plane ofS2.Let Lrepresent thetransformation fromS;toS2andL’thetransformation fromS;to S3withyandy’associated withBandB’.Then fromEq.(7.11), O OOY O'—'OO P-‘COO1/—i/I3 L=‘gt’ (7.16) and 2/’ —i/'5} —i/'5} 0'1, Mi’, _ III 1+(I_ (I_ 0 U: rﬂ, rIé2 1/ 52,2 I (7.17) II I Bx!/‘yr 1 ﬂlyl -V/3y (V-1)"? l+(l’“1)F 0 O 0 O l. Weassume thatthecomponents ofB’aresmall andonlyneedberetained toﬁrst order giving viamatrix multiplications ofEq.(7.16) andEq.(7.17) 4 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity O I-QCOri/’ —)’7/.5 —i/’i6'§i LI! =LIL = Y I -l/V15;,i//ii/’ﬂ§,i r ()0 O 0 Since L”isnotsymmetric, itmustcorrespond toarotation andaboost. Weshall write thevelocity ofS3asobserved byS1asB”. Since theoff-diagonal elements corresponding tothezaxisarezero. thisro- tation isabout anaxisperpendicular tothexyplane. Theboost from S1toS3is denoted byB”,andweassume thatB’issmall compared toBandalsosmall compared tothespeed oflight(y’wl).Then, toﬁrstorder, thenonvanishing components of73"are(Since thevelocity perpendicular toxissmall wecanig- noretoﬁrstorder thedistinction among y.y’,andy”) I 13,1’=ti. /ii’= '3”=52, and y”=r. (7-19) andEq.(7.18) becomes yll _}/Ilﬂél _}/Ilﬂgzl "N _y//5'” I I. '\-' _)/"ﬂit ylrﬁgﬂy 1 O O 0‘<\o —-coo Inthisapproximation, apure Lorentz transformation from S3toS1(theinverse transformation) would correspond toalarge boost inthex”axisof—;8;’ anda small boost inthey”axisof—/8;’. TheLorentz boost forthattransfonnation yr! yr/ﬂip y//ﬁg’! 0 /1II I! II__1 0 1.3-1=y5‘ YE.O’ >51 . (7.20)yllﬂgl (yll _ 1 0 O 0 0 l Finally, therotation matrix induced bytherotation from S1toS3,after some algebraic simpliﬁcation andthedropping ofhigher-order tenns inB”,isfound tobe 1 O O 0ﬁll O 1 —l—’ OR=L”|.3_1 = 1/ )'8 . 0-(1/-1)% 1 0 O O O 1 Comparison withEq.(4.44) shows thatRimplies S3isrotated withrespect toS1 about thezaxisthrough aninﬁnitesimal angle: 7.3 Velocity Addition andThomas Precession 285 ﬂu /1 '_1AQ=(y—l)7;- =ﬁyﬁ . (7.22) Thespatial rotation resulting fromthesuccessive application oftwononparallel Lorentz transformations hasbeen declared every bitasparadoxical asthemore frequently discussed apparent violations ofcommon sense, suchastheso-called “twin paradox." Butthepresent apparent paradox hasimportant applications, es- pecially inatomic physics, andtherefore hasbeenabundantly veriﬁed experimen- tally. Consider aparticle moving inthelaboratory system withavelocity vthatis notconstant. Since thesystem inwhich theparticle isatrestisaccelerated with respect tothelaboratory, thetwosystems should notbeconnected byaLorentz transformation. Wecancircumvent thisdifﬁculty byafrequently usedstratagem (elevated bysome tothestatus ofanadditional postulate ofrelativity). Weimagine aninﬁnite number ofinertial systems moving uniformly relative tothelaboratory system, oneofwhich instantaneously matches thevelocity oftheparticle. The particle isthusinstantaneously atrestinaninertial system thatcanbeconnected to thelaboratory system byaLorentz transformation. Itisassumed thatthisLorentz transformation willalsodescribe theproperties oftheparticle anditstruerest system asseenfromthelaboratory system. Suppose nowthatS1isthelaboratory system, while S2andS3aretwoofthe instantaneous restsystems atimeAtapartintheparticle's motion. ByEq.(7.22), thelaboratory observer willseeachange intheparticle’s velocity inthistime, Av,which hasonlyay-component ﬁg,’c=Av.Since theinitial xaxishasbeen chosen along thedirection ofv=fie,thevector oftheinﬁnitesimal rotation in thistimecanbewritten as Aan=—(y-1)”-‘E5-Y (7.23) Hence, iftheparticle hassome speciﬁc direction attached toit(such asaspin vector), itwillbeobserved fromthelaboratory system thatthisdirection precesses withanangular velocity d9. vxa=——=— — i—— .24 wdt0/1)U2 (1) where aistheparticle’s acceleration asseenfromS1.Equation (7.24)isfrequency encountered intheformittakes when vissmall enough thatycanbeapproxi- mated (using ywl+5-52) as co=i%5(a xv). (7.25) Ineither form, toisknown astheThomas precession frequency. 86 7.4IChapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity VECTORS AND THE METRIC TENSOR Wewillusethenotation thatthecoordinates, which neednotbeCartesian, are written asxi‘where xo=ctisthetimecoordinate, andx1,x2.x3arethespace coordinates. Thischange innotation isneeded tobeconsistent withthedevelop- ments inthefollowing sections. Consider anarbitrary one-dimensional curve in4-dimensional spacetime. 73', described byaparameter A,where foragiven Athecoordinates ofapoint of thecurve canbewritten asx°()t), xl(Jl),x2()t), x3(7l). Inintroductory textsa4- vector, v,isdeﬁned bythiscurve asanarrow whose tailislocated atanevent A onthecurve andwhose head isatanevent Bonthecurve where HA3='P3—PA. However, instead ofdeﬁning thevector attwopoints, wecanusetheparameter It,which isameasure ofthelength along thecurve from AtoB,bywriting d'Pv_A3 -(Elmo. (7.26) Such a4-vector isatangent vector tothecurve. Weadopt thenotation thatthe components ofvectors arewritten withsuperscripts suchasv0,vi,v2,:13.Inspite ofthewaywedrawtangent vectors, theydonothaveanyextension inspacetime. Thearrows wedrawsimply helpusvisualize thevector. Ateachpoint along the curve, thetangent vector hasadirection andamagnitude. Forcurves thatare timelike, theproper time, t,isusually chosen astheparameter Jl.Thelaboratory coordinates arethenxo=ct('c), x’=x(t), x2=y(t), x3=z("c), andthe tangent tothecurve isthefour-velocity, u,ofaparticle traveling along thecurve 'P.Equation (7.26) becomes dct d‘uo=H=ye, u‘=%-=yv' (7.27) where v‘=dx‘/dz‘isthenormal three-velocity withv2=(v")2 +(v>)2 +(u‘)2. Weshallassume thatGreek letters cantakeonthevalues 0-3andLatin letters thevalues 1-3.Repeated indices aresummed. Since the4-velocity ofaparticle is deﬁned overarange oftheparameter it,there isaninﬁnite setof4-velocities for theparticle, oneforeachvalue ofJt.Suchasetofvectors istermed avector ﬁeld. Some common examples ofvector ﬁelds aregiven inTable 7.1. Weassume thatthecomponents ofany4-vector canbeexpressed bytheval- uesofthevector’s projections along asetofbasis vectors, en,e1,e1,e3,andthat thecoordinates aremeasured along thedirection given bythebasisvectors. Such asystem iscalled acoordinates basis.* Cartesian. spherical, andcylindrical co- ordinate systems, among many possible systems, canhave such abasis set.The position ofapoint onthecurve 'P('c) canbewritten as 'P(t) =x”“(t)e,,,, (7.28) *The choice ofacoordinate basis isarbitrary butavoids some complications. Forthisintroductory chapter wewillassume thateachbasis vector hesinthedirection ofiLsincreasing coordinate. 7.4 Vectors andtheMetric Tensor 287 TABLE 7.1 Examples ofVector Fields Time Space Name Portion Portion (Magnitude)2 Type l Coordinate ct r 1.2::—r2 spacelike, null,ortimelike ll l Velocity ya yv C2 timelike l Momentum — p m2c2 timelrke"‘-"1 Force Y%=yr_(FNeW(0m3n)2 »pa¢e11k= FREli. Current density ypc yj p262 ‘ timelrlte where repeated Greek indices, oneraised andonelowered, aresummed from0 to3.Inparticular, the4-velocity given inEq.(7.27) becomes 41>4#u=Z;=713%,, =u/‘e,,,. (7.29) Themagnitude ofthe4-velocity isascalar whose values canvaryaswe change it.Thissetofmagnitudes isanexample ofascalar ﬁeld. Toconvert a 4-vector ﬁeldtoascalar ﬁeld, weneedwhat iscalled afunctional,* which can convert apairofvectors intoascalar function ateachpoint inspacetime. Inother words, wewishtodeﬁne thescalar product oftwovectors orvector ﬁelds. This conversion ofa4-vector ﬁeld(ortwodifferent vector ﬁelds) toascalar ﬁeldis anexample ofamapping. Ifboththevectors arethesame, thenthisscalar would bethesquare ofthelength ofthevector, andwhen thevectors aredifferent, it iscalled thescalar product ofthevectors. Such afunctional iscalled themet- rictensor, g.lThemetric tensor functional canbeconsidered asamachine with twoslotsintowhich youcaninsert twovectors toproduce ascalar (real-valued function). That is, s(u.v)=g(v.u)=u-v. ('1-30) isthescalar product. Inparticular ifthebasisvectors areinserted intothemetric, gqﬁ =g(8q,6)3) =ea -65. Thego,/3arethecomponents ofthemetric tensor associated withthebasis vec- torsea.Forexample, consider atwo-dimensional Minkowski space withcoordi- nates ctandxandavector v=(a,b).Then g(v,v)=a2—b2andgm=1, 811=-1- Theformofthegagisdeﬁned bytheformfortheinterval. Thissuggests that weconsider small displacements. Iftherelative displacement vector between two *Ahinctional isatfunction whose arguments arethemselves functions. lWeusethesame notation fortensors in4-space aswedofor4-vectors. 8 Chapter 7TheClassical Mechanics oftheSP8ClEJl Theory ofRelativity points issmall, itcanbewritten as d;=Ax°‘e,,,. (7.32) Recasting Eq.(7.32) inthelanguage ofEq.(7.4’), weseeforMinkowski coordi- nates (As)2=dz-<1;=Ax°‘Ax'5e,, -6;;=g0,;;Ax°‘Axﬁ =rem)’—(Ax?—<A>»>’—(A1)? Inthelimit ofrnﬁnitesitnal displacements thiscanbewritten as dsz=g.,,,,.1x"¢1x§, 0.32’) which holds foranymetric tensor. Themetric tensor foraMinkowski coordinate system, using the+——— signconvention, hasthefollowing tensor representa- tion* OQO'—' CO O000 -100 5': -10 -1. (7.33) Thescalar product oftwovectors inthiscoordinate system is u-v=u°’vBg,,/5 =u0vo —ulvl —u2v2 —u3u3. (7.34) Itisstraightforward toshow thatinanycoordinate system, thesquare ofthe magnitude ofthefour-velocity is u-u=02. (7.35) The4-momentum canbedeﬁned fromEq.(7.27) p=mu, (7.36) where themass, m,isascalar. Sothelength squared ofthefour-momentum is p-p=m2c2, (7.37) orfrom Eqs.(7.27) and(7.34), E2 p_P=m2c2 =m2c2y2 _m2v2y2 =CT__P2 (738) *The notation used forthedisplay oramatrix 1S[J,while tortensors ()willbeused asitwas mChapter 5.Matrices areused forrelating different coordinate frames while tensors arephysical geometric objects. 7.5I7.5 l-Forms andTensors 289 where pisthelength ofthe3-momentum. ThislastformofEq.(7.38) isoften written as E2=m2C4+pit-2. 0.38’) Therelativistic kinetic energy, T,isdeﬁned as r=E-mC2=mc2(y-1) (7.39) =,/(mc2)2 +pzcz —mcz. (7.39’) For,8<<I,apower series expansion gives T=gmvz+0(,s“). (7.40) Since p=myv, Eq.(7.39) shows thatthekinetic energy ofabodywithﬁniterest masstends toinﬁnity asthespeed approaches thatoflight(as/3—>1,y-—>oo). Inother words, ittakes aninﬁnite amount ofenergy toincrease thespeed ofa mass particle (oraspace ship) from anyvelocity lessthanctocitself. Thisis another proof thatitisimpossible toattain orexceed thespeed oflightstarting from anyﬁnite speed lessthanc. 1-FORMS AND TENSORS* Suppose weinsert onlyone4-vector intothemetric tensor inEq.(7.30). We would produce anobject thatcould beWritten asum=g,,,5u5. Forexample. in thetwo-dimensional Minkowski space, ifu°‘hascomponents (a,b),thenuahas components (a,—b). Thisgeometric object, ua,iscalled a1-form or,inanolder notation, acovariant vector. Intheolder notation thevector itself wascalled a conrravariant vector. Ifthevector isthought ofasadirected line,thel-forrn isa setofnumbered surfaces through which thevector passes asisshown inFig.7.3. Itisanother functional (machine) similar tog,except itconverts avector toa linear real-valued scalar function. That is,if17isal-form (ﬁeld) andvissome vector (ﬁeld), thequantity denoted by(11,v)isanumber thattellsushowmany surfaces of11arepierced byv.Foreach vector ﬁeld V,there isanassociated 1- fortn. V,,suchthat(V,,,V)=V-Visthescalar contraction orthesquare ofthe magnitude ofV. Thegradient isanexample ofa1-form since, ifweconsider acurve 79,param- eterized by1,where A=Oat'P()andtakeascalar function, f,deﬁned along the curve. -_3 _i_ Hi a.f-@f<1><>t>)- MP0-vax, (1.41) *The material inSections 7.5and7.6ISnotneeded forSection 7.7TheSection order hasbeenchosen forcontinuity ofideas. Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity '7 surfaces positive sense of1; FIGURE 7.3Avector vbetween twoneighbonng points andal-form 11.Thepiercing ofnbyvproduces anumber given by(11,v).thenumber (including fractions) ofsurfaces pierced So .. 36,,=deg= (7.42) Weoften write either 3,ordtoindicate thegradient ofascalar. Several ex- amples ofvectors, 1-forms, scalar products. andmetrics fromrelativity andother areasofphysics aregiven inTable 7.2. Thegradient ofthecoordinates, co“,deﬁned as co“=dx“, (7.43) provides asetofbasis 1-fomis since (wa,05)=5%’. (T44) TABLE 7.2 Examples ofVectors and1-forms SYSTEM Vectors: l-forms. Scalar Metric (Contra\ ariant (Covariant Contraction Components) Components) Euclidean (dx,dy,dz) (dx,dy.dz) £112+dyz+431 100 Cartesian (x.y.z) 010 001 Euclidean (dr.d9,a¢) (4,-.r2d9. dr2+r2deg 10 0 Spherical r2sing911¢) +r2sin26d¢2 0r2 0 00r2sinze Solid-state r(lattice vector) k(reciprocal vector) r-k varies Quﬂnlum lhﬁofy Ill(Rel) (JI(bﬁl) (J11) |I')(JI Special theoryot (tdz.dr) (cm.—dr) 8ml-drz 000 relativity —l O 0 (Mirikowski) -10 OC)I—- Q O 0 0 -—l 7.5 1-Forms andTensors 291 andanyl-form 77canbewritten as 77=naw". (7.45) ltfollows that (77:31!) =mi (7.46) andforanyvector, v (77,v)=nan“. (7.47) Thisgives ustwoways tocalculate thescalar product oftwovectors vandii. Ifwedeﬁne theinverse metric by gaﬁgﬁy =5; (7.48) orinindex-free notation by g“s'=gs"=1. (7-48’) wecanconvert vectors (u"‘)to1-forms (ua)andconversely as u,,,=gapup and u“=g°’5u;;. (7.49) Wecantherefore writefortwo4-vectors uandv(ortheycould betwo1-forms), it-u=g(u,v)=gaﬁu“ vﬂ=ii“va=u,,,v;;g°"3. (7.34') Since each1-form hasaunique associated vector, wecould usethesame symbol forboth. Thedifference isimportant onlywhen considering components. Interms ofthetwo-dimensional example thatwepreviously considered (Mirikowski spacetime) withctandxasthecoordinates). ifthevector uhas components (a,b)andthevector vhascomponents (c,d),thelastthreeterms of thepreceding equation canbewritten as g...w“v" =<1>(a><<.~> +(-1>(1=><d> =ac-r>a. uaua =(a)(c) +(b)(—d) =ac—bd, and u..vtg"" =<a><c><1> +<1>><d><—1> =ac—bd- ltmayhelptoconsider therelationship between avector andal-form from a more general point ofviewusing theMirikowski two-dimensional space asan Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity example. Avector Vintwo-dimensional space withbasis vectors e1ande2can bewritten as V=V1e1+ vie; Ingeneral, itisnotnecessary thatanyofthebasisvectors benormalized (e|-er75 1,E2-e2751)orthattheybeorthogonal (e1-cgqé0).Thismeans thatthe magnitude ofthescalar product isnotconveniently obtained from asimple sum ofsquares 2 v-v=Zv'v1=(V1)2e1-e1+ VlV2(81-B2+92'81)+(V2)28g~e2y,]=l 2 72Zv‘v',i=l anditdoes nothave thevalue \/(V1)2 +(V2)2. Onewaytoobtain themagni- tudeofthevector istodeﬁne thedualspace withbasis vectors mland0:2(cf. Eq.(7.43)). which havetheproperties e1-ml =60] ~61=e2-w2=w2-e2=l and£1-w2= 002-er =e2-w1=w'-e2=0. Wesaythatthevector basis, e,,isorthonormal tothe1-form basisco’.Thel-form, v.corresponding tothevector Vmaybewritten as v=viml+D2602. Thisvector hasa(magnitLIde)2 of (magnitude)2 =v~V=V-U=Vlv1+ V2172. When wewanttorequire anobject tobeexpressed interms ofitscoordinate basis vectors wewillwrite withaRoman letter (e.g.,u)anduseGreek letters when it istobeexpressed interms ofthebasis 1-forms (e.g., 71).Thissame approach provides thescalar product oftwovectors VandUinterms oftheir associated 1-forms vanduas scalarproduct= V-u =v-U =u-V =U-v =V1u1+V2u2 =v1Ul+vgU2. These results areeasily generalized tomore dimensions, tospaces thathave anindeﬁnite metric, andeven tomore general spaces, suchasthose discussed in Section 7.11. Forexample, inafour-dimensional Mirikowski space, thel-form, v,associated withthevector V,isvg=V0,v1=—V1, v2=—V2, v3=—V3, sothesquared length ofthevector Vis v°v<,+Vlv1+ V2»;+vb);=v°v°-v1v1- V7'V2—V3V3. 7.5 1-Forms andTensors 293 TheLorentz transformations canbeexpressed intenns ofthebasisvectors. If weletx”,x1,x2,x3bethecoordinates inaframe Sandx"‘/=x°"(x0, xl,x2,x3) bethetransformed coordinates intheframe S’,thentheLorentz transformation canbewritten as x°"=t“',;xﬂ and X“=t",,,x/", (7.50) where Lat,’istheinverse transformation ofL°/,9. Thebasis vectors transform as ea:=t¢’,,,/eﬂ and ea=tﬁ',,,e,,/. (7.51) Anyvector transforms asv=v"ea =v/3,85’, so(17,v)=nav“ =17,,»v°‘J.This means thatI-forms transform as17=1),,10°‘=17,,/10”’, anditfollows that 01°"=Ldgwﬁ and to“=L°‘5'wﬁ', (7.52) so v""=L°"¢;v'6 and v°‘=L“/1/v5’, (7.53) and mu=LEW andit=t'“’.m'- (1.54) Toconvert vectors, sumonthesecond (lowered) index ofthetransformation ma- trix.Toconvert 1-forms, sumontheﬁrst(raised) index. Intensor notation, vectors arecolumns. while I-fonns arerows. Scalars, vectors and1-forms aresimple examples ofgeometric objects called tensors. Atensor isafunctional intowhich weinsert pvectors andn1-forms toproduce amapping ontoascalar. Wedescribe atensor bysaying thatithasa rank given bythenumbers nandp,where nisthenumber of1-forms insertions possible andpisthenumber ofpossible vector insertions. Atensor, Q,with n 71l-form slotsandpvector slotsiswritten asQofrank(P).Atensor Hofrank isafunctional intowhich wecaninsert n1-forms or,2»,....,3andpvec- torsu.v,...,wtoproduce ascalar. Forexample, theenergy momentum vector (E/c, p)isatensor ofrank since contracting itwith a1-form produces a scalar.7An example ofanordinary second-rank tensor isthequadrupole tensor of rank Although thecomponents of1-forms arewritten withtheirindices down, the number ofl-form slotsiswritten astheupper ofthetwonumbers usedtogivethe rankofatensor. Thisisbecause incomponent notation theobject generated will have thatnumber ofindices tobecontracted with 1-forms. Forexample, ifSisa tensor ofrankG), S(rr(,,w°', Apwﬁ, vyey) =aaA.,gv7'S(w°‘, mp,ey)=S°‘5,,o},)t,gv”, (7.55) Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity where theS‘*5,,arecalled thecomponents ofthetensor Sinthechosen coordinate frame. Theoutput ofSisascalar (seeEq.(7.55)), soifwerepeat thiscalcula- tioninanother Lorentz frame, weobtain thetransformation lawforthetensor components under acoordinate transformation, s°"*“',' =s°"’,,L“',,L5’;LY,,,. (7.56) Themetric tensor canbeused toconvert indices from vector to1-form or1-form tovector; forexample, Sam, =g5,,S°"’,,. (7.57) Hence, anytensor ofrank canbeconverted bythemetric tensor, without lossofinformation, toanyarrangement oftensor and1-form indices desired as longasthetotalnumber ofindices (n+p)isconserved. Allofthese objects are different coordinate fonns ofthesame geometric object (tensor). Consider ourtwo-dimensional example with avector, u,whose components are(a,b)andal-form, 0',withcomponents (c,d).Ifweexamine atensor Wof rank then, from Eq.(7.55), W(a,u)=W“/;a,,u'9 =w%¢a+w°1@b+Wlqda +What». Physically, byusing setsofvectors, u’s,andI-forms, a’s,andmeasuring the value ofthescalar ﬁeldW(a, u),thevalues ofthecomponents ofW"¢; canbe determined inoneframe. And from Eq.(7.56), specialized tothenumber and typeofcomponents, thevalues inallinertial frames areknown. InaMinkowski space withpseudo-Cartesian coordinates, thecomponents ofthetensor Wofrank (1)canbeconverted toacorresponding tensor ofrank using themetric tensor inEq.(7.33) {goo=1,g11=ggg=g33=-1}andtheexpression inEq.(7.57) togivethefollowing relations: W00=800W00 =W00. W01=800W°1 =W01, W1o=3uWlo=-W10. and W11=31lWll =_Wll- Given anytwovectors, wecanconstruct asecond-rank tensor bytheoperation called tensor product, T=u®v.Thetensor product isamachine whose output isanumber when thetwovectors andthetwo1-forms areinserted (u®v)(u-, /\)=(0-,u)()t, v). (7.58) Thecomponents ofthetensor product are T“=u“v'8. (7.59) Inourtwo-dimensional example ofvector uwithcomponents (a,b)andvector vwithcomponents (c,d),Eq.(7.59) becomes written intensor form 7.5 l-Forms andTensors 295 5_acad (Ta)_(bc r>a)‘ Thisprocess canbecontinued andcould include I-forms aswellasvectors; for example, twovectors (u,v)anda1-fonn (0')would bewritten asu®v®0'. Other useful operations include thegradient, contraction, thedivergence, and thewedge product. First, letusconsider thegradient operation. Weused dfor thegradient operation onscalars. Forahigher-rank tensor, thegradient isoften denoted byV.Inthree-dimensional Cartesian space, Vistheoperator aaav='- '—k—,‘ax+J8y+ az which mayalsobewritten as _a a a 8'-‘1ra+”w+"3m Returning to4-dimensions, anexample ofamore general case, letSbearank tensor, thenbydeﬁnition, VS(u, v,w,§)=3§S(u, v,w)withthevectors u,v,w heldﬁxed, and 8S Vs(u, V,W’ = (saﬁyudvﬁwy) = €5uWL|ﬁwy =.Saﬁyjé-5uQvﬁu)Y _ (7.60) Thatis,thegradient operates onlyonthecoefﬁcients inthedeﬁnition ofthetensor, notontheincluded vector ﬁelds. Since thevectors and1-forms inEq.(7.60) are arbitrary andconstant, wecanrewrite thepreceding as BS 5t($~m») =Tf}?-r‘ =s.,r~,...s<’. 0.60’) where the£5deﬁne thedirection ofthegradient, andthelastequality shows clearly thatthederivative doesnotoperate onthevector given by$5. InMinkowski spacetime, contracting theenergy momentum vector (E/c, p) withthecharge-current l-form (pc,—J)produces thescalar (Ep—p-J).This ideacanbeextended toreduce therankofatensor byaprocess called contraction. Thecontraction operation canbeperformed onanytensor whose totalrank(sum ofvector andl-form indices) isequal toorgreater than2.Todothis,enterabasis vector inoneslotandthecorresponding l-form basis inanother slotandsum overthebasis, thereby producing alower-rank tensor. Forexample, consider the 4-index tensor whose components areRa,/9”. Wecanform atwo-index tensor bytheinserting abasis I-form intotheﬁrstslotofthetensor deﬁnition, andthe related basis vector inthethird slot,andsumming overthebasis set.Formally, R(e,,,, u,w“.v)=M(u, v), (7.61) Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity orincomponent form M,,,,u”v" =R,,,,f',,u“r."’, (7.62) which canbewritten as Mm, =R,,,,,_°‘,,. (7.62’) Inthree-dimensional Cartesian space, thedivergence ofavector Visthescalar quantity V-V=%+3%+87:‘,while in4-dimensional space the4-divergence is13%‘:.InMinkowski spacetime the4-divergence operator isoften denoted bythe same symbol, V,initalics, orbyElwhose components are I]—V—m°‘—1—-"‘_“_“8,0.’ with (ozthel-form basic components. Forexample, thecontinuity equation in electromagnetic theory is 31'“ 9(p¢') .Hp .i=l:l- =V- =i V- =— V~ =0. axe J JBet+Jat+J Theoperator V2(sometimes written asI12)iscalled thed’A|embertian andis 32 2 2 2 l:l2=V2=V-V=g"wa 8=1 —(8 +8 6x“Bx“ c28t2 8x2 8y2 fizz where thelastequality istheexpression inMinkowski space withCartesian co- ordinates. The4-divergence operator ontensors reduces therankofthetensor by 1.Forspacetime tensors, thedivergence iswritten asV-Sand,considering asan example atensor Swithaslotfora1-form andthree vector slots, El-S(H, U)=Vv.S'(u, U)=V-S(w", u.1),80)=Sagyﬂuﬁvy. (7.63) Thatis,thegradient ofEq.(7.60) istaken along abasis direction, andthena contraction isformed between thisdirection andoneofthel-form slots inthe tensor. Incomponent form, thisreduces to vasaﬂy 1 Saﬁy’a- Theﬁnaltensor operator weneed isthewedge product, alsocalled thebrvector orbiform, which is u/\v=u®v—v®u, (7.64) where thetensor product, ®,wasdeﬁned inEq.(7.58). Thewedge product isan antisymmetric vector product. Incomponent form, Eq.(7.64) becomes 7.6I7.6 Forces ll‘)theSpecial Theory; Electromagnetism 297 (u/\v)°¢’=tr“vﬁ-v°‘u'8. (7.64') Successive /\operations canbestrung together justlikethe®operator. The wedge product isuseful whenever wedealwith antisymmetric expressions. In particular, when welookattheelectromagnetic ﬁeldinthenextsection, wewill discover thatthefundamental ﬁeld tensor, called Faraday, canbeexpressed in terms ofthewedge product. Consider thetwo-dimensional example usedpreviously, where u=ale,+ M262 andv=vlel +D282. Thewedge product inEq.(7.64-’) hascomponents W=uAvgivenby W_ ulvl—v'u' u'v2—u2v' _ 0 ulvz-vluz u2v1—-v2u' u2v2—-v2u2 -'u2vl——u2ul 0 ' Although theexamples given above assumed acertain combination of1-form slotsandvector slots, wemust stress thatthemetric tensor canbeusedtoproduce atensor withindices inanydesired position. FORCES INTHE SPECIAL THEORY; ELECTROMAGNETISM Thepreceding material hasbeen concemed with thekinematics ofthespecial theory. Thedynamics ofthetheory follows from theassumption thatNewton’s lawsarecorrect forobjects atrestintherestframe oftheobserver, nearly correct forobjects moving slowly relative tothespeed oflight, andrequire generaliza- tions tocovariant equations. Thecorrect generalization ofthethree-velocity tothe four-velocity wasgiven inEq.(7.27). Sowemust generalize theforce law, _d(mv')F‘_T , (7.65) toacovariant form. Since Maxwel]’s equations areassumed tobeacorrect description, weshall brieﬂy consider acovariant reformulation ofelectromagnetic theory asaguide forthecorrect form oftheforce laws ofmechanics. Thevector andscalar elec- tromagnetic potentials forrn afour-vector A“=(¢/c,A). Ifthepotentials satisfy theLorentz condition (inSIunits), which isthevanishing ofthefour-divergence oftheelectromagnetic potential 4-vector, 8A“ 3115l:l'A=V'A=8'xTL=V'A‘l'I/l()8()5=0, theyseparately satisfy thewave equations oftheform (where uoso =1/c2) EIZA=V2A=i'E’2—A -VZA=noj (7.67a)c23t2 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity forthespace components andforthetimecomponent 132¢02¢=v2¢=-2—2 -v2¢=5. (7.67b)C3t S0 Interms ofqbandA,theLorentz forceisF=e{—V¢ +5%+%[vx(V><A)]}. Thissuggests thatweshould generalize theLorentz force lawto up,_8(u"A,,) _iii F_‘ll370“ at 0'68) Forthethree-momentum, p3,andthree-velocity, v,Eq.(7.68) becomes <1%=e(E+v><B), (7.6s') withEtheelectric ﬁeld,Bthemagnetic ﬁeld, andetheelectric charge. Thegeo- metric approach istodeﬁne atensor F,named Faraday, whose components will betheelectromagnetic ﬁeldtensor andwrite, withuthe4-velocity, di=eF(u). (7.69)dr Incomponent notation, thisbecomes dn %=e1~"",,ul‘. (7.70) Thisproduces Maxwel1’s equations, provided (according toEq.(7.68)) F“);is given by 0 Ex E), Ez Q ‘_ Ex 0 CB: F5_E CB7 0 B. (7.71)1*". CI El CB); —CBx 0 InMinkowski space, theindices areraised andlowered bythemetric tensor (Eq.(7.33)), so 0-E, -E, -E, aﬁ 1 Ex 0 *CBz CB); / E? CBX 0 and 0 E, E), E2 _ 1E; 0 —CBz H F“_-E, CB1 0-¢B,, ' 0'71) .7.6 Forces intheSpecial Theory; Electromagnetism 299 TheFaraday tensor canbewritten inatleasttwodifferent ways using either the tensor product, Eq.(7.58), orthewedge product, Eq.(7.64), as F=Fapdxa ®(lxﬂ =%Fapd.1'a /\dJt"8. Thelatter expression explicitly shows theantisymmetry. Wecanwrite Maxwell’s equation intheirnormal component form using geo- metric notation: VF=0 and V-F=J, (7.72) where Jisthe4-current density withcomponents (pc,j),where pisthecharge density andjisthethree-current density. Theﬁrstofthese equations produces (using three-dimensional notation) V-B=0and8B/8t +VxE=0,while the second gives V-E=p/soand(1/(:2) 8E/81 —VxB=—/.tqj. Following theguide provided bythecovariant formulation ofelectromagnetic theory, theproper generalization ofNewton’s second law,Eq.(7.65), is I-L ‘git=K”, (7.73) where K"isa4-vector force, known astheMinkowskiforce. Thespatial compo- nents ofK“arenotthecomponents oftheforce inEq.(7.65), butrather theyare quantities thatreduce totheF‘as,5—>0.Theexact formclearly results from theLorentz transformation properties oftheforces present. Some aspects ofthe 4-force arelisted inTable 7.l. Thegeneral question (which cannot beuniquely resolved) is,I-lowdoweﬁnd theproper relativistic expression forforce? Electromagnetism isusedtojustify the special theory, soweshould expect noproblem withit.Aswesawintheprevious paragraphs, thisistrivial forelectromagnetic forces because thespecial theory and theLorentz transformations areconstructed tomake Maxwel1’s electromagnetic theory covariant. Forexample, theelectromagnetic force isgiven byEq.(7.68) as 3u,,A" dAMK!‘ ——q -' , withqthecharge ontheparticles andAMthecomponents ofthefour-potential given by(¢/c,A). Note that¢isthescalar potential andAisthethree- dimensional electromagnetic vector potential. Sotheordinary force, F,,and thespatial component oftheMinkowski electromagnetic force, K,,arerelated by F‘,=Ki‘/l —52. (7.75) What about other forces? Twomethods arecommonly usedtodeduce acceptable transformation properties offorces andhence thecorrect relativistic form ofthe forces. Theﬁrstmethod istoargue thatthere areonly fourfundamental forces in nature—gravitational, weak nuclear, electromagnetic, andstrong nuclear. Acor- 00 7.7 IChapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity rectrelativistic theory must provide valid expressions forthese fourforces. These expressions, ifstated incovariant form, willautomatically provide thetransfor- mation properties oftheforces. Inthisapproach, since weunderstand electro- magnetic forces, itremains toﬁndexpressions fortheother three fundamental forces inacovariant form insome frame andassume thisiscorrect inallinertial frames. Itisassumed thetransformations involve noterms thatvanish inthecho- senframe; forexample, there isnoneed toarbitrarily addterms proportional to (v/c)3. Thisprogram hasbeen carried outfortwooftheremaining three forces (weak nuclear andstrong nuclear) andforweak gravitational forces. ltfailscom- pletely forstrong gravitational effects. Itisbeyond thescope ofthepresent text toprobe more deeply intothisquestion. Thesecond approach ofdetermining thecorrect relativistic force istosimply deﬁne force asbeing thetimerateofchange ofthemomentum. Then wewrite div.-=F 7.76 dt . () where thep,inEq.(7.76) issome relativistic generalization oftheNewtonian momentum thatreduces tomv,inthelimitofsmall ,3.Thesimplest generalization istheonegiven inEq.(7.36). Thissecond approach hasthusfarfailed toproduce anyresults other thanthose predicted bytheﬁrstapproach. RELATIVISTIC KINEMATICS OFCOLLISIONS AND MANY-PARTICLE SYSTEMS Theformulations oftheprevious sections enable ustogeneralize relativistically thediscussion ofSection 3.11onthetransformation ofcollision phenomena be- tween various systems. Thesubject isofconsiderable interest inexperimental high-energy physics. While theforces between elementary particles areonlyim- perfectly known, andarecertainly farfrom classical, solongastheparticles in- volved inareaction areoutside theregion ofmutual interaction theirmean motion canbedescribed byclassical mechanics. Further, themain principle involved in thetransformations—-conservation ofthefour-vector ofmomentum—is valid in bothclassical andquantum mechanics. Theactual collision orreaction istaken as occurring atapoint—or inside averysmall black box—and welookonlyatthe behavior oftheparticles before andafter. Because oftheimportance tohigh-energy physics, thisaspect ofrelativistic kinematics hasbecome anelaborately developed ﬁeld. Itisimpossible togivea comprehensive discussion here. Allthatwecandoisprovide some oftheim- portant tools, andciteafewsimple examples thatmay illustrate theﬂavor of thetechniques employed. Although many collision experiments involve colliding beams, weshall, forsimplicity, conﬁne ourattentions toproblems where oneof theparticles isatrestinthelaboratory frame. Thegeneralization tobothparticles moving inthelaboratory frame isstraightforward. 7.7 Relativistic Kinematics ofCollisions andMany-Particle Systems 301 Thenotion ofapoint designated asthecenter ofmass obviously presents dif- ﬁculties inaLorentz-invariant theory. Butthecenter-of-mass system canbesuit- ablygeneralized astheLorentz frame ofreference inwhich thetotalspatial linear momentum ofallparticles iszero.ThatsuchaLorentz frame canalways befound follows from thetheorem thatthetotalmomentum 4-vector istimelike forasys- temofmass points. Onesuch frame isthecenter-of-momentum frame. This isaframe inwhich thecomponents ofthespatial momentum oftheinitial particles addtozero. Such aframe obviously exists. Letusdeﬁne EandpinEq.(7.36) tobe I1 IX E=Z1:E, and p=Z131), (7.77) I I where thesumisover theparticles involved. Theleft-hand sideofEq.(7.38) becomes Zm,m,c2 —Zmrmsy/,y,(v, -v.,). (7.78) r,S rr This clearly ispositive (hint: separate thenegative terms inwhich r=s),so itispossible toﬁndaframe inwhich thethree-momentum, p,equals zero. The Lorentz system, inwhich thespatial components ofthetotalmomentum arezero, istermed thecenter-of-momentum system, ormore loosely, andsomewhat incor- rectly, asthecenter-of-mass system, andwillbedesignated bytheabbreviation “C-O-M system.” Asanexample, letusconsider aparticle ofmass m1andmomentum plinthe x-direction, which suffers ahead-on collision withaparticle ofmass mgatrestin anexpei-imenter’s frame (called thelaboratory frame). Theinitial 4-momentum is P”=ilmll’+mac.miw‘.0.0). (7.79) Thelength squared ofmomentum hasthemagnitude p”p,, =(mf+mg+2m1ym2)c2. (7.79’) When components aregiven, weshallfollow thepractice ofdenoting theprimed frame byprimes ontheindices. Thetwoparticles aredenoted bysubscripts 1 and2respectively. IntheC-O-M system, thetotalmomentum is (Emit/1' +mi/51¢, 0.0,0). (7-80) since bydeﬁnition thespace partofthemomentum vanishes, ml)/{Bic +mg)/éﬁéc =0, (7.81) Chapter 7TheClassical Mechanics oftheSpecial Theoiy ofRelativity where BiandB5arethevelocities ofm1andM2,respectively, intheC-O-M frame. Theboost, B’,needed togofrom thelaboratory totheC-O-M frame, hasthe value at=-B’- <7-81') Since allvelocities areparallel, thevelocity addition formula Eq.(7.15) gives thevelocity Biofmass mlintheC-O-M system interms ofB’anditsvelocity B=v/cinthelaboratory frame, /_/3_/5,,5,_---1__B5,. (7.82) Thetotalsquared momentum intheC-O-M frame given inEq.(7.80) canbe rewritten using theresults ofEqs.(7.81) and(7.82) as p"mi=?€m%ﬂ2I;l__5'?/2)C2. (7.33) Equating Eqs.(7.79’) and(7.83) gives asingle equation thatcanbesolved for theboost velocity )8’.There aretworealroots, oneofwhich corresponds tothe physically meaningful caseof)3’<1. Since thespatial momentum intheC-O-M frame iszero, there isclearly more energy, po,inthisframe thaninthelaboratory frame.* Theexcess energy inthe C-O-M frame, AE,isobtained bysubtracting thetimecomponent ofEq.(7.79) fromthetimecomponent ofEq.(7.80). Thetotalmomentum fourvector isconserved, which automatically implies both conservation ofspatial linear momentum andconservation oftotal energy (including restmass energy). Ourmajor tools formaking useoftheconserva- tionprinciple areLorentz transformations toandfromtheC-O-M system, and theformation ofLorentz invariants (world scalars) having thesame value inall Lorentz frames. Since energy andmomentum arecombined intooneconservation law,therelativistic results aremoreeasily obtained thanthenonrelativistic results ofprevious chapters. Thetransformations between laboratory system andC-O-M system aremerely special cases oftheLorentz transformation. Asanexample oftheuseofLorentz invariants, letusconsider areaction ini- tiated bytwoparticles thatproduces another setofparticles with masses mr, r=3,4,5,....IntheC-O-M system, thetransformed totalmomentum is P“,=(r:'/C,0,0,0). (7.84) Itisoften convenient tolookontheC-O-M system astheproper (orrest)system ofacomposite mass particle ofmass M=E’/c2.T Thesquare ofthemagnitude of *For asingle particle, theenergy hasamtmmtim value, ITIC2, mtherestframe TheC-O-M frame is nottherestframe ofeither particle. lAlthough itiscustomaiy inhigh-energy physics touseunitsinwhich c=l,itseems morehelpful inanintroductory exposition suchasthistoretain thepowers ofcthroughout. 7.7 Relativistic Kinematics ofCollisions andMany-Particle Systems 303 Pmustbeinvariant inallLorentz systems andconserved inthereaction. Hence, wehave I E12 P,,P#=P,,,P# =2;=M28. (7.85) Butfortheinitial particles, P“P“canbeevaluated as P,,P#=(mi+m§)¢2-2p,,,p§. (7.86) Theenergy intheC-O-M system, orequivalent mass M,istherefore given in terms oftheincident particles as E’2EM26‘=(mi+m§)c“+2(E1E2 -czpi-pg). (7.87) Suppose nowthat, oneparticle, say2,wasinitially stationary inthelaboratory system. Since thenpg=0andE2=H1262, theC-O-M energy becomes 5'2EM284=(mi+m§)¢4+2m2¢2E;. (7.88) Iftheexcess ofE|overtherestmass energy bedenoted byT1,[cf.Eq.(7.39)] thatis,thekinetic energy, thiscanbewritten E’2EM204=(m1+m2)%“ +2m2¢.~2r,. (7.89) Itisclearthattheavailable energy intheC-O-M system increases onlyslowly withincident kinetic energy. Even inthe“ultrarelativistic" region, where theki- neticenergy ofmotion isverylargecompared totherestmassenergy, E’increases onlyasthesquare rootofT1. Theeffect oftheproportionally small amount ofincident energy available in theC-0-M system isshown dramatically interms ofthethreshold energies. Itis obvious thatthelowest energy atwhich areaction (other thanelastic scattering) is possible iswhen thereaction products areatrestintheC-O-M system. Anyﬁnite kinetic energy requires ahigher E’orequivalently higher incident energy. The totalfour-momentum intheC-O-M system afterthereaction, denoted byPM hasthemagnitude atthreshold given by 2 PMP7,,=C2 mr), (7.90) I’ which, byconservation ofmomentum, must bethesame asEq.(7.85). Fora stationary target, theincident energy ofmotion asthreshold isthen given asa consequence ofEq.(7.89) by 04 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity 2 (2%) —(m1+m2)2 Tl _ r m1c2 2m1m2 IftheQvalue ofthereaction isdeﬁned as* Q=[Zm,-(ml+milC’, (7.91)I‘ thisthreshold energy becomes T1=_Q2+2Q(m1+m2)¢’2 U92)m|c2 2m|m2C4 ' ' Acommon illustration oftheapplication ofEq.(7.92) isthehistoric production ofanantiproton, 13,bythereaction, involving aproton p, P-l-n—>]J+l1+P"l"P, where nisanucleon, either neutron orproton. Themasses ofallparticles involved arenearly equal at938MeVequivalent restmassenergy andweselect Q=2mc2. Equation (7.92) thensaysthattheincident particle kinetic energy atthreshold must be T1=6mc2=5.63GeV, which is3times theenergy represented byQ!If,however, thereaction wasini- tiated bytwonucleons incident oneach other with equal andopposite velocity, thenthelaboratory system isthesame astheC-O-M system. Allofthekinetic energy isavailable inthiscasetogointoproduction oftheproton-antiproton pair, andeachoftheincident particles atthreshold needhaveakinetic energy ofmo- tionequivalent toonlythemassofoneproton, 938MeV. Itisnowonder somuch effort hasbeen putintoconstructing colliding beam machines! Another instructive example ofathreshold calculation isphotomeson produc- tion,say,bythereaction )1+p=>3“+K+, (7.93) where ystands foranincoming photon. Forthepl11‘p0S€S ofclassical mechanics, aphoton isazero-mass particle with spatial momentum Opandenergy 0pC.T In calculating Q,themass miofthephoton iszero: Q=(mzo+mK+-m,)¢2=749MeV. *Qherehastheopposite signtotheconvention adopted inEq.(3.112). lThe square ofthemagnitude ofthephoton momentum four-vector iszero, sothevector canbe described as“lightlike ”TheC-O-M theorem ISimperiled onlyifalloftheparticles arephotons, and eventhenonlyifthephotons aregoing inthesame direction. 7.7 Relativistic Kinematics ofCollisions andMany-Particle Systems 305 Equation (7.92) isrewritten forareaction involving anincident photon as Q2+2Qm2¢2T=0-=_i-.l P‘ 2mgc2 From thevalue ofQandtherestmass energy mgoftheproton, thethreshold energy forthereaction Eq.(7.93) isthen T1=1.05 G6‘/, which isonlyslightly higher thanQ. Wecanalsoeasily ﬁndtheenergy ofthereaction products inthelaboratory system atthreshold. TheC-O-M system istherestsystem forthemass M,with P0’=Mc.Inanyother system, thezeroth component ofthe4-vector isP0= Mcy.Butinthelaboratory system 1P“=gust+E2)=-2-(E1+mzr-'2). where thelastform holds onlyforastationary target particle. Hence, theC-O-M system moves relative tothelaboratory system suchthat E1+m2¢2y= . Butatthreshold allthereaction products areatrestintheC-O-M system sothat M=Zm,,andtherefore7' 2 y=-L+lm‘+mi)“ (threshold). (7.95)Zm,-c2 I‘ Thekinetic energy ofthesthreaction product inthelaboratory system isthen 1}=m,t-2(y -1). (7.96) Thus, theantiproton atthreshold hasakinetic energy T5=mcz=938MeV. In contrast, theK'l'meson emerges atthreshold with494MeV. InSection 3.11, thekinematic transformations ofatwo-body nonrelativistic collision wereinvestigated. Eq.(3.117’)gives thereduction inenergy ofaninci- dentparticle afterelastic scattering fromastationary target, asafunction ofthe scattering angle intheC-0-M system. Thederivation oftherelativistic analog provides another interesting example ofthemethods ofrelativistic kinematics. UseofLorentz invariants hereisnotparticularly helpful; instead direct Lorentz transformations aremade between thelaboratory andC-O-M systems. Figure 7.4 illustrates therelations oftheincident andscattered spatial momentum vectors in both systems. Theincident andscattered momentum vectors deﬁne aplane, in- 06 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity P’;X P3 6 6G O Q Z P1 pl Pl»P4 (a)Center-of-momentum system (b)Laboratory system FIGURE 7.4 Momentum vectors forrelativistic elastic scattering inC-O-M andlabora- toryLorentz frames. variant inorientation under Lorentz transformation, heretaken tobethexzplane withtheincident direction along thezaxis.Because thecollision iselastic. the masses oftheincident particle, mi,andofthestationary target, mg,remain un- changed; thatis,m3=m1,m4=mg.Primes onthevectors denote C-O-M values, unprimed vectors areinthelaboratory system. Todistinguish clearly between be- foreandafter thescattering, theindexes 3and4willberetained forthevectors after scattering. Wehave onlytoremember that3denotes thescattered incident particle, and4therecoiling target particle. Components oftheseparate particle 4-vectors willalways havetwoindices: theﬁrstfortheparticle, thesecond forthe component. TheLorentz transformation from thelaboratory totheC-O-M system isde- ﬁned bytheyofEq.(7.94) withMgiven byEq.(7.89): y= E1+m2¢2 = Ti+(mi+m2)¢2 _(797) \/2m2(_.2E1 +(m?+m%)C4 \/2m2(_-2T1 +(mi+m2)2¢4 Thequantity 5canbefound from y,ormore directly byarguments similar to those used toobtain y.IntheC-O-M system, thespatial partofthetotalmomen- tumfour-vector ISzero; inanyother system, thespatial partisMcﬂy. However, inthelaboratory system thespatial partisp1.Hence, byEq.(7.94) Bmust be glV€l'1 ﬂS p1C ]J|C = = . 7.98 B E1-i-mgcz T1+(m1 -l-m2)t‘2 ( ) Because Bisalong thezaxis, theLorentz transformation takes (with ,8,=,6}.= 0)theform given byEq.(7.11), andthecomponents ofpl‘!intheC-O-M system aregiven by 7.7 Relativistic Kinematics ofCollisions andMany-Particle Systems 307 /_ 3/_ _ﬂEl P1-P1-Z" P1 T I -E—'=pl’=y(5-apt). <7-99>C C After thecollision, pgisnolonger along thezaxis, butsince thecollision is elastic, itsmagnitude isthesame asthatofpa.If(9istheangle between pgand theincident direction, asinSection 3.11, thenthecomponents ofpgintheC-O-M system are I 1 _ I E pl=1);S1119, pg=picosé), pg’=p§”= (7.100) Thetransformation back tothelaboratory system isthesame Lorentz transfor- mation butwithrelative velocity -B.Hence, thecomponents ofpg,are Pi=18’=PiSiﬂ@ 155'Pi=i/(p§"—i6i>§")=1/(1/1<=<>S® +-—;—‘ , , E’ ,pg=;/(pg -l-ﬂpg)=y(:L+ﬂp1C0S@). (7.101) IfE1andpiaresubstituted inthelastof (7.101), from Eqs.(7.99) weobtain, afteralittlesimpliﬁcation, anexpression fortheenergy ofthescattered particle intenns ofitsincident properties: E,=E1-y2)6(1— cos®)(p1c -512,). (7.102) InEq.(7.102), yand['3must beexpressed terms oftheincident quantities through Eqs.(7.97) and(7.98), resulting intherelation 22 2 -E='”m”"’_. 7.103 l’,5(Pl(' 51)2m2El+(m,+m,)c, () With thehelpoftherelation between p1andE1,Eq.(7.38’), thiscanbewritten 2 _ _m2T1(T1+2m1¢2) 1041/ﬂ(pi¢ ,5El)— i—~ii—2m2T1+(m1+m2)2C2- (7-) Some further algebraic manipulation thenenables ustorewrite Eq.(7.102) as -T3=1-liﬁgu -coso), (7.105)Ti (1+P)2+2/>51 08 Chapter 7TheClassical Mechanics oftheSpecial Theoiy ofRelativity where p=mi/mg,asinSection 3.11forelastic scattering, and£1isthekinetic energy oftheincident particle inunits oftherestmass energy, 8.Z (7.1...) Equation (7.105) istherelativistic counterpart ofEq.(3.1l7’).Itiseasytosee thatEq.(7.105) reduces tothenonrelativistic caseas5|—>0,andthatifp=l (equal masses), therelativistic corrections cancel completely. Equation (7.105) implies thattheminimum energy afterscattering, inunits ofm1c2, isgiven by _ (1—p)2(£3)ITLlIl —£1(1+p)2+2p8l- (7-l07) Inthenonrelativistic limit, theminimum fractional energy afterscattering is 8min 1- 2 % =(ﬁ) ;£1<<1, (7.108) which isawell-known result, easily obtained fromEq.(3.117').Equation (7.108) saysthatinthenonrelativistic region aparticle ofmass micannot losemuch ki- netic energy through scattering from amuch heavier particle, thatis,when p<<l, which clearly agrees withcommon sense. However, intheultrarelativistic region, when pS|>>1,theminimum energy afterscattering isindependent of£1: <->22(T3)lﬂlll = i P81 >>l. (7-log) Since thecondition on81isequivalent torequiring T1>>711262, itfollows from Eq.(7.109) thatsuch aparticle canlosealarge fraction ofitsenergy even when scattered byamuch heavier particle. Thisbehavior isunexpected, butitshould be remembered thatforparticles atthese energies, traveling veryclose tothespeed oflight, evenaslight change invelocity corresponds toalargechange inenergy. Finally, wemayeasily obtain therelation between thescattering angles inthe C-O-M andlaboratory system bynoting that(ﬁrst index particle, second compo- nent) tam?=531=-—-355‘-(9—ﬁ,,T-. (7.110)P33 y(cos(9+33%) ByEq.(7.36), I I 5E1?°=%Ep;, (7.111) sothattan1?canalsobewritten 7.8I7.8 Relativistic Angular Momentum 309 an=1/(¢°S@+»5/131) Interms ofinitial quantities, Eqs.(7.99) show that(7.112) -"5P1) +=—_i5,;i-Pic P1-TL"Ga1-1‘O:/'\-ta(7.113) Thiscanbefurther reduced byemploying therelations (cf.Eq.(7.98)) % = (7114) pl _T 11126 E_ _m|(m1+ m2)C4 +m2c2T1_ 7.115 15111 (ml+m2)c2+T1 ( ) Theﬁnalexpression fortan29canthenbewritten as siné)tan29=a , .16 1/[<=<>s®+/>s(p.8|)l (7') where 300.51)isthefunction _l+P(1+-$1)(1+£|)+p , (7.117) 800,51) andy,byEq.(7.97), takes theform 1+5 + rpe>= ‘”y" ./(1+p)2+2p51' Again, inthenonrelativistic region, yandgtendtounity, andEq.(7.116) re- duces toEq.(3.107). Thecorrection function g(p,51)never really amounts to much, approaching theconstant limitpas£1becomes verylarge. Theimportant factor affecting thetransformed angle isy,which ofcourse increases indeﬁnitely as81increases. Itdoes notaffect thebounds oftheangular distribution, when Q-)=0orrt,butitspresence means thatatother angles 19isalways smaller than itwould benonrelativistically. TheLorentz transformation from C-O-M tothe laboratory system, which does notaffect thetransverse component ofthemo- mentum, thusalways tends todistort thescattered angular distribution intothe forward direction.(7.118) RELATIVISTIC ANGULAR MOMENTUM InChapter 1,itwasproven thatthenonrelativistic angular momentum obeys an equation ofmotion much likethatforthelinear momentum, butwith torques 0 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity replacing forces. Itwasshown thatforanisolated system obeying thelawof action andreaction thetotal angular momentum isconserved, andthatinthe C-0-M system itisindependent ofthepoint ofreference. Allofthese statements havetheirrelativistic counterparts, attimes involving some additional restrictions. Forasingle particle, letusdeﬁne anantisymmetric tensor ofrank in Minkowski space using theformalism ofEq.(7.64) m=xAp (7.119) whose elements would be m‘“’=x“p" -x"p“. (7.120) The3x3subtensor m”clearly corresponds, aswasseeninSection 5.1,with thespatial angular momentum oftheparticle. Anequation ofmotion form“"can befound bytaking itsderivative with respect tothepartic1e’s proper time and making useofEq.(7.73) giving d£=u/\p+x/\K=x/\K, (7.121) where theﬁrsttermvanishes bytheantisymmetry ofthewedge product andKis theMinkowski force. Incomponent notation, Eq.(7.121) becomes d#1’git=x“K"-ﬁx“. (7.122) Thissuggests wedeﬁne therelativistic generalization ofthetorque by N=xAK, (7.123) whose components are Nl” =x“K" —x"K“. (7.124) Thus, mobeys theequations ofmotion dmE=N, (7.125) whose component form is #9 %=1»/#1", (7.126) withEq.(1.11) asthenonrelativistic limiting form. Forasystem involving acollection ofparticles, atotal angular momentum 4-tensor canbedeﬁned (analogously tothetotallinear momentum 4-vector) as M=Zm, (7.127).8 7.8 Relativistic Angular Momentum 311 orincomponent form M11"=Zmf", (7.128).5 where theindex sdenotes thesthparticle. Itismore difﬁcult toform anequation ofmotion forMbecause eachparticle hasitsownproper time. (Forthesame rea- son,wedidnotattempt iteven forP.)Nevertheless, plausible arguments canbe given fortheconservation ofMunder certain circumstances. Ifthesystem iscom- pletely isolated andtheparticles donotinteract witheachother orwiththeoutside world (including ﬁelds), thenmforeachparticle isconserved byEq.(7.126), and therefore Misalsoconserved. Even iftheparticles interact, buttheinteraction takes place onlythrough binary collisions atapoint, therestillcould beconser- vation ascanbeseenfromthefollowing argument. lnstantaneously when thetwo particles collide theyaretraveling together andhave thesame proper time. In other words, their world lines cross andtheyshare thesame event. Onecanthere- forewrite anequation ofmotion oftheform ofEq.(7.126) forthesumoftheir angular momenta. Iftheimpulsive forces ofcontact areequal andopposite—as wewould expect from conservation oflinear momentum inthecollision-then thesumoftheimpulsive torques cancel. Hence relativistic angular momentum is alsoconserved through such collisions. Note thatunlike thenonrelativistic case covariance requires thattheinteractions areassumed tobeinstantaneous point collisions. Therelativistic angular momentum obeys thesamekindoftheorem regarding translation ofthereference point asdoesitsnonrelativistic counterpart. Inthedef- inition, Eq.(7.120) orEq.(7.128), thereference point (really reference “event”) isthearbitrary origin oftheLorentz system. Withrespect tosome otherreference event a),thetotalangular momentum is Mtao=Zrx.—<11.)Ap. (7.129)S =M(0) —a;_AP (7.130) Asinthenonrelativistic case, thechange intheangular momentum components isequal totheangular momentum, relative totheorigin. thatthewhole system would have ifitwere located ata1. InChapter 1,oneparticular reference point played animportant role-—the cen- terofmass. Wecanﬁndsomething similar here, atleastinoneLorentz frame, by examining thenature ofthemixed timeandspace components ofM‘“’,namely, M0-’=—M10. Bydeﬁnition, insome particular Lorentz frame, these components aregiven by MW=z(x?p_{ -rip?) (7.131)Y Jx{E,=C£ fps‘-"C? . A 7.9IChapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity IntheC-O-M frame. thetotallinear momentum p=Zp,vanishes, andM01in thisframe hastheform MW=-tZ (7.133).\ Ifthesystem issuch thatthetotalangular momentum isconserved, asdescribed above, thenalong withothercomponents Mtsconserved andhence ZxiES=constant. S Conservation oftotallinear momentum means thatE=ZE,isalsoconserved. Itistherefore possible todeﬁne aspatial point RJ, Zx.’E.R=‘i, 7.134 S associated withthesystem, which isstationary intheC-O-M coordinate frame. Inthenonrelativistic limit, where toﬁrstapproximation ES=m,c2, Eq.(7.134) reduces totheusual deﬁnition, Eq.(1.21). Thus, ameaningful center ofmass (sometimes called center ofenergy) canbedeﬁned inspecial relativity only in terms oftheangular-momentum tensor, andonlyforaparticular frame ofrefer- ence. Finally, itshould benoted thatbyEq.(7.130) thespatial partoftheangular momentum tensor, M,isindependent ofreference point intheC-O-M system. exactly asinthenonrelativistic case. Except forthespecial caseofpoint collisions. wehave sofarcarefully skirted theproblem ofﬁnding themotion ofarelativistic particle given theMinkowski forces. Tothismore general problem weaddress ourselves inthenextsection. within thenominal framework oftheLagrangian formulation. THE LAGRANGIAN FORMULATION OFRELATIVISTIC MECHANICS Having established theappropriate generalization ofNewton’s equation ofmotion forspecial relativity, wecannowseektoestablish aLagrangian formulation ofthe resulting relativistic mechanics. Generally speaking, therearetwowaysinwhich thishasbeenattempted. Onemethod makes nopretense atamanifestly covariant formulation andinstead concentrates onreproducing, forsome particular Lorentz frame, thespatial partoftheequation ofmotion, Eq.(7.76). Theforces F,mayor maynotbesuitably related toacovariant Minkowski force. Theothermethod sets outtoobtain acovariant Hamilton’s principle andensuing Lagrange‘s equations inwhich space andtimearetreated incommon fashion ascoordinates inafour- dimensional conﬁguration space. Thebasis fortheﬁrstmethod isattimes quite shaky, especially when theforces arenotrelativistically wellformulated. Most of 7.9 TheLagrangian Formulation ofRelativistic Mechanics 313 thetime. however, theequations ofmotion soobtained, while notmanifestly co- variant, arerelativi stically correct forsome particular Lorentz frame. Thesecond method, ontheotherhand, seems clearly tobetheproper approach, butitquickly runsintodifﬁculties thatrequire skillful handling iftheyaretobesolvable, even forasingle particle. Forasystem ofmore thanoneparticle, itbreaks down almost fromthestart.Nosatisfactory formulation foraninteracting multiparticle system exists inclassical relativistic mechanics except forsome fewspecial cases. Thissection follows theﬁrstmethod, seeking toﬁndaLagrangian thatleadsto therelativistic equations ofmotion interms ofthecoordinates ofsome particular inertial system. Within these limitations there isnogreat difﬁculty inconstruct- ingasuitable Lagrangian. Itistruethatthemethod ofSection (1.4), deriving the Lagrangian from D’Alembert’s principle, willnotwork here. While theprinciple itself remains valid inanygiven Lorentz frame, thederivation there isbased on p,=m,v,,which isnolonger valid relativistically. Butwemayalsoapproach the Lagrangian fonnulation fromthealtemative route ofHamilton’s principle (Sec- tion2.1)andattempt simply toﬁndafunction Lforwhich theEuler-Lagrange equations, asobtained fromthevariational principle r 81=Sf2Ldt=0. (7.135) It agree withtheknown relativistic equations ofmotion, Eq.(7.76). Asuitable relativistic Lagrangian forasingle particle acted onbyconservative forces independent ofvelocity would be* 1.=-ml,/1 -51-v, (7.136) where Visthepotential, depending onlyuponposition, and,82=v2/c2, withv thespeed oftheparticle intheLorentz frame under consideration. Thatthisisthe correct Lagrangian canbeshown bydemonstrating thattheresultant Lagrange equations, dE_££_0dz8v’ Bx’_’ agree withEq.(7.76). Since thepotential isvelocity independent vioccurs only intheﬁrsttermof(7.136) andtherefore 8L mv‘$2‘/ ip!. Theequations ofmotion derived fromtheLagrangian (7.136) arethen *Wedonotchoose L=mC2‘/ l—(/1—B2—Vbecause wewant hinEq.(7.139) tobethetotal energy Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity d mvi _ 8V_ dt/1..'52 Zlxi which agree with(7.76). Note thattheLagrangian isnolonger L=T—Vbutthat thepartial derivative ofLwithvelocity isstillthemomentum. Indeed, itisthis lastfactthatensures thecorrectness oftheLagrange equations, andwecould have worked backward from Eq.(7.137) tosupply atleast thevelocity dependence of theLagrangian. Wecanreadily extend theLagrangian (7.136) tosystems ofmany particles andchange from Cartesian toanydesired setofgeneralized coordinates q.The canonical momenta, "P,willstillbedeﬁned byF’, 8LPi=5?, sothattheconnection between cyclic coordinates andconservation ofthecone- sponding momenta remains justasinthenonrelativistic theory. Further, justasin Section (2.7), ifLdoes notcontain thetimeexplicitly, there exists aconstant of themotion h=qi1>,~-L. (7.139) However, theidentiﬁcation ofhwiththeenergy for,say,aLagrangian ofthefonn ofEq.(7.136) cannot proceed along thesame route asinSection (2.7). Notethat LinEq.(7.136)isnotatallahomogeneous function ofthevelocity components. Nonetheless, direct evaluation ofEq.(7.139) from Eq.(7.136) shows thatinthis casehisindeed thetotalenergy: 11: +m¢2,/1-52+v, which, oncollecting terms, reduces to mC2 2 h=i——+V=T+V+mc=E. (7.140) ./1-51 Thequantity histhusagain seentobethetotalenergy E,which istherefore a constant ofthemotion under these conditions. Theintroduction ofvelocity-dependent potentials produces noparticular difﬁ- culty hereandcanbeperformed inexactly thesame manner asinSection 1.5for nonrelativistic mechanics. Thus, theLagrangian forasingle particle ofcharge, q, inanelectromagnetic ﬁeldis L=-m@2./1- 192-q¢+qA-v. (7.141) 7.9 TheLagrangian Formulation ofRelativistic Mechanics 315 Note thatthecanonical momentum isnolonger mu;there arenowadditional terms arising from thevelocity dependent partofthepotential: P"=mui+qAi. (7.142) Thisphenomenon isnotarelativistic oneofcourse; exactly thesame additional tenn wasfound intheearlier treatment (cf.Eq.(2.47)). Theformulation of Eq.(7.141) isnotmanifestly covariant. Butwecanconﬁdently expect thatthe results willhold inallLorentz frames asaconsequence oftherelativistic co- variance oftheLorentz force derivable from thevelocity dependent potential in Eq.(7.141). Almost alloftheprocedures devised previously forthesolution ofspeciﬁc mechanical problems thuscanbecarried overintorelativistic mechanics. Afew simple examples willbeconsidered herebywayofillustration. 1.Motion under aconstant force; hyperbolic motion. Itwillbenolossofgener- alitytotakethexaxisasthedirection oftheconstant force. TheLagrangian is therefore L=-mcz,/1 -52-max, (7.143) where ,6isX/candaistheconstant magnitude oftheforce perunitmass. Either from Eq.(7.143) ordirectly onthebasis ofEq.(7.76), theequation ofmotion is easily found tobe d )3 _a dt ,/1_52 —c‘ Theﬁrstintegration leads to )3 at+a i/1-52: C ’8__ at+a ‘/02+(at+002, where ozisaconstant ofintegration. Asecond integration overtfrom 0totand xfrom x9tox,OI‘ /’ (at’+a)dt’x—x0—c ?—————,2 / 2_ 0‘/0+(at +0!) leads tothecomplete solution 6 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity x—xo=§[\/c2+(al+a)2—\/c2+a2]. (7.144) Iftheparticle starts atrestfromtheorigin sothatxq=0andU0=0=a,then Eq.(7.144) canbewritten as 22 4C .22__‘-’x+——- -ct-—,a a2 which istheequation ofahyperbola inthex,tplane. (Under thesame conditions thenonrelativistic motion isofcourse aparabola inthex,zplane). Thenonrela- tivistic limit isobtained from Eq.(7.144) byconsidering (at+a') small compared toc;theusual freshman-physics formula forxasafunction oftistheneasily obtained, recognizing thatinthislimitor-—>v0. Themotion described inthisexample arises inreasonably realistic situations. Itcorresponds, forexample, totheacceleration ofelectrons torelativistic speeds inthelaboratory system bymeans ofaconstant andunifonn electric ﬁeld. The illustration considered nextismore academic, butisofinterest asanexample of thetechniques employed. 2.Therelativistic one-dimensional harmonic oscillator: TheLagrangian inthis caseisoftheform ofEq.(7.136) with vo)=%aR. 01%) Since Listhen notexplicitly afunction oftime andVisnotvelocity depen- dent, thetotalenergy Eisconstant. Equation (7.140) maynowbesolved forthe velocity xas 1dx2 m2c4F =l—fr_W2. (7.146) Forthemoment, weshall postpone substituting intheparticular form ofl/(x) andgeneralize theproblem slightly toinclude anypotential sharing thequalita- tivecharacteristics ofEq.(7.145). Thus, letussuppose thatV(x) isanypoten- tialfunction symmetric about theorigin andpossessing aminimum atthatpoint. Then providing Eliesbetween V(0) andthemaximum ofV,themotion willbe oscillatory between limits x=—bandx=+b,determined by V(:l:b) =E. Theperiod oftheoscillatory motion is,byEq.(7.146), tobeobtained from b r=5f-——ii——. ampCO 1__ "I204 V(E-van‘ 7.9 TheLagrangian Formulation ofRelativistic Mechanics 317 Equation (7.147), when specialized totheparticular Hooke’s lawform (7.145) forV(x), canbeexpressed interms ofelliptic integrals. Weshallinstead examine theﬁrst-order relativistic corrections when thepotential energy isalways small compared totherestmass energy mcz. Achange ofnotation ishelpful. Theenergy Ecanbewritten as E=mc2(l+5) sothathere El/J‘) =1+£ -KX2=1-l-K(b2 -'-'.X2), (7.148)mC where k Totheorder (1cb2)2, theperiod, Eq.(7.147) thenreduces to h r1éf---511‘-_ l1--54i(b2-13)]. (7.150)04/21c(b2 —x2) Theintergral inEq.(7.150) canbeevaluated byelementary means, mostsimply bychanging variable through x=bsin<15;theﬁnalresult is 21: 2 /" 3kbz Note thattheexpression infront ofthebracket isto,thenonrelativistic period of theharmonic oscillator. Inspecial relativity, theperiod oftheharmonic oscillator isthusnotindependent oftheamplitude; instead. there isanamplitude dependent correction given approximately byoto 3 Av Ar 3kbz 3-—=—-—:———-=-. 7.151v0 1:0 l6mc2 88 ( ) 3.Motion ofacharged particle inaconstant magnetic ﬁeld. Inprinciple, we should startfromaLagrangian oftheformofEq.(7.141) withthescalar potential ¢=0andAappropriate toaconstant magnetic ﬁeld(Eq.5.106). ButWeknow suchaLagrangian corresponds totheLorentz force onthecharged particle of charge q,given by F=q(v XB) (7.152) (cf.Eq.1.60). Hence, theequation ofmotion mustbe d—P=q(vxB)=iq,XB). (7.153)dt my 7.10 IChapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity Thenature oftheforce, Eq.(7.152), isclearly suchthatthemagnetic ﬁelddoes nowork ontheparticle: F-v=O.Hence, Emust beaconstant, asalsop andybyEq.(7.38’). Further, byEq.(7.152), thereisnocomponent oftheforce parallel toB,andthemomentum component along thatdirection mustremain constant. Itistherefore nolossofgenerality toconsider themotion onlyinthe plane perpendicular toBandtoletprepresent theprojection ofthetotallinear momentum ontothatplane. Equation (7.153) thensaysthatthevector p(whose magnitude isconstant) isprecessing around thedirection ofthemagnetic ﬁeld withafrequency s2=Q (7.154)my referred toasthecyclotron frequency. Inthenonrelativistic limit y-->l.This agrees withthecyclotron resonance expression found insolid statephysics texts. Because yisconstant, thevelocity vector intheplane isalsoofconstant mag- nitude androtating with thesame frequency. Theparticle must therefore move uniformly inacircular orbit intheplane withangular speed S2.Since thecentrifu- galforce, F,equals muz/r, itfollows thatthemagnitude ofthelinear momentum intheplane mustbegiven by p=myr§2. Combining thisexpression withEq.(7.154) leads totherelation between thecir- cleradius andthemomentum: r=qlB. (7.155) Theradius ofcurvature intowhich theparticle motion isbentdepends onlyupon theparticle properties through theratio p/q(=Br).which issometimes called themagnetic rigidity oftheparticle. Note thatwhile Q(Eq.(7.154)) shows rela- tivistic corrections through thepresence ofy,therelation between randpisthe same bothrelativistically andnonrelativistically. Recall thatinbothEqs. (7.154) and(7.155) pisthemagnitude ofthemomentum perpendicular toB.butincalcu- lating yWemust useboththeperpendicular andparallel components toﬁnd;3.* COVARIANT LAGRANGIAN FORMULATIONS TheLagrangian procedure asgiven above certainly predicts thecorrect relativistic equations ofmotion. Yetitisarelativistic formulation only“inacertain sense." *The Larmor precession frequency ar|_ofEq.(SI04)hasanextra factor of2,andcorresponds tothe precession ofamagnetic moment inaconstant magnetic ﬁeld Thisisaphysically different casefrom thatofthecyclotron resonance ofacharged particle moving ataconstant speed inamagnetic ﬁeld 7.10 Covariant Lagrangian Formulations 319 Noeffort hasbeenmade tokeeptotheidealofacovariant four-dimensional form forallthelaws ofmechanics. Thus, thetime thasbeen treated asaparameter entirely distinct fromthespatial coordinates, while acovariant formulation would require thatspace andtimebeconsidered asentirely similar coordinates inworld space. Clearly some invariant parameter should beused, instead oft,totrace the progress ofthesystem point inconﬁguration space. Further, theexamples ofLa- grangian functions discussed intheprevious section donothaveanyparticular Lorentz transformation properties. Hamilton’s principle must itself bemanifestly covariant, which canonlymean inthiscasethattheaction integral must beaworld scalar. Iftheparameter ofintegration isaLorentz invariant, thentheLagrangian function itself must beaworld scalar inanycovariant formulation. Finally, in- stead ofbeing afunction ofx,and25,,theLagrangian should beafunction of thecoordinates inMinkowski space andoftheir derivatives withrespect tothe invariant parameter. Weshall consider primarily asystem ofonlyoneparticle. Tirenatural choice oftheinvariant parameter insuchasystem would seemtobetheparticle’s proper time 1:.Butthevarious components ofthegeneralized velocity, u”,must then obey therelation u-u=u,,u" =02, (7.35) which shows theyarenotindependent. Therefore, weshallinstead assume the choice ofsome Lorentz-invariant quantity 0withnofurther speciﬁcation thanthat itbeamonotonic function oftheprogress oftheworld point along theparticle’s world line.Forthepurpose ofthisdiscussion, asuperscript prime willbeusedto denote differentiation withrespect to0: x...Edi"d9’ while adotovertheletter indicates differentiation withrespect tot.Asuitably covariant Hamilton’s principle must therefore appear as 92 at=5faA(x“.x'”)d6, (7.156)l where theLagrangian function Amust beaworld scalar andthe(x"‘,x’")means afunction ofalloranyofthese. Note thatthisformulation includes what would haveordinarily beencalled “time-dependent Lagrangians,” because Aisconsid- eredafunction ofxo.TheEuler—Lagrange equations corresponding toEq.(7.156) GIG d 8A BAE —W=0. (7.157) Theproblem istoﬁndtheform ofAsuchthatEqs.(7.157) areequivalent tothe equations ofmotion, Eq.(7.73). Onewayofseeking Aistotransform theaction integral from theusual integral overttooneover0,andtotreatthetimetappearing explicitly intheLagrangian 0 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity notasaparameter butasanadditional generalized coordinate. Since 8mustbea monotonic function oftasmeasured insome Lorentz frame, wehave dx‘ dx'd0 x" Hence, theaction integral istransformed as lg 1 92 _xi] 0 ]=/_ L(X],l,iJ)dt=--/ LxJ.c--6 x'd9. 1| C91 X, Itwould seem therefore thatarecipe forasuitable Aisgiven bytherelation 14> IJ A(x“,x'”)=x—L x",cx—0 . (7.159)C X’ TheLagrangian obtained thiswayishowever astrange creature, unlike any Lagrangian wehave sofarmet.Note thatnomatter what thefunctional form of L,thenewLagrangian Aisahomogeneous function ofthegeneralized velocities intheﬁrstdegree: A(x", ax”"‘) =aA(x",x'“). (7.160) Thisisnotaphenomenon ofrelativistic physics perse;itisamathematical conse- quence ofenlarging conﬁguration space toinclude rasadynamical variable and using some other parameter tomark thesystem-point’s travel through thespace. ALagrangian obeying Eq.(7.l60)isoften called (somewhat misleadingly) aho- mogeneous Lagrangian andthecorresponding “homogeneous” problem ofthe calculus ofvariations requires special treatment. Themostserious oftheresulting difﬁculties willarise intheHamiltonian formulation, butwecanglimpse some of them bynoting thatinconsequence theenergy function h,according toEq.(2.53), isidentically zero. Itfollows from Eu1er’s theorem onhomogeneous functions that ifAishomogeneous toﬁrstdegree inx"‘,then HAA=X,“ Wecanthenshow (cf.Derivation 10attheendofthischapter) thatasaresult the function Aidentically satisﬁes therelation d BA BA Thus, ifanythreeoftheLagrangian Eqs.(7.157) aresatisﬁed, itwillfollow, solely asaconsequence ofthehomogeneous property ofA,thatthefourth issatisﬁed identically. Being thusforewamed totread carefully, sotospeak, letuscarry outthistrans- formation forafreeparticle. From Eq.(7.136), the“relativistic” but“nonc0vari- 7.10 Covariant Lagrangian Formulations 321 ant”Lagrangian forthefreeparticle is L=-mc\/c2 —i'Ji,-. Bythetransformation ofEq.(7.159), apossible covariant Lagrangian isthen A=—mc, /x”‘x"‘. (7.162) With thisLagrangian, theEuler—Lagrange equations areequivalent to d I_ mcx =O as Theparameter 9mustbeamonotonic function oftheproper timersothatderiva- tives withrespect to6arerelated tothose interms ofraccording to dxlzizﬁu _d9 d0' Hence, theLagrangian equations correspond to d mcu d(mu) z(W)=?=°» which areEqs.(7.73) forafreeparticle. Aswehaveseenabove, thefourth of these equations saysthatthekinetic energy Tisconserved, which isindeed not newbutcanbederived from theother three equations. Wehavethusbeen ledtoacovariant Lagrangian procedure thatworks, atleast forasingle freeparticle. butonlyinatortuous fashion. Theelaborate superstruc- turecanbegreatly simpliﬁed however byafewboldpragmatic steps. Firstofall, wecanavoid using 0andwork interms oftheproper time‘Cdirectly byaproce- dureintroduced inaslightly different context byDirac. Theconstraint onthegen- eralized velocities interms ofr,Eq.(7.35), isnotatmedynamical constraint on themotion; rather itisageometric consequence oftheWayinwhich 1'isdeﬁned. Equation (7.35) saysineffect thatwecannot roam overthefullfour-dimensional uspace; weareconﬁned toaparticular three-dimensional surface inthespace. Dirac calls relations such asEq.(7.35) weak equations. Wecanwith impunity treat u”asunconstrained quantities, andonlyafter alldifferentiation operations havebeencarried out,needthecondition ofEq.(7.35) beimposed. Certainly the procedure would haveworked above forthefreeparticle Lagrangian. There would have been nodifference if6were setequal to1:from thestartandEq.(7.35) ap- plied onlyinthelaststep.Thecovariant Lagrange equations canwiththisproviso therefore bewritten directly interms of1': d BA BA— —— —-—=. .3dr(ﬁuv) 8x” 0 (716 ) Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity Secondly, itisnotasacrosanct physical lawthattheaction integral inHamil- ton’s principle must have thesame value whether expressed interms oftorin terms of6(orr).Itneedn’t begiven bytheprescription ofEq.(7.159). Allthat isrequired isthatAbeaworld scalar (orfunction ofaworld scalar) thatleads to thecorrect equations ofmotion. Itdoesn’t havetobehomogeneous toﬁrstdegree inthegeneralized velocities. Forexample, asuitable Aforafreeparticle would clearly bethequadratic expression A=gmtnu". (1.164) Many other possibilities areavailable.* WeshalluseEq.(7.162) forthe“kinetic energy” partoftheLagrangian inallsubsequent discussions; many present and future headaches willthereby beavoided. Iftheparticle isnotfree, butisacted onbyexternal forces, theninteraction terms have tobeadded totheLagrangian ofEq.(7.164) thatwould leadtothe corresponding Minkowski forces. Very little canbesaidatthistime about the additional terms, other thantheymust beLorentz-invariant. Forexample, ifG” were some (external) four-vector, thenGﬁx“would besuitable interaction term. Ifinsome particular Lorentz frame G1=maandallother components vanish, thenwewould haveanexample ofaconstant force suchasdiscussed inthepre- vious section. Ingeneral, these tenns willrepresent theinteraction oftheparticle withsomeextemal ﬁeld.Thespeciﬁc formwilldepend uponthecovariant formu- Iation oftheﬁeldtheory. Wehaveonlyoneexample ofaﬁeldalready expressed inacovariant way—the electromagnetic ﬁeld—and itisinstructive therefore to examine theLagrangian foraparticle inanelectromagnetic ﬁeld. Asuitable Lagrangian caneasily beseentobe A(x”,ll”)=%mu,,,u" +qu"'A,,(x*). (7.165) Thecorresponding Lagrange’s equations arethen d qdA" 8 z;""“">='7+a;a(‘1"“"#)’ which areexactly thegeneralized equations ofmotion Eq.(7.73), with the Minkowski force K,.onacharged particle, Eq.(7.74). Note thatagain the“me- chanical momentum” four-vector pl‘differs fromthecanonical momentum P“: *Ingeneral, Acanhavetheform mf(u,,u"), where f(y)isanyfunction ofysuchthat Elf 1 17>".=.2=5“ lnEq.(7164), wehave usedj(u,,u") =%u,,u". Thechoice f("vl4v) =—c~/vim" corresponds toEq(7162) 7.10 Covariant Lagrangian Formulations 323 8ApIL=ﬁ=mu“+qA#=pIL+qA“ u byatenn linear intheelectromagnetic potential. Thecanonical momentum, P, conjugate toxoisnow E 1-Po=—+q9=—E, C 1. C\ where Eisthemechanical energy andEisthetotalenergy oftheparticle, E+ qgb.Thus, themomentum conjugate tothetimecoordinate isproportional tothe total energy. Asimilar conjugate connection between these twoquantities will recur laterinnonrelativistic theory. Theconnection between themagnitude ofthe spatial “mechanical” momentum andtheenergy Eisstillgiven byEq.(7.38’). From Eq.(7.166), itisseenthatthecanonical momenta conjugate toxformthe components ofaspatial Cartesian vector 'Prelated topby P=p+qA. (7.167) lnterms of"P,Eq.(7.100) canberewritten as E2=(‘P-qA)2+m2c4, (7.168) which isauseful relation between theenergy Eandthecanonical momentum vector P. Theinteraction termintheLagrangian ofEq.(7.165) isanexample ofavector ﬁeldinteraction (asisalsoatermoftheformG,,x“). Wecould alsohaveasim- plescalar ﬁeldinteraction where thetermadded totheLagrangian would besome world scalar 1/r(x"‘). Ormore complicated invariant interaction terms canbecre- atedinvolving anextemal tensor ﬁeld. Thenature ofsuchLagrangians properly stems from thephysical ﬁeldtheory involved andcannot concern usfurther here. Sofarwehave spoken onlyofsystems comprising asingle mass particle. Mul- tiparticle systems introduce newcomplications. Oneobvious problem isﬁnding aninvariant parameter todescribe theevolution ofthesystem—each particle in thesystem hasitsownproper time.Withalittlethought, however. wecould imag- ineways ofsolving thisdifﬁculty. Forexample, theproper timeassociated with theC-O-M system involves asymmetric treatment ofallthep3.l1iClCS andmight prove suitable. Wecould alsoinclude inthepicture interactions oftheparticles withextemal ﬁelds verymuch aswasdoneforasingle particle. Thegreatstum- bling block however isthetreatment ofthetypeofinteraction thatissonatural andcommon innonrelativistic mechanics—dircct interaction between particles. Atﬁrstsight, itwould seem indeed thatsuchinteractions areimpossible in relativistic mechanics. Tosaythattheforce onaparticle depends upon thepo- sitions orvelocities ofother particles atthesame time implies propagation of effects withinﬁnite velocity from oneparticle toanother—“act.ion atadistance.” Inspecial relativity, where signals cannot travel faster than thespeed oflight, 7.11 IChapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity action-at-a-distance seems outlawed. Andinacertain sense thisseems tobethe correct picture. Ithasbeenproven thatifwerequire certain properties ofthesys- temtobehave inthenormal way(such asconservation oftotallinear momentum), thentherecanbenocovariant direct interaction between particles except through contact forces. There have been many attempts inrecent years togetaround this“no- interaction" theorem. After all,wehave seen thatelectromagnetic forces can beexpressed covariantly, andastatic electric ﬁeld gives risetotheCoulomb lawofattraction, which hasthesame form asthesupposedly harmed Newtonian gravitational attraction. Some ofthese attempts have ledtoapproximately covari- antLagrangians, correct through orders of1.12/c2. Others involve formulations of mechanics atvariance with ournonnal structures; most forexample cannot be stated interms ofasimple Harnilton’s principle. INTRODUCTION TOTHE GENERAL THEORY OFRELATIVITY Thus farwehave been careful tousethetenn “special theory ofrelativity” and nottointroduce thetenn “special relativity,” bywhich weendeavored tomake clear thatitisthetheory thatisspecial, nottherelativity. Thespecial theory uses ideal inertial frames thatareassumed toexist over allofspacetime. The general theory notonlyremoves thatrequirement, butalsohasaspacetime whose nature ispartofthesolution tothequestion ofmotion. Toparaphrase JohnA. Wheeler: “Matter tellsspace howtobend, andspace returns thecompliment by telling matter howtomove.” Thegeneral theory isoften interpreted interms of non-Euclidean geometry, soterms likegeodesic (paths ofshortest distance) and curvature ofspacetime areoften used.Inthisbriefsection wecanonlyoutline the formalism ofthegeneral theory toshow howthefulltensor notation isused. Fiveprinciples guided Einstein inthedevelopment ofthegeneral theory: 1.Mach ’sprinciple—the special theory used inertial frames. E.Mach ob- served thatNewtonian inertial frames were notrotating withrespect tothe ﬁxedstars. Thissuggests Mach’s principle, whereby inertial properties are determined bythepresence ofother bodies intheuniverse. Principle ofequivalence—whereby thegravitational mass foreach body in theuniverse canbeconsistently anduniversally chosen toequal itsinertial mass. Tothebestaccuracy ofallexperiments performed todate. theratio ofthegravitational mass (themass thatappears inNewton’s force lawfor gravity) totheinertial mass (themass thatappears inthesecond law) of anyobject isindependent ofboththetotalmass andofthecomposition of theobject. This means thatnolocal experiments candistinguish nonrotat- ingfreefallinagravitational ﬁeldfromunifonn motion intheabsence of anygravitational ﬁelds. Likewise, local experiments cannot distinguish be- tween being atrestinauniform gravitational ﬁeldandundergoing uniform acceleration intheabsence ofanygravitational ﬁeld(thatis,inarocket).2. 7.11 Introduction totheGeneral Theory ofRelativity 325 3.Principle ofc0variance—in thespecial theory, allinertial observers are equivalent. Thegeneral theory extends thisideabypostulating theprinciple ofcovariance. Thisprinciple isthatallobservers, inertial ornot,observe the same lawsofphysics. Thatmeans thelawsofphysics canbeexpressed in terms oftensors, since tensors aregeometric objects deﬁned independent of anycoordinate system. 4.Correspondence principle—in weak gravitational ﬁelds with velocities small compared tolight, thegeneral theory should make predictions that approximate thepredictions ofgravitational behavior inNewtonian me- chanics. Asgravitational ﬁelds gotozero, thecorrespondence principle states thepredictions ofthegeneral theory should approach those ofthe special theory. 5.Principle ofminimal gravitational coupling—this principle postulates that noterms explicitly containing thecurvature should beadded inmaking the transition from thespecial theory tothegeneral theory. Newton’s ﬁrstlawtellsusthatintheabsence ofextemal force bodies move along straight lineswithout acceleration. Thepreceding guiding principles sug- gestthatinthegeneral theory, objects willmove along thegeodesics ofspacetime. Forexample, letusconsider afamily ofgeodesics thatstartoutparallel. Ifgrav- itational effects intheregion under consideration areuniform, thegeodesics will remain parallel. Ifthere isanonuniform gravitational ﬁeld, thegeodesics should starttoapproach orrecede. Thechange inseparation, orgeodesic deviation, isthe proper measure ofthegravitational ﬁeld. Near Earth's surface, weoften assume thegravitational ﬁeldisuniform oversmall regions. Thus, weassume twofalling bodies released sidebysidefallparallel. Anexperiment forlarger separations or longer falltimes measures thenonuniformity ofEarth’s gravitational ﬁeld. Toillustrate this,letusconsider anexample oftwoballsseparated horizontally byadistance, d,which aredropped atthesame timefrom thesame height high above Earth. Veryclose toeither ball,andneglecting thegravitational massofthe balls, local experiments willgiveresults thatallow ustotreatthelocal region as aninertial frame. Locally, gravity canbemade tovanish byachoice ofcoordinate frame. Letuschoose thislocal free-fall frame forourobservations. Locally this satisﬁes theconditions foraninertial frame. However, astheballs falltoward Earth, their separation, d,decreases. This change inseparation, rather thanthe falltoward Earth, isthelocal measure ofthegravitational effect ofEarth since it cannotbeeliminated byachoice offrame. Thisisreﬂected bythegeneral theory statement thatonly thetides (differential effects) arerealgravitational effects. Anyother gravitational effects canbelocally eliminated byfreely falling. Now consider twogeodesics asshown inFigure 7.5.Wecandeﬁne twovector ﬁelds atanypoint. Oneﬁeld, denoted byu,gives the4-velocity ofmotion along thegeodesic, while theother ﬁeld, denoted by5,gives theseparation tothenext geodesic. Weassume atsome time. 1',there were testparticles atthehead andtail ofthe5vector. Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity 5 ll FIGURE 7.5 Tangent vector, u,anddeviation vector. 5. Weshallusetheproper timeatthetailofthedeviation vector andhavethe head point towhere theother testparticle isatthattime. Ingeneral, asthemotion progresses, theproper time oftheﬁrsttestparticle willnotbethesame proper time fortheother testparticle. Astraightforward calculation, intheNewtonian limit, fortheexample oftwofalling balls, gives forthespace components of5 perpendicular tothedirection toward Earth’s center, d21 Ti =R5’, (7.169) where Rdepends uponthedistance toEarth’s center andotherphysical constants. Equation (7.169) saystheacceleration inthesepwation oftwogeodesics ispro- portional totheirseparation. Atwo-dimensional example isthegeodesics onthe surface ofasphere. Consider twoinitially parallel geodesics onasphere. These geodesics willmeet after theyhave traveled one-quarter ofthecircumference of thesphere. Forthiscase, Eq.(7.169) hasR=1/a2, where aistheradius ofthe sphere. Ifweanalyze thisproblem inthree ormore dimensions, therelative accelera- tioniswritten asD2§/dsz where dsisthelength ofthetravel along thegeodesic andweuseaDforthederivative since ourcoordinate system iscompletely arbi- trary. Thetwrsts andturns inthecoordinate system cancause changes inthecom- ponents ofEeven ifitsmagnitude isnotchanging. Ashedeveloped more ofthe theory, Einstein discovered thatthemathematicians—in particular, Riemann- hadalready developed themathematical tools needed. Themetric serves therole ofpotentials andderivatives ofthemetric givethegeometric forces. Since the derivatives ofthemetric arenottensors, acombination ofthederivatives andthe metric must beused. There arealsoproblems introduced bythefreedom ofusing anycoordinate system. Some ofthechanges areduetophysical forces andothers areduetothechoice ofthecoordinate system inanalogy totheCoriolis effect in arotating coordinate system. Thecorrect expression forthedeviation ofgeodesic motion isprovided byatensor named Riemann. Itisconstructed ofl.inear com- binations ofsecond derivatives ofthemetric contracted withthemetric. Riemann hasslotsforthree vectors andoneslotforasingle one-fonn. Ifweputthetangent vector intothesecond andfourth slotsandthedeviation vector intothethirdslot. 7.11 Introduction totheGeneral Theory ofRelativity 327 Riemann produces V“Vu§+Riemann(. ..,u.é.u)=O, (7.170) wherevuvu=Incomponent notation, Eq.(7.170) is 425" ax!‘ dxsF +Rap)”; $57 F =O. (7.171) Ifwecontract Riemann onslots land3,weproduce atensor called Ricci, deﬁned as Ricci(u. v)=Riemann(w°‘, u,ea,v), (7.172) whose components are R,,_,,=R°‘,,(,,,. (7.173) Another critical contraction produces thecurvature scalar, called R R=Ricci(w°‘, ea)=Ra“. (7.174) Ofallthese possible contractions ofRiemann, onlyonetensor ofrank retains allthedifferential symmetries ofRiemann. Thattensor iscalled Einstein (denoted byG)andisdeﬁned as c=Ricci-%gR, (7.175) withcomponents 0,“,=Rm,-%g,,WR. (7.176) Using Ttodenote thestress-energy tensor, Einstein’s ﬁeldequations make Ein- stein proportional toT. G=kT. (7.177) These equations forWeak gravitational ﬁelds andforspeeds much lessthan lightapproach Newtonian gravitational theory, andfornogravitational ﬁelds pro- duce theresults ofthespecial theory. They alsocorrectly predict allthemeasured ﬁrst-andsecond-order corrections tothespecial theory ofrelativity inexperi- ments thusfarperfonned. Inaddition, thetheory predicts theexistence ofgravita- tional waves frommoving masses. Although thesewaves havenot,atthiswriting, been directly observed, measured changes intheperiods ofseveral binary star systems areconsistent withtheexistence ofsuchradiation existing. Soon after Einstein proposed Eqs. (7.177), astronomers pointed outthatthe solutions ofthese equations were notconsistent withtheir observation ofastatic universe thatwasneither expanding norcontracting. Einstein modiﬁed theequa- tions byadding atermthatWasproportional tothemetric tensor. Theconstant of 8 Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity proportionality, called thecosmological constant, wasdenoted byAgiving G+Ag=kT. (7.178) Soon after that,astronomers decided thattheobservational datashowed that theuniverse wasexpanding andthecosmological constant wasnotneeded, and mostphysicists dropped thetemi.Einstein saidthatthecosmological constant was hisgreatest mistake. However, theearly21stcentury observational dataondistant galaxies suggests thattheuniverse isaccelerating asitexpands. Thiswould re- introduce thecosmological constant intotheﬁeldequations. Thecurrent terminol- ogy,since thiswould beaA<0,istorefer tothecosmological constant as“dark energy," since itisapositive contribution totheright-hand sideofEq.(7.178). DERIVATIONS 1.Consider amechanical system ofnparticles, withaconservative potential consisting oftenns dependent only upon thescalar distance between pairs ofparticles. Show explicitly thattheLagrangian forthesystem when expressed incoordinates derived byaGalilean transformation differs informfromtheoriginal Lagrangian onlybya tennthatisatotaltimedenvative ofafunction oftheposition vectors. Thisisaspecial caseofinvariance under apointtransformation (cf.Derivation l0,Chapter l). 2.Obtain theLorentz transformation inwhich thevelocity isataninﬁnitesimal angle d6 counterclockwise fromthexaxis,bymeans ofasimilarity transformation applied to Eq(7.16). Show directly thattheresulting matrix isorthogonal andthattheinverse matrix isobtained bysubstituting —vforv. 3.TheEinstein addition lawcanalsobeobtained byremembermg thatthesecond ve- locity isrelated directly tothespace components ofafour-velocity, which maythen betransformed back totheinitial system byaLorentz transfonnation. Ifthesecond system ismoving withaspeed ii’relative totheﬁrstinthedirection oftheir zaxes, while athirdsystem ismoving relative tothesecond withanarbitrarily oriented ve- locity v”,show bythisprocedure thatthemagnitude ofthevelocity vbetween the ﬁrstandthirdsystem isgiven by \/Ii /1_ﬂr2 /1__ﬂrr2 "'3andthatthecomponents ofvare ﬁ_n;’\/1—ﬁ'2 5_ﬁ;f\/1-5” ﬁ_a'+r;' "1+r’r2" "I+ri'ii2" “1+r'ﬂ2' Here [if=vjf/c, andsoforth. 4.Show thatthemagnitude ofthevelocity ofthepreceding exercise between theﬁrst andthethirdsystems canbegiven 1l'1general by Derivations 329 ﬁ2 :(B! +BH)2 _(Bl XBI!)2 (|+B:_Br/')2 ' Show thatthematrix Rdeﬁned byEq(7.21) hastheform ofaspatial rotation bydoing thematrix multiplication, andbyexamining theproperties ofthe3><3submatrix with elements RU.Prove thatthere cannot betworotation matrices suchthatEq.(7.21) is satisﬁed; thatis,Risunique. Finally, show thatlcansimilarly beuniquely factored intoarotation andapureLorentz transformation inthefonn L=P"R’. Show thattoeachplane wave there isassociated acovariant four-vector involving the frequency andthewave number. From theconsequent transformation equations ofthe components ofthefour-vector, derive theDoppler-effect equations. From thetransformation properties oftheworld acceleration, show thatthecompo- nents oftheacceleration aaregiven interms ofthetransformed acceleration a’ina system momentarily atrestwithrespect totheparticle bytheformulas ar= ax ax= ay ar= al X (1__ﬂ2)3/2' Y 1__52’ Z 1_ﬂ2‘ thexaxisbeing chosen inthedirection oftherelative velocity. Byexpanding theequation ofmotion, Eq.(7.73), withEq.(7.36) forthemomentum show thattheforce isparallel totheacceleration Only when thevelocity iseither parallel orperpendicular totheacceleration. Obtain expressions forthecoefﬁcients oftheacceleration inthese twocases. Intheolder literature, these coefﬁcients were known asthelongitudinal andtransverse masses, respectively. Ageneralized potential suitable foruseinacovariant Lagrangian forasingle particle U=—A;_,,(x”')u}‘uv Where AM,stands forasymmetric world tensor ofthesecond rank andu"arethe components oi‘theworld velocity. IftheLagrangian ismade upofEq.('7.I64)minus Z/I,obtain theLagrange equations ofmotion. What istheMinkowski force‘? Give the components oftheforce asobserved insome Lorentz frame. Show thatifAsatisﬁes theLagrange equations, itidentically satisﬁes Eq.(7161) onthebasis ofthehomogeneity ofA,byexplicitly fanning thetotalderivative with respect to6thatoccurs intheequation. lnspecial relativity, itisnotnecessarily obvious thatthevelocity ofsystem Bas observed insystem Aisthenegative ofthevelocity vector ofsystem Aobserved in system B.From theorthogonality properties ofL,prove thatthetwovectors have thesame magnitude andareinI"actthenegative ofeach other. Forsimplicity, apure Lorentz transformation maybeassumed, although thiscondition isnotnecessary for theproof. Asetoftransformations aresaidtohave thegroup property iftheypossess thefol- lowing fourcharacteristics: Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity 0Thetransformation equivalent totwosuccessive transformations (‘1iroduct” of transformations) isamember oftheset. 0Theproduct operation obeys theassociative law. 0Theidentity transformation isamember oftheset. 0Theinverse ofeach transformation inthesetisalsoamember oftheset. Prove thatthesetsoffullLorentz transformations andofrestricted Lorentz transfor- mation have (separately) thegroup property. EXERCISES 13.Show bydirect multiplication ofthevector fonn oftheLorentz transformation, Eqs.(7.9), that r'2—c2/2 =r2—cztz. 14.Arocket oflength loinitsrestsystem ismoving with constant speed along thez axisofaninertial system. Anobservei attheorigin ofthissystem observes theap- parent length oftherocket atanytimebynoting thezcoordinates thatcanbeseen fortheheadandtailoftherocket Howdoesthisapparent length varyastherocket moves fromtheextreme leftoftheobserver totheextreme right? Howdothese re- sultscompare withmeasurements intherestframe oftheobserver? (Note: observe, notmeasure). 15.Abeam ofparticles moving withuniform velocity collides withacollection oftarget particles thatareatrestinaparticular system. Let00bethecollision cross section observed inthissystem. Inanother system. theincident particles haveanormalized velocity B1andthetarget particles anormalized velocity B2.If0istheobserved cross section inthissystem, show that 2 a=a0‘ll— . (—B2) Remember thatcollision ratemust beinvariant under aLorentz transformation. 16.Fora“close” satellite ofEarth (semimajor axisapproximately theradius ofEarth) calculate numencally thevalue oftheThomas precession rate.Compare theresult with theprecession rateinduced intheorbit because oftheoblate ﬁgure ofEarth. Assume thesatellite orbital plane isinclined at30°totheequator. 17.Twoparticles withrestmasses mlandmgareobserved tomove along theobserver’s zaxistoward eachother withspeeds v1andv2,respectively. Upon collision, theyare observed tocoalesce intooneparticle ofrestmass mgmoving withspeed v3relative totheobserver. Findm3andv3interms ofm1,m2,v1.andU2.Would itbepossible l"ortheresultant particle tobeaphoton, thatis,m-4=0,ifneither mlnormgarezero? 18.lnthe5disintegration considered inExercise I7,Chapter l,theelectron hasamass equivalent toarestenergy of0.511 MeV, while theneutrino hasessentially nomass Exercises 331 What arethetotalenergies carried away bytheelectron andneutrino? What fraction of thenuclear mass isconverted intokinetic energy (including theelectron restenergy)? Ameson ofmass mnatrestdisintegrates intoameson ofmass ml)andaneutnno of effectively zeromass Show thatthekinetic energy ofmotion oftheti.meson is T_(mar —mp.)2c2 — 2m” . Arr+meson ofrestmass l39.6 MeV collides withaneutron (restmass 939.6 MeV) stationary inthelaboratory system toproduce aK+meson (restmass494MeV) and aAhyperon (restmass 1116 MeV). What isthethreshold energy forthisreaction in thelaboratory system? Aphoton maybedescribed classically asaparticle ofzeromass possessing never- theless amomentum h/it=hv/c, andtherefore akinetic energy hv.ifthephoton collides withanelectron ofmass matrest,itwillbescattered atsome angle 6witha newenergy hii’.Show thatthechange inenergy isrelated tothescattering angle by thefonnula 1 .9it-3»=21¢SH12:2-, where ii‘=h/mc, isknown astheCompton wavelength. Show alsothatthekinetic energy oftherecoil motion oftheelectron is I—*+to[Q/_\ /-\>5’>-3"\_/\_/‘L’.<.'.=‘.s=5)IQQ roserelive Aphoton ofenergy Ecollides atangle 0withanother photon ofenergy E.Prove that theminimum value ofE.‘permitting formation ofapairofparticles ofmass mis 2m2c4 5'"- Thetheory ofrocket motion developed inExercise I3,Chapter 1.nolonger applies in therelativistic region, inpartbecause there isnolonger conservation ofmass. Instead, alltheconservation laws"arecombined intotheconservation oftheworld momentum; thechange ineach component oftherocket’s world momentum inaninﬁnitesimal timedzmust bematched bythevalue ofthesame component ofpvforthegases ejected bytherocket inthattimeinterval. Show thatiithere arenoexternal forces acting ontherocket, thedifferential equation foritsvelocity asafunction ofthemass 1S dv v2mg’;-+d(l—c—2)—0, where aistheconstant velocity oftheexhaust gases relative totherocket. Verify that thesolution canbeputintheform Chapter 7TheClassical Mechanics oftheSpecial Theory ofRelativity 20/r. l+(%) mgbeing theinitial mass oftherocket. Since mass isnotconserved, what happens to themass thatislost? Aparticle inhyperbolic motion starts from theongin att=0.Find thetimetosuch thatifaphoton isemitted from theorigin after to,itwillnever catch upwith the particle. Aparticle ofrestmass m,charge q,andinitial velocity voenters auniform electric ﬁeldEperpendicular tovg.Find thesubsequent trajectory oftheparticle andshow thatitreduces toaparabola asthelimit cbecomes inﬁmte. Show thattherelativistic ﬂlO[lOl'l ofaparticle inanattractive inverse-square lawof force isaprecessing ellipse. Compute theprecession oftheperihelion ofMercury resulting from thiseffect. (The answer, about 7”percentury, ismuch smaller than theactual precession of43”percentury thatcanbeaccounted forcoirectly onlyby general relativity. Theother planets produce aprecession greater than5.000” per century.) Starting fromtheequation ofmotion (7.73), derive therelativistic analog ofthevirial theorem, which states thatformotions bounded inspace andsuch thatthevelocities involved donotapproach indeﬁnitely close tot,then Z5+T=—F-r, where L9istheform theLagrangian takes intheabsence ofexternal forces. Note that although neither L0norTcorresponds exactly tothekinetic energy innonrelativistic mechanics, their sum, L+T,plays thesame roleastwice thekinetic energy inthe nonrelativistic virial theorem, Eq.(3.26). Lete|ande2bethebasis vectors foraCartesian coordinate system inatwo- dimensional Euclidean space thatcontains acrystal whose lattice vectors area=e1 andb=e1+82.Usetheunderlying Euclidean geometry todetermine thattherecip- rocallattice vectors areA=e1—e1andB=e2.Using thea,bpairasbasisvectors, determine themetric tensor gnecessary forAandBtobethe1-forms asdeﬁned by Eqs.(7.34’) and(7.49). Using Maple orMathematica calculate theLorentz transformation matrix inEq.(7.l7) thenwithout assuming thatthevelocities intheframe S’aresmall, ﬁndtheexact Lorentz boost from StoS",(generalization ofEq.(7.20)) andtherotation (general- ization ofEq.(7.21)). Show thatyourresults reduce toEqs.(7.20) and(7.2l). Using Maple orMathemaiica orasimilar program calculate theEinstein ﬁeldequa- tions forspherical coordinates assuming Tm,=0everywhere except possibly for r=O,where thecoordinate system isundeﬁned. Themost general spherical static metric corresponds toaninterval given by dsz=e"(')c2 dtz—em’) dr2—r2(d92 +sin20d¢2), where r,6,and¢correspond totheusual three-dimensional spherical coordinates Solve these equations using anintegration constant mtoobtain theSchwarzchild so-t Exercises 333 lution forapoint source ofmass m.Asyouwilldiscover, these coordinates have a singularity atr=2m.Show thatthisisacoordinate singularity (asmgularity deter- mined bythechoice ofcoordinates) rather thanaphysical singularity byexamining thecomponents ofRiemann asrcrosses 2m. Toshow thattheword “relativity” iiithespecial theory ofrelativity does nothave its ordinary meaning, consider adiskrotating inaninertial frame about anaxisﬁxed at itscenter andperpendicular tothedisk. Mounted ontheedge ofthediskaremirrors arranged sothatlight emitted tangentially from apoint onthediskisreﬂected tan- gentially around thediskback tothestarting location. Compare thebehavior oflight emitted inthedirection ofrotation (assumed clockwise) tothebehavior oflightemit- tedintheopposite direction. Nowconsider apulse oflightemitted byasource onthe axisandusedtosynchronize theclocks ontheperimeter. Since clocks arecommonly synchronized bylight anddistance inthespecial theory (elapsed time=distance/c), what doesthissayabout theabsolute sense ofrotation inthespecial theory? Show thatthespace components ofEq.(7.68) areidentical tothecomponents inthe equation onthepreceding line. CHAPTER 3348.1ITheHamilton Equations ofMotion TheLagrangian formulation ofmechanics wasdeveloped largely intheﬁrsttwo chapters, andmostofthesubsequent discussion hasbeeninthenature ofappli- cation, butstillwithin theframework oftheLagrangian procedure. Inthischap- terweresume theformal development ofmechanics, tuming ourattention toan alternative statement ofthestructure ofthetheory known astheHamiltonian for- mulation. Nothing newisadded tothephysics involved; wesimply gainanother (andmore powerful) method ofworking withthephysical principles already es- tablished. TheHamiltonian methods arenotparticularly superior toLagrangian techniques forthedirect solution ofmechanical problems. Rather, theusefulness oftheHamiltonian viewpoint liesinproviding aframework fortheoretical exten- sions inmany areas ofphysics. Within classical mechanics itforms thebasis for further developments, suchasHamilton-Jacobi theory, perturbation approaches andchaos. Outside classical mechanics, theHamiltonian formulation provides much ofthelanguage withwhich present-day statistical mechanics andquantum mechanics isconstructed. Weshall assume inthefollowing chapters thattheme- chanical systems areholonomic andthattheforces aremonogenic, thatis,derived either from apotential dependent upon position only, orfrom velocity-dependent generalized potentials ofthetypediscussed inSection 1.5. LEGEN DRETRANSFORMATIONS AND THE HAMILTON EQUATIONS OFMOTION intheLagrangian formulation (nonrelativistic), asystem withndegrees offree- dompossesses nequations ofmotion oftheform d 3L 3L — ,—— =0. 8.1 dt(391) aql () Astheequations areofsecond order, themotion ofthesystem isdetermined for alltime only when 2ninitial values arespeciﬁed, forexample, thenq,’sandn 43$ataparticular time ti,orthennq,-’sattwotimes, tiandt2.Werepresent thestateofthesystem byapoint inann-dimensional conﬁguration space whose coordinates arethengeneralized coordinates q,-andfollow themotion ofthe system point intimeasittraverses itspathinconﬁguration space. Physically, in theLagrangian viewpoint asystem withnindependent degrees offreedom isa 8.1 Legendre Transformations andtheHamilton Equations ofMotion 335 problem innindependent variables q,-(t),andti,appears onlyasashorthand for thetimederivative ofq,.Allncoordinates must beindependent. IntheHamil- tonian formulation there canbenoconstraint equations among thecoordinates. Ifthencoordinates arenotindependent, areduced setofmcoordinates, with m<n,must beused fortheformulation oftheproblem before proceeding with thefollowing steps. TheHamiltonian formulation isbased onafundamentally different picture. Weseektodescribe themotion interms ofﬁrst-order equations ofmotion. Since thenumber ofinitial conditions determining themotion must ofcourse stillbe2n, theremustbe2nindependent ﬁrst-order equations expressed interms of2ninde- pendent variables. Hence, the2nequations ofthemotion describe thebehavior ofthesystem point inaphase space whose coordinates arethe2nindependent variables. Inthusdoubling oursetofindependent quantities, itisnatural (though notinevitable) tochoose halfofthem tobethengeneralized coordinates q,-.As weshall see,theformulation isnearly symmetric ifwechoose theother halfof thesettobethegeneralized orconjugate momenta p,already introduced bythe deﬁnition (cf.Eq.(2.44)): 3L( ,',1) .p,= (nosumonj) (8.2) where thejindex shows thesetofq’sand¢§’s.Thequantities (q,p)areknown asthecanonical vari'ables.* From themathematical viewpoint, itcanhowever beclaimed thattheq’sand 4’shavebeentreated asdistinct variables. InLagrange’s equations, Eq.(8.1), the partial derivative ofLwithrespect toq,-means aderivative taken withallother q’s andall¢}’sconstant. Similarly, inthepartial derivatives withrespect tocj,theq’s arekeptconstant. Treated strictly asamathematical problem, thetransition from Lagrangian toHamiltonian formulation corresponds tochanging thevariables in ourmechanical functions from (q,c},t)to(q,p,t),where pisrelated toqand 4byEqs.(8.2). Theprocedure forswitching variables inthismanner isprovided bytheLegendre transformation, which istailored forjustthistypeofchange of variable. Consider afunction ofonly twovariables f(x,y),sothatadifferential off hastheform df=udx+vdy, (8.3) where 3f 3f=— =—. 8.4 “ 8x’ U 3y () *Unless otherwise speciﬁed. inthisandsubsequent chapters thesymbol pwillbeused onlyforthe conjugate orcanonical momcnnim. When theforces arevelocity dependent, thecanonical momentum willdiffer from thecorresponding mechanical momentum (cf.Eq.(247)). Chapter 8TheHamilton Equations ofMotion Wewishnowtochange thebasisofdescription fromx,ytoanewdistinct setof variables u,y,sothatdifferential quantities areexpressed interms ofthediffer- entials duanddy.Letgbeafunction ofuandydeﬁned bytheequation g=f-ux. (8.5) Adifferential ofgisthengiven as dg=df -udx —xdu, or,by(8.3), as dg=vdy-xdu, which isexactly intheform desired. Thequantities xandvarenowfunctions of thevariables uandygiven bytherelations as as=--, =-, .6x Bu vEly (8) which aretheanalogues ofEqs.(8.4). TheLegendre transfonnation sodeﬁned isusedfrequently inthennodynarnics. Theﬁrstlawofthennodynamics relates thedifferential change inenergy, a’U,to thecorresponding change inheatcontent, dQ,andthework done, dW: dU=dQ-a’W. (8.7) Foragasundergoing areversible process, Eq.(8.7)canbewritten as dU=TdS -PdV, (8.8) where U(S,V)iswritten asafunction oftheentr0PY» S,andthevolume, V, where thetemperature, T,andthegaspressure, P,aregiven by T=% and P=—%%. (8.9) Theenthalpy, H(S,P)isgenerated bytheLegendre transformation H=U+PV, (8.10) which gives dH=TdS+VdP. (8.11) where an anr=_ =_-.as and V3P 8.1 Legendre Transformations andtheHamilton Equations ofMotion 337 Additional Legendre transformations, F=U—TS (8.12) G=H-TS, generate theHelmholtz freeenergy, F(T,V),andtheGibbs freeenergy, G(T, P). Thetransformation from (q,4,t)to(q,p,t)differs from thetypeconsidered inEqs. (8.3) to(8.12) only inthatmore thanonevariable istobetransformed. Webegin bywriting thedifferential oftheLagrangian, L(q,4,t),as at at atdL Zid 1+ i,d I;+idt- aq,‘Iaq,‘Zat Thecanonical momentum wasdeﬁned inEq.(2.44) aspi=8L/8a,; substituting thisintotheLagrange equation (8.1), weobtain 8L'=—, 8.14 Pi aqi ( ) soEq.(8.13) canbewritten as . .3L , db=indqi+11.dqi+§d¢- (8-13) TheHamiltonian H(q,p,t)isgenerated bytheLegendre transformation H(qsp:t)=élpJ *L(q,é, t): which hasthedifferential . . 3LdH=q,dp,—p,dq;— (8.16) where thetermp,dq;isremoved bytheLegenche transformation. Since dHcan alsobewritten as H 8H H 8q, Zip, 81‘ weobtain the2n+1relations -flqr-am __aHpi-aqt(8.18) BL 8H-—— =—. (8.19)8r 8: Chapter 8TheHamilton Equations ofMotion Equations (8.18) areknown asthecanonical equations ofHamilton; theyconsti- tutethedesired setof2nﬁrst-order equations ofmotion replacing thensecond- order Lagrange equations.* TheﬁrsthalfofHarnilton’s equations givetheq,’sasfunctions of(q,p,i). They form therefore theinverse oftheconstitutive equations (8.2), which deﬁne themomenta p,asfunctions of(q,Q,t).Itmaytherefore besaidthattheyprovide nonewinformation. lnterms ofsolving mechanical problems bymeans ofthe canonical equations, thestatement iscorrect. Butwithin theframework ofthe Hamiltonian picture. where H(q,p.t)issome given function obtained nomatter how, thetwohalves ofthesetofHamiltonian equations areequally independent andmeaningful. Theﬁrsthalfsayshow4depends onq,p,andt;thesecond says thesame thing forIi. Ofcourse, theHamiltonian Hisconstructed inthesame manner. andhasiden- tically thesame value, ash,theenergy function deﬁned inEq.(2.53). Butthey arefunctions ofdifferent variables: LiketheLagrangian, hisafunction ofq,4 (andpossibly t),while Hmustalways beexpressed asafunction ofq,p(and possibly t).Itistoemphasize thisdifference infunctional behavior thatdiffer- entsymbols have been given tothequantities even though theyhave thesame numerical values. Nominally, theHamiltonian foreachproblem must beconstructed viatheLa- grangian formulation. Theformal procedure callsforalengthy sequence ofsteps: l.With achosen setofgeneralized coordinates, q,-,theLagrangian L(q,,4},,t) =T—Visconstructed. 2.The conjugate momenta aredeﬁned asfunctions ofq,,4.,andtby Eqs.(8.2). 3.Equation (8.15) isusedtoformtheHamiltonian. Atthisstagewehavesome mixed function ofq,,c],,p,,andt. 4.Equations (8.2) aretheninverted toobtain Q,asfunctions of(q,p,t).Pos- sible difﬁculties intheinversion willbediscussed below. 5.Theresults oftheprevious steparethenapplied toeliminate qfrom Hso astoexpress itsolely asafunction of(q,p,t). Now weareready tousethel-lamiltonian inthecanonical equations ofmotion. Formany physical systems itispossible toshorten thisdrawn~out sequence quite appreciably. Ashasbeen described inSection 2.7,inmany problems the Lagrangian isthesumoffunctions each homogeneous inthegeneralized veloc- *Canonical isused herepresumably inthesense ofdesignating asimple, general setofstandard equations. Itappears thattheierrnwasﬁrstintroduced byC.G.J.Jacobi in1837 (Campres rendui de I‘/lcadémie derSciences dcParis. 5.p61)butinaslightly different context referring toanapplication ofI-lamilton’s equations ofmotion toperturbation theory. Although thetennrapidly gained common usage, thereason foritsintroduction apparently remained obscure eventocontemporaries. By1879. only45years after Hamilton explicitly introduced hisequations, Thomson (Lord Kelvin) andTa.ii were moved bytheadjective “canonical” toexclaim “Why ithasbeen socalled would behardto say” 8.1 Legendre Transformations andtheHamilton Equations ofMotion 339 itiesofdegree O,l,and2,respectively. Inthatcase, Hbytheprescription of Eq.(8.15) isgiven by(cf.Eqs.(2.53) and(2.55)) H=(lap! _L=érpr —lL0(q1, t)'l'Ll(qr: 0&1: “l”L2(qr» Uékém] (nosumoniinthesquare brackets) where L0isthepartoftheLagrangian thatis independent ofthegeneralized velocities, L1represents thecoefﬁcients ofthepart oftheLagrangian thatishomogeneous in4,intheﬁrstdegree, andLgisthepart thatishomogeneous in4,inthesecond degree. Further, iftheequations deﬁning thegeneralized coordinates don't depend ontime explicitly, thenLgzjkejm =T (thekinetic energy), andiftheforces arederivable from aconservative potential V(that is,work isindependent ofthepath), then L0=-V.When both these conditions aresatisﬁed, theHamiltonian isautomatically thetotalenergy: H=T+V=E. (8.21) Ifeither Eq.(8.20) or(8.21) holds, thenmuch ofthealgebra insteps3and4above iseliminated. Wecanattimes gofurther. Inlarge classes ofproblems, ithappens thatL2isa quadratic function ofthegeneralized velocities andL1isalinear function ofthe same variables withthefollowing speciﬁc functional dependencies: L(ql1 élr t)=LOW, t)+élal (Q, + t)s where thea,’sandtheT,’sarefunctions oftheq’sandt. Thealgebraic manipulations required insteps 2-5canthenbecarried out,at least formally, once andforall.Toshow this,letusform theti,-’sintoasingle colunm matrix q.Under thegiven assumptions theLagrangian canbewritten as Lam.I)=Lo(q-0+<ia+éftrq. (8.23) where thesingle rowmatrix hasbeenwritten explicitly asthetranspose ofa single column matrix, q.Hereaisacolumn matrix, andTisasquare n><nmatrix (much likethecorresponding matrix introduced inSection 6.2).Theelements of bothareingeneral functions ofqandt.Toillustrate thisformalism, letusconsider thespecial casewhere q,={x,y,z}andTisdiagonal. Wewould thenwrite 1_ 1 mO0:2 m 5qrq=5<r>>z> omor»=;tr2+>>’+z*> (8.2%)OOm 2 and qa=(2)72) a).=axri+ayjr+azi=a-t. (8.24b) az Chapter 8TheHamilton Equations ofMotion Inthisnotation theHamiltonian. H=tip—L,becomes H=Fm»—=0-iirq—Lo- <8-24¢) Theconjugate momenta, considered asacolumn matrix p,isthen, byEq.(8.2), given as p=Tc]+a, (8.25) which canbeinverted (step 4)tothecolumn vector q=T-1(p-a). (8.263) This steppresupposes thatT‘1exists, which itnormally does byvirtue ofthe positive deﬁnite property ofkinetic pnergy. Thecorresponding equation forqis 5;=(p-5)r~‘. (8.26b) Toobtain thecorrect functional form fortheHamiltonian, Eqs. (8.26) must be usedtoreplace qandfr,yielding theﬁnalfonn fortheHamiltonian: Htq.p,0=to-in“(p-a)-Lo(q-1). (8.21) IftheLagrangian canbewritten intheformofEq.(8.23), thenwecanimme- diately skiptheintervening steps andwrite theHamiltonian asEq.(8.27). The inverse matrix T‘lcanusually mosteasily beobtained straightforwardly as TT-1=_°, (8.28)ITI where Tcisthecofactor matrix whose elements (T¢),k are(—1)j‘l'l‘ times the determinant ofthematrix obtained bystriking outthejthrowandthekthcolumn ofT. Intheexample Eq.(8.24a), these threematrices aregiven explicitly by m0O 5 0 r=[0 m0], r-'= 0-0. and OOm () iIII 00 andthedeterminant |T|=m3.Itiseasytoseethatfortheusual casewhen Tis diagonal, thenT'lisalsodiagonal withelements thatarejustthereciprocals of thecorresponding elements ofT.O3’-‘O oo5~ 05,0swo 8.1 Legendre Transformations andtheHamilton Equations ofMotion 341 Anumber ofexercises inapplying thisformalism tovarious mechanical sys- temswillbefound intheproblems attheendofthechapter. Twoverysimple examples areconsidered herebecause theyillustrate some important aspects of thetechnique. Firstconsider thespatial motion ofaparticle inacentral force ﬁeld, using spherical polar coordinates (r,6.¢)forthegeneralized coordinates. Thepotential energy issome function V(r) andthekinetic energy is Z T=-mzi=gm+r2sml0&2+rzéz). (s.2s') Clearly theHamiltonian hastheformofEq.(8.21) andcorresponds tothetotal energy T+V.Since Tisdiagonal theform ofHis,byinspection, 1 2Pei piH010, Pr,P6,P¢)= Z; (I7, +rT+ -I-l/(T). (8.29) NotethattheHamiltonian would haveadifferent functional formifthegener- alized coordinates were chosen tobetheCartesian coordinates x,oftheparticle. Ifwemake thatchoice, thenthekinetic energy hasthefonn T mvz m.i,:i, -2-" 2’ sothattheHamiltonian isnow H(x..1>.) =+1/tr). (8.30) m Itissometimes convenient toform thecanonical momenta p,conjugate tox,into avector psuchthattheHamiltonian canbewritten as Ht».P.)=+v<~/x.x'>- <8-31> Wecanofcourse takethecomponents ofprelative toanycoordinate system wedesire, curvilinear spherical coordinates, forexample. Butitisimportant notto confuse, say,pgwiththe0component ofp,designated as(p)9.Theformer isthe canonical momentum conjugate tothecoordinate 9;thelatter isthe9component ofthemomentum vector conjugate totheCartesian coordinates. Dimensionally. itiscleartheyarequiteseparate quantities; pgisanangular momentum, (p)@isa linear momentum. Whenever avector isusedfrom hereontorepresent canonical momenta itwillrefer tothemomenta conjugate toCartesian position coordinates. Forasecond example, letusconsider asingle (nonrelativistic) particle ofmass mandcharge qmoving inanelectromagnetic ﬁeld. ByEq.(1.63), theLagrangian forthissystem is L=T—V=%mv2—q¢+qA-v. where thescalar potential term. —q¢, istheL0term oftheLagrangian asex- pressed inEq.(8.22) andthevector potential term, qA-v,istheL1term. Chapter 8TheHamilton Equations ofMotion Using Cartesian position coordinates asgeneralized coordinates, theLa- grangian canalsobewritten as L= +qA.x".-—q¢, (8.32) where thepotentials ¢andAareingeneral functions ofx,andthetime. There isnowalinear term inthegeneralized velocities such thatthematrix ahastheelements qA,. Because ofthislinear term inV,theHamiltonian isnot T+V.However, itisstillinthiscasethetotalenergy sincethe“potential” energy inanelectromagnetic ﬁeld isdetermined by¢alone. Thecanonical momenta, either byEq.(8.2) orEq.(8.25), are Pi=mi: +qA,. (8-33) andtheHamiltonian (cf.Eq.(8.27)) is —A —AH= Q +q¢, (334) 2m which isthetotalenergy oftheparticle. Again, themomenta p,canbeformed intoavector pandHwritten as H=gin»-qA>’+q¢. (8.35) m andremembering thatprefers onlytomomenta conjugate tox,. Itisclear thatHamilton’s equations ofmotion donottreatthecoordinates and momenta inacompletely syrmnetric fashion. Theequation forphasaminus sign thatisabsent intheequation forzj.Considerable ingenuity hasbeen exercised indevising nomenclature schemes thatresult inentirely syrmnetric equations, orcombine thetwosetsintoone.Most ofthese schemes have only curiosity value, butonehasproved tobeanelegant andpowerful toolformanipulating the canonical equations andallied expressions. Forasystem ofndegrees offreedom, weconstruct acolumn matrix 1|with2n elements suchthat 771=qr, 77i+n =P1; iE71- (8-36) Similarly. thecolumn matrix 8H/81;hastheelements H 8H an(@_)=_. (.g) <8-31> an.8!-1» 611.~+,,61>: Finally, letIbethe2n><2nsquare matrix composed offourn><nzeroandllllll matrices according tothescheme I=|:_01 (8.38a) 8.2 I8.2 Cyclic Coordinates andConservation Theorems 343 withthefollowing transpose matrix, which isitsinverse i=H-01], (8.38b) which means -~ 10II=I]=1=[01], (8.380) SO .- 1=-1=F‘ (ma) and F=-1, (8.38e) andthedeterminant is |||.-=+1. (s.38f) Here0isthen><nmatrix allofwhose elements iszero, and1isthestandard nxnunitmatrix. Hamilt0n’s equations ofmotion canthenbewritten incompact formas 3H'=—. 3.39 nIan () Fortwocoordinate variables, thishastheexpanded form _é1 = -P2 at-10 <11* ‘8“‘°)I52 9—1 qz where usewasmade ofEqs. (8.37) and(8.18). This method ofdisplaying the canonical equations ofmotion willbereferred toasHamilton’s equations inma- trixorsymplectic* notation. Insubsequent chapters weshallfrequently employ thismatrix formoftheequations.~§-v-§-Nl—lDC GO O©O'-* O©'—*O CYCLIC COORDINATES AND CONSERVATION THEOREMS According tothedeﬁnition given inSection 2.6,acyclic coordinate qJisonethat does notappear explicitly intheLagrangian; byvirtue ofLagrange’s equations *The termsymplectic comes from theGreek for“intertwined,” particularly appropriate forHam.Llton’s equations where 4ismatched withaderivative withrespect topandpsimilarly withthenegative of aqderivative H.Weyl ﬁrstintroduced theterminI939inhisbook TheClassical Groups. 44 Chapter 8TheHamilton Equations ofMotion itsconjugate momentum [7]isthenaconstant. Butcomparison ofEq.(8.14) with Eq.(8.16) hasalready toldusthat ,at anr>)=—=--—-aqi 341 Acoordinate thatiscyclic willthusalsobeabsent from theHamiltonian.* Con- versely ifageneralized coordinate doesnotoccur inH,theconjugate momentum isconserved. Themomentum conservation theorems ofSection 2.6canthusbe transferred totheHamiltonian formulation withnomore thanasubstitution ofH forL.Inparticular, theconnection between theinvariance orsymmetry proper- tiesofthephysical system andtheconstants ofthemotion canalsobederived in terms oftheHamiltonian. Forexample, ifasystem iscompletely self-contained, withonlyintemal forces between theparticles, thenthesystem canbemoved as arigid ensemble without affecting theforces orsubsequent motion. Thesystem issaidtobeinvariant under arigid displacement. Hence, ageneralized coordinate describing sucharigid motion willnotappear explicitly intheHamiltonian, and thecorresponding conjugate momentum willbeconserved. Iftherigidmotion is atranslation along some particular direction, thentheconserved momentum isthe corresponding Cartesian component ofthetotallinear (canonical) momentum of thesystem. Since thedirection isarbitrary, thetotalvector linear momentum is conserved. Therigid displacement maybearotation, from whence itfollows that thetotalangular momentum vector isconserved. Evenifthesystem interacts with external forces, theremaybeasymmetry inthesituation thatleadstoaconserved canonical momentum. Suppose thesystem issymmetrical about agiven axisso thatHisinvariant under rotation about thataxis.Then Hobviously cannot in- volve therotation angle about theaxisandtheparticular angle variable mustbea cyclic coordinate. Itfollows, asinSection 2.6,thatthecomponent oftheangular momentum about thataxisisconservedfl Theconsiderations concerning hinSection 2.7have already shown thatifL (andinconsequence ofEq.(8.15), alsoH)isnotanexplicit function oft,then Hisaconstant ofmotion. Thiscanalsobeseendirectly fromtheequations of motion (8.18) bywriting thetotaltimederivative oftheHamiltonian as dH_E)H, +8H ,+8H dz_Elq,q' 8p,p' 8tl Inconsequence oftheequations ofmotion (8.18), theﬁrsttwosums ontheright cancel eachother, andittherefore follows that dH 8H 8L—=—=-——. 8.41dz 8: 8: () *This conclusion alsofollows from thedeﬁnition ofEq(8.15). forHdiffers from —Lonlybyp,11,. which doesnotinvolve q,explicitly. lThe relation between conservation laws. symmetry oftheLagrangian, (andtheHamrltoruan) ofthe system iscalled N0ether’s theorem. Theformal proof isgiven inSection 137. 8.2 Cyclic Coordinates andConservation Theorems 345 Thus iftdoesn’t appear explicitly inL,itwillalsonotbepresent inH,andH willbeconstant intime. Further, itwasproved inSection 2.7thatiftheequations oftransformation that deﬁne thegeneralized coordinates (1.38), rm=rm(q1=---»qn;t)- donotdepend explicitly upon thetime, andifthepotential isvelocity indepen- dent,thenHisthetotalenergy, T+V.Theidentiﬁcation ofHasaconstant ofthe motion andasthetotalenergy aretwoseparate matters, andtheconditions sufﬁ- cientfortheonearenotenough fortheother. Itcanhappen thattheEqs.(1.38) doinvolve timeexplicitly butthatHdoesnot.Inthiscase, Hisaconstant of themotion butitisnotthetotalenergy. Aswasalsoemphasized inSection (2.6), theHamiltonian isdependent bothinmagnitude andinfunctional fonn upon the initial choice ofgeneralized coordinates. FortheLagrangian, wehave aspeciﬁc prescription, L=T—V,andachange ofgeneralized coordinates within that prescription maychange thefunctional appearance ofLbutcannot alteritsmag- nitude. Ontheother hand, useofadifferent setofgeneralized coordinates inthe deﬁnition fortheHamiltonian, Eq.(8.15), mayleadtoanentirely different quan- tityfortheHamiltonian. Itmaybethatforonesetofgeneralized coordinates H isconserved, butthatforanother itvaries intime. Toillustrate some ofthese points inasimple example, wemayconsider a somewhat artiﬁcial one-dimensional system. Suppose apoint massmisattached toaspring, offorce constant k,theother endofwhich isﬁxedonamassless cart thatisbeing moved unifomtly byanextemal device withspeed vq(cf.Fig.8.1). lfwetakeasgeneralized coordinate theposition xofthemass particle inthe stationary system, thentheLagrangian ofthesystem isobviously 02 k L(x,;t, 1)=T-v='1;-Eu-v9t)2. (8.42) (Forsimplicity, theorigin hasbeenchosen sothatthecartpasses through itat t=0.)Thecorresponding equation ofmotion isclearly mié=—k(x —vot). _iL6<?>\——’E>E>’0' FIGURE 8.1Aharmonic oscillator ﬁxedtoauniformly moving cart. Chapter 8TheHamilton Equations ofMotion Anobvious wayofsolving thisequation istochange theunknown tox"(r) deﬁned as x’=x—vot, (8.43) andnoting that36’=56,theequation ofmotion becomes mi’=—kx’. (8.44) From Eq.(8.43), x’isthedisplacement oftheparticle relative tothecart; Eq.(8.44) saysthattoanobserver onthecarttheparticle exhibits simple har- monic motion, aswould beexpected ontheprinciple ofequivalence inGalilean relativity. Having looked atthenature ofthemotion, letusconsider theHamiltonian formulation. Since xistheCartesian coordinate oftheparticle, andthepotential does notinvolve generalized velocities, theHamiltonian relative toxisthesum ofthekinetic andpotential energies, thatis,thetotalenergy. Infunctional form theHamiltonian isgiven by 2 H(x, p,2‘)=T+V=L+E(x—v0t)2. (8.45)2m 2 TheHamiltonian isthetotalenergy ofthesystem, butsince itisexplicitly afunc- tionoft,itisnotconserved. Physically thisisunderstandable; energy mustﬂow intoandoutofthe“external physical device” tokeep thecartmoving uniformly against thereaction oftheoscillating particle.* Suppose nowweformulated theLagrangian fromthestartinterms oftherel- ative coordinate x’.Thesame prescription gives theLagrangian as ‘/2 2 kI2 Lot’,)2’)=%+mm,+%- (8.46) Insetting upthecorresponding Hamiltonian, wenotethere isnowaterm linear inx’,withthesingle component ofabeing mvo. ThenewHamiltonian isnow (p’—mvQ)2 kx’2 mug I1 I_ _ H(X.11)— 2m +2 2- (3-47) Notethatthelasttermisaconstant involving neither x’norp’;itcould, ifwe wished, bedropped from H’without affecting theresultant equations ofmotion. Now H’isnotthetotalenergy ofthesystem, butitisconserved. Except forthe lastterm, itcanbeeasily identiﬁed asthetotalenergy ofmotion oftheparticle relative tothemoving cart.ThetwoHamiltonian’s aredifferent inmagnitude. *Put another way, themoving canconstitutes atime-dependent constraint ontheparticle, andthe force oftheconstraint doesdowork inactual (notvirtual) displacement ofthesystem. 8.3I8.3 Routh’s Procedure 347 Cm k k @..~»..Q ,0-——->- Z-> @....@k It C111 cm ta) (b) FIGURE 8.2 Vibrating dumbbell under twoconditions: (a)freely oscillating, and(b)os- cillating withmassmgkeptataconstant velocity time dependence, andfunctional behavior. Butthereader caneasily verify that bothleadtothesame motion fortheparticle. Additional insight intotheproblem ofthemass cartpreviously discussed can begained byconsidering adumbbell oftwomasses connected byaspring of constant k.Weshallconsider thecasewhere thecenter ofmass ofthedumbbell isinconstant motion ataspeed viialong thedirection determined bythespring andallow oscillations ofthemasses onlyalong thisdirection. Thisisshown in Fig.8.2,where C-O-M denotes thecenter ofmass. Thedumbbell ismade tovibrate while itscenter ofmasshasaninitial velocity vi).Itwillcontinue withthisvelocity withunifonn translational motion. This translational motion willhave noeffect ontheoscillations. Themotion ofthe center ofmass andthemotion relative tothecenter ofmass separate astheydo intheKepler problem. Once themotion isstarted, energy isconserved andthe Hamiltonian isthetotalconserved energy. Thesituation isdifferent ifthemass m2moves attheconstant speed vi;since aperiodic force isapplied. Thecenter ofmass andthemass mithenoscillate relative tomg.Since achanging external force mustbeapplied tothesystem tokeepmgattheconstant velocity U0,the Hamiltonian isnolonger conserved, noristheHamiltonian thetotalenergy. ROUTH'S PROCEDURE Ithasbeenremarked thattheHamiltonian formulation isnotparticularly helpful inthedirect solution ofmechanical problems. Often wecansolve the2nﬁrst- order equations onlybyeliminating some ofthevariables, forexample, thep variables, which speedily leads back tothesecond-order Lagrangian equations of motion. Butanimportant exception should benoted. TheHamiltonian procedure isespecially adapted tothetreatment ofproblems involving cyclic coordinates. Letusconsider thesituation inLagrangian formulation when some coordinate, sayq,,,iscyclic. TheLagrangian asafunction ofqandqcanthenbewritten L=L(qi.---.qn-i; cii,--“tin; I)- 4 Chapter 8TheHamilton Equations ofMotion Allthegeneralized velocities stilloccur intheLagrangian andingeneral willbe functions ofthetime. Westillhave tosolve aproblem ofndegrees offreedom, eventhough onedegree offreedom corresponds toacyclic coordinate. Acyclic coordinate intheHamiltonian formulation, ontheother hand, trulydeserves itsal- ternative description as“ignorable,” forinthesame situation p,,issome constant Ol,andHhastheform H=H(qls---»qn—li pi.---.11»-1; vat)- Ineffect, theHamiltonian nowdescribes aproblem involving onlyn-1coordi- nates, Which maybesolved completely ignoring thecyclic coordinate except as itismanifested intheconstant ofintegration 0:,tobedetennined fromtheinitial conditions. Thebehavior ofthecyclic coordinate itself withtimeisthenfound by integrating theequation ofmotion ,_8H q”_8a' Theadvantages oftheHamiltonian formulation inhandling cyclic coordinates maybecombined withtheLagrangian conveniences fornoncyclic coordinates by amethod devised byRouth. Essentially, wecarry outamathematical transforma- tionfromtheq,qbasistotheq,pbasisonlyforthose coordinates thatarecyclic, obtaining theirequations ofmotion intheHamiltonian form, while theremain- ingcoordinates aregoverned byLagrange equations. Ifthecyclic coordinates are labeled q,+1, ...,qn, thenanewfunction R(known astheRouthian) maybe introduced, deﬁned as fl R(¢11,---iqni él,---Jisl P9-l-la---spit; 2 plql _Ls i=t+I which isequivalent towriting R(qla'-'iqI|'; él:"-iéd‘; .p-9+1:--raplli HcytI(P.i+l» ---tPu)_Lnonryrl(¢1l» ~»-iqt:(fl,-~~r (8-49) Itiseasytoshow forthesnonignorable coordinates, theLagrange equations d8R BRi:1!~--15.1 aresatisﬁed. while forthen—signorable coordinates, Hamilton’s equations apply as BR BR—=—' =0, d —-=', '= 1,..., . 8.51 aql p, an apt q, ls+ n () Asimple, almost trivial, example mayclarify Routh’s procedure andthephys- icalsigniﬁcance ofthequantities involved. Consider theKepler problem investi- 8.4 I8.4 TheHamiltonian Formulation ofRelativistic Mechanics 349 gated inSection 3.7,thatofasingle particle moving inaplane under theinﬂuence oftheinverse-square central force f(r)derived fromthepotential V(r)=—k/r". TheLagrangian isthen _'"-2 2'2 kL_—i-(r +r6)+;;. Asnoted before, theignorable coordinate is9,andiftheconstant conjugate mo- mentum isdenoted bypg,thecorresponding Routhian (8.49) is 2_ _pa l_2kR(I",7',p9)-—'2T1?—E7Il!" —;;. Physically weseethattheRouthian istheequivalent one-dimensional potential V'(r) minus thekinetic energy ofradial motion. Applying theLagrange equation (8.50) tothenoncyclic radial coordinate r, weobtain theequation ofmotion (3.11) ..P3 nk_r—é'r-”_—3+’m—+f-0. (8.52) Applying Hamilton's equation (8.51) tothecyclic variable 6.weobtain thepair ofequations pa=0and L2=é. (8.53)M7’ whose solution isthesame asEq.(3.8), pg=mrzé ==l=constant. Typically, Routh’s procedure doesnotaddtothephysics oftheanalysis pre- sented earlier inChapter 3,butitmakes theanalysis more automatic. Incompli- cated problems withmany degrees offreedom, thisfeature canbeaconsiderable advantage. itisnotsurprising therefore thatRouth’s procedure ﬁndsitsgreatest usefulness inthedirect solution ofproblems relating toengineering applications. Butasafundamental entity, theRouthian isasterile hybrid, combining some of thefeatures ofboththeLagrangian andtheHamiltonian pictures. Forthedevel- opment ofvarious formalisms ofclassical mechanics, thecomplete Hamiltonian formulation ismore fruitful. THE HAMILTONIAN FORMULATION OFRELATWISTIC MECHANICS AswiththeLagrangian picture inspecial relativity, twoattitudes canbetaken to theHamiltonian formulation ofrelativistic mechanics. Theﬁrstmakes nopretense atacovariant description butinstead works insome speciﬁc Lorentz orinertial frame. Time asmeasured intheparticular Lorentz frame isthennottreated ona Chapter 8TheHamilton Equations ofMotion common basis withother coordinates butserves, asinnonrelativistic mechanics, asaparameter describing theevolution ofthesystem. Nonetheless, iftheLa- grangian thatleads totheHamiltonian isitselfbased onarelativistically invariant physical theory (forexample, Maxwell’s equations andtheLorentz force), then theresultant Hamiltonian picture willberelativistically correct. Thesecond ap- proach ofcourse attempts afullycovariant description oftheHamiltonian picture, butthedifﬁculties thatplagued thecorresponding Lagrangian approach (cf.Sec- tion7.9)areevenﬁercer here.Weshallconsider thenoncovariant method ﬁrst. Forasingle-particle Lagrangian oftheformofEq.(7.136), 1.=-mt-2,/1-51 -v, wehavealready shown thattheHamiltonian (intheguise oftheenergy function h)isthetotalenergy ofthesystem: H=T+V. Theenergy Tcanbeexpressed interms ofthecanonical momenta p,(Eq.7.139) through Eq.(7.38):* T2=pzcz +m2c4, sothatasuitable formfortheHamiltonian is H=‘Ip202+m2c4 +V. (8.54) When thesystem consists ofasingle particle moving inanelectromagnetic ﬁeld, theLagrangian hasbeen given as(cf.Eq.(7.141)) L=—mr.-2,/1 —ﬂ2+qA-v—q¢. TheterminLlinear inthevelocities doesnotappear explicitly intheHamiltonian (cf.Eq.(8.54)), aswehaveseen,whereas theﬁrsttermleads totheappearance of TintheHamiltonian. Thus, theHamiltonian isagain thetotalparticle energy: H=T+q¢. (3.55) Forthissystem, thecanonical momenta conjugate totheCartesian coordinates of theparticle aredeﬁned by(cf.Eq.(7.142)) pl=muI+qA|‘ sothattherelation between Tandp’isgiven byEq.(7.168), andtheHamiltonian hastheﬁnalform *Inthissection weuseTforthemotion energy (pc)plustherestenergy (mcz) toavoid confusing it withthetotalenergy T+V 8.4 TheHamiltonian Formulation ofRelativistic Mechanics 351 H=,/(p-qA)2c2 +m2c4+q¢. (8.56) Itshould beemphasized again thatphereisthevector ofthecanonical momenta conjugate totheCartesian position coordinates oftheparticle. Wemayalsonote that(H—q¢)/c isthezeroth component ofthe4-vector mu”+qA" (cf.Eqs.(7.27), (7.38’), and(7.166)). While theHamiltonian (8.56) isnotex- pressed incovariant fashion, itdoeshaveadeﬁnite transformation behavior under aLorentz transformation asbeing, insome Lorentz form, thezeroth component ofa4-vector. Inacovariant approach totheHamiltonian formulation, timemustbetreated in thesamefashion asthespace coordinates; thatis,timemustbetaken asoneofthe canonical coordinates having anassociated conjugate momentum. Thefounda- tions ofsuchanextension ofthedimensionality ofphase space caninfactbecon- structed eveninnonrelativistic mechanics. Following thepattem ofSection 7.10, theprogress ofthesystem point along itstrajectory inphase space canbemarked bysome parameter 6,andr“released,” sotospeak, toserve asanadditional co- ordinate. Ifderivatives withrespect to6aredenoted byasuperscript prime, the Lagrangian inthe(qr,...,q,,;t)conﬁguration space is(cf.Eq.(7.l59)) I A(q,q’,r,r’)=Ht.(q,%,r). (8.57) Themomentum conjugate totisthen an_ ,atpt=57 —L+I Ifwemake explicit useoftheconnection rj=q’/1', thisrelation becomes __€i§£_L_-‘i__p,_L Z,84!_q,aqi_H. (ass) Themomentum conjugate tothetime“coordinate” istherefore thenegative ofthe ordinary Hamiltonian.’-‘ While theframework ofthisderivation iscompletely non- relativistic, theresult isconsistent withtheidentiﬁcation ofthetimecomponent of the4-vector momentum withE/c.Ascanbeseenfromthedeﬁnition, Eq.(8.2), ifqismultiplied byaconstant oi,thentheconjugate momentum isdivided byoi. Hence, thecanonical momentum conjugate toctisH/c. *The remaining momenta areunchanged bytheshiftfrom tto9,ascanbeseenbyevaluating the corresponding derivative 8A_t,8L__l, 6L1 _ liq,’Taqj'aq1'""' Chapter 8TheHamilton Equations ofMotion Thus, there seems tobeanatural route available forconstructing arelativis- tically covariant Hamiltonian. Buttheroute tums outtobemined withbooby traps. Itwillberecalled thatthecovariant Lagrangian usedtostart;theprocess, Eq.(7.159) orEq.(8.57), ishomogeneous inﬁrstdegree inthegeneralized ve- locities q’,andforsuchaLagrangian therecipe described above forconstructing theHamiltonian formulation breaks down irreparably. IfLisoftypeL|,thecor- responding Hamiltonian, callitH6(q,t,p,p,),isidentically zero! Fortunately, theredoesnotseem tobeanycompelling reason whythecovari- antLagrangian hastobehomogeneous intheﬁrstdegree, atleast forclassical relativistic mechanics. Ithasalready been seenthatforasingle freeparticle the covariant Lagrangian A(x“, u“)=émupu” leads tothecorrect equations ofmotion. Ofcourse thefour-velocity components, Lip‘,arestillnotallindependent, buttheconstraint canbetreated asa“weak con- dition” tobeimposed onlyafiferallthedifferentiations havebeencarried through. There isnownodifficulty inobtaining aHamiltonian fromthisLagrangian, by thesame route asinnonrelativistic mechanics; theresult isclearly H,=pig (8.59)2m Forasingle particle inanelectromagnetic ﬁeld, acovariant Lagrangian hasbeen found previously: (cf.Eq.(7.l65))* A(x“,Lip’)=§mu,,u'* +qii#A,,(xr), (7.147) withthecanonical momenta (cf.Eq.(7.l67)), pp=mu,i +qA,,. (7.149) Inthecorresponding Hamiltonian, thetermlinear inupdoesnotappear ex- plicitly intheHamiltonian, andtheremaining L3partinterms ofthecanonical momenta is _.A ll_Ari I-i;=--l__-_(””‘I“)(Pq). (8.60)2m BothHamiltonians, Eqs.(8.59) and(8.60), areconstant, withthesame value. —mc2/2, buttoobtain theequations ofmotion itisthefunctional dependence on the4-vectors ofposition andmomenta thatisimportant. Withasystem ofone particle, thecovariant Hamiltonian leads toeight ﬁrst-order equations ofmotion *The Legendre transfomratron process isreversible: Given aHamiltonian wecanobtain thecorre- sponding Lagrangian (cfDerivation 1)Butthedifﬁculties alsoarise meither direction Ifagnen I-lamiltoruan 1spostulated tobehomogeneous inﬁrstdegree inthemomenta, theiritisnotpossible to trndancqu1valentLag"rang:ian 8.5I8.5 Derivation ofHamilton's Equations fromaVariational Principle 353 dx” 8H; dp“ HHC= . =— . 8.61dt 8p" dt 8x” () Weknow thatthese equations cannot beallindependent. Thespace parts of Eqs.(8.6l) obviously leadtothespatial equations ofmotion. Weshould expect therefore thattheremaining twoequations tellusnothing new,exactly asinthe Lagrangian case.Thiscanbeveriﬁed byexamining thev=0equations insome particular Lorentz frame. Oneofthem istheconstitutive equation forp0: I u0=_3é‘%=%(p0_qA0) or 1 H’ p°=-0"+q¢>=—‘. (8.62) c c ageneral conclusion thathasbeen noted before. Theother canbewritten as ‘_dP°__l3_”1/1_,32 dt Cat dH 28H¢—-=,/ ———. 8.63dz I38: () Aswiththecovariant Lagrangian formulation, wehave theproblem ofﬁnding suitable covariant potential terms intheLagrangian todescribe theforces other thanelectromagnetic. Inmultiparticle systems weareconfronted infullmeasure with thecritical difﬁculties ofincluding interactions other thanwith ﬁelds. In Hamiltonian language, the“no-interaction” theorem already referred toinSec- tion7.10saysthatonlyintheabsence ofdirect particle interactions canLorentz invariant systems bedescribed interms oftheusual position coordinates andcor- responding canonical momenta. Thescope oftherelativistic Hamiltonian frame- work istherefore quite limited andsoforthemost partweshall conﬁne ourselves tononrelativistic mechanics.OI‘ DERIVATION OFHAMll.TON'S EQUATIONS FROM AVARIATIONAI. PRINCIPLE Lagrange’s equations have been shown tobetheconsequence ofavariational principle, namely, theHamilton’s principle ofSection 2.1.Indeed, thevariational method isoften thepreferable oneforderiving Lagrange’s equations, foritis applicable totypes ofsystems notusually included within thescope ofmechanics. Itwould besimilarly advantageous ifavariational principle could befound that 4 Chapter 8TheHamilton Equations ofMotion leads directly totheHamilton’s equations ofmotion. Hamilton’s principle, z 6IE8f2Ldr=O, (8.64) f1 lends itself tothispurpose, butasformulated originally itrefers topaths incon- ﬁguration space. Theﬁrstmodiﬁcation therefore isthattheintegral mustbeeval- uated overthetrajectory ofthesystem point inphase space, andthevaried paths must beintheneighborhood ofthisphase space trajectory. Inthespirit ofthe Hamiltonian formulation, bothqandpmustbetreated asindependent coordi- nates ofphase space, tobevaried independently. Tothisendtheintegrand inthe action integral, Eq.(8.64), must beexpressed asafunction ofbothqandp,and theirtimederivatives, through Eq.(8.15). Equation (8.64) thenappears as 7. 61=8f9(pa.-H<q.p.»>)d1=o. <8-65>F1 Asavariational principle inphase space, Eq.(8.65) issometimes referred toas themodiﬁed Hamilton ’sprinciple. Although itwillbeused most frequently in comiection withtransformation theory (seeChapter 9),themaininterest inithere istoshow thattheprinciple leads toHarnilton’s canonical equations ofmotion. Themodiﬁed Hami1ton’s principle isexactly oftheform ofthevariational problem inaspace of2ndimensions considered inSection 2.3(cf.Eq.(2.14)): rz t1=@f r<q.i.p.r>.:>dr=o. cm)ti forwhich the2nEuler—Lagrange equations are d Bf df .—— ——=0 =l,..., 8.67 <11(a@.) aq. ’ " () d Hf 8f _-—_ ——=0 =1,...,. 8.68 d1‘(9P )91¢ J n () J i Theintegrand fasgiven inEq.(8.65) contains q1-onlythrough thep,q,term, andqjonlyinH.Hence, Eqs.(8.67) leadto _ 3H Z-lq] Ontheother hand, there isnoexplicit dependence oftheintegrand inEq.(8.65) on13,-.Equations (8.68) therefore reduce simply to 3H'-i =O. 8.70‘I1 apj ( ) 8.5 Derivation ofHamilton’s Equations from aVariational Principle 355 Equations (8.69) and(8.70) areexactly Hamilton’s equations ofmotion. Eqs. (8.18). TheEuler—Lagrange equations ofthemodiﬁed Hamilton’s principle are thusthedesired canonical equations ofmotion. This derivation ofHamilton’s equations from thevariational principle isso brief astogivetheappearance ofasleight-of-hand trick. Onewonders whether something extra hasbeen sneaked inwhile wewere being misdirected bythe magician’s patter. Isthemodiﬁed Hamilton’s principle equivalent toHamilton’s principle, ordoes itcontain some additional physics? Thequestion islargely ir- relevant; theprimary justiﬁcation forthemodiﬁed Hamilton’s principle isthatit leads tothecanonical equations ofmotion inphase space. After all,nofurther argument wasgiven forthevalidity ofHamilton's principle than thatitcorre- sponded totheLagrangian equations ofmotion. SolongasHamiltonian canbe constructed, theLegendre transformation procedure shows thattheLagrangian andHamiltonian formulations, andtherefore their respective variational princi- ples,havethesame physical content. Onequestion thatcanberaised however iswhether thederivation putslimita- tions onthevariation ofthetrajectory thatarenotpresent inHamilton’s principle. Thevariational principle leading totheEuler-Lagrange equations isformulated, asinSection 2.2,such thatthevariations oftheindependent variables vanish at theendpoints. lnphase space, thatwould require 8q,=0and6p,=0atthe endpoints, whereas Hamilton’s principle requires onlythevanishing ofSq;un- derthesame circumstances. Alookatthederivation asspelled outinSection 2.2 willshow however thatthevariation isrequired tobezeroattheendpoints only inorder togetridoftheintegrated terms arising from thevariations inthetime derivatives oftheindependent variables. While theffunction inEq.(8.66) that corresponds tothemodiﬁed Hamilton’s principle, Eq.(8.65), isindeed afunc- tionof4}],there isnoexplicit appearance ofp,.Equations (8.68) andtherefore (8.70) follow from Eq.(8.65) without stipulating thevariations ofpJattheend points. Themodiﬁed Hamilton’s principle, withtheintegrand Ldeﬁned interms oftheHamiltonian byEq.(8.19), leads toHamilton’s equations under thesame variation conditions asthose inHamilton’s principle.* Nonetheless, thereareadvantages torequiring thatthevaried paths inthemod- iﬁedHamilton’s principle retum tothesame endpoints inbothqandp,forwe thenhave amore generalized condition forHamilton’s equations ofmotion. As withHamilton’s principle, ifthere isnovariation attheendpoints wecanadda totaltimederivative ofanyarbitrary (twice-differentiable) function F(q,p,t)to theintegrand without affecting thevalidity ofthevariational principle. Suppose. forexample. wesubtract from theintegrand ofEq.(8.65) thequantity ‘Itmaybeobjected thatqandpcannot bevaried independently, because thedeﬁning Eqs.(8.2) link pwithqandrjWecould notthenhaveavariation ofq(and1})without acorresponding variation of p.Butthisentire objection iscompletely atvariance withtheintent andthespirit ortheHamiltonian picture Once theHamiltonian fnnnulation hasbeensctup,Eqs(8.2)jbrm nopartofitThemomenta havebeen elevated tothestatus ofindependent vanables, onanequal basis withthecoordinates and connected withthem andthetimeonlythrough themedium oftheequations ofmorion themselves and notbyanyaprion deﬁning rclationship 8.6IChapter 8TheHamilton Equations ofMotion 5>dt(q1P1- Themodiﬁed Hamilton’s principle would thenread I2 5f(-12% —H(¢/.11. r))dr=0- (8-71)It Herethefintegrand ofEq.(8.66) isafunction of15,anditiseasily veriﬁed that theEuler—Lagrange equations (8.67) and(8.68) withthisfagain correspond to Hamilton’s equations ofmotion, Eqs.(8.18). Yettheintegrand inEq.(8.71) is nottheLagrangian norcanitingeneral besimply related totheLagrangian bya point transformation inconﬁguration space. Byrestricting thevariation ofbothq andptobezeroattheendpoints, themodiﬁed Hamilton’s principle provides an independent andgeneral wayofsetting upHamilton’s equations ofmotion with- outaprior Lagrangian formulation. Ifyouwill, itdoes away withthenecessity ofalinkage between theHamiltonian canonical variables andacorresponding Lagrangian setofgeneralized coordinates andvelocities. Thiswillbeveryimpor- tanttousinthenextchapter where weexamine transformations ofphase space variables thatpreserve theHamiltonian formoftheequations ofmotion. Therequirement ofindependent variation ofqandp,soessential fortheabove derivation, highlights thefundamental difference between theLagrangian and Hamiltonian formulations. Neither thecoordinates q,northemomenta p,are tobeconsidered thereasthemore fundamental setofvariables; bothareequally independent. Onlybybroadening theﬁeldofindependent variables fromnto2n quantities areweenabled toobtain equations ofmotion thatareofﬁrstorder. In asense, thenames “coordinates” and“momenta” areunfortunate, fortheybring tomind pictures ofspatial coordinates andlinear, oratmost, angular momenta. A wider meaning must nowbegiven totheterms. Thedivision intocoordinates and momenta corresponds tonomore thanaseparation oftheindependent variables describing themotion intotwogroups having anahnost symmetrical relationship toeachother through Hamilton’s equations. THE PRINCIPLE OFLEAST ACTION Another variational principle associated with theHamiltonian fonnulation is known astheprinciple ofleast action. Itinvolves anewtypeofvariation, which weshall calltheA-variation, requiring detailed explanation. Inthe8-variation process used inthediscussion ofHamilton’s principle inChapter 2,thevaried path inconﬁguration space always terminated atendpoints representing the system conﬁguration atthesame timet;andT2asthecorrect path. Toobtain Lagrange’s equations ofmotion, wealsorequired thatthevaried path return tothesame endpoints inconﬁguration space, thatis,6q,(r1) =5q,(12)=()_ TheA-variation islessconstrained; ingeneral, thevaried path over which an integral isevaluated mayendatdifferent times thanthecorrect path, andthere 8.6 ThePrinciple ofLeast Action 357 maybeavariation inthecoordinates attheendpoints. Wecanhowever usethe same parameterization ofthevaried path asinthe8-variation. Inthenotation ofSection 2.3,afamily ofpossible varied paths isdeﬁned byfunctions (cf.Eq. (2.15)) qt(r,cw)=q.(r,0)+am(I), (8.72) where ctisaninﬁnitesimal parameter thatgoestozeroforthecorrect path.Here thefunctions 17,-donotnecessarily havetovanish attheendpoints, either theorig- inalorthevaried. Allthatisrequired isthattheybecontinuous anddifferentiable. Figure 8.3illustrates thecorrect andvaried pathforaA-variation inconﬁguration space. Letusevaluate theA-variation oftheaction integral: I2 22+Al‘; Z2 Afrm5/ L(oz)dt -/L(0)dt, (8.73)t1 21-l-At; fl where L(a) means theintegral isevaluated along thevaried pathandL(0) corre- spondingly refers totheactual pathofmotion. Thevariation isclearly composed oftwoparts. Onearises fromthechange inthelimits oftheintegral; toﬁrst-order inﬁnitesimals, thispartissimply theintegrand ontheactual pathtimes thediffer- enceinthelimits intime.Thesecond partiscaused bythechange intheintegrand onthevaried path,butnowbetween thesametimelimits astheoriginal integral. Wemaytherefore write theA-variation oftheaction integral as F2 I2 AI Ldt=L(t2)Atg —L(z1)At1 +f 6Ldt. (8.74) P1 It Here thevariation inthesecond integral canbecarried outthrough aparame- terization ofthevaried path, exactly asforHamilton’s principle except thatthe q,l i A?1,+A12:2 ‘ Bq '2 (w=0) (a) I] [1+Al’l It ‘71 FIGURE 8.3 TheA-variation inconﬁguration space. 358 Chapter 8TheHamilton Equations ofMotion variation inq,does notvanish attheendpoints. Theendpoint terms arising in theintegration bypans must beretained, andtheintegral tennontheright appears aS *1 '1ataat 8L 2 X1 21 aqz dt 8511 ql aqi qr1 ByLagrange’s equations thequantities inthesquare brackets vanish, andtheA- variation therefore takes theform 12Af Ldr=(LA: +p,5q,)|f. (8.75) Yr InEq.(8.75), Sq,refers tothevariation inq,attheoriginal endpoint times r1and :2.Wewould liketoexpress theA-variation interms ofthechange Aq,between q,attheendpoints oftheactual pathandq,attheendpoints ofthevaried path, including thechange inendpoint times. Itisclear from Fig.8.3thatthese two variations areconnected bytherelation* Aq; =Sq; +égAL Hence, Eq.(8.75) canberewritten as P2 AfLdr=(LA:-p,r},At +p,aq,)|fn OI‘ F; 2 AfLdt=(,1,Aq,-HAr)|1. (s.77)n Toobtain theprinciple ofleastaction, werestrict ourfurther considerations by three important qualiﬁcations: 1.Only systems areconsidered forwhich L,andtherefore H,arenotexplicit functions oftime, andinconsequence Hisconserved. 2.Thevariation issuch thatHisconserved onthevaried pathaswellason theactual path. 3.Thevaried paths arefurther limited byrequiring thatAq;vanish attheend points (butnotAt). *Equation (876)maybedenved formally from theparameter form. Eq.(8.72), ofthevaried path Thus, attheupper endpoint wehave A11,(2)=qt(I2+M2,“) —4110210)=11102+A12-0)-17102.0) +l1"'l¢(t+A12)» which toﬁrstorder insmall quantities orandA22IS 541(2) =¢li(3) N2+51?»(2). which iswhat Eq.(876)predicts 8.6 ThePrinciple ofLeast Action 359 Thenature oftheresultant variation maybeillustrated bynoting thatthevaried pathsatisfying these conditions might verywelldescribe thesame curve incon- ﬁguration space astheactual path. Thedifference willbethespeed withwhich thesystem point traverses thiscurve; thatis.thefunctions q,(t)willbealtered in thevaried path. Inorder thentopreserve thesame value oftheHamiltonian atall points onthevaried path, thetimes oftheendpoints must bechanged. Vﬁth these three qualiﬁcations satisﬁed, theA-variation oftheaction integral, Eq.(8.77), reduces to Y2 Al La‘:=-H(Ar; —An). (8.78) fr Butunder thesame conditions, theaction integral itself becomes tg I2 IL<1t=f l7t5lrdt_ H0:—n).fr fl theA-variation ofwhich is I I A/2Ldr=Afzp,q,at-H(At; -Ati). (8.79)t1 I1 Comparison ofEqs.(8.78) and(8.79) ﬁnally gives theprinciple ofleast action?‘ '2 AIpg},at=0. (8.80)fr Bywayofcaution, notethatthemodiﬁed Hamilton’s principle canbewritten inaform withasuperﬁcial resemblance toEq.(8.80). Ifthetrajectory ofthesys- tempoint isdescribed byaparameter 6,asinSections 7.10and8.4,themodiﬁed Hamilton’s principle appears as 9 sI2(p,a,-H)t’d9 =o. (s.s1)91 Itwillberecalled (cf.footnote onp.351) thatthemomenta p,donotchange under theshiftfrom tto9,andthatrj,-t’=qf.Further, themomentum conjugate totis—H. Hence, Eq.(8.8!) canberewritten as 92II-l-1 5faZp,q;ae =0, (s.s2)I1:] where thasbeendenoted byq,,+1. There should however benoconfusion be- tween Eq.(8.82) andtheprinciple ofleastaction, Equations (8.82) involve phase ‘The urtegral inEq(8.80) isusually referred tointheolder literature astheaction, oraction integral, andtheﬁrstedition ofthisbook followed thesame practice. Itisnowcustomary torefertotheintegral ll'lHamilton’s pnnciple astheaction, andwehaveaccepted thisusage here. Sometimes themtegral in Eq.(8.80) isdesignated astheabbreviated acnon 360 Chapter 8TheHamilton Equations ofMotion space of(2n+2)dimensions, asisindicated bytheexplicit surmnation toi= n+1,whereas Eq.(8.80) isintheusual conﬁguration space. Butmost important, theprinciple ofleast action isinterms ofaA—variat1on forconstant H,while Eq.(8.82) employs the8-variation, andHinprinciple could beafunction oftime. Equation (8.82) isnothing more thanthemodiﬁed Hamilton’s principle, andthe absence ofaHamiltonian merely reﬂects thephenomenon thattheHamiltonian vanishes identically forthe“homogeneous problem.” Theleastaction principle itself canbeexhibited inavariety offonns. Innon- relativistic mechanics, ifthedeﬁning equations forthegeneralized coordinates do notinvolve thetimeexplicitly, thenthekinetic energy isaquadratic function of the4},’s(cf.Eq.(1.71)): T=%M,t<q>q,-a. (8.83) When inaddition thepotential isnotvelocity dependent, thecanonical momenta arederived fromTonly,andinconsequence p,¢j,=2T. Theprinciple ofleast action forsuchsystems cantherefore bewritten as z AI2Tdt =O. (8.84) f1 If,further, there arenoexternal forces onthesystem, as,forexample, arigid body withnonetapplied forces, thenTisconserved along withthetotalenergy H.The leastaction principle thentakes thespecial form A(tg —t|)=0. (8.85) Equation (8.85) states thatofallpaths possible between twopoints, consistent withconservation ofenergy, thesystem moves along thatparticular pathforwhich thetimeoftransit istheleast(more strictly, anextremum). Inthisform theprinci- pleofleast action recalls Fermat’s principle ingeometrical optics thatalightray travels between twopoints along such apaththatthetimetaken istheleast. We discussed these considerations inSection 10-8 oftheSecond Edition when we considered theconnection between theHamiltonian fommlation andgeometrical optics. InSection 7.4wediscussed theinﬁnitesimal interval inametric space giving theinterval as dsz=g#,,dx”'dx" 0.32’) where gm,wasthemetric ofapossibly curvilinear space anddszwastheinterval traversed fordisplacements given bydx“. Wecandosomething entirely similar herewhenever Tisoftheform ofEq.(8.83). Aconﬁguration space istherefore constructed forwhich theMJkcoefﬁcients form themetric tensor. Ingeneral, the 8.6 ThePrinciple ofLeast Action 361 space willbecurvilinear andnonorthogonal. Theelement ofpathlength inthe space isthendeﬁned by(cf.Eq.(7.33’)) tap)’=M,/.dq,an (8.86) sothatthekinetic energy hastheform 1dp 2T=—— , .872(dz) (8) orequivalently 4/1at=_. (asst/27" ) Equation (8.88) enables ustochange thevariable intheabbreviated action integral from ttop,andtheprinciple ofleast action becomes I2 P2 Af Ta’t=0=Af ,/T/2dp, T1 P1 or,ﬁnally Afm,/H-V(q)dp =0. (2.89)Pr Equation (8.89) isoften called Jacobi ’sform oftheleast action principle. Itnow refers tothepathofthesystem point inaspecial curvilinear conﬁguration space characterized byametric tensor withelements MJk.Thesystem point traverses thepathinthisconﬁguration space withaspeed given byJﬁ. Ifthere areno forces acting onthebody, Tisconstant, andJacobi’s principle saysthesystem point travels along theshortest pathlength intheconﬁguration space. Equiva- lently stated, themotion ofthesystem isthensuch thatthesystem point travels along thegeodesics oftheconﬁguration space. Note thattheJacobi form oftheprinciple ofleast action isconcemed withthe path ofthesystem point rather thanwithitsmotion intime. Equation (8.89) isa statement about theelement ofpathlength dp;thetimenowhere appears, since Hisaconstant andVdepends upon q,only. Indeed, itispossible tousethe Jacobi fom1oftheprinciple tofurnish thedifferential equations forthepath,bya procedure somewhat akintothatleading toLagrange’s equations. Intheformof Ferrnat’s principle, theJacobi version oftheprinciple ofleastaction ﬁndsmany fmitful applications ingeometrical optics andinelectron optics. Togointoany detail herewould leadustoofaraﬁeld. Ahostofother similar, variational principles forclassical mechanics canbe derived inbewildering variety. Togiveoneexample outofmany, theprinciple ofleast action leads immediately toHertz’s principle ofleast curvature, which states thataparticle notunder theinﬂuence ofexternal forces travels along the Chapter 8TheHamilton Equations ofMotion path ofleast curvature. ByJacobi’s principle such apath must beageodesic, andthegeometrical property ofminimum curvature isoneofthewell-known characteristics ofageodesic. Ithasbeen pointed outthatvariational principles in themselves contain nonewphysical content, andtheyrarely simplify thepractical solution ofagiven mechanical problem. Their value lieschieﬂy asstarting points fornewformulations ofthetheoretical structure ofclassical mechanics. Forthis purpose, Hamilton’s principle isespecially fruitful, andtoalesser extent, soalso istheprinciple ofleast action. DERIVATIONS 1.(a)Reverse theLegendre transfomiation toderive theproperties ofL(q;, Q,,1)from H(q,,p,.t),treating thez},asindependent quantities. andshow thatitleads to theLagrangian equations ofmotion. (b)Bythesame procedure ﬁndtheequations ofmotion interms ofthefunction I/(Pi 1}»ti=_l3lql "H(q- I7~7)- 2.Ithasbeenpreviously noted thatthetotaltimederivative ofafunction ofq,andt canbeadded totheLagrangian without changing theequations ofmotion. What does such anaddition dotothecanonical momenta andtheHamiltonian? Show thatthe equations ofmotion interms ofthenewHamiltonian reduce totheoriginal Hamilton's equations ofmotion 3.AHaimltonian-like formulation canbesetupinwhich :1;andg5,aretheindependent variables witha“Hamiltonian” 014,, ;5,,t).[Here p,isdeﬁned interms ofq,,1},in theusual manner] Starting from theLagrangian formulation, show indetail howto construct G(1i, .)5,.r),andderive thecorresponding “Hamilton's equation ofmotion.” 4.Show thatifA,aretheeigenvalues ofasquare matrix. thenifthereciprocal matnx exists ithastheeigenvalues A71. 5.Verify thatthematrix] hastheproperties given inEqs.(8.38c) and(8.38e) andthat itsdeterminant hasthevalue +1. 6.Show thatHamilton’s principle canbewritten as 2 8f [2H(‘ﬂ. t)+1|Ii|]dt =0. l 7.Verify thatbothI-Iamiltonians, Eq.(8.45) andEq.(8.47). leadtothesame motion as described byEq.(8.44). 8.Show thatthemodiﬁed Hamilton’s principle. intheform ofEq.(8.71). leads toHamil- ton’s equations ofmotion. 9.Ifthecanonical variables arenotallindependent, butareconnected byauxiliary con- ditions ofthefomi ¢k(qi- Pi»I)=0» Exercises 363 show thatthecanonical equations ofmotion canbewiitten 8H 39¢]; . 3H 31/rk _ i + A‘'H = 9 i’ A H Z _ 9 an ;I‘311: q‘ Ba,+;kiiq. p’ where theAkaretheundetermined Lagrange multipliers. Theformulation ofthe Hamiltonian equations inwhich tisacanonical variable isacaseinpoint, since a relation exists between p,,+1 andtheother canonical variables: H(ql~----qn+l3 PM-->Pn)+Pn+l =0- Show thatasaresult ofthese circumstances the2n+2Hamilton’s equations ofthis formulation canbereduced tothe2nordinary Hamilton’s equations plusEq.(8.41) andtherelation k_dr ‘as’ Note thatwhile these results arereminiscent oftherelativistic covariant Hamiltonian formulation, theyhavebeenarrived atentirely within theframework ofnonrelativistic mechanics. Assume thattheLagrangian isapolynomial inifofnohigher order thanquadratic. Convert the2nequations (8.2) and(8.14) _BL __BL p'—a¢l|’ p'—aql, into2nequations for4},and1},interms ofqandp,using thematrix form oftheLa- grangian. Show thatthese arethesameequations aswould beobtained fromHamil- ton’s equations ofmotion. EXERCISES Aparticle isconﬁned toaone-dimensional box. Theends oftheboxmove slowly towards themiddle. Byslowly wemean thespeed oftheendsissmall when compared tothespeed oftheparticle. Solve thefollowing using Lagrangian formulation andthen using theHamiltonian. (a)ifthemomentum oftheparticle ispgwhen thewalls areadistance xoapart, ﬁnd themomentum oftheparticle atanylatertime assuming thecollisions withthe wallareperfectly elastic. Alsoassume themotion isnonrelativistic atalltimes. (b)When thewalls areadistance xapart, what average extemal force must beapplied toeachwallinorder tomove itataconstant speed‘? Write theproblem ofcentral force motion oftwomass points inHamiltonian forinu- lation, eliminating thecyclic variables. andreducing theproblem toquadratures. Formulate thedouble-pendulum problem illustrated byFig.1.4,interms oftheHamil- tonian andHainilton‘s equations ofmotion. Itissuggested thatyouﬁndtheHamilto- nianbothdirectly from LandbyEq.(8.27). 4 Chapter 8TheHamilton Equations ofMotion 14TheLagrangian forasystem canbewritten as L=022+11%+way+fyzii+gy-k,/12+yz, where a,b,c,f.g,andkareconstants. What istheHamiltonian" What quantities are conserved‘? Adynamical system hastheLagrangian -2. 47 .. L=11%+‘Z2 +l<1qf+k2qiq2.a-l-bq] where a,b,kl,andkgareconstants. Findtheequations ofmotion intheHamiltonian formulation. AHamiltonian ofonedegree offreedom hastheform 2 k2 H=L—bqpe_"” +biq2e_“’(a +be_°") +-1-.20: 2 2 where a,b,oz,andkareconstants. (a)FindaLagrangian corresponding tothisHamiltonian. (b)Findanequivalent Lagrangian thatisnotexplicitly dependent ontime. (c)What istheHamiltonian corresponding tothissecond Lagrangian, andwhat is therelationship between thetwol-Iarniltonians? FindtheHamiltonian forthesystem described inExercise 19ofChapter 5andobtain Hamilton’s equations ofmotion forthesystem. Useboththedirect andthematrix approach mﬁnding theHamiltonian. Repeat thepreceding exercise except thistimeallow thependulum tomove inthree dimensions, thatis,aspring-loaded spherical pendulum. Either thedirect orthematrix approach maybeused. Thepoint ofsuspension ofasimple pendulum oflength landmass misconstrained to move onaparabola z=axzinthevertical plane. Derive aHamiltonian governing the motion ofthependulum anditspoint ofsuspension. Obtain theHamilton’s equations ofmotion. Z I m ' x Obtain Hamilton’s equations ofmotion foraplane pendulum oflength lwithmass point mwhose radius ofsuspension rotates uniformly onthecircumference ofaverti- calcircle ofradius a.Describe physically thenature ofthecanonical momentum and theHamiltonian. Exercises 365 21.(a)Thepoint ofsuspension ofaplane simple pendulum ofmass mandlength lis constrained tomove along ahorizontal track andisconnected toapoint onthe circumference ofauniform ﬂywheel ofmass Mandradius athrough amass- lessconnecting rodalsooflength a,asshown intheﬁgure. Theﬂywheel rotates about acenter ﬁxed onthetrack. FindaHamiltonian forthecombined system and determine Hamilton’s equations ofmotion. a I m (b)Suppose thepoint ofsuspension were moved along thetrack according tosome function oftimex=f(t),where xreverses atx=i2a(relative tothecenter of theﬂywheel). Again, ﬁndaHamiltonian andHamilton’s equations ofmotion. 22.Forthearrangement described inExercise 21ofChapter 2,ﬁndtheHamiltonian of thesystem, ﬁrstinterms ofcoordinates inthelaboratory system andtheninterms ofcoordinates intherotating systems. What aretheconservation properties ofthe l-lamiltonians, andhowaretheyrelated totheenergy ofthesystem? 23.(a)Aparticle ofmass mandelectric charge emoves inaplane under theinﬂuence ofacentral force potential V(r) andaconstant uniform magnetic ﬁeldB,perpen- dicular totheplane, generated byastatic vector potential A=%Bxr. FindtheHamiltonian using coordinates intheobserver’s inertial system. (b)Repeat part(a)using coordinates rotating relative totheprevious coordinate sys- temabout anaxisperpendicular totheplane withanangular rateofrotation: eBco=——m 24.Auniform cylinder ofradius aanddensity pismounted soastorotate freely around avertical axis.Ontheoutside ofthecylinder isarigidly ﬁxed uniform spiral orhelical track along which amass point mcanslidewithout friction. Suppose aparticle starts 66 Chapter 8TheHamilton Equations ofMotion 25. 26. 27. 28.atrestatthetopofthecylinder andslides down under theinﬂuence ofgravity. Using anysetofcoordinates, arrive ataHamiltonian forthecombined system ofparticle andcylinder, andsolve forthemotion ofthesystem. Suppose thatmtheprevious exercise thecylinder isconstrained torotate uniformly withangular frequency co.SetuptheHamiltonian fortheparticle inaninertial system ofcoordinates andalsoinasystem ﬁxed intherotating cylinder. Identify thephysical nature oftheHamiltonian ineach caseandindicate whether ornottheHarniltonians areconserved. Aparticle ofmass mcanmove inonedimension under theinﬂuence oftwosprings connected toﬁxed points adistance aapart (seeﬁgure). Thesprings obey Hooke’s lawandhavezerounstretched lengths andforceconstants k1andk2,respectively. 1 (1 I- 4 It, "’ kl A (a)Using theposition oftheparticle from oneﬁxed point asthegeneralized co- ordinate, ﬁndtheLagrangian andthecorresponding Hamiltonian. Istheenergy conserved? IstheHamiltonian conserved? (b)Introduce anewcoordinate Qdeﬁned by /620 Q=q-l7S1n60t‘, b=m What istheLagrangian interms ofQ?What isthecorresponding Hamiltonian? Istheenergy conserved? IstheHarniltoinan conserved? (a)TheLagrangian forasystem ofonedegree offreedom canbewritten as m-2-2 - ~ 22L=-2-(q sincot+qqwsin2wt +qcu). What isthecorresponding Hamiltonian? Isitconserved? (b)Introduce anewcoordinate deﬁned by Q=qsinwt. FindtheLagrangian interms ofthenewcoordinate andthecorresponding Hamil- tonian. IsHconserved? Consider asystem ofparticles interacting witheach other through potentials depend- ingonlyonthescalar distances between them andacted upon byconservative central forces from aﬁxed point. Obtain theHamiltonian oftheparticle with respect toa setofaxes, withorigin atthecenter offorce, which isrotating around some axisin aninertial system with angular velocity w.What isthephysical signiﬁcance ofthe Hamiltonian inthiscase? Isitaconstant ofthemotion’? Exercises 367 29.Obtain theHamiltonian ofaheavy symmetrical topwithonepoint ﬁxed, andfrom it theHamilton’s equations ofmotion. Relate thesetotheequations ofmotion discussed inSection 5.7and,inparticular, showhowthesolution maybereduced toquadratures. Also usetheRouthian procedure toeliminate thecyclic coordinates. 30.InExercise 16ofChapter 1,there isgiven thevelocity-dependent potential assumed in Weber’s electrodynamics. What istheHamiltonian forasingle particle moving under theinﬂuence ofsuchapotential? 31.Treat thenutation ofa“fast” topasanexample ofsmall oscillations about steady motion, hereprecession atconstant 9.Findthefrequency ofnntation. 32.Asymmetrical topismounted sothatitpivots about itscenter ofmass. Thepivot in tumisﬁxed adistance rfrom thecenter ofahorizontal diskfreetorotate about a vertical axis.Thetopisstarted withaninitial rotation about itsﬁgiue axis,which is initially atanangle 90tothevertical. Analyze thepossible nutation ofthetopasa caseofsmall oscillations about steady motion. 33.Two mass points, miandm2,areconnected byastring thatactsasaHool<e's-law spring withforce constant k.Oneparticle isfreetomove without friction onasmooth honzontal plane surface, theother hangs vertically down from thestring through a holeinthesnrface. Find thecondition forsteady motion inwhich themass point on theplane rotates uniformly atconstant distance fromthehole.Investigate thesmall oscillations intheradial distance fromthehole,andinthevertical height ofthesecond particle. 34.Apossible covaii antLagrangian forasystem ofoneparticle interacting withaﬁeldis A=%mu;(u;, +Div(x,,,)ml,,, where D,“(xn)isanantisymmetric ﬁeldtensor andmM,istheantisymmetric angular momentum tensor, mm=m(x;_u,, —xvul). What arethecanonical momenta? What isthecorresponding covariant Hamiltoman? 35.Consider aLagrangian oftheform L=%m(i2 —-co2x2)e7", where theparticle ofmassmmoves inonedirection. Assume allconstants areposi- tive. (a)Findtheequations ofmotion. (b)Interpret theequations bygiving aphysical interpretation oftheforces acting on theparticle. (c)Findthecanonical momentum andconstruct theHamiltonian. IsthisHamiltonian aconstant ofthemotion? (d)Ifinitially x(0) =0anddx/dr =0,what isx(t)astapproaches large values? CHAPTER 9.1I 368Canonical Transformations When applied inastraightforward manner, theHamiltonian formulation usually does notmaterially decrease thedifﬁculty ofsolving anygiven problem inme- chanics. Wewind upwit.hpractically thesame differential equations tobesolved asareprovided bytheLagrangian procedure. Theadvantages oftheHamiltonian formulation lienotinitsuseasacalculational tool,butrather inthedeeper in- sight itaffords intotheformal structure ofmechanics. Theequal status accorded tocoordinates andmomenta asindependent variables encourages agreater free- dominselecting thephysical quantities tobedesignated as“coordinates” and “momenta.” Asaresult weareledtonewer, more abstract ways ofpresenting thephysical content ofmechanics. While often ofconsiderable helpinpractical applications tomechanical problems, thesemoreabstract formulations areprimar- ilyofinterest toustoday because oftheiressential roleinconstructing themore modern theories ofmatter. Thus, oneoranother ofthese formulations ofclassical mechanics serves asapoint ofdeparture forbothstatistical mechanics andquan- tumtheoty. Itistosuch formulations, arising asoutgrowths oftheHamiltonian procedure, thatthisandthenextchapter aredevoted. THE EQUATIONS OFCANONICAL TRANSFORMATION There isonetypeofproblem forwhich thesolution oftheHamilton’s equations is trivial. Consider asituation inwhich theHamiltonian isaconstant ofthemotion, andwhere allcoordinates q,arecyclic. Under these conditions, theconjugate momenta p,-areallconstant: pl ials andsince theHamiltonian cannot beanexplicit function ofeither thetimeorthe cyclic coordinates, itmaybewritten as H=H(a1, ...,a,,). Consequently, theHamilton’s equations forti,aresimply . 3H qt=8?=wt, (9-1) 9.1 TheEquations ofCanonical Transformation 369 where theco,’sarefunctions oftheoz,-’sonlyandtherefore arealsoconstant in time. Equations (9.1) havetheimmediate solutions qt=wit +ﬁle where thefl,‘sareconstants ofintegration, determined bytheinitial conditions. Itwould seem thatthesolution tothistypeofproblem, easyasitis,canonly beofacademic interest, foritrarely happens thatallthegeneralized coordinates arecyclic. Butagiven system canbedescribed bymore thanonesetofgeneral- izedcoordinates. Thus, todiscuss motion ofaparticle inaplane, wemayuseas generalized coordinates either theCartesian coordinates ql=xv Q2=ya ortheplane polar coordinates q1=r. q2=9- Bothchoices areequally valid, butoneoftheother setmaybemore convenient fortheproblem under consideration. Notethatforcentral forces neither xnory iscyclic. while thesecond setdoescontain acyclic coordinate intheangle 6.The number ofcyclic coordinates canthusdepend uponthechoice ofgeneralized co- ordinates, andforeach problem there maybeoneparticular choice forwhich all coordinates arecyclic. Ifwecanﬁndthisset,theremainder ofthejobistrivial. Since theobvious generalized coordinates suggested bytheproblem willnotnor- mally becyclic, wemust ﬁrstderive aspeciﬁc procedure fortransforming from onesetofvariables tosome other setthatmaybemore suitable. Thetransfomiations considered intheprevious chapters have involved going from onesetofcoordinates q,-toanewsetQ,bytransfonnation equations ofthe form Qt=QI(qs t)- (9-3) Forexample, theequations ofanorthogonal transformation, orofthechange from Cartesian toplane polar coordinates, have thegeneral fonn ofEqs. (9.3). Ashasbeen previously noted inDerivation I0ofChapter 1,suchtransformations areknown aspoint transformations. ButintheHamiltonian fOl'mtllﬂlIlO1'l themo- menta arealsoindependent variables onthesamelevelasthegeneralized coordi- nates. Theconcept oftransformation ofcoordinates musttherefore bewidened to include thesimultaneous transformation oftheindependent coordinates andmo- menta, q,,p,,toanewsetQ,.P,,with(invertible) equations oftransformation: Qt=Qt(q,r1.r). P.=P.(q,P.r)- (9-4) Thus, thenewcoordinates willbedeﬁned notonlyinterms oftheoldcoordi- nates butalsointenns oftheoldmomenta. Equations (9.3) maybesaidtodeﬁne Chapter 9Canonical Transformations apoint transformation ofconﬁguration space; correspondingly Eqs.(9.4)deﬁne apoint transformation ofphase space. Indeveloping Hamiltonian mechanics, onlythose transformations canbeofin- terest forwhich thenewQ,Parecanonical coordinates. Thisrequirement willbe satisﬁed provided thereexists some function K(Q,P,t)suchthattheequations ofmotion inthenewsetareintheHamiltonian form - 8K - 8K Qt—-‘BE, P!—"aQ'- Thefunction Kplays theroleoftheHamiltonian inthenewcoordinate set.* Itisimportant forfuture considerations thatthetransformations considered be problem-independent. That istosay,(Q.P)must becanonical coordinates not onlyforsome speciﬁc mechanical systems, butforallsystems ofthesame num- berofdegrees offreedom. Equations (9.5)mustbetheformoftheequations of motion inthenewcoordinates andmomenta nomatter what theparticular initial form ofH.Wemayindeed beincited todevelop aparticular transformation from (q,p)to(Q,P)tohandle, say,aplane harmonic oscillator. Butthesame trans- formation must thenalsoleadtoHamilton’s equations ofmotion when applied. forexample, tothetwo-dimensional Kepler problem. AswasseeninSection 8.5,ifQ,andP,aretobecanonical coordinates, they must satisfy amodiﬁed Hamilton’s principle thatcanbeputintheform 6fitsQ.-K(Q.P.om=0. on1| (where summation overtherepeated index iisimplied). Atthesame timetheold canonical coordinates ofcourse satisfy asimilar principle: I2 6frm.-Ho.p.o><1»=0- (9-1)Ii Thesimultaneous validity ofEqs.(9.6)and(9.7)doesnotmean ofcourse thatthe integrands inboth expressions areequal. Since thegeneral form ofthemodiﬁed Hamilton’s principle haszerovariation attheendpoints, bothstatements willbe satisﬁed iftheintegrands areconnected byarelation oftheform . - dF A-(Piqi_H)=PiQi_K+Ti?- Here Fisanyfunction ofthephase space coordinates with continuous second derivatives, andAisaconstant independent ofthecanonical coordinates andthe time.Themultiplicative constant Aisrelated toaparticularly simple typeoftrans- formation ofcanonical coordinates known asascale transformation. *lthasbeen remarked inajocular veinthatifHstands fortheHamiltoiuan, Kmust stand forthe Kamiltonianl Ofcourse, Kisevery bitasmuch tiHamiltonian asH,butthedesignation isoccasionally El.convenient substitute forthelonger term“transformed Hamiltonian ” 9.1 TheEquations ofCanonical Transformation 371 Suppose wechange thesizeoftheunitsusedtomeasure thecoordinates and momenta sothatineffect wetransform themtoaset(Q’,P’)deﬁned by Q:=/J-qi» P,’=vPi- Then itisclear l-lamilton’s equations intheform ofEqs. (9.5) willbesatisﬁed foratransformed Hamiltonian K’(Q',P’)=;i.vH (q,p).Theintegrands ofthe corresponding modiﬁed Hamilton’s principles are,alsoobviously, related as /Mp.-it—H)=PIQ;-K’. <9-10> which isoftheform ofEq.(9.8) withIt=/.iv.With theaidofsuitable scale trans- fonnation, itwillalways bepossible toconﬁne Ourattention totransformations ofcanonical coordinates forwhich A=1.Thus, ifwehave atransfonnation of canonical coordinates (q,p)—>(Q’,P’)forsome Ayé1,thenwecanalways ﬁndanintermediate setofcanonical coordinates (Q,P)related to(Q’,P’)bya simple scale transformation oftheform (9.9) suchthatp.valsohasthesame value A.Thetransformation between thetwosetsofcanonical coordinates (q,p)and (Q,P)willsatisfy Eq.(9.8), butnowwithA=1: . - dFplql—H=PlQl_K+2T' (9-11) Since thescale transformation isbasically trivial, thesigniﬁcant transformations tobeexamined arethose forwhich Eq.(9.11) holds. Atransformation ofcanonical coordinates forwhich It¢lwillbecalled an extended canonical transfomzation. Where A=1,andEq.(9.11) holds, wewill speak simply ofacanonical transformation. Theconclusion oftheprevious para- graph maythenbestated assaying thatanyextended canonical transformation canbemade upofacanonical transformation followed byascale transforma- tion. Except where otherwise stated, allfuture considerations oftransformations between canonical coordinates willinvolve onlycanonical transformations. Itis alsoconvenient togiveaspeciﬁc name tocanonical transformations forwhich the equations oftransformation Eqs.(9.4)donotcontain thetimeexplicitly; theywill becalled restricted canonical transformations. ThelasttermontherightinEq.(9.11) contributes tothevariation oftheac- tionintegral onlyattheendpoints andwilltherefore vanish ifFisafunction of (q,p,t)or(Q,P,2‘)oranymixture ofthephase space coordinates since these have zerovariation attheendpoints. Further, through theequations oft:ransfor- mation, Eqs.(9.4)andtheirinverses Fcanbeexpressed partly interms oftheold setofvariables andpartly ofthenew.Indeed, Fisuseful forspecifying theexact formofthecanonical transfonnation onlywhen halfofthevariables (beside the time) arefrom theoldsetandhalfarefrom thenew. Itthenacts, asitwere, as abridge between thetwosetsofcanonical variables andiscalled thegenerating function ofthetransformation. Toshow howthegenerating function speciﬁes theequations oftransforma- tion,suppose Fweregiven asafunction oftheoldandnewgeneralized space 72 Chapter 9Canonical Transformations coordinates: F=F1(q, Q.r)- (912) Equation (9.1l)thentakes thefonn . - dFp.q,—H=P.-Q,-K+T‘ - 3F1 BF; _ BF1 -=P —K— — — . 9.13 IQ! + +aq'qI+aQ!Q| ( ) Since theoldandthenewcoordinates, q,andQ,,areseparately independent, Eq.(9.13) canholdidentically onlyifthecoefﬁcients ofQ,andQ,~each vanish: Nm=5% muna 3171 P,=—i, (9.l4lJ 8Q. ) leaving ﬁnally 8FK=H+3f (mm) Equations (9.14a) arenrelations deﬁning thep,asfunctions ofqJ,QJ,andt. Assuming theycanbeinverted, theycould thenbesolved forthenQ,-’sintenns ofqJ,[J],andt,thusyielding theﬁrsthalfofthetransformation equations (9.4). Once therelations between theQ,-‘sandtheoldcanonical variables (q,p)have been established, theycanbesubstituted intoEqs.(9.l4b) sothattheygivethen P,’sasfunctions ofqJ,pJ,andt,thatis,thesecond halfofthetransformation equations (9.4). Tocomplete thestory, Eq.(9.140) provides theconnection be- tween thenewHamiltonian, K,andtheoldone,H.Wemustbecareful toread Eq.(9.14c) properly. FirstqandpinHareexpressed asfunctions ofQandP through theinverses ofEqs.(9.4). Then theq,in8F1/81‘ areexpressed intenns ofQ,Pinasimilar manner andthetwofunctions areadded toyieldK(Q,P,t). Theprocedure described shows how. starting from agiven generating function F1,theequations ofthecanonical transformation canbeobtained. Wecanusually reverse theprocess: Given theequations oftransformation (9.4), anappropriate generating function F1maybederived. Equations (9.4)areﬁrstinverted toex- press p,andP,asfunctions ofq,Q,andt.Equations (9.l4a, b)thenconstitute acoupled setofpartial differential equations thancanbeintegrated, inprinciple, toﬁndF1providing thetransformation isindeed canonical. Thus, F1isalways uncertain towithin anadditive arbitrary function oftalone (which doesn’t affect theequations oftransformation), andthere mayattimes beother ambiguities. Itsometimes happens thatitisnotsuitable todescribe thecanonical transfor- mation byagenerating function ofthetypeF1(q, Q,1).Forexample, thetrans- formation maybesuchthatp,cannot bewritten asfunctions ofq,Q,andt,but TABLE 9.1 Properties9.1 TheEquations ofCanonical Transformanon 373 rather willbefunctions ofq,P,andr.Wewould thenseekagenerating func- tionthatisafunction oftheoldcoordinates qandthenewmomenta P.Clearly Eq.(9.13) mustthenbereplaced byanequivalent relation involving rather than Q,-.Thiscanbeaccomplished bywriting FinEq.(9.11) as F=B@PM—Q£- (%$ Substituting thisFinEq.(9.11) leads to - d m%—H=—QB—K+EH@Pﬂ- 9%) Again, thetotal derivative ofF2isexpanded andthecoefﬁcients ofzj,andP, collected, leading totheequations _L5PI * aq‘ 7 3172 Q! * -5-Fa I(9.17a) (9.17b) with K=H+ (9.l7c)8t Asbefore, Eqs.(9.17a) aretobesolved forP;asfunctions ofqJ,pJ,andttocor- respond tothesecond halfofthetransformation equations (9.4). Theremaining halfofthetransformation equations isthenprovided byEqs.(9.17b). Thecorresponding procedures fortheremaining twobasic types ofgenerating functions areobvious, andthegeneral results aredisplayed inTable 9.1. ltistempting tolook upon thefourbasic types ofgenerating functions as being related toeach other through Legendre transformations. Forexample, the oftheFourBasic Canonical Transformations Generating Function Generating Function Derivatives Trivial Special Case F=F1(q. Q.t) E P=_ﬁ aqt I3% ﬁ=%Qn Q=m. ﬁ=—a Pr: F=F2(q. P.r)—QzPt3F; aF2W if Q'-rt;F=-P, =-, P~= ' aq‘ I 2qt: Q: qr 1P1 Pr: F=F3(P~Qit)'l'qtP1 qtE p._ aF3 apt t 3Q:=_ F3=PiQt~ Q:=_qn Pl=_Pl F=F4(P,P,t)'l'qrPr_Q1PiEa Q1=Pr: Pt=_qt Pi3F qt=" Qlzﬁ F4=P:Pz= Chapter 9Canonical Transformations transition from F1toF2isequivalent togoing from thevariables q,Qtoq,P withtherelation 8F;-—-=i. 9.18 Pl 8Q‘ ( ) 'x\\'\s‘\s‘1\\s\\tne tomreqxmeo ‘torabegenore transi oi-matron oitnebasis vafiznales, asdescribed inSection 8.1,andinanalogy toEq.(8.5) wewould set F2(q, Pvt)=Fl(q> Q11)+PIQI! (9-19) which isequivalent toEq.(9.15) combined withEq.(9.12). Alltheother deﬁning equations forthegenerating functions cansimilarly belooked on,incombina- tionwithEq.(9.12) asLegendre transformations from F1,withthelastentry in Table 9.1describing adouble Legendre transformation. Theonlydrawback to thispicture isthatitmight erroneously leadustobelieve thatanygiven canoni- caltiansfonnation canbeexpressed interms ofthefourbasic types ofLegendre transformations Listed inTable 9.1.Thisisnotalways possible. Some transfor- mations arejustnotsuitable fordescription interms ofthese orother elementary forms ofgenerating functions, ashasbeennoted above andaswillbeillustrated inthenextsection withspeciﬁc examples. Ifwetrytoapply theLegendre trans- formation process, wearethenledtogenerating functions thatareidentically zeroorareindeterminate. Forthisreason, wehavepreferred todeﬁne eachtype ofgenerating function relative toF,which issome unspeciﬁed function of2n independent coordinates andmomenta. Finally, notethatasuitable generating function doesn’t havetoconform to oneofthefourbasic types forallthedegrees offreedom ofthesystem. Itis possible, andforsome canonical transformations necessary. touseagenerating function thatisamixture ofthefourtypes. Totakeasimple example, itmaybe desirable foraparticular canonical transformation withtwodegrees offreedom tobedeﬁned byagenerating function oftheform F'(qi, P2,P1,Q2,f)- (9-20) Thisgenerating function would berelated toFinEq.(9.11) bytheequation F=F'(¢1i, P2,Pi.Q2.!) —QiPi -P112112, (9-21) andtheequations oftransformation would beobtained from therelations _aP Q_8F 8F’ 8F’=--a P=-—- an) Q2 am 2 aQ2 ( 9.2I9.2 Examples ofCanonical Transformations 375 with 8F’K=H —. 9.23 +at () Speciﬁc illustrations aregiven inthenextsection andintheexercises. EXAMPLES OFCANONICAL TRANSFORMATIONS Thenature ofcanonical transfoiinations andtheroleplayed bythegenerating function canbestbeillustrated bysome simple yetimportant examples. Letus consider, ﬁrst,agenerating function ofthesecond typewiththeparticular fonn F2=qrPi (9-24) found incolumn 3ofTable 9.1.From Eqs.(9.17), thetransformation equations are 3F P: =i ZPM aq! 3172 Q:—TR" —qt. K=H. (9.25) Thenewandoldcoordinates arethesame; hence F2merely generates theidentity transformation (cf.Table 9.1). Wealsonote, referring toTable 9.1,thatthepar- ticular generating function F1=p,Q,generates anidentity transformation with negative signs; thatis,Q,=—q;,P,=-p,. Amore general typeoftransformation isdescribed bythegenerating function F‘2:'_fI(q]s-~-sqrl; t)-PM where thef,maybeanydesired setofindependent functions. ByEqs.(9.17b), thenewcoordinates Q,-aregiven by 8F Q.=5,%=f.(q,.....q,.; 0 <9-21>I Thus, withthisgenerating function thenewcoordinates depend onlyupon the oldcoordinates andthetimeanddonotinvolve theoldmomenta. Such atrans- formation istherefore anexample oftheclass ofpoint transformations deﬁned byEqs.(9.3). Inorder todeﬁne apoint transformation, thefunctions f,mustbe independent andinvertible, sothattheq1-canbeexpressed interms oftheQ,. Since thef,areotherwise completely arbitrary, wemayconclude thatallpoint transformations arecanonical. Equation (9.170) fumishes thenewHamiltonian interms oftheoldandofthetimederivatives ofthef,-functions. Chapter 9Canonical Transformahons NotethatF2asgiven byEq.(9.26) isnottheonlygenerating function leading tothepoint transformation speciﬁed bythef,-.Clearly thesame point transfor- mation isimplicit inthemore general form Fz=ﬁ(q1..__,qn; r)P.+g(q1,-_-,qn; 1), (9-28) where g(q,t)isany(differentiable) function oftheoldcoordinates andthetime. Equations (9.27), thetransformation equations forthecoordinates, remain unal- tered forthisgenerating function. Butthetransformation equations ofthemo- menta differ forthetwoforms. From Eqs.(9.l7a), wehave pl=E=%P,+ lag, (9.29)aqj aqj 6(1) using theform ofF2given byEq.(9.28). These equations maybeinverted togive Pasafunction of(q,p),most easily bywriting them inmatrix notation: 8f 8g ,p_a-qP+ﬁ. (9.29) Herep,P,and8g/Elq aren-elements ofsingle-column matrices, and8f/Hq isa square matrix whose ijthelement isBf,-/Bqj. Intwodimensions, Eq.(9.29’) can bewritten as EE *’_g[121] 3111 3112 [P1] aql= + .P2 3f; 3f) P2 Bi 3¢11 33¢! _E 2 Itfollows thatPisalinear function ofpgiven by at" 8gP: — ——. 9.30 iaqi[Pat] <>Intwodimensions, (9.30) becomes E311-1 3_g P1_3611 342 P1_341[P2]_[% Um] (9.31)8q1 aq 3612 2 Thus, thetransformation equations (9.27) forQareindependent ofganddepend onlyupon thej}(q,r),butthetransformation equations (9.29) forPdodepend upontheformofgandareingeneral functions ofboththeoldcoordinates and momenta. Thegenerating function given byEq.(9.26) isonlyaspecial caseof Eq.(9.28) forwhich g=O,with correspondingly specialized transformation equations forP. 9.3 I9.3 TheH;-1fITIOﬁl(. Oscillator 377 Aninstructive transformation isprovided bythegenerating function oftheﬁrst kind, F1(q,Q,t),oftheform FI=qkQt- Thecorresponding transformation equations, from (9.l4a, b)are BF P1=8*: =Qt, (9-323) 511 BF;P=———- =—. 9.32b r 8Q! qr ( ) Ineffect, thetransformation interchanges themomenta andthecoordinates; the newcoordinates aretheoldmomenta andthenewmomenta areessentially theold coordinates. Table 9.1shows thattheparticular generating function oftypeF4= p,P,produces thesametransformation. These simple examples should emphasize theindependent status ofgeneralized coordinates andmomenta. They areboth needed todescribe themotion ofthesystem intheHamiltonian formulation. The distinction between themisbasically oneofnomenclature. Wecanshiftthenames around withatmostnomore thanachange insign.There isnolonger present in thetheory anylingering remnant oftheconcept ofq,asaspatial coordinate and p,asamasstimes avelocity. Incidentally, wemayseedirectly fromI-lamilton‘s equations, , 8H _3HP: aqf 1 qt apt 1 thatthisexchange transfonnation iscanonical. Ifq,issubstituted for12,,theequa- tionsremain inthecanonical formonlyif-p,issubstituted forq,. Atransformation thatleaves some ofthe(q,p)pairs unchanged, andinter- changes therest(with asignchange), isobviously acanonical transformation of a“mixed” form. Thus, inasystem oftwodegrees offreedom, thetransformation Q1=qt, P1=P1. Q2=P2, P2=—q2, isgenerated bythefunction F=q1P1+612Q2, (9-33) which isamixture oftheF1andF2types. THE HARMONIC OSCILLATOR Asaﬁnalexample, letusconsider acanonical transformation thatcanbeused to solve theproblem ofthesimple harmonic oscillator inonedimension. Iftheforce Chapter 9Canonical Transformations constant isk.theHamiltonian forthisproblem interms oftheusual coordinates is 2 k2 H=£7+%. (9.34a) Designating theratiok/mby:02,Hcanalsobewritten as lH=—(p2 +mzwzqz). (9.34b)2m Thisform oftheHamiltonian, asthesumoftwosquares, suggests atransfor- mation inwhich Hiscyclic inthenewcoordinate. Ifwecould ﬁndacanonical transformation oftheform p=f(P)cos Q, (9.35a) q=ESmQ, (9.35b)ma) thentheHamiltonian asafunction ofQandPwould besimply 2P 2P K=H=L2(cos2 Q+sinzQ)=l), (9.36)2m 2m sothatQiscyclic. Theproblem istoﬁndtheformoftheyetunspeciﬁed function f(P)thatmakes thetransformation canonical. Ifweuseagenerating function of theﬁrstkindgiven by 2 F1=imgicotQ, (9.37) Eqs.(9.14) thenprovide theequations oftransformation, 8Fp=T11=mcoq cotQ, (9.38a) 2 P=lg= (9.38b)5Q 2s1n Q Solving forqandp,wehave* q=‘l2isinQ, (9.39a)mm *Itcanbeargued thatFgdoes notunambiguously specify thecanonical transformation, because in solving Eq(938b)forqwecould have taken thenegative square rootinstead ofthepositive rootas (implied) inEqs.(9-39). However, thetwocanonical transfoririations thusderived from F1differ only trivially; ashiftinorbyJrcorresponds togoing from onetransformation totheother. Nonetheless, it should bekeptinmind thatthetiansforinations derived from agenerating function mayatlimes be double-valued orevenhavelocal singularities. 9.3TheHarmonic Oscillator 379 p=\/2pmw cosQ, (9.39b) andcomparison withEq.(9.35a) evaluates f(P): f(P) =~/2ma>P. (9.40) Itfollows thenthattheHamiltonian inthetransformed variables is H=a>P. (9.41) Since theHamiltonian iscyclic inQ,theconjugate momentum Pisaconstant. It isseenfrom Eq.(9.41) thatPisinfactequal totheconstant energy divided byw: EP=——. w Theequation ofmotion forQreduces tothesimple form Q_8H_ _aP_°” withtheimmediate solution Q=mt+ix, (9.42) where oiisaconstant ofintegration ﬁxed bytheinitial conditions. From Eqs. (9.39), thesolutions forqandpare /2Eq=Z2 sin(a>t +oz), (9.43a)ma) p=x/2mE cos(a>t +av). (9.43b) Itisinstructive toplotthetimedependence oftheoldandnewvariables asis shown inFig.9.1.Weseethatqandposcillate (Fig. 9.1a, b)whereas QandP arelinear plots (Fig. 9.1d, e).Theﬁgure alsoshows thephase space plots forp versus q(Fig. 9.lo)andforPversus Q(Fig. 9.lt). Fig.9.1cisanellipse withthe following semimajor axes(fortheqandpdirections, respectively): l2E a= iz and b='\/2mE, ma) where misthemass oftheoscillator, witsfrequency, andEtheoscil1ator’s en- ergy. Thearea, A,ofthisellipse inphase space is EA=1rab=2L.w Chapter 9Canonical Transformations P (2mE) 1/2 P E -l 1 <9Jrr 2irrt C t (3) id) 4 (2E/mw2)'/7 Q Zrrr I l‘ (2 JTT 21:1:I (b) (9) I7 /(2mE)'/2 P Ew 0 ‘\q (£91/2 ma’ an 240-r Q (c) (fl FIGURE 9.1 Theharmonic oscillator intwocanonical coordinate systems. Draw- ings(a)-(c) show theq,psystem and(d)-(f) show theP,Qsystem. When weinvoke quantum mechanics, wewrite E=hm,where it=h/221, andh isPlanck’s constant. Thecoordinate andmomentum qandpcanbenormalized as I mwz 1 P=i and =i—. q V25q P ~/2mE tomake thephase space plotofp’versus q’acircle ofareaJT.This normalized form willbeuseful inSection ll.lonchaos. 9.4I9.4 TheSymplectic Approach toCanonical Transformations 381 Itwould seem thattheuseofcontact tiansformations tosolve theharmonic oscillator problem issimilar to“cracking apeanut with asledge hammer.” We haveherehowever asimple example ofhowtheHamiltonian canbereduced toa form cyclic inallcoordinates bymeans ofcanonical transformations. Discussion ofgeneral schemes forthesolution ofmechanical problems bythistechnique will beresen/ed forthenextchapter. Forthepresent, weshallcontinue toexamine the formal properties ofcanonical transformations. THE SYMPLECTIC APPROACH TOCANONICAL TRANSFORMATIONS Another method oftreating canonical transformations, seemingly unrelated tothe generator formalism, canbeexpressed intenns ofthematrix orsymplectic for- mulation ofHamilton’s equations. Bywayofintroduction tothisapproach, letus consider arestricted canonical transformation, thatis,oneinwhich timedoesnot appear intheequations oftransformation: Q1 2 QI(q> P)’ P.=P1-(9.11) (9-44) Weknow thattheHamiltonian function doesnotchange insuchatransformation. Thetimederivative ofQ,,onthebasis ofEqs.(9.44), istobefound as -3Qi. 3Q».3Q»3H 3Q,31"!'= + = — . (9.45)Q‘ Zlqjqj 8p]P] 3qJElpj 8p]8qJ Ontheother hand, theinverses ofEqs.(9.44), Q) Z P): P] P): enables ustoconsider H(q,p,i)asafunction ofQandPandtoformthepartial derivative '11 ~ 8Ha6=9Hapi+ q’. (9.47)8P, zip]8P, 3:1]8P, Comparing Eqs.(9.45) and(9.47), itcanbeconcluded that . 8H Q! — ﬁe I thatis,thetransformation iscanonical, onlyif <9)-(9)1(e)-<991 5611 q,p 3Pi Q,P 319.1 ,,_p 3P1 Q,P Chapter 9Canonical Transformations Thesubscripts onthederivatives aretoremind usthatontheleft-hand sideof these equations Q;isconsidered asafunction of(q,p)(cf.Eqs.(9.44)), while ontheright-hand sidethederivatives areforqjandpJasfunctions of(Q,P)(cf. Eqs.(9.46)). Asimilar comparison ofP,withthepartial ofHwithrespect toQJ leads totheconditions <9)-ta)»is)-ta)1 aqj q_;, 3Q: Q_p 3171 qyp 3Qi Q,p ThesetsofEqs.(9.48) together aresometimes known asthe“direct conditions” fora(restricted) canonical transformation. Thealgebraic manipulation thatleads toEqs. (9.48) canbeperfonned ina compact andelegant manner ifwemake useofthesymplectic notation forthe Hamiltonian formulation introduced above attheendofSection 8.1.If1|isa column matrix with the2nelements q,,p,,then Ha.mi1ton’s equations canbe written, itwillberemembered, asEq.(8.39) .__]8H11-an. where Iistheantisymmetric matrix deﬁned inEq.(8.38a). Similarly thenewset ‘*<<>~:~<.%eme~z-e%~y%‘-;-€r<:-<.-<<-i-;.*‘-;;-e.?.='<:iez~ae<s,*m<s"/?~r%r§§:"%:=§\—~e.}-..4..= canonical transformation theequations oftraiisfonnation(9.-1-hue: isi-i §=nm- Analogously toEq.(9.45) wecanseektheequations ofmotion fezIran ables bylooking atthetimederivative ofatypical element of§ -%=—', ','=1,...,2n. 91 an,77] 5J Lnmatrix notation, thistimederivative canbewritten as §=Ma where MistheJacobian matrix ofthetransformation withelemeim 3M1] = L . 3711' Making useoftheequations ofmotion for1;,Eq.(9.50) becomes - 8§=MI (9.52) Now, bytheinverse transformation Hcanbeconsidered asafunction of§,and thederivative withrespect to11,evaluated as 8H 8H8;i =ii] 8771 39]3771, 9.4 TheSymplectic Approach toCanonical Transformations 383 or,inmatrix notation* nu3H 3H-5?-ME. (9.53) Thecombination ofEqs.(9.52) and(9.53) leads totheformoftheequations ofmotion foranysetofvariables §transforming, independently oftime, from the canonical set1;: . ~au;=M|ME. (9.54) Wehave theadvantage ofknowing from thegenerator formalism thatforare- stricted canonical transformation theoldHamiltonian expressed interms ofthe newvariables serves asthenewHamiltonian: -an ,§=]—éZ. (9.54) Thetransformation, Eq.(9.49), willtherefore becanonical ifMsatisﬁes thecon- dition aw M|M=|. (9.55) ThatEq.(9.55) isalsoanecessary condition forarestricted canonical transforma- tioniseasily shown directly byreversing theorder ofthestepsoftheproof. Note thatforanextended time-independent canonical transformation, where K=AH, thecondition ofEq.(9.55) would bereplaced by I\I MJM=xi. (9.56) Equation (9.55) maybeexpressed invarious forms. Multiplying fromtheright bythematrix inverse toMleads to M]=ith-1, (9.57) (since thetranspose oftheinverse istheinverse ofthetranspose). Theelements ofthematrix equation (9.57) willbefound tobeidentical withEqs.(9.48a) and (9.48b). IfEq.(9.57) ismultiplied byIfrom theleftand—]from theright, then byvirtue ofEq.(8.38e) wehave lM=f»"i*‘i. *Readers ofSection 7.5willhaverecognized thatEq.(9.50) isthestatement that1|transforms con- travanantly (asavector) under thetransformation, andEq.(9.53) saysthatthepartial derivative ofH withrespect totheelements of1;transfonns covarianily (orasal-form) (cf.Eqs.(7.50) and(7.54)). 84 Chapter 9Canonical Transformations or Av M]M =I. (9.58) Equation (9.55), oritsequivalent version, Eq.(9.58), isspoken ofasthesym- plectic condition foracanonical transformation, andthematrix Msatisfying the condition issaidtobeasymplectic matrix. These concepts maybecome more obvious ifwedisplay thedetails oftheIand Mmatrices corresponding tothemixed generating function F=F2(q1, P1)+ F1(qg, Q2)ofEq.(9.33). Thevariables 1|and§arecolumn vectors given by q1 Q1 =qz d =Q2 1' Pl an g P1' P2 P2 Thetransfomation =M-i|(cf.Eq.(9.5())) ismade bythefollowing Mmatrix: inagreement with theexpressions obtained bydifferentiating theresults ofthe generating function withrespect totime(cf.Column 3,Table 9.1).Hamilton’s equations forthetransformed variables =l%(Eq.(9.54')) areexpressed as follows independent ofthegenerating function F ifizli’where -P,=an/at, for4,and(2andQ,=ari/ag, forg3andg4.Note thatMdepends onFwhereas Jdoesnot(cf.Eq.(8.38a)). Thisformalism isnot applicable toallcases. Forexample, asimple Mmatrix carmot bewritten forthe harmonic oscillator example discussed inSection 9.3. Foracanonical transformation thatcontains thetimeasaparameter, thesimple derivation given forthesymplectic condition nolonger holds. Nonetheless, the symplectic condition remains anecessary andsufﬁcient condition foracanonical transformation even ifitinvolves thetime. Itispossible toprove thegeneral validity ofthesymplectic conditions forallcanonical transformations bystraight- forward, albeit lengthy, procedures resembling those employed forrestricted canonical transformations. Instead weshall takeadifferent tack, onethattakes-IQ-lO~[\)_| ooo»-coo I11 0»-cooo»-0.vQ-.oQ..]\)>-I "a-%- .N... re-to§)u-1CO QC i ©O©>-1 ©©I—©-to 9.4 TheSymplectic Approach toCanonical Transformations 385 advantage oftheparametric formofthecanonical transformations involving time. Acanonical transformation oftheform §=§(11,I) (9-59) evolves continuously astimeincreases from some initial value tg.Itisasingle- parameter instance ofthefamily ofcontinuous transformations ﬁrststudied sys- tematically bythemathematician Sophus Lieandassuchplays adistinctive role inthetransformation theory ofclassical mechanics. Ifthetransformation 1;—>§(t) (9.60a) iscanonical, thensoobviously isthetransformation 11->§(ro)- (9-60b) Itfollows thenfromthedeﬁnition ofcanonical transformation thatthetransfor- mation characterized by §(l0)->§(l) (9-60¢) isalsocanonical. Since toinEq.(9.60b) isaﬁxedconstant, thiscanonical trans- formation satisﬁes thesymplectic condition (9.58). Ifnowthetransformation of Eq.(9.600) obeys thesymplectic condition, itiseasytoshow (cf.Derivation 13) thatthegeneral transformation Eq.(9.60a) willalso. Todemonstrate thatthesymplectic condition doesindeed holdforcanonical transformations ofthetypeofEq.(9.600), weintroduce thenotion ofaninﬁnites- imalcanonical transformation (abbreviated I.C.T.), aconcept thatwillprove to bewidely useful. Asinthecaseofinﬁnitesimal rotations, such atransformation isoneinwhich thenewvariables differ fromtheoldonlybyinftnitesimals. Only ﬁrst-order terms inthese inﬁnitesimals aretoberetained inallcalculations. The transformation equations canthenbewritten as Q:=ql+sqn (9-613) Pr=Pr+8])!1 orinmatrix form §=1)+81;. (9.61c) (Here 8:1,and8p,donotrepresent virtual displacements butaresimply thein- ﬁnitesimal changes inthecoordinates andmomenta.) Aninﬁnitesimal canonical transformation thusdiffers only inﬁnitesimally from theidentity transformation discussed inSection 9.1.Inthegenerator formalism, asuitable generating func- tionforanI.C.T. would therefore be F2=qrPr+6G(q~ P,1‘). (9-62) 8 Chapter 9Canonical Transformations where eissome inﬁnitesimal parameter ofthetransformation, andGisany(dif- ferentiable) function ofits2n+larguments. ByEq.(9.l7a), thetransformation equations forthemomenta aretobefound from 8F2 3G=_=P _ pl aq] J-+63% OI‘ 8G8p]EP]—pj=-6 (9.63a) 41 Similarly. byEq.(9.17b),thetransformation equations forQJaredetermined by therelations 8F2 8GQJ— —qJ+€ Since thesecond tennis already linear ine,andPdiffers from ponlybyanin- ﬁnitesimal, itisconsistent toﬁrstorder toreplace P,inthederivative function by pJ.Wemaythenconsider Gasafunction ofq,ponly(andpossibly t).Following theusual practice, wewillrefertoG(q,p)asthegenerating ﬁmcrion ofthein- ﬁnitesimal canonical transformatton, although strictly speaking thatdesignation belongs onlytoF.Thetransformation equation forQ,cantherefore bewritten as 8G8q}=6 (9.63b) P1 Bothtransformation equations canbecombined intoonematrix equation 8G8=—. 9.63 1|6lan (c) Anobvious example ofaninﬁnitesimal canonical transformation would bethe transformation ofEq.(9.60c) when tdiffers from tobyaninﬁnitesimal t: §(t0) —>§(zg+dt), (9.64) withdtastheinﬁnitesimal parameter e.Thecontinuous evolution ofthetrans- formation §(27,t)from§(n,to)means thatthetransformation §(tQ) —>§(t)can bebuiltupasasuccession ofsuchI.C.T.’s instepsofdt.Itwilltherefore sufﬁce toshow thattheinﬁnitesimal transformation, Eq.(9.64), satisﬁes thesymplectic condition (9.58). Butitfollows from thetransformation equations (9.63) thatthe Jacobian matrix ofanyI.C.T. isasymplectic matrix. Bydeﬁnition theJacobian matrix (9.51) foraninﬁnitesimal transformation is 1 r d§ 681)ME—- =1+-Q, an an 9.4 TheSymplectic Approach toCanonical Transformations 387 orbyEq.(9.630) azaM=1+e]i. (9.65) Bnan Thesecond derivative inEq.(9.65) isasquare, symmetric matrix withelements alc _azc Bnan UBmﬁnf Because oftheantisymmetrical property ofI,thetranspose ofMis __ 2 M=1—e,a G]. (9.66)61181] Thesymplectic condition involves thevalue ofthematrix product ~ ale aleM]M= (1+€I——-)1 (1—ei—|).61;81p 81;8-r| Consistent toﬁrstorder inthisproduct is ~ ale aleMlM=l-fflﬁl-léﬁl =l, thus demonstrating thatthesymplectic condition holds foranyinﬁnitesimal canonical transformation. Bythechain ofreasoning wehave spun out,itthere- forefollows thatanycanonical transfonnation, whether ornotitinvolves timeas aparameter, obeys thesymplectic conditions, Eqs.(9.55) and(9.58). Thesymplectic approach, forthemost part,hasbeen developed independently ofthegenerating function method, except inthetreatment ofinﬁnitesimal canon- icaltransfonnations. They areofcourse connected. Weshall sketch later, forex- ample, aproof thatthesymplectic condition implies theexistence ofagenerating function. Butthecomiection islargely irrelevant. Botharevalidwaysoflooking at canonical transfomiations, andbothencompass alloftheneeded properties ofthe transformations. Forexample, either thesymplectic orthegenerator formalisms canbeused toprove thatcanonical transformations have thefourproperties that characterize agroup (cf.Appendix B). l.Theidentity transformation iscanonical. 2.Ifatransformation iscanonical, soisitsinverse. 3.Twosuccessive canonical transformations (thegroup “product” operation) deﬁne atransformation thatisalsocanonical. 4.Theproduct operation isassociative. 9.5 IChapter 9Canonical Transformations Weshall therefore befreetouseeither thegenerator orthesymplectic approach atwill,depending onwhich leads tothesimplest treatment atthemoment. POISSON BRACKETS AND OTHER CANONICAL INVARIANTS ThePoisson bracket oftwofunctions u,vwithrespect tothecanonical variables (q,p)isdeﬁned as Bu3v Bu8v=—— ———. 9.67 [u,v]q‘p aqrHP: aptaqr ( ) Inthisbilinear expression wehaveatypical symplectic structure, asinHamilton’s equations, where qiscoupled with p,andpwith —-q.ThePoisson bracket thus lends itself readily tobeing written inmatrix form, where itappears as »-4 88 [u.v]1|=8-21 (9.68) Thetranspose signisused ontheﬁrstmatrix ontheright-hand sidetoindicate explicitly thatthismatrix must betreated asasingle-row matrix inthemulti- plication. Onmostoccasions thisspeciﬁc reminder willnotbeneeded andthe transpose signmaybeomitted. Suppose wechoose thefunctions u,voutofthesetofcanonical variables (q,p)themselves. Then itfollows trivially from thedeﬁnition, either asEq.(9.67) or(9.68), thatthese Poisson brackets have thevalues lg)» qk]q,p =0=[Pp q/<lq,,>. and re.P/<1“,=at=-tp..qu,.,.. <9-69> Wecansummarize therelations ofEqs. (9.69) inoneequation byintroducing asquare matrix Poisson bracket, in,1}],whose lmelement is[n1,17",].Equa- tions (9.69) canthenbewritten as [TbTilt,=l- (9-70) Now letustakeforu,vthemembers ofthetransformed variables (Q,P),or §,deﬁned intenns of(q,p)bythetransformation equations (9.59). Thesetof allthePoisson brackets thatcanbeformed outof(Q,P)comprise thematrix Poisson bracket deﬁned as rt.in=%r 9.5 Poisson Brackets andOther Canonical Invariants 389 Butwerecognize thepartial derivatives asdeﬁning thesquare Jacobian matrix of thetransformation, sothatthePoisson bracket relation isequivalent to A4 [L£11;=MlM- (9-71) Ifthetransformation 1;->§iscanonical, thenthesymplectic condition holds andEq.(9.71) reduces to(cf.Eq.(9.58)) andconversely, ifEq.(9.72) isvalid, thenthetransformation iscanonical. Poisson brackets ofthecanonical variables themselves, such asEqs. (9.70) or(9.72), arereferred toasthefundamental Poisson brackets. Since wehave from Eq.(9.70) that [Q£1;=I. (9-73) Eq.(9.72) states thatthefundamental Poisson brackets ofthe§variables havethe same value when evaluated withrespect toanycanonical coordinate set.Inother words, thefundamental Poisson brackets areinvariant under canonical transfor- mation. WehaveseenfromEq.(9.71) thattheinvariance isanecessary andsufﬁ- cient condition forthetransformation matrix tobesymplectic. Theinvariance of thefundamental Poisson brackets isthusinallways equivalent tothesymplectic condition foracanonical transformation. Itdoesnottakemany more stepstoshowthatallPoisson brackets areinvariant under canonical transformation. Consider thePoisson bracket oftwofunctions u,vwithrespect tothensetofcoordinates, Eq.(9.68). Inanalogy toEq.(9.53), thepartial derivative ofvwithrespect to1;canbeexpressed interms ofpartial derivatives withrespect to§as ﬂ=g@an Bé (thatis,thepartial derivative transforms asal-form). Inasimilar fashion, 57¢173'; 511—=M——= M. an 8;6; Hence thePoisson bracket Eq.(9.68) canbeWritten E’);8v 5'71 ~du ll‘,vl-n=—'H15; =iM|M&- Ifthetransformation iscanonical, thesymplectic condition intheform of Eq.(9.55) holds, andwethenhave 5'a[u,v],,=£13; E[u.111;. (9.74) Chapter 9Canonical Transformations Thus, thePoisson bracket hasthesame value when evaluated withrespect toany canonical setofvariables—-all Poisson brackets arecanonical invariants. Inwrit- ingthesymbol forthePoisson bracket, wehave sofarbeen careﬁil toindicate by thesubscript thesetofvariables interms ofwhich thebrackets aredeﬁned. So longasweuseonlycanonical variables thatpractice isnowseentobeunneces- sary, andweshall ingeneral dropthesubscript.* Thehallmark ofthecanonical transformation isthatHamilton’s equations of motion areinvariant inform under thetransformation. Similarly, thecanonical in- variance ofPoisson brackets implies thatequations expressed interms ofPoisson brackets areinvariant inform under canonical transformation. Asweshall see,we candevelop astructure ofclassical mechanics, paralleling theHamiltonian for- mulation, expressed solely interms ofPoisson brackets. Historically thisPoisson bracket formulation, which hasthesame form inallcanonical coordinates, was especially useful forcarrying outtheoriginal transition fromclassical toquantum mechanics. There isasimple “correspondence principle" thatsaysthattheclas- sicalPoisson bracket istobereplaced byasuitably deﬁned commutator ofthe corresponding quantum operators. Thealgebraic properties ofthePoisson bracket aretherefore ofconsiderable interest. Wehavealready usedtheobvious properties lu,u]=0, (9.75a) [u,v]=—[v,u]. (antisyrmnetry) (9.75b) Almost equally obvious arethecharacteristics [au+bv,wj=a[u.w]+b[v,w], (linearity) (9.750) where aandbareconstants, and [uv,w]=[u,w]v+u[v,w]. (9.75d) Oneother property isfarfromobvious, butisveryimportant indeﬁning the nature ofthePoisson bracket. Itisusually given intheform ofJacobi ’siden- tity,which states thatifu,v,andwarethree functions withcontinuous second derivatives, then lu.lv,wll+lv.lw.Ml]+[w,[wtvll=0; (9-75¢) thatis,thesumofthecyclic permutations ofthedouble Poisson bracket ofthree functions iszero.There seems tobenosimple wayofproving Jacobi’s identity for thePoisson bracket without lengthy algebra. However, itispossible tomitigate thecomplexity ofthemanipulations byintroducing aspecial nomenclature. We *Note thatforascale transformation, oranextended canonical transformation, where thesymplectic condition takes ontheform ofEq(956),thenPoisson brackets donothave thesame values inall coordinate systems That rsoneofthereasons scale transformations areexcluded from theclass of canonical transformattons thatareuseful toconsider. 9.5POISSOH Brackets andOther Lanonical lnvariants 391 shallusesubscripts onu,v.w(orfunctions ofthem) todenote partial derivatives bythecorresponding canonical variable. Thus, u—au and v—av ‘Tm,’ ”'an.an,‘ Inthisnotation thePoisson bracket ofuandvcanbeexpressed as [u,v]=u,J,]v]. Here J1],asusual, issimply theijthelement of].Intheproof, theonlyproperty of]thatweshallneedisitsantisyrmnetry. Nowletusconsider theﬁrstdouble Poisson bracket inEq.(9.75e): lu,lv.wll=urlqlv, wl,=H|Jrj(vkJk1wz);- Because theelements Jk;areconstants, thederivative withresect to17doesn’t act onthem, andwehave [11,lv,wll=l1|J,,(vk-Ykrwz, +Pk;Jklwl)- (9-76) Theother double Poisson brackets canbeobtained from Eq.(9.76) bycyclic pemiutation ofu,v,w.There arethussixtemts inall,eachbeing afourfold sum overdummy indices i,j.k,andZ.Consider thetenn inEq.(9.76) involving a second derivative ofw: -7:;J/outv/<w1,~ Theonlyother second derivative ofwwillappear inevaluating thesecond double Poisson bracket in(Eq.9.75e): [vs[wt 74]]=vkJkI(w] J]|“r)I- Here theterminthesecond derivative inwis J],Jk;u,vkwJ;. Since theorder ofdifferentiation isimmaterial, wfj=wJ1,andthesumofthe twoterms isgiven by (J1; 'l'Jjz)Jkl7-Hvkwl] =0, byvirtue oftheantisymmetry ofJ.Theremaining fourterms arecyclic permuta- tionsandcansimilarly bedivided intwopairs, oneinvolving second derivatives ofuandtheother ofv.Bythesame reasoning, each ofthese pairs sums tozero, andJacobi’s identity isthusveriﬁed. IfthePoisson bracket ofu,vislooked onasdeﬁning a“product” operation ofthetwofunctions, then Jacobi’s identity isthereplacement fortheassocia- Chapter 9Canonical Transformations tivelawofmultiplication. Recall thattheordinary multiplication ofarithmetic is associative; thatis,theorder ofasequence ofmultiplications isimmaterial: a(bc) =(ab)c. Jacobi’s identity saysthatthebracket “product” isnotassociative andgives theeffect ofchanging thesequence of“multiplications.” Brackets thatsatisfy Eqs.(9.75), together withtheexpression [u,,uj]=Zcﬁuk. (9.77) It constitute agenerally noncommunitive algebra called aLiealgebra. ForPoisson brackets inthree-dimensional space, either thestructure constants cf-‘jareallzero oronlyonetermintheright-hand sideofEq.(9.77) exists foranypairofindices. Examples ofthiswillbegiven later, andamore detailed discussion ofLiealgebras isgiven inAppendix B. Poisson bracket operation isnottheonlytypeof“product” familiar tophysi- ciststhatsatisﬁes theconditions foraLiealgebra. Itwillbelefttotheexercises toshow thatthatvector product oftwovectors, v[A,B]->AxB, (9.78a) andthecommutator oftwomatrices, M[A,B]—>AB—BA, (9.78b) satisfy thesame Liealgebra conditions asthePoisson bracket. Itisthislastthat makes itfeasible toreplace theclassical Poisson bracket bythecommutator ofthe quantum mechanical operators. Inother words, the“correspondence principle” canwork onlybecause boththePoisson bracket andcommutator arerepresenta- tions ofaLiealgebra “product.”* There areother canonical invariants besides thePoisson bracket. One, mainly ofhistorical interest now, istheLagrange bracket, denoted by{u,v}.Suppose u andvaretwofunctions outofasetof2nindependent functions ofthecanonical variables. Byinversion, thecanonical variables canthenbeconsidered asfunc- tionsofthesetof2nfunctions. Onthisbasis, theLagrange bracket ofuandv withrespect tothe(q,p)variables isdeﬁned as ‘Ofcourse, wemust notmistake themathematical acceptability ofthisversion ofthecorrespondence pnncrple withitsphysical necessity. Themtroduction ofthequantum cotmnutation relations wasa great actofphysical discovery bythepioneers ofquantum mechanics. Allweshow hereisthatthere isasimilarity inthemathematical structure ofthePoisson bracket formulation ofclassical mechanics andthecommutation relation version ofquantum mechanics Theformal correspondence isthat l[u,vj—>—(uv —vu)th where ontheleftu,vareclassical functions andontherighttheyarequantum operators. 9.5 Poisson Brackets andOther Canonical Invariants 393 _84.81>. an8q.- 979 {”"’}“*"*'a7 3vBuav’ (') or,inmatrix notation. 5?]an{u,v},]=E] (9.80) Proof ofthecanonical invariance oftheLagrange bracket parallels thatforthe Poisson bracket. IfforuandvWetaketwomembers ofthesetofcanonical variables, thenwe obtain thefundamental Lagrange brackets: i‘l1>qJlqp=0={P:>Pjiqp {q,.1>;}qp =51j, (931) or.inmatrix notation, {'11,nl=I. (9-82) TheLagrange andPoisson brackets clearly stand insome kindofinverse rela- tionship toeachother, buttheprecise fonnofthisrelation issomewhat compli- cated toexpress. Letu,,i=1,...,2n,beasetof2nindependent functions of thecanonical variables, toberepresented byacolumn (orrow)matrix u.Then {u,u}isthe2n><2nmatrix whose ijthelement is{u,,uJ},withasimilar descrip- tionfor[u,u].Thereciprocal character ofthetwobrackets manifests itself inthe relation {u,u}[u, u]=-1. (9.83) Ifforuwechoose thecanonical setitself, 1),then Eq.(9.83) obviously fol- lows from thefundamental bracket formulas, Eqs.(9.70) and(9.82), andthe properties ofI.Theproof forarbitrary uisnotdifficult ifwritten interms of thematrix deﬁnitions ofthebrackets andisreserved fortheexercises. While theproperties oftheLagrange andPoisson brackets parallel each other in many aspects, notethattheLagrange brackets donotobey Jacobi’s identity. Lagrange brackets therefore donotqualify asa“product” operation inaLie algebra. Another important canonical invariant isthemagnitude ofavolume element in phase space. Acanonical transformation 1)—>Q,’transforms the2n-dimensional phase space withcoordinates 17,toanother phase space withcoordinates ;,.The volume element (dn)=dq1r1q2-- -dqndm ---div» transforms toanewvolume element -T: ~--dpn. Chapter 9Canonical Transformatlons Asiswellknown, thesizes ofthetwovolume elements arerelated bythe absolute value oftheJacobian determinant ||M||; (I1?)=||M|l(dn)- Forexample, inthetwo-dimensional transformation ﬁom 1;,=q,pto;',=Q,P, thisexpression becomes 3 3P 2£.%8QaP But,bytaking thedeterminant ofbothsides ofthesymplectic condition, Eq.(9.58), wehave wWn=m ow) Thus, inarealcanonical transformation theJacobian determinant isi1,andthe absolute value isalways unity, proving thecanonical invariance ofthevolume element inphase space. Itfollows, also,thatthevolume ofanyarbitrary region 111 phase space, a=f fan ow) isacanonical invariant. lnourtwo-dimensional example, theinvariant isd1;= dqdpandJ1: fdq dp. Thevolume integral inEq.(9.86) istheﬁnalmember ofasequence ofcanon- icalinvariants known astheintegral invariants ofPoincare’, comprising integrals oversubspaces ofphase space ofdifferent dimensions. Theother members ofthe sequence carmot bestated assimply asJ,,,andbecause theyarenotneeded for thefurther development ofthetheory, theywillnotbediscussed here. Finally, theinvariance ofthefundamental Poisson brackets nowenables usto outline aproof thatthesymplectic condition implies theexistence ofagenerat- ingfunction, asmentioned attheconclusion oftheprevious section. Tosimplify considerations, weshall examine onlyasystem withonedegree offreedom; the general method oftheproof canbedirectly extended tosystems withmany de- grees offreedom?‘ Wesuppose thattheﬁrstoftheequations oftransformation, Q=Q(qv P)’ P: *lntheliterature, theconnection between thesymplectic approach andthegenerator formalism is sometimes referred toastheCaratheodory theorem. 9.5 Poisson Brackets andOther Canonical Invariants 395 isinvertable soastogivepasafunction qandQ,say P=¢(<1-Q)- (9-37) Substitution inthesecond equation oftransformation gives Passome function ofqandQ,say P=1//(61,Q)- (9-33) Insuchacase,wewould expect thetransformation tobegenerated byagenerating function oftheﬁrstkind,* F1,withEqs.(9.87) and(9.88) appearing as p=”%f;@, P=-%<q,Q>. (9.89) IfEq.(9.89) holds, thenitmustbetruethat 59¢ 310—=— . 9.90 3Q aq () Conversely, ifwecanshowthatEq.(9.90) isvalid, thentheremustexistafunction F1suchthatpandParegiven byEqs.(9.89). Todemonstrate thevalidity ofEq.(9.90), wetrytolookonallquantities as functions ofqandQ.Thus. weofcourse have theidentity E=13Q ' butifEq.(9.87) besubstituted intheﬁrsttransfomation equation, Q=Q(q,¢(q.Q)), (9-91) thepartial derivative canalsobewritten QJQ228Q‘3paQ’ sothatwehave therelation "£‘i= 992 apag 1. (. ) Inthesame spirit weevaluate thePoisson bracket _aQaP aPaQ *Ofcourse, ittheQtransfonnauon equation 15nottnvertable, asintheidentity transformation, then wewould inven thePequation andbeledtoagenerating function ofthesecond kind. 9.6IChapter 9Canonical Transformations Thederivatives ofParederivatives of(0fromEq.(9.88) considered asafunction ofqandQ(q, p).Hence, thePoisson bracket canbewritten 8Q8(08Q 6Q(61,b 811/8Q) [.Pl= — + ,.Q Ba3Q3P91>3118Qdq or,consolidating tenns, as _31// 3Q3Q_3Q3Q _3Q§§Q [Q’P]_6Q(9q3P Bpaq) BPBQ’ andtherefore 3Qaw1=-——. 9.93 apaq <) Combining Eqs.(9.92) and(9.93), wehave "£8i__@ﬁHp6Q_HpM Since thepartial derivative ofQwithrespect topisthesame onbothsides ofthe equation, thatis,theothervariable being heldconstant isqinbothcases, andsince thederivative doesn’t vanish (elsetheQequation could notbeinverted), itfollows thatEq.(9.90) must betrue.Thus, from thevalue ofthefundamental Poisson bracket [Q,P],which wehaveseenisequivalent tothesymplectic condition, we areledtotheexistence ofagenerating function. Thetwoapproaches tocanonical transformations, though arrived atindependently, arefullyequivalent. EQUATIONS OFMOTION, INFINITESIMAL CANONICAL TRANSFORMATIONS, AND CONSERVATION THEOREMS INTHE POISSON BRACKET FORMULATION Almost theentire framework ofHamiltonian mechanics canberestated interms ofPoisson brackets. Asaresult ofthecanonical invariance ofthePoisson brack- ets,therelations soobtained willalsobeinvariant inform under acanonical transformation. Suppose, forexample, welookforthetotaltimederivative of some function ofthecanonical variables andtime, u(q,p,t),byuseofHamil- ton’sequations ofmotion: du Ziu_ Bu,Ziu Elu3H Bu8H Bu-T +, + Z T + 9 ataq,-‘I’ ap/" at3q,3p, ap,aq.- at OI’ du 8u 9.6 Equations ofMotion 397 interms ofthesymplectic notation, thederivation ofEq.(9.94) would run du_3u,+8u_6u]iiH+3u dz“a1," ai"aﬁ an at’ from whence Eq.(9.94) follows, byvirtue of(9.68). Equation (9.94) may be looked onasthegeneralized equation ofmotion foranarbitrary function Min thePoisson bracket formulation. Itcontains I-1amilton’s equations asaspecial casewhen foruwesubstitute oneofthecanonical variables él =[qia H]: I51 =[PH H1’ or,insymplectic notation, ii=in.Hl- (9.95b) ThatEq.(9.95b) isidentical withHamilton’s equations ofmotion maybeseen directly from theobservation thatbythedeﬁnition ofthePoisson bracket, Eq.(8.39), wehave in.Hl=i (9.96) sothatEq.(9.95b) issimply another wayofwriting Eq.(8.31). Another familiar property maybeobtained fromEq.(9.94) bytaking uasHitself. Equation (9.94) thensaysthat dH_8H dt_6t’ aswasobtained previously inEq.(8.41). Note thatthegeneralized equation ofmotion iscanonically invariant; itisvalid inwhatever setofcanonical variables q,pisused toexpress thefunction uorto evaluate thePoisson bracket. However, theHamiltonian usedmust beappropriate totheparticular setofcanonical variables. Upon transforming toanother setof variables byatime-dependent canonical transformation, wemustalsochange to thetransformed Hamiltonian K. Ifuisaconstant ofthemotion, thenEq.(9.94) saysitmusthavetheproperty [11,H]= (9.97) Allfunctions thatobey Eq.(9.97) areconstants ofthemotion, andconversely the Poisson bracket ofHwithanyconstant ofthemotion mustbeequal totheexplicit timederivative oftheconstant function. Wethushaveageneral testforseeking andidentifying theconstants ofthesystem. Forthose constants ofthemotion not Chapter 9Canonical Transformations involving thetimeexplicitly, thetestofEq.(9.97) reduces torequiring thattheir Poisson brackets withtheHamiltonian vanish, thatis,[H,u]=0.* Iftwoconstants ofthemotion areknown, theJacobi identity provides apossi- blewayforobtaining further constants. Suppose uandvaretwoconstants ofthe motion notexplicitly functions oftime. Then ifwinEq.(9.75e) istaken tobeH, theJacobi identity says [H.[14,vi]=0; thatis,thePoisson bracket ofuandvisalsoaconstant intirne. Even when theconserved quantities depend upon timeexplicitly, itcanbeshown withabit more algebra (cf.Exercise 30)thatthePoisson bracket ofanytwoconstants ofthe motion isalsoaconstant ofthemotion (Poisson’s theorem). Repeated application oftheJacobi identity inthismanner caninprinciple leadtoacomplete sequence ofconstants ofthemotion. Quite often, however, theprocess isdisappointing. ThePoisson bracket ofuandvfrequently turns outtobeatrivial function ofu andvthemselves, oreven identically zero. Still, thepossibility ofgenerating new independent constants ofmotion byPois$on’s theorem should bekeptinmind. ThePoisson bracket notation canalsobeusedtoreforinulate thebasic equa- tions ofaninﬁnitesimal canonical transfonnation. Asdiscussed above (Sec- tion9.4),suchatransformation isaspecial caseofatransformation thatisa continuous function ofaparameter, starting from theidentity transformation at some initial value oftheparameter (which may, forconvenience, besetequal tozero). Iftheparameter issmall enough tobetreated asaﬁrst-order inﬁnitesi- mal,thenthetransfonned canonical variables differ onlyinﬁnitesimally from the initial coordinates: §=1|+81] (9.98) withthechange being given interms ofthegenerator Gthrough Eq.(9.63c): 3G( ) 81]=G] Now, bythedeﬁnition (9.68) ofthePoisson bracket, itfollows that in.ul=1 (9.99) (cf.Eq.(9.96)), arelation thatremains valid when thePoisson bracket isevaluated intenns ofanyother canonical variables. Ifuistaken tobeG,itisseenthatthe equations oftransfomiation foraninﬁnitesimal canonical transformation canbe *lnview ofthe“correspondence principle” between theclassical Poisson bracket andthequantum commutator, itisseenthatthisstatement corresponds tothewell-known quantum theorem thatcon- served quantities Oommute withtheHamiltonian. 9.6 Equations ofMotion 399 written as 691=€[1i,G]. (9.100) Consider nowaninﬁnitesimal canonical transforination inwhich thecontin- uousparameter ist(aswasdone inproving thesymplectic condition) sothat e=dt,andletthegenerating function GbetheHamiltonian. Thentheequations oftransformation forthisI.C.T. become, byEq.(9.100), an=dt[1],H]=ipdt=an. (9.101) These equations statethatthetransformation changes thecoordinates andmo- menta atthetimertothevalues theyhaveatthetimer+dt.Thus, themotion of thesystem inatimeinterval drcanbedescribed byaninﬁnitesimal contact trans- formation generated bytheHamiltonian. Correspondingly, thesystem motion in aﬁnite timeinterval from tototisrepresented byasuccession ofinﬁnitesimal contact transformations, which, aswehaveseen, isequivalent toasingle ﬁnite canonical transformation. Thus, thevalues ofqandpatairytimetcanbeob- tained fromtheirinitial values byacanonical transformation thatisacontinuous function oftime. According tothisview, themotion ofamechanical system cor- responds tothecontinuous evolution orunfolding ofacanonical transformation. Inaveryliteral sense, theHamiltonian isthegenerator ofthesystem motion with time. Conversely, theremustexistacanonical transformation fromthevalues ofthe coordinates andmomenta atanytimettotheirconstant initial values. Obtain- ingsuchatransformation isobviously equivalent tosolving theproblem ofthe system motion. Atthebeginning ofthechapter itwaspointed outthatamechan- icalproblem could bereduced toﬁnding thecanonical transformation forwhich allmomenta areconstants ofthemotion. Thepresent considerations indicate the possibility ofanalternative solution bymeans ofthecanonical transformation for which boththemomenta andcoordinates areconstants ofthemotion. These two suggestions willbeelaborated inthenextchapter inorder toshow howformal solutions maybeobtained foranymechanical problem. implicit tothisdiscussion hasbeen analtered wayoflooking atacanonical transformation andtheeffect itproduces. Thenotion ofacanonical transforma- tionwasintroduced asachange ofthecoordinates usedtocharacterize phase space. Ineffect, weswitched fromonephase space T]withcoordinates (q,p)to another, ;,withcoordinates (Q,P).Ifthestateofthesystem atagiven timewas described byapoint Ainonesystem, itcould alsobedescribed equally well bythetransfomied point A’(cf.Fig.9.2).Anyfunction ofthesystem variables would have thesame value foragiven system conﬁguration whether itwasde- scribed bythe(q,p)setorbythe(Q,P)set.Inother words, thefunction would havethesame value atA’asatA.Inanalogy tothecorresponding description oforthogonal transformations, wemaycallthisthepassive viewofacanonical transformation. Chapter 9Canonical Transformations '7 K Phase Phase -A‘.(Q,P) space space A-<q.p> Z’ FIGURE 9.2 Thepassive view ofacanomcal transformation. Incontrast, wehave spoken ofthecanonical transformation generated bythe Hamiltonian asrelating thecoordinates ofonepoint inphase space tothose of another point inthesame phase space. From thisviewpoint, thecanonical trans- fonnation accomplishes, inthemathematician’s language, amapping oftl1epoints ofphase space ontothemselves. Ineffect, wehaveanactive interpretation ofthe canonical transformation as“moving” thesystem point fromoneposition, with coordinates (q,p),toanother point, (Q,P),inphase space (cf.Fig.9.3).Of course. thecanonical transformation initselfcannot move orchange thesystem conﬁguration. What 1tdoesisexpress oneconﬁguration ofthesystem interms of another. With some classes ofcanonical transfonnation, theactive viewpoint is nothelpful. Forexample, thepoint transformation fromCartesian coordinates to spherical polar coordinates isacanonical transformation ofthepassive type,and an“active” interpretation of1twould border onthell.ld1CI'011S. Theactive viewpoint isparticularly useful fortransfonnations depending con- tinuously onasingle parameter. Ontheactive interpretation, theeffect ofthe transformation isto“move” thesystem point continuously onacurve inphase space astheparameter changes continuously. When thegenerator oftheassoci- atedI.C.T. istheHamiltonian, thecurve onwhich thesystem point moves isthe trajectory ofthesystem inphase space. B Phase (Q’P) spacc A(4,P) FIGURE 9.3 Theactive view ofacanonical transformation. 9.6 Equations ofMotion 401 IfwePosethequestion, Howdoesafunction change under acanonical trans- formation? theanswer depends onwhether weshould takeanactive orapassive point ofview. From thepassive point ofview, thefunction changes inform, orin functional dependence, butitdoesnotchange invalue. Thisisbecause ingeneral thefunction, callitU,hasadifferent functional dependence on(Q,P)thanit doeson(q,p).Itsvalue however remains thesame atthecorresponding points l/(qo, P0)andU(Q0,Po)Since Q0=Q(qo. Po)andPo=P(qo,P0),S0both setsofcoordinates refer tothesame physical location inphase space butusedif- ferent coordinates todescribe thephase space. Incontrast tothis,ifweconsider thecanonical transformation fromanactive point ofview, then wearetalking about atranslation ofthesystem from point Atopoint B,fromposition (qA,pA)toposition (qg,pg).From thispoi.nt of view, thefunction U(q,p)does notchange itsfunctional dependence upon po- sition andmomentum, rather itchanges itsvalues asaresult ofreplacing the values (qA,pA)by(qg,pl-3)inthefunction U(q,p),There arethentwodistinct phase spaces, oneusing (q,p)andtheother using (Q.P).Thetransformation formalism usesthenotation (q,p)forthevariables atpoint Aand(Q,P)forthe variables atpoint B.Thisisanalogous toapassive rotation incoordinate space corresponding totherotation ofthecoordinate axesrelative toastationary ob- ject,andanactive rotation corresponding torotating anobject relative toaﬁxed coordinate system. Weshallusethesymbol 8todenote achange inthevalue ofafunction under an“active” inﬁnitesimal canonical transformation: Bu=u(B) —u(./1), (9.102) where ofcourse AandBwillbeinﬁnitesimally close. Using thematrix notation forthecanonical variables, thechange inthefunction value under anI.C.T. would bedeﬁned as Bu=u('|1+51])—u(1]). Expanding inaTaylor series andretaining terms inﬁrst-order inﬁnitesimals, we have, byvirtue ofEq.(9.63c), at 512aGBu—-57,51] —€-55] Recalling thedeﬁnition ofthePoisson bracket, Eq.(9.68), weseethatthechange canbewritten as Bu=e[u, G]. (9.103) Animmediate application ofEq.(9.103) istotakeforuoneofthephase space coordinates themselves (orthematrix ofthecoordinates). Wethen have, by Eq.(9.100), 81;=e[1),G]=51). 4 Chapter 9Canonical Transformations Ofcourse, thisresult isobvious fromthedeﬁnition ofthepoint Binrelation to A;the“change” inthecoordinates fromAtoBisjusttheinﬁnitesimal difference between theoldandnewcoordinates. These considerations must begeneralized somewhat intalking about the “change intheHamiltonian.” Recall thatthedesignation “Hamiltonian” doesnot mean aspeciﬁc function, thesame inallcoordinate systems. Rather itrefers to thatfunction which inthegiven phase space deﬁnes thecanonical equations of motion. Where thecanonical transformation depends upon thetime, thevery meaning of“Hamiltonian” isalsotransformed. Thus, H(.A) goesovernotinto H(.A')butintoK(A'), andH(A)willnotnecessarily havethesame value as K(A’).Insuchacase,weshallmean byZ-)Hineffect thedifference inthevalue oftheHamiltonian under thetwointerpretations: 3H=H(B) —K(A'). (9.104) Where thefunction itselfdoesnotchange under thecanonical transfonnation the twofonns forthechange, Eqs.(9.102) and(9.104), areidentical since u(A') = u(A). Ingeneral, Kisrelated toHbytheequation 6FK=H+—.at where foranI.C.T. thegenerating function isgiven byEq.(9.62) intenns ofG. Since onlyGinthatequation canbeanexplicit function oftime, thevalue ofthe newHamiltonian isgiven by 6G 6GK(.A') =H(A') +e— =H(A) +6—,8t 8: andthechange intheHamiltonian is 8G Following along thepaththatledfromEq.(9.103), weseethat8Hisgiven by 8H=e[H,G]—e§a—(ti. (9.106) From thegeneralized equation ofmotion, Eq.(9.106), withGasu,itfollows ﬁnally thatthechange inHis dGH=— ——. 9.107 8 ed’ ( ) IfGisaconstant ofthemotion, Eq.(9.107) saysthatitgenerates aninﬁnites- imalcanonical transformation thatdoesnotchange thevalue oftheHamiltonian. Equivalently, theconstants ofthemotion arethegenerating functions ofthose inﬁnitesimal canonical transformations thatleave theHamiltonian invariant. Im- 9.6 Equations ofMotion 403 plied inthisconclusion isaconnection between thesymmetry properties ofthe system andconserved quantities, aconnection thatissimplest toseeforconstants ofthemotion notexplicitly depending upontime.Thechange intheHamiltonian under thetransfonnation isthensimply thechange inthevalue oftheHamil- tonian asthesystem ismoved from conﬁguration Atoconﬁguration B.Ifthe system issymmetrical under theoperation thatproduces thischange ofconﬁg- uration, thentheHamiltonian willobviously remain unaffected under thecorre- sponding transformation. Totakeasimple example, ifthesystem issymmetrical about agiven direction, thentheHamiltonian willnotchange invalue ifthesys- temasawhole isrotated about thatdirection. Itfollows thenthatthequantity that generates (through anI.C.T.) sucharotation ofthesystem mustbeconserved. Therotational symmetry ofthesystem implies aparticular constant ofthemo- tion.Thisisnottheﬁrstinstance ofaconnection between constants ofthemotion andsymmetry characteristics. Weencountered itpreviously (Sections 2.6,8.2) inconnection withtheconservation ofgeneralized momenta. Here, however, the theorem ismore elegant, andmore complete, foritembraces allindependent con- stants ofthemotion andnotmerely theconserved generalized momenta. Themomentum conservation theorems appear nowasaspecial caseofthe general statement: Ifacoordinate q,-iscyclic, theHamiltonian isindependent ofq;andwillcertainly beinvariant under aninﬁnitesimal transformation that involves adisplacement ofq,-alone. Consider, now, atransformation generated bythegeneralized momentum conjugate toq,-: G(q.P)=Pi- (9,108) ByEqs.(9.63a andb),theresultant inﬁnitesimal canonical transformation is Sq] =685]", 8p,=0, (9.109) thatis,exactly therequired inﬁnitesimal displacement ofqiandonlyq,-.Weread- ilyrecognize thisasthefamiliar momentum theorem: Ifacoordinate iscyclic, its conjugate momentum isaconstant ofthemotion. Theobservation thatadisplace- ment ofonecoordinate alone isgenerated bytheconjugate momentum maybe putinaslightly expanded form. Ifthegenerating function ofanI.C.T. isgiven by GI=(ITI): =Jzrﬂr. (9110) thentheequations oftransformation asobtained from Eq.(9.630) appear as 3G 877k =5-Iks E11 =5-[ks-Ilr5rs =€JksJls- s Byvirtue oftheorthogonality of],these reduce ﬁnally to 811/< =€5k1; (9.111) 404 Chapter 9Canonical Transformations thatis,adisplacement ofanycanonical variables :7;alone isgenerated interms of theconjugate variable intheformgiven byEq.(9.110). Ofcourse, ifnlisq,-,G fromEq.(9.110) isjust pi,andif1;;ispi,Gisthen—q,-. Asaspeciﬁc illustration ofthese concepts, letusconsider again theinﬁnites- imalcontact transformation ofthedynamical variables thatproduces arotation ofthesystem asawhole byanangle d6.Thephysical signiﬁcance ofthecorre- sponding generating function camiot depend upon thechoice ofinitial canonical coordinates,* anditisconvenient touseforthispurpose theCartesian coordinates ofallparticles inthesystem. Norwilltherebeanylossingenerality iftheaxesare sooriented thattheinﬁnitesimal rotation isalong thezaxis.Foraninﬁnitesimal counterclockwise rotation ofeachparticle, thechange intheposition vectors isto befound fromtheinﬁnitesimal rotation matrix ofEq.(4.69). Witharotation only about thezaxis, thechanges intheparticle coordinates are 8x;=-y,d6,8y,-=x,~d9,81,-=0. (9.ll2a) Theeffect ofthetransformation onthecomponents oftheCartesian vectors formed bythemomenta conjugate totheparticle coordinates issimilarly given by 5Pix=-Pryd9. 5Piy==Pixd9, 5Piz=0- (9-1121)) Comparing these transformation equations withEqs.(9.63a andb),itisseenthat thecorresponding generating function is G=Iipiy -yipix, (9-113) withd6astheinﬁnitesimal parameter e.Foradirect check, notethat 3G 3G 316;=d9i =—yid9, 5pix =—d6 -—— =—p,'y d6, apix axi 3G 8G8)’;=d9i =1165116, 5p,'y =—d9 ‘ =pix(19, any an agreeing withEqs. (9.112). Thegenerating function (9.113) inaddition hasthe physical signiﬁcance ofbeing thez-component ofthetotalcanonical angular mo- mentum: G=L,E(r;><p,-),. (9.114) Since thezaxiswasarbitrarily chosen, wecanstate thatthegenerating function corresponding toaninﬁnitesimal rotation about anaxisdenoted bytheunitvector *This canmost easily beseenfrom thecanonically invariant Eq.(9.100). Thechange inthecanonical variable 11,-remains thesamenomatter inwhatsetofcanonical variables Gisexpressed. 9.6 Equations ofMotion 405 nis G=L-n. (9.115) Note thatthecanonical angular momentum asdeﬁned heremaydiffer from themechanical angular momentum. Iftheforces onthesystem arederivable from velocity-dependent potentials, thenthecanonical momentum vectors p,-arenot necessarily thesame asthelinear momentum vectors, andLinEqs.(9.114) and (9.115) maynotbethesame asthemechanical angular momentum. Theresult obtained hereistherefore ageneralization oftheconclusion given inSection 2.6 thatthemomentum conjugate toarotation coordinate isthecorresponding com- ponent ofthetotal angular momentum. Theproof presented there wasrestricted tosystems with velocity-independent potentials. Byvirtue ofEqs. (9.108) and (9.109), wecannowconclude thatthemomentum conjugate toageneralized co- ordinate thatmeasures therotation ofthesystem asawhole about anaxisnisthe component ofthetotalcanonical angular momentum along thesame axis.Justas theHamiltonian isthegenerator ofadisplacement ofthesystem intime, sothe angular momentum isthegenerator ofthespatial rotations ofthesystem. Ithasalready beennoted thatonthe“active” interpretation acanonical trans- formation depending upon aparameter “moves” thesystem point along acon- tinuous trajectory inphase space. Since theﬁnite transformation canbelooked onasthesumofaninﬁnite succession ofinﬁnitesimal canonical transformations, each corresponding toaninﬁnitesimal displacement along thecurve, itshould therefore bepossible formally toobtain theﬁnite transformation byintegrating theexpression fortheinﬁnitesimal displacements. Wecandothisbynoting that eachpoint onthetrajectory inphase space corresponds toaparticular value of theparameter, which weshall callor,starting from theinitial system conﬁgura- tiondenoted byoz=0.Ifuissome function ofthesystem conﬁguration, thenu willbeacontinuous function ofaalong thetrajectory ,u(a), withinitial value un=u(0). (Forsimplicity. weshall consider uasnotdepending explicitly upon time.) Equation (9.103) fortheinﬁnitesimal change ofuonthetrajectory canbe written as 8u=da[u, G], orasadifferential equation inthevariable oz: d_”=[u,0]. (9.116)da Wecangetu(ix),andtherefore theeffect oftheﬁnite canonical transformation, byintegrating thisdifferential equation. Aformal solution maybeobtained by expanding u(or)inaTaylor series about theinitial conditions: dul +a2d2u‘ +oz3d3u| + duo 2!doz20 3ld0130 'u(0z) =u0+rx 406 Chapter 9Canonical Transformations ByEq.(9.116), wehave du ___ = aG a ddl0[Mlo thezerosubscript meaning thatthevalue ofthePoisson bracket istobetaken at theinitial point, or=0.Repeated application ofEq.(9.116), taking [u,G]itself asafunction ofthesystem conﬁguration, gives dzu W =[lut G]! G]: andtheprocess canberepeated togivethethird derivative ofuandsoon.The Taylor series foru(a)thusleads totheformal series solution 2 3 aw)=at+am.G10+%rru.G1.G10+%rrru.G1.G1.G10+---.(9.117) Ifforuwetakeanyofthecanonical variables Q,withuothestarting setofvari- ables 17,,thenEq.(9.115)isaprescription forﬁnding thetransformation equations oftheﬁnite canonical transformation generated byG. Itisnotdifﬁcult toﬁndspeciﬁc examples showing thatthisprocedure actu- allyworks. Suppose forGwetakeLz,sothattheﬁnalcanonical transformation should correspond toaﬁnite rotation about thezaxis.Thenatural parameter to useforozistherotation angle. Foru,letustakethex-coordinate oftheithparticle inthesystem. Either bydirect evaluation ofthePoisson brackets orbyinference fromEqs.(9.112a),itiseasytoseethat [Xrul-z]= —Yi, [Yul-zl =Xi, (9-113) where capital letters havebeen usedtodenote thecoordinates aftersome rotation 9,thatis,theﬁnalcoordinate. Theinitial coordinates, thatis,before rotation, are asusual represented bylowercase letters. Itfollows thenthat l:Xi» L210 =_)’i, LI], Z =_-xiv LZ]1 LZ:l» = —": andsoon.Theseries representation forX,-thusbecomes 62 63 64 Xi=xi"‘yi6_xi?+)'i¥+xi$—"' _I0104 603_ET+ﬁ—--. —§+--a ' 9.6 Equations ofMotion 407 Thetwoseries willberecognized astheexpansion forthecosine andsine,re- spectively. Hence, theequation fortheﬁnite transformation ofX,is X,-=xicos0 —y,-sin6, which isexactly what wewould expect fortheﬁnite rotation ofavector counter- clockwise about thezaxis. Foranother example, letusconsider thesituation when G=Handthepa- rameter isthetime. Equation (9.116) thenreduces totheequation ofmotion for u: du '5 '“[uv H]: withtheformal solution ,2 ,3 14(1)=Mo+Flu.H10+51114. H].Hlo+5111”, H],H],H10+'---(9-119) Here thesubscript zerorefers totheinitial conditions att=0. Letusapply thisprescription tothesimple problem ofone-dimensional motion withaconstant acceleration a,forwhich theHamiltonian is 2 H=57—max, withuastheposition coordinate x.ThePoisson brackets needed inEq.(9.119) areeasytoevaluate directly orfromthefundamental brackets: ix.H1=5.m llx.Hl,Hl =%lP.Hl =11- Because thislastPoisson bracket isaconstant, allhigher-order brackets vanish identically andtheseries temrinates, withthecomplete solution being given by 1 :2x—_=_xO+_I3l+i__ m 2 Remembering thatpg/m=v0,thiswillberecognized asthefamiliar elementary solution totheproblem. Itmaybefeltthatwhat wehave done hereisatourdeforce, amere virtuoso performance. There isforce totheobjection. Wewould notpropose thefonnal se- riessolution, Eq.(9.119), asthepreferred method forsolving realistic problems inmechanics. Itissurely oneofthemost recondite procedures wecanconceive of forsolving theeasiest offreshman physics problems! Nonetheless, thetechnique provides insights intothestructure ofclassical mechanics asbased oncanoni- caltransfonnation theory. Theseries expansion shows directly thatinﬁnitesimal 9.7IChapter 9Canonical Transformations canonical transfonnations cangenerate ﬁnite canonical transformations, depend- ingonaparameter. andthusleadtosolutions totheequations ofmotion. Ofpar- ticular interest fortherelation between classical andquantum mechanics isthe observation thattheseries inEqs.(9.117) or(9.119) bearafamily resemblance totheseries foranexponential. ThenestofPoisson brackets inthenthtermcan beconsidered asthenthrepeated application (from theright!) oftheoperator [,G],orthenthpower oftheoperator. Equation (9.119), forexample, could symbolically bewritten as 11(1)=ue (9.120) Theexponential heremeans nomore thanitsseries representations andthesym- bolHisusedtoindicate theoperator [,H].What wehavehereisveryremi- niscent oftheHeisenberg picture inquantum mechanics where theu(t)become time-varying operators, whose timedependence isgiven interms ofexp[i Ht/h] insuchamanner astoleadtothesame equation ofmotion, Eq.(9.94). (The additional factor i/iiarises outofthecorrespondence between theclassical Pois- sonbracket andthequantum commutator.) ThePoisson bracket formulation of mechanics isthustheclassical analog oftheHeisenberg picture ofquantum me- chanics. THE ANGULAR MOMENTUM POISSON BRACKET RELATIONS Theidentiﬁcation ofthecanonical angular momentum asthegenerator ofarigid rotation ofthesystem leads toanumber ofinteresting andimportant Poisson bracket relations. Equations (9.103) forthechange ofafunction uunder anin- ﬁnitesimal canonical transformation (onthe“active” view) isalsovalid ifuis taken asthecomponent ofavector along aﬁxed axisinordinary space. Thus, if Fisavector function ofthesystem conﬁguration, then(cf.Eq.(9.116)) BF;=doz[F;, G]. Note thatthedirection along which thecomponent istaken must beﬁxed, thatis, notaffected bythecanonical transformation. Ifthedirection itself isdetemrined intenns ofthesystem variables, thenthetransformation changes notonlythe value ofthefunction butthenature ofthefunction, justaswiththeHamiltonian. Withthisunderstanding thechange inavector Funder arotation ofthesystem about aﬁxed axisn,generated byL-n,canbewritten invector notation (cf.Eq. (9.115)) 8F=d9[F,L-n]. (9.121) Toputitinother words, Eq.(9.121) implies thattheunitvectors i,j,kthatform thebasis setforFarenotthemselves rotated byL-n. 9.7 TheAngular Momentum Poisson Bracket Relations 409 Thewords describing whatismeant byEq.(9.121) mustbechosen carefully foranother reason. What isspoken ofistherotation ofthesystem under theI.C.T., notnecessarily therotation ofthevector F.Thegenerator L-ninduces aspatial rotation ofthesystem variables, notforexample ofsome external vector suchasa magnetic ﬁeldorthevector oftheacceleration ofgravity. Under whatconditions thendoes L-ngenerate aspatial rotation ofF?Theanswer isclear-—when Fis afunction onlyofthesystem variables (q,p)anddoes notinvolve anyextemal quantities orvectors notaffected bytheI.C.T. Only under these conditions doesa spatial rotation imply acorresponding rotation ofF.Weshalldesignate suchvec- torsassystem vectors. Thechange inavector under inﬁnitesimal rotation about anaxisnhasbeengiven several times before (cf.Eq.(2.50) andEq.(4.75)): dF=nd6 xF. Forasystem vector F,thechange induced under anI.C.T. generated byL-ncan therefore bewritten as 8F=d19[F,L-n] =nd6 xF. (9.122) Equation (9.122) implies animportant Poisson bracket identity obeyed byallsys- temvectors: [F,L-n]=nxF. (9.123) NotethatinEq.(9.123)thereisnolonger anyreference toacanonical transforma- tionoreventoaspatial rotation. Itissimply astatement about thevalue ofcertain Poisson brackets foraspeciﬁc class ofvectors and,assuch, canbeveriﬁed by direct evaluation inanygiven case. Suppose, forexample, wehadasystem ofan unconstrained particle andused theCartesian coordinates asthecanonical space coordinates. Then theCartesian vector piscertainly asuitable system vector. Ifn istaken asaunitvector inthezdirection, thenbydirect evaluation wehave lPx»xpy—ypxl=-Pyr 1129-xpy—ypxl=pi. [P2,xpy_Ypxl =0- Theright-hand sidesofthese identities isclearly thesame asthecomponents of nxp,aspredicted byEq.(9.123). Ontheother hand, suppose thatinthesame problem wetriedtouseforFthe vector A=%(rxB)where B=Biisaﬁxed vector along thexaxis.Thevector Awillberecognized asthevector potential corresponding toauniform magnetic ﬁeldBinthex-direction. AsAdepends uponavector external tothesystem, we would expect itnottoﬁtthecharacteristics ofasystem vector andEq.(9.123) should notholdforit.Indeed, weseethatthePoisson brackets involved arehere 0 Chapter 9Canonical Transformations [0,xpy—)’Pxl=0. l:%ZB1-xpy —ypx]=0. [—%yB.xpy —mt]=—%Br. whereas thevector nxAhasinstead thecomponents (—%Bz, 0,0). Therelation (9.123) maybeexpressed invarious notations. Perhaps themost advantageous isaform using theLevi—Civita density toexpress thecross product (cf.Eq.(4.77’)). Theithcomponent ofEq.(9.123) forarbitrary nthencanbe written [Fi.Lj"j] =6ijk"jF1<, (9-124) which implies thesimple result [F,', Lj] =€,'jkFk. (9.125) Analternative statement ofEq.(9.125) istonotethatifl,m,narethree indices incyclic order, then [F;,Lm]=F,,, l,m,nincyclic order. (9.l25’) Another consequence ofEq.(9.123) relates tothedotproduct oftwosystem vectors: F-G.Being ascalar, such adotproduct should beinvariant under rota- tion,andindeed thePoisson bracket ofthedotproduct withL~niseasily shown tovanish: [F-G,L-n] =F-[G,L-n]+G-[F,L-n] =F~nxG+G-nxF =F-nx G+F-G xn =0. (9.126) Themagnitude ofanysystem vector therefore hasavanishing Poisson bracket withanycomponent ofL. Perhaps themostfrequent application ofthese results arises fromtaking Fto bethevector Litself. Wethenhave [L,L-n]=nxL, (9.127) ll-1',Lj]=5ijkLk, (9-123) and [L2,L-11]=0. (9.129) 9.7 TheAngular Momentum Poisson Bracket Relations 411 Anumber ofinteresting consequences follow fromEqs.(9.127) [P1L-I1]=I1XP lPi.Lj] ='5ijkPk- IfLxandLyareconstants ofthemotion, Poisson’s theorem thenstates that [Lx,Ly]=Lzisalsoaconstant ofthemotion. Thus, ifanytwocomponents of theangular momentum areconstant, thetotalangular momentum vector iscon- served. Asafurther instance, letusassume thatinaddition toL,andLybeing conserved there isaCartesian vector ofcanonical momentum pwithpzacon- stant ofthemotion. NotonlyisL2conserved butwehave twofurther constants ofthemotion: [PuLxl=Py and [P2, Ly] Z—pX1 thatis,bothLandpareconserved. Wehavehereaninstance inwhich Poisson’s theorem doesyield newconstants ofthemotion. Note, however, thatifpx,py, andL2were thegiven constants ofthemotion, thentheirPoisson brackets are lpx.Pyl=0. lpx» Lzl=_Pyr [Pyr Lz] :Px- Herenonewconstants canbeobtained fromPoisson’s theorem. Recall from thefundamental Poisson brackets, Eqs. (9.69), thatthePois- sonbracket ofanytwocanonical momenta must always bezero. But, from Eq.(9.128), L;doesnothaveavanishing Poisson bracket withanyoftheother components ofL.Thus, while wehave described Lasthetotalcanonical angular momentum byvirtue ofitsdeﬁnition asr,~xpi(summed over allparticles), notwocomponents ofLcansimultaneously becanonical variables. However, Eq.(9.129) shows thatanyoneofthecomponents ofL,anditsmagnitude L,can bechosen tobecanonical variables atthesame time.* *Ithasbeenremarked previously thatthecorrespondence between quantum andclassical mechanics is suchthatthequantum mechanical commutator goesoveressentially intotheclassical Poisson bracket ash—>0.Much oftheformal structure ofquantum mechanics appears asaclosecopyofthePoisson bracket formulation ofclassical mechanics. Alltheresults ofthissection therefore haveclose quantum analogs. Forexample, thefactthattwocomponents ofLcannot besimultaneous canonical momenta appears asthewell-known statement thatL,andLjcannot have simultaneous eigenvalues. ButL2 andanyL,canbequantized together. Indeed, most ofthese relations areknown farbetter intheir quantum formthanasclassical theorems. 9.8IChapter 9Canonical Transformations SYMMETRY GROUPS OFMECHANICAL SYSTEMS Ithasalready been pointed outthatcanonical transformations form agroup. Canonical transformations thatareanalytic functions ofcontinuous parameters form groups thatareLiegroups. ALiegroup withcontinuous parameters, 9;, hasassociated withitaﬁatvector space whose basis vectors, u,-,constitute aLie algebra satisfying thepreviously given condition onthePoisson bracket [u;,uj]=Z30,-jkuk. (9.77) k Theelements, Q(9,-), oftheassociated Liegroup arerelated totheelements of theLiealgebra by Q(9;)=exp 26114,") . (9.130) Thedeﬁnitions ofLiegroups andLiealgebras areconsidered inmore detail in Appendix B. InChapter 4oftheﬁrsttwoeditions ofthistext,anextensive discussion was given ofthePauli matrix representation oftherotational group inthree dimensions where thePauli matrices thatform thebasis, 01 0—i 10 “"=10’ ">'=i0’ “i=0-1 arebothherrnitian (thematrix isequal toitsowntranspose complex conjugate) andunitary (thetranspose complex conjugate ofthematrix istheinverse). These matrices have theproperties* [<Ii,<1j]= 2i<T1< fori,j,andkacyclic permutation ofx,y,andz.Thestructure constants arethus c,--k=2ie,-~k and0-2=l,theunit2x2matrix. TheEuler anlescanbeused J J 1 _g_ astheparameters thatgenerate thegroup elements. Forarotation 1nthey-zplane wehave, forexample, r\>¢ol\>°> I\)%t\)%0 9 cos— zs1n— Q(9) =lcosi +i0x sing = isin cos *Some physicists deﬁne aLiealgebra with theexpression [u;,uJ]=i2,,c1yk“k instead of Eq.(9.77). Thismakes thestructure constants inthefollowing discussion real.Many mathematicians omitthei=~/-1inthedeﬁnition. Thepresent textfollows thelatter convention. 9.8 Symmetry Groups ofMechanical Systems 413 Inthisformalism, vectors arerepresented by2x2matrices oftheform V VZ Vx '_ (””_ w+wy -n ’ andarotation isperformed byasimilarity transformation vt./My =Q<@)v<..y,.)Q*(@). where Qlistheadjoint, orcomplex conjugate transpose ofthematrix Q. The2x2matrices Qareunitary with determinant +1,sotheyconstitute a representation ofthespecial unitary group intwodimensions, SU(2). Theset ofunitary 2x2matrices withdeterminant either +1or-1hastwice asmany elements (both inﬁnite innumber), which fonnthefullunitary group U(2)intwo dimensions. Thisgroup of2x2rotatation matrices hasthesame properties as thegroup oftheassociated inﬁnitesimal canonical transformations (I.C.T.). Itis customary towork primarily with theI.C.T.’s astheyareeasier tohandle. The Liegroups ofI.C.T.’s whose generators aretheconstants ofthemotion ofthe system areknown asthesymmetry groups ofthesystem for,aswehave seen, such transformations leave theHamiltonian invariant. Finding thesyrmnetry groups of asystem goes alongwaytoward solving theproblem ofitsclassical motion and isevencloser toasolution ofthequantum-mechanical problem. Asystem withspherical symmetry isinvariant under rotation about anyaxis,so itcanberepresented bythegroup SU(2) asdiscussed above. Ofmore practical use isthesetoftheusual 3X3rotation matrices withdeterminant +1,which represent thespecial rotation group inthree dimensions R(3) ESO(3). Thevector Lis conserved insuchasystem inaccord withouridentiﬁcation ofthecomponents of Lasthegenerators ofspatial rotations. Forthegroup oftransformations generated byL,-,Eq.(9.128) shows thatthestructure constants areci1-"=e,-J-k,anditisthis relationship thatstamps thegroup asbeing therotation group inthree dimensions. Thus, thematrix generators M,-ofinﬁnitesimal rotations, Eqs.(4.79), havebeen seentoobey thecommutation relations, Eq.(4.80), [MnMjl=€ijkMk, (4-30) thatis,with thesame structure constants asforL,-.Thequantities L;andM,- aredifferent physically; thebrackets inEqs.(9.125) and(4.80) refer todifferent operations (although they share thesame signiﬁcant algebraic properties). But theidentity ofthestructure constants forL,-andM,-(cf.Eqs.(9.128) and(4.80)) shows thattheyhavethesamegroup structure, thatofSO(3). Forthebound Kepler problem, wehave seen (Section 3.9)thatthere exists inaddition toLanother conserved vector quantity, A,theLaplace-Runge—Lenz vector deﬁned byEq.(3.82) kA=pxL-Tl. am)r Chapter 9Canonical Transformations ThePoisson bracket relations ofthecomponents ofAwiththemselves andwith thecomponents ofLcanbeobtained inastraightforward manner. Since Aclearly qualiﬁes asasystem vector, weimmediately have thebracket relations [A;,Lj] =€,'jkAk. (9.131) ThePoisson brackets ofthecomponents ofAamong themselves cannot beob- tained byanysuchsimple stratagem, butafterafairamount oftedious manipula- tionitisfound that* 2k[A1,A2]=_(P2_ L3. (9.132) Thequantity ontheright intheparentheses willberecognized as2mH, which hastheconserved value 2mE. Ifwetherefore introduce anewconstant vector D deﬁned as A AD=4 E4 9.133 \/—2mE t/2m|E| ( ) (note thatEisnegative forbound motionl), thenthecomponents ofDsatisfy the Poisson bracket relation [D1,D21=L3- Bycyclically permuting theindices, thecomplete setofPoisson brackets follows immediately. Thus, thecomponents ofLandDtogether form aLiealgebra forthe bound Kepler problem, withstructure constants tobeobtained fromtheidentities. [L1,Ljl=Ezjkl-k. (9-123) [D1,L11=5ijkDk» (9-134) and [D,-,Dy-]=e,-J-kLk. (9.135) Anexamination ofthefundamental matrices forrotation willshow thatthe symmetry group forthebound Kepler problem istobeidentiﬁed withthegroup offour-dimensional realproper rotations, called thespecial orthogonal group of dimension 4,which isusually designated asSO(4) orR(4). Such atransfonnation preserves thevalue ofthescalar quadratic form xpxﬂ, where allthexy,arereal. Anorthogonal transformation infourdimensions has10conditions onthe16el- *Some reduction inthelength ofthederivation isobtained byidentifying pxLasasystem vector C, andﬁrstevaluating thePoisson brackets [C1,(pxL)2]and[C1,r/r]making useofthefundamental Poisson brackets andEqs.(9.125) totheutmost. 9.8 Symmetry Groups ofMechanical Systems 415 ements ofthematrix withdetemrinant zkl,soonly6areindependent. Bylooking ontheinﬁnitesimal transformation asbeing made upofasequence ofrotations in thevarious planes, wecaneasily obtain thecorresponding sixgenerators. Three ofthemarerotations inthethreedistinct x,--x1-planes andsocorrespond totheM; generators ofEqs.(4.79), except thatthere areadded zeros inthezeroth rowand column. Theremaining three generate inﬁnitesimal rotations intheX9-X1 planes. Thus, thegenerator matrix foraninﬁnitesimal rotation inthexo-x1 plane would be CDCDl—‘© CDCDCD>—' OOCDCD COCON1= (9.136) withN2andN3given incorresponding fashion. Direct matrix multiplication shows thatthese sixmatrices satisfy thecommutator (orLiebracket) relations [MnMjl=sij/<Mk. [Ni>Mj1= 61;/<Nk. [NixNj1= 5ijkMk» with structure constants c,-y-I‘ =e,-J-k. Since these arethesame asthePoisson bracket relations, Eqs.(9.128), (9.134), and(9.135), theidentiﬁcation ofthesym- metry group ofthebound Kepler problem withR(4)isthusproven. Note thatfortheKepler problem withpositive energy (thatis,scattering) Ais stillaconstant ofthemotion,* buttheappropriate reduced realvector, instead of D,isCdeﬁned as c=_2A_E, (9.137)‘\/ "1 andthePoisson bracket relations forLandCarenow [Li»Lj1= 5ijkLk, [C,-,Ly-]=e,~y-kCk. (9.138) [C1,Cjl=_5ijkLk- These structure constants arethesame asfortherestricted Lorentz group, which must therefore bethesymmetry group forthepositive energy Kepler problem—in nonrelativistic mechanics. Wemust notreadanykinship ofphysical ideas intothis happenstance. TheKepler problem doesnotcontain inittheseedofthebasic con- ceptions ofspecial relativity; itispurely aproblem ofnonrelativistic Newtonian mechanics. That thesymmetry group mayinvolve aspace ofhigher dimension thanordinary space iscomrected with thefactthatthesymmetry weseek here *The arguments ofSection 3.9areindependent ofthesignofeither Eortheforce constant k. Chapter 9Canonical Transformations isoneinthesix-dimensional phase space. Thesymmetry group consists ofthe canonical transformations inthisspace thatleave theHamiltonian unchanged. It should notbesurprising therefore thatthegroup canbeinterpreted interms of transformations ofspaces ofmore thanthree dimensions. Thetwo-dimensional isotropic harmonic oscillator isanother mechanical sys- temforwhich asymmetry group iseasily identiﬁed. InCartesian coordinates, the Hamiltonian forthissystem maybewritten as 1 1 H=-003+mlwzﬁ) +—0>%+mzwzyzr. (9.139)2m 2m 3 Asitdoesn’t depend ontimeexplicitly, theHamiltonian isconstant andisequal tothetotalenergy ofthesystem. Thezaxisisanaxisofsymmetry forthesystem, andhence theangular momentum along thataxis(which isinfactthetotalangular momentum) isalsoaconstant ofmotion: L=xpy—ypx. (9.140) Further constants ofthemotion exist forthisproblem thatcanbewritten ascom- ponents ofasymmetrical two-dimensional tensor Adeﬁned as A,-y-=#(PiPj +m2w2x,-xj). (9.141) Ofthethree distinct elements ofthetensor, thediagonal terms maybeidentiﬁed astheenergies associated with theseparate one-dimensional motions along the xandyaxes, respectively. Physically, asthere isnocoupling between thetwo motions, thetwoenergies must separately beconstant. Alittle more formally, it isobvious from thewayinwhich Hhasbeen written inEq.(9.139) thatA111 andA22eachhave avanishing Poisson bracket withH.Theoff-diagonal element ofA, l A12=A21=Err.» +m2w’xy>. (9.142) isalittlemore difﬁcult torecognize. Thatitisaconstant ofthemotion mayeasily beseenbyevaluating thePoisson bracket withH.Inrelation totheseparate xand ymotions, A11andA22arerelated totheamplitudes oftheoscillations, whereas A12isdetemrined bythephase difference between thetwovibrations. Thus, the solutions forthemotion canbewritten as 2Ax=J4‘; sin(wt+01),"'10) /2A .y=is srn(wt +62),ma) 9.8 Symmetry Groups ofMechanical Systems 417 anditthenfollows from Eq.(9.142) that A12 =\/A1114}; COS(92 —91). (9.143) Thetrace oftheAtensor isthetotalenergy oftheharmonic oscillator. Outof theelements ofthematrix, wecanform twoother distinct constants ofthemotion, which itisconvenient towrite intheform AA 1st=£2-‘ =—<p.py +m’w’xy>. (9.144)2w 2mm A-A 1s2=% =Q[pi-pi+m2a)2(y2 -13)]. (9.145) Tothese wemayaddathirdconstant ofthemotion fromEq.(9.140): L1S3=5=5(x11y —)’Px)- (9-146) Thequantities S;plusthetotal energy Hform fouralgebraic constants ofthe motion notinvolving time explicitly. Itisclear thatnotallofthem canbeinde- pendent, because inasystem oftwodegrees offreedom there canatmost beonly three suchconstants. Weknow thattheorbit fortheisotropic harmonic oscillator isanellipse andthreeconstants ofthemotion areneeded todescribe theparam- etersoftheorbitintheplane—say, thesemimajor axis,theeccentricity, andthe orientation oftheellipse. Thefourth constant ofmotion relates tothepassage of theparticle through aspeciﬁc point atagiven timeandwould therefore beexplic- itlytimedependent. Hence, there must exist asingle relation connecting S,-and H.Bydirect evaluation itiseasytoshow that* 2 2 2 H2S1 +S2 +S3 =W. Bystraight forward manipulation ofthePoisson brackets, wecanverify that thethree S;quantities satisfy therelations [S,‘,Sj]=e,-J-kSk. (9.148) These arethesame relations asforthethree-dimensional angular momentum vec- tor,orforthegenerators ofrotation inathree-dimensional space. Thegroup of transformations generated bySimaytherefore beidentiﬁed withR(3) orSO(3). Actually, there issome ambiguity intheidentiﬁcation. *Anequivalent fomi ofthecondition Eq.(9.147) isthatthedeterminant ofAisLzwz/4. Itwillbe recalled thatsimilarly inthecaseoftheKepler problem, thecomponents ofthenewvector constant ofmotion Awere notallindependent oftheother constants ofthemotion. There exist indeed two relations linking A,L,andH,Eqs.(3.83) and(3.87). Chapter 9Canonical Transformations There isahomomorphism (inthiscase,a2to1mapping) between theorthog- onalunimodular group SO(3) alsocalled therotation group R(3) inthree dimen- sions andtheunitary unimodular group* SU(2) intwodimensions. Ittums out thatSU(2) isheremore appropriate. Toglimpse atthecircumstances justifying thischoice, notethatEq.(9.147) suggests there isathree-dimensional space, each point ofwhich corresponds toaparticular setoforbital parameters. Foragiven system energy, Eq.(9.147) saystheorbit “points” inthisspace lieonasphere. Theconstants S,-generate three-dimensional rotations onthissphere; thatis,they change oneorbit intoanother orbit having thesame energy. Itmaybeshown that S1generates atransformation thatchanges theeccentricity oftheorbit andthat foranygiven ﬁnaleccentricity wecanﬁndtwotransformations leading toit.Itis thisdouble-valued quality ofthetransformation thatindicates SU(2) rather than SO(3) isthecorrect symmetry group forthetwo-dimensional harmonic oscilla- £01‘. Forhigher dimensions, thestructure constants oftheLiealgebras oftheSO(n) rotation groups andtheSU(n) unitary groups arenolonger identical, andaclear- cutseparation between thetwocanbemade. Forthethree-dimensional isotropic harmonic oscillator, there isagain atensor constant ofthemotion deﬁned by Eq.(9.141), except thattheindices nowrunfrom 1to3.Thedistinct components ofthistensor, together withthecomponents ofLnowsatisfy Poisson bracket rela- tions withtherather complicated structure constants thatbelong toSU(3). Indeed, itispossible toshow thatforthen-dimensional isotropic harmonic oscillator the symmetry group isSU(n). Ithaspreviously been pointed outinSection 3.9thatthere exists aconnection between theexistence ofadditional algebraic constants ofthemotion—and there- foreofhigher-symmetry groups—and degeneracy inthemotions ofthesystem. InthecaseoftheKepler andisotropic harmonic oscillator problems, theaddi- tional constants ofthemotion arerelated toparameters oftheorbit. Unless the orbit isclosed, thatis,themotion isconﬁned toasingle curve, wecanhardly talkofsuchorbital parameters. Only when thevarious components ofthemo- tionhave commensurate periods willtheorbit beclosed. Theclassic example isthetwo-dimensional anisotropic oscillator. When thefrequencies inthexand ydirections arerational fractions ofeach other, theparticle traverses aclosed Lissajous ﬁgure. Butifthefrequencies areincommensurate, themotion ofthe particle isspace-ﬁlling orergodic, eventually coming asclose asdesired toany speciﬁc point intherectangle deﬁned bytheenergies ofmotion inthetwodirec- tions (ergotic hypothesis). Attempts atﬁnding complicated (andperhaps complex) symmetry groups forincommensurate systems, applicable toallproblems ofthe same number ofdegrees offreedom, havenotyetproved fruitful. Weshallhave occasion inSection 13.7toconsider further therelation between symmetry and invariance when wediscuss Noether’s theorem which gives aformal proof ofthe relation between invariance andconserved quantities. *Amatrix isunitary ifitsinverse isitstranspose complex conjugate, andaunimodular matrix isone whose determinant is+1. 9.9I9.9 Liouville’s Theorem 419 |.lOUVlLl.E'S THEOREM Asaﬁnal application ofthePoisson bracket formalism, weshall brieﬂy discuss afundamental theorem ofstatistical mechanics known asLiouville’s theorem. While theexact motion ofanysystem iscompletely determined inclassical me- chanics bytheinitial conditions. itisoften impracticable tocalculate anexact solution forcomplex systems. Itwould beobviously hopeless, forexample, to calculate completely themotion ofsome 1023molecules inavolume ofgas.In addition, theinitial conditions areoften onlyincompletely known. Wemaybe abletostatethatattimetoagiven mass ofgashasacertain energy, butwecan- notdetemiine theinitial coordinates andvelocities ofeachmolecule. Statistical mechanics therefore makes noattempt toobtain acomplete solution forsystems containing many particles. Itsaim,instead, istomake predictions about certain average properties byexamining themotion ofalarge number ofidentical sys- tems. Thevalues ofthedesired quantities arethencomputed byforming averages overallthesystems intheensemble. Allthemembers oftheensemble areaslike theactual systems asourimperfect knowledge permits, buttheymayhave anyof theinitial conditions thatareconsistent withthisincomplete information. Since each system isrepresented byasingle point inphase space, theensemble ofsys- tems corresponds toaswarm ofpoints inphase space. Liouville’s theorem states thatthedensity ofsystems intheneighborhood ofsome given system inphase space remains constant intime. Thedensity, D.asdeﬁned above canvarywithtimethrough twoseparate mechanisms. Since itisthedensity intheneighborhood ofagiven system point, there willbeanimplicit dependence asthecoordinates ofthesystem (q,-,pi)vary withtime, andthesystem point wanders through phase space. There mayalsobe anexplicit dependence upon time. Thedensity maystillvarywithtimeevenwhen evaluated ataﬁxed point inphase space. ByEq.(9.94), thetotaltimederivative ofD,duetobothtypes ofvariation withtime, canbewritten as dD 3D Tr—l:D1H]+ W, (9-149) where thePoisson bracket arises from theimplicit dependence, andthelasttenn from theexplicit dependence. Theensemble ofsystem points moving through phase space behaves much like aﬂuidinamultidimensional space, andtherearenumerous similarities between ourdiscussion oftheensemble andthewell-known notions ofﬂuiddynamics. In Eq.(9.149), thetotalderivative isaderivative ofthedensity aswefollow themo- tionofaparticular bitoftheensemble “ﬂuid” intime. Itissometimes referred to asthematerial orhydrodynamic derivative. Ontheother hand, thepartial deriva- tiveisatﬁxed (q,p);itisasifwestation ourselves ataparticular spotinphase space andmeasure thetime variation ofthedensity astheensemble ofsystem points ﬂows byus.These twoderivatives correspond totwoviewpoints frequently usedinconsidering ﬂuidﬂow. Thepartial derivative ataﬁxed point inphase space isinlinewiththeEulerian viewpoint thatlooks onthecoordinates solely asiden- Chapter 9Canonical Transformations tifying apoint inspace. Thetotalderivative ﬁtsinwiththeLagrangian picture inwhich individual particles arefollowed intime; thecoordinates ineffect rather identify aparticle thanapoint inspace. Basically, ourconsideration ofphase space hasbeen more liketheLagrangian viewpoint; thecollection ofquantities (q,p)identiﬁes asystem anditschanging conﬁguration withtime. Consider aninﬁnitesimal volume inphase space surrounding agiven system point, withtheboundary ofthevolume formed bysome surface ofneighboring system points atthetimet=O.Note thatthesurface ofthevolume isone- dimension lessthanthevolume. Inthecourse oftime, thesystem points deﬁning thevolume move about inphase space, andthevolume contained bythem will takeondifferent shapes astimeprogresses. Thedashed curve inFig.9.4indicates theevolution oftheinﬁnitesimal volume withtime. Itisclear thatthenumber ofsystems within thevolume remains constant, forasystem initially inside can never getout.Ifsome system point were tocross theborder, itwould occupy at some timethesame position inphase space asoneofthesystem points deﬁning theboundary surface. Since thesubsequent motion ofasystem isuniquely deter- mined byitslocation inphase space ataparticular time, thetwosystems would travel together from there on.Hence, thesystem cannever leave thevolume. By thesame token, asystem initially outside cannever enter thevolume. Ithasbeenshown thatontheactive picture ofacanonical transformation, the motion ofasystem point intimeissimply theevolution ofacanonical transfor- mation generated bytheHamiltonian. Thecanonical variables (q,p)attimelg,as shown inFig.9.4,arerelated tothevariables attimet1byaparticular canonical transformation. Thechange intheinﬁnitesimal volume element about thesystem point overthetimeinterval isgiven bythesame canonical transformation. Now, Poincaré’s integral invariant, Eq.(9.86), saysthatavolume element inphase space isinvariant under acanonical transformation. Therefore, thesizeofthevolume element about thesystem point cannot varywithtime. Thus, boththenumber ofsystems intheinﬁnitesimal region, dN,andthe volume, dV.areconstants, andconsequently thedensity P ,-~I // \\ ,--'q(r2).p(r2) lI /( / \\ /.._. \:\I FIGURE 9.4Motion ofavolume intwo-dimensional phase space.‘Ir Derivations 421 dND=— dV must alsobeconstant intime, thatis, dD_=Q, dz which proves Liouville’s theorem. Analtemative statement ofthetheorem follows fromEq.(9.149) as 3DW=—[D. H]. (9.150) When theensemble ofsystems isinstatistical equilibrium, thenumber ofsys- tems inagiven state must beconstant intime, which istosaythatthedensity ofsystem points atagiven spotinphase space does notchange withtime. The variation ofDwithtimeataﬁxedpoint corresponds tothepartial derivative with respect tot,which therefore must vanish instatistical equilibrium. ByEq.(9.150), itfollows thattheequilibrium condition canbeexpressed as [D.H]=O. Wecanensure equilibrium therefore bychoosing thedensity Dtobeafunction ofthose constants ofthemotion ofthesystem notinvolving timeexplicitly, for thenthePoisson bracket with Hmust vanish. Thus, forconservative systems D canbeanyfunction oftheenergy, andtheequilibrium condition isautomatically satisﬁed. Thecharacteristics oftheensemble willbedetermined bythechoice of function forD.Asanexample, onewell-known ensemble, themicrocanonical ensemble, occurs ifDisconstant forsystems having agiven narrow energy range andzerooutside therange. Theconsiderations havebeen presented heretoillustrate theusefulness ofthe Poisson bracket formulation inclassical statistical mechanics. Further discussion ofthese points would carry usfaroutside ourﬁeld. DERIVATIONS 1.Oneoftheattempts atcombining thetwosetsofHamilton’s equations intoonetriesto takeqandpasforming acomplex quantity. Show directly fromHamilton’s equations ofm0ti0n thatforasystem ofonedegree offreedom thetransformation Q=q+iP. P=Q* isnotcanonical iftheHamiltonian isleftunaltered. Canyouﬁndanother setofcoordi- nates Q’,P’thatarerelated toQ,Pbyachange ofscaleonly,andthatarecanonical? Chapter 9Canonical Transformations 2Show thatthetransformation forasystem ofonedegree offreedom, Q=qcosot —psina, P=qsina +pCOSol, satisﬁes thesymplectic condition foranyvalue oftheparameter or.Findagenerating function forthetransfonnation. What isthephysical signiﬁcance ofthetransformation fora=0?Foror=rt/2?Does yourgenerating function work forbothofthese cases. InSection 8.4some oftheproblems oftreating timeasoneofthecanonical variables arediscussed. Ifweareabletosidestep these difﬁculties, show thattheequations of transformation inwhich tisconsidered acanonical variable reduce toEqs.(9.14) if infactthetransformation does notaffect thetimescale. Show directly thatthetransformation 1.Q=log(Zs1np), P=qcotp iscanonical. Show directly thatforasystem ofonedegree offreedom thetransformation 2 2‘111 all P= -_, P=_ 1Z Qarctan P 2<+a2q2) iscanonical, where orisanarbitrary constant ofsuitable dimensions. Thetransformation equations between twosetsofcoordinates are Q=l0s(1+q1/260511). P=2(l+q1/2 cosp)q1/2 sinp. (a)Show directly fromthese transformation equations thatQ,Parecanonical vari- ablesifqandpare. (b)Show thatthefunction thatgenerates thisuansfonnation is F3=-(eQ—1)2tanp. (a)Ifeachofthefourtypes ofgenerating functions existforagiven canonical trans- formation, usetheLegendre transformation toderive relations between them. (b)Findagenerating function oftheF4typefortheidentify transformation andof theF3typefortheexchange transfonnation. (c)Foranorthogonal point transformation ofqinasystem ofndegrees offreedom, show thatthenewmomenta arelikewise given bytheorthogonal transfonnation ofann-dimensional vector whose components aretheoldmomenta plusagradi- entinconﬁguration space. Prove directly thatthetransformation Derivations 423 9 10 11 12. 13. 14.Q1=q1» P1=P1-2112- Q2=P2.P2=—2qi—112 iscanonical andﬁndagenerating function. (a)Forasingle particle show directly (that is,bydirect evaluation ofthePoisson brackets), thatifuisascalar function onlyofr2,p2,andr-p,then [u,L]=0. (b)Similarly show directly thatifFisavector function, F=ur+vp+w(r xp), where u,v,andwarescalar functions ofthesame typeasinpart(a),then [Fr-.Ljl=61'jkFk- Findunder what conditions Q=Q.P=ﬁx’.x where ozand)3areconstants, represents acanonical transformation forasystem of onedegree offreedom, andobtain asuitable generating function. Apply thetransfor- mation tothesolution ofthelinear harmonic oscillator. Determine whether thetransformation Q1=mm. Pw=ﬂ;13+Lqz-<11 Q2=q1+q2. P2= —(q2+q1)q2—q1 iscanonical. Show thatthedirect conditions foracanonical condition aregiven immediately by thesymplectic condition expressed intheform |M=M“t Thesetofrestricted canonical transformations hasagroup-property. Verify thisstate- ment once using theinvariance ofHamilton’s principle under canonical transforma- tion(cf.Eq.(9.11)), andagain using thesymplectic condition. Prove thatthetransformation Q1=11%. Q2=q256¢P2. P1C05P2—242 -P=——————= P= - 1 ZqlCOSP2 2S111P2241 iscanonical, byanymethod youchoose. Findasuitable generating function thatwill leadtothistransformation. 424 Chapter 9Canonical Transformations 15.(a)Using thefundamental Poisson brackets ﬁndthevalues ofozandBforwhich the equations Q=q°‘cosﬂp, P=q“sinﬁp represent acanonical transformation. (b)Forwhatvalues ofozand)3dothese equations represent anextended canonical transfomiation‘? Findagenerating function oftheF3form forthetransformation. (c)Onthebasisofpart(b),canthetransformation equations bemodiﬁed sothatthey describe acanonical transformation forallvalues of)3? 16.Forasymmetric rigidbody, obtain formulas forevaluating thePoisson brackets nJo¢M.wJoaw where 6,¢,and11/aretheEuler angles, andfisanyarbitrary function oftheEuler angles. 17.Show thattheJacobi identity issatisﬁed ifthePoisson bracket signstands forthe commutator oftwosquare matrices: [A,B]=AB—BA. Show alsothatforthesame representation ofthePoisson bracket that [A,BC]=[A,BIC+B[A.C]- 18.Prove Eq.(9.83) using thesymplectic matrix notation fortheLagrange andPoisson brackets. 19.Verify theanalog oftheJacobi identity forLagrange brackets, ill".vl alvtwl alw, H}3 i -i =08w + Ziu + 3v ’ where u,v,andwarethree functions interms ofwhich the(q,p)setcanbespeciﬁed. 20.(a)Verify thatthecomponents ofthetwo-dimensional matrix A,deﬁned byEq. (9.141), areconstants ofthemotion forthetwo-dimensional isotropic harmonic oscillator problem. (b)Verify thatthequantities S,-,i=l,2,3,deﬁned byEqs.(9.144), (9.145), (9.146), havetheproperties stated inEqs.(9.147) and(9.148). EXERCISES 21.(a)Foraone-dimensional system withtheHamiltonian 21H=L-_.22q2 Exercises 425 showthatthereisaconstant ofthemotion PqD=———H. 2 1‘ (b)Asageneralization ofpart(a),formotion inaplane withtheHamiltonian H=lpl"—ar"", where pisthevector ofthemomenta conjugate totheCartesian coordinates, show thatthereisaconstant ofthemotion D=p—'r-H1.fl (c)Thetransformation Q=Jtq,p=APisobviously canonical. However, thesame transformation withttimedilatation, Q=Jtq,p=AP,t’=Jtzt,isnot.Show that,however, theequations ofmotion forqandpfortheHamiltonian inpart(a) areinvariant under thistransformation. Theconstant ofthemotion Dissaidtobe associated withthisinvariance. Forthepoint transformation inasystem oftwodegrees offreedom, Q1=11%. Q2=<11+42, ﬁndthemost general transformation equations forP1andP2consistent withtheover- alltransformation being canonical. Show thatwithaparticular choice forP1andP2 theHamiltonian _ 2 H= +P2+(q1+q2)2 canbetransformed tooneinwhich both Q1andQ2areignorable. Bythismeans solve theproblem andobtain expressions forq1,q2,pl,andp2asfunctions oftime andtheirinitial values. Byanymethod youchoose, show thatthefollowing transformation iscanonical: 1 .X=-(w/2P1$1HQ1+P2), Px='0£(\/2P1c°SQl_Q2)~or 2 1 oz _ >’=;(~/2P1¢0SQ1+Q2). Py=—§(\/2P1$1I1Q1-P2). where aissome ﬁxed parameter. Apply thistransformation totheproblem ofaparticle ofcharge qmoving inaplane thatisperpendicular toaconstant magnetic ﬁeldB.Express theHamiltonian forthis problem inthe(Q;,P,-)coordinates letting theparameter oztaketheform B u2=L.c From thisHamiltonian, obtain themotion oftheparticle asafunction oftime. 426 Chapter 9Canonical Transformations 24.(a)Show thatthetransformation _ p—iaq= ,P=iQ ‘D+mq 2ia iscanonical andﬁndagenerating function. (b)Usethetransfonnation tosolvethelinear harmonic oscillator problem. 25.(a)TheHamiltonian forasystem hastheform 1l 24H=-— . 2(112+Pq) Findtheequation ofmotion forq. (b)Findacanonical transformation thatreduces Htotheform ofaharmonic oscilla- tor.Show thatthesolution forthetransformed variables issuchthattheequation ofmotion found inpart(a)issatisﬁed. 26.Asystem ofnparticles moves inaplane under theinﬂuence ofinteraction forces derived frompotential tenns depending onlyuponthescalar distances between parti- cles. (a)Using plane polar coordinates foreachparticle (relative toacommon origin), identify theformoftheHamiltonian forthesystem. (b)Findagenerating function forthecanonical transformation thatcorresponds toa transformation tocoordinates rotating intheplane counterclockwise withauni- formangular ratew(thesameforallparticles). What arethetransformation equa- tionsforthemomenta? (c)What isthenewHamiltonian‘? What physical signiﬁcance canyougivetothe difference between theoldandthenewHamiltonians? 27.(a)Intheproblem ofsmall oscillations about steady motion, show thatatthepoint ofsteady motion alltheHamiltonian variables 1|areconstant. Ifthevalues for steady motion arenosothat1|=rm+§,show thattothelowest nonvanishing approximation theeffective Hamiltonian forsmall oscillation canbeexpressed as How.o=ésss. where Sisasquare matrix withcomponents thatarefunctions of119only. (b)Assuming allfrequencies ofsmall oscillation aredistinct, letMbeasquare 2nx 2nmatrix formed bythecomponents ofapossible setofeigenvectors (forboth positive andnegative frequencies). Only thedirections oftheeigenvectors are ﬁxed, nottheirmagnitudes. Show thatitispossible toapply conditions tothe eigenvectors (ineffect ﬁxing their magnitudes) thatmake MtheJacobian matrix ofacanonical transformation. (c)Show thatthecanonical transformation sofound transforms theeffective Hamil- tonian tothefonn H=iw]-qjpj, where wjisthemagnitude ofthenormal frequencies. What aretheequations of motion inthissetofcanonical coordinates? Exercises 427 (d)Finally, show that .2 . F_ .P.+l I.2 2-H’ 2% 4%” leads toacanonical transformation thatdecomposes HintotheHamiltonians for asetofuncoupled linear harmonic oscillators thatoscillate inthenormal modes. Acharged particle moves inspace withaconstant magnetic ﬁeldBsuchthatthe vector potential, A,is A=%@xﬂ (a)IfvjaretheCartesian components ofthevelocity oftheparticle, evaluate the Poisson brackets [12,-,vj], i#j= 1,2,3. (b)Ifp,-isthecanonical momentum conjugate toxi.alsoevaluate thePoisson brack- GIS [Xnvj], [Pr.vjl, [Xi.15jl, lPi,Pjl- Thesemimajor axisaoftheelliptical Kepler orbit andtheeccentricity earefunctions ofﬁrstintegrals ofthemotion, andtherefore ofthecanonical variables. Similarly, the mean anomaly ¢E(1)(I—T)=l/i—€SlIl1// isafunction ofr,6,andtheconjugate momenta. Here Tisthetimeofperiapsis passage andisaconstant ofthemotion. Evaluate thePoisson brackets thatcanbe formed ofa,e,¢,w,andT.There areinfactonlyninenonvanishing distinct Poisson brackets outofthese quantities. (a)Prove thatthePoisson bracket oftwoconstants ofthemotion isitselfaconstant ofthemotion evenwhen theconstants depend upontimeexplicitly. (b)Show thatiftheHamiltonian andaquantity Fareconstants ofthemotion, then thenthpartial derivative ofFwithrespect tormustalsobeaconstant ofthe motion. (c)Asanillustration ofthisresult, consider theuniform motion ofaﬁ-eeparticle of mass m.TheHamiltonian iscertainly conserved, andthere exists aconstant ofthe motion F=x—Lt.m Show bydirect computation thatthepartial derivative ofFwith t,which isa constant ofthemotion, agrees with[H,F]. 31Show bytheuseofPoisson brackets thatforaone-dimensional harmonic oscillator there isaconstant ofthemotion udeﬁned as _ _ ku(q,p,t)=ln(p+rmwq)—rwt, a)=—. m 4 Chapter 9Canonical Transformations What isthephysical signiﬁcance ofthisconstant ofthemotion? Asystem oftwodegrees offreedom isdescribed bytheHamiltonian H=q1P1— <I2P2—aqf+bq§- Show that P-411F1=-lil andF2=q1q2Q2 areconstants ofthemotion. Arethereanyother independent algebraic constants of themotion‘? Cananybeconstructed fromJacobi’s identity‘? Setupthemagnetic monopole described inExercise 28(Chapter 3)inHamiltonian fonnulation (youmaywanttousespherical polarcoordinates). Bymeans ofthePois- sonbracket formulation, show thatthequantity Ddeﬁned inthatexercise iscon- served. Obtain themotion intimeofalinear harmonic oscillator bymeans oftheformal solution forthePoisson bracket version oftheequation ofmotion asderived from Eq.(9.116). Assume thatattimet=0theinitial values arex0andpg. Aparticle moves inonedimension under apotential mk x Findxasafunction oftime,byusing thesymbolic solution ofthePoisson bracket formfortheequation ofmotion forthequantity y=x2.Initial conditions arethatat t=0,x=x0,andv=0. (a)Using thetheorem conceming Poisson brackets ofvector functions andcompo- nents oftheangular momentum, show thatifFandGaretwovector functions of thecoordinates andmomenta only,then [F-L,G-L]=L-(GXF)+L;Lj[F},Gj]. (b)LetLbethetotal angular momentum ofarigid body with onepoint ﬁxed and letLL,beitscomponent along asetofCartesian axesﬁxed intherigid body. By means ofpart(a)ﬁndageneral expansion for [Lﬂa LU]! I1’: v=11 293' (Hint: Choose forFandGunitvectors along theuandvaxes.) (c)From thePoisson bracket equations ofmotion forL“derive Eu1er’s equations of motion forarigidbody. Setuptheproblem ofthespherical pendulum intheHamiltonian formulation, using spherical polar coordinates fortheq,-.Evaluate directly interms ofthese canonical variables thefollowing Poisson brackets: Exercises 429 38 39 40. 41[LXQ ll-y» Ll]! [LZY LI]! showing thattheyhavethevalues predicted byEq.(9.128). Why isitthatpgandp¢ canbeused ascanonical momenta, although theyareperpendicular components of theangular momentum‘? InSection 9.7,itisshown thatifanytwocomponents oftheangular momentum are conserved, thenthetotalangular momentum isconserved. Iftwoofthecomponents areidentically zero, thethird must beconserved. From thisitwould appear tofollow thatinanymotion conﬁned toaplane, sothatthecomponents oftheangular mo- mentum intheplane arezero, thetotalangular momentum isconstant. There appear tobeanumber ofobvious contradictions tothisprediction; forexample, theangular momentum ofanoscillating spring inawatch, ortheangular momentum ofaplane diskrolling down aninclined plane allinthesame vertical plane. Discuss theforce of theseobjections andwhether thestatement ofthetheorem requires anyrestrictions. (a)Show from thePoisson bracket condition forconserved quantities thatthe Laplace—Runge—Lenz vector A, kA=:pXL-E , r isaconstant ofthemotion fortheKepler problem. (b)Verify thePoisson bracket relations forthecomponents ofAasgiven by Eq.(9.131). Consider asystem thatconsists ofarigidbody inthree-space withonepoint ﬁxed. Using cylindrical coordinates ﬁndthecanonical transformation corresponding tonew axesrotating about thez-axis withanarbitrary time-dependent angular velocity. Ver- ifythatyourproposed solution iscanonical. Westartwithatimeindependent Hamiltonian H0(q,p)andimpose anexternal oscil- lating ﬁeldmaking theHamiltonian H=H,,(q, p)——:-rsinwt where sandwaregiven constants. (a)How arethecanonical equations modiﬁed? (b)Findacanonical transfonnation thatrestores thecanonical formoftheequations ofmotion anddetermine the“new” Hamiltonian. (c)Giveapossible physical interpretation oftheimposed ﬁeld. CHAPTER 410.1 IHamilton-Jacobi Theory and Action-Angle Variables Ithasalready beenmentioned thatcanonical transformations maybeusedtopro- videageneral procedure forsolving mechanical problems. Twomethods have been suggested. IftheHamiltonian isconserved, thenasolution could beobtained bytransfomiing tonewcanonical coordinates thatareallcyclic, thereby provid- ingnewequations ofmotion withtrivial solutions. Analtemative technique isto seekacanonical transfomration from thecoordinates andmomenta, (q,p),atthe time t,toanewsetofconstant quantities, which maybethe2ninitial values, (qr),pg),att=0.With suchatransformation, theequations oftransformation relating theoldandnewcanonical variables areexactly thedesired solution ofthe mechanical problem: q=q(qo.P0,I), P=P(qo,Po.I)- Theygivethecoordinates andmomenta asafunction oftheirinitial values andthe time. Thislastprocedure isthemore general one,especially asitisapplicable, in principle atleast, evenwhen theHamiltonian involves thetime. Weshalltherefore begin ourdiscussion byconsidering howsuchatransfonnation maybefound. THE HAMILTON-IACOBI EQUATION FOR HAMlLTON'S PRINCIPAL FUNCTION Wecanautomatically ensure thatthenewvariables areconstant intimebyrequir- ingthatthetransformed Hamiltonian, K,shall beidentically zero, forthenthe equations ofmotion are 3K . 5?l=Qi—0, BK . Aswehave seen, Kmust berelated totheoldHamiltonian andtothegenerating function bytheequation 8FK=H+—,82 10.1 TheHamilton-Jacobi Equation forHamilton’s Principal Function 431 andhence willbezeroifFsatisﬁes theequation 8FH(q, p,t)+E=O. (10.2) Itisconvenient totakeFasafunction oftheoldcoordinates q,-,thenewconstant momenta P,-,andthetime; inthenotation oftheprevious chapter wewould desig- natethegenerating function asF2(q,P,t).Towrite theHamiltonian inEq.(10.2) asafunction ofthesame variables, usemaybemade oftheequations oftransfor- mation (cf.Eq.(9.17a)), ._fEZ pl_3%‘, sothatEq.(10.2) becomes 8F 8F BFHqhnqqm-i,Hq-Qt +-_3=0. nanq 8q 8t 8i n Equation (10.3), known astheHamilton-Jacobi equation, constitutes apartial differential equation in(n+1)variables, q1,...,qn;t,forthedesired generating function. Itiscustomary todenote thesolution F2ofEq.(10.3) bySandtocall itHamilton ’sprincipal function. Ofcourse, theintegration ofEq.(10.3) onlyprovides thedependence onthe oldcoordinates andtime; itwould notappear totellhowthenewmomenta are contained inS.Indeed, thenewmomenta have notyetbeen speciﬁed except that weknow theymustbeconstants. However, thenature ofthesolution indicates howthenewP,-’saretobeselected. Mathematically Eq.(10.3) hastheformofaﬁrst-order partial differential equa- tioninn+1variables. Suppose thereexists asolution toEq.(10.3) oftheform F2ES=S(q1,...,q,,; a1,...,a,,+1;t), (10.4) where thequantities 011,...,oz,,+1 aren+1independent constants ofintegration. Such solutions areknown ascomplete solutions oftheﬁrst-order partial differen- tialequation.* Oneoftheconstants ofintegration, however, isinfactirrelevant to thesolution, foritwillbenoted thatSitself does notappear inEq.(10.3); only itspartial derivatives withrespect toqortareinvolved. Hence, ifSissome so- lution ofthedifferential equation, thenS+oz,where ozisanyconstant, mustalso beasolution. Oneofthen+1constants ofintegration inEq.(10.4) mustthere- foreappear onlyasanadditive constant tacked ontoS.Butbythesame token, anadditive constant hasnoimportance inagenerating function, since onlypar- tialderivatives ofthegenerating function occur inthetransfonnation equations. *Equation (10.4) isnottheonly typeofsolution possible forEq.(10.3). Themost general form ofthesolution involves oneormore arbitrary functions rather thanarbitrary constants. Noristhere necessarily aunique solution oftheform(10.4). There maybeseveral complete solutions forthegiven equation. Butallthatisimportant forthesubsequent argument isthatthere existacomplete solution. Chapter l0Hamilton-Jacobi Theory andAction-Angle Variables Hence, forourpurposes acomplete solution toEq.(10.3) canbewritten inthe form »5'=$(q1,.--,qn; <11,-...<1n;r), (10-5) where noneofthenindependent constants issolely additive. Inthismathematical garb, Stallies exactly with thedesired form foranF2typeofgenerating func- tion,forEq.(10.5) presents Sasafunction ofNcoordinates, thetimet,andn independent quantities oz,-.Wearetherefore atliberty totakethenconstants of integration tobethenew(constant) momenta: P,-=01;. (10.6) Such achoice doesnotcontradict theoriginal assertion thatthenewmomenta areconnected withtheinitial values ofqandpattimeto.Thentransformation equations (9.17a) cannowbewritten as p,=éﬁqit), (10.7)aqi where q,orstand forthecomplete setofquantities. Atthetimeto,these constitute nequations relating thenoz’swiththeinitial qandpvalues, thusenabling usto evaluate theconstants ofintegration interms ofthespeciﬁc initial conditions of theproblem. Theother halfoftheequations oftransfonnation, which provide the newconstant coordinates, appear as Qt=a»= (10.8)at Theconstant ,6’scanbesimilarly obtained from theinitial conditions, simply by calculating thevalue oftheright sideofEq.(10.8) att=towiththeknown initial values ofq,-.Equations (10.8) canthenbe“turned inside out”tofurnish q1-in terms ofct,)9,andtr qj=q;(<1.B.r). (10-9) which solves theproblem ofgiving thecoordinates asfunctions oftimeandthe initial conditions.* After thedifferentiation inEqs.(10.7) hasbeenperfonned, *Asamathematical point, itmaybequestioned whether theprocess of“turning inside out”isfeasible forEqs.(10.7) and(10.8), thatis,whether theycanbesolved fora,-andq,,respectively. Thequestion hinges onwhether theequations ineach setareindependent, forotherwise theyareobviously not sufﬁcient todetermine thenindependent quantities 01,-orq,asthecasemaybe.Tosimplify the notation, letS0,symbolize members ofthesetofpartial derivatives ofSwithrespect to01,-,sothat Eq.(10.8) isrepresented byﬂ=Sq.That thederivatives SO,in(10.8) form independent functions oftheq’sfollows directly from thenature ofacomplete solution totheHamilton-Jacobi equation; indeed thisiswhat wemean bysaying thenconstants ofintegration areindependent. Consequently, theJacobian ofSo,withrespect toq,-cannot vanish. Since theorder ofdifferentiation isimmaterial, thisisequivalent tosaying thattheJacobian ofSqwithrespect toor;cannot vanish, which proves the independence ofEqs.(10.7). 10.1 TheHamilton—lacobi Equation forHamilton's Principal Function 433 Eqs.(10.9) maybesubstituted fortheq’s,thusgiving themomenta piasfunctions ofthea,)5,andt: Pi=Pz(¢v,19.I)- (10-10) Equations (10.9) and(10.10) thus constitute thedesired complete solution of Hamilton’s equations ofmotion. Hamilton’s principal function isthusthegenerator ofacanonical transforma- tiontoconstant coordinates andmomenta; when solving theHamilton-Jacobi equation, weareatthesame timeobtaining asolution tothemechanical prob- lem.Mathematically speaking, wehaveestablished anequivalence between the 2ncanonical equations ofmotion, which areﬁrst-order differential equations, to theﬁrst-order partial differential Hamilton-Jacobi equation. Thiscorrespondence isnotrestricted toequations governed bytheHamiltonian; indeed, thegeneral theory ofﬁrst-order partial differential equations islargely concemed withthe properties oftheequivalent setofﬁrst-order ordinary differential equations. Es- sentially, theconnection canbetraced tothefactthatboththepartial differential equation anditscanonical equations stemfromacommon variational principle, inthiscaseHamilton’s modiﬁed principle. Toacertain extent, thechoice ofthe01,-’sasthenewmomenta isarbitrary. We could justaswellchoose anynquantities, y,-,which areindependent functions of thea,-constants ofintegration: Vi=)'r(¢1i, ---,dn)- (10-11) Bymeans ofthese deﬁning relations, Hamilton’s principal function canbewritten asafunction ofq,-,y,-,andt,andtherestofthederivation thengoesthrough unchanged. Itoften proves convenient totakesome particular setofy,-’sasthe newmomenta, rather than theconstants ofintegration thatappear naturally in integrating theHamilton-Jacobi equation. Further insight intothephysical signiﬁcance ofHamilton’s principal function Sisfumished byanexamination ofitstotaltimederivative, which canbecom- puted from theformula if -it '.+9.5 dt_Bq,q’ at’ since theP;’sareconstant intime. ByEqs.(10.7) and(10.3), thisrelation canalso bewritten (IS Z=p,q,- —H=L, (10.12) sothatHamilton’s principal function differs atmost from theindeﬁnite timeinte- graloftheLagrangian onlybyaconstant: S=fLdt+constant. (10.13) 10.2 IChapter 10Hamilton-Jacobi Theory andAction-Angle Variables Now, Hamilton’s principle isastatement about thedeﬁnite integral ofL,andfrom itweobtained thesolution oftheproblem viatheLagrange equations. Herethe same action integral, inanindeﬁnite form, fumishes another wayofsolving the problem. Inactual calculations, theresult expressed byEq.(10.13) isofnohelp, because wecarmot integrate theLagrangian withrespect totimeuntilq,-andp,» areknown asfunctions oftime, thatis,untiltheproblem issolved. When theHamiltonian does notdepend explicitly upon thetime, Hamilton’s principle function canbewritten inthefonn S(q, oz,t)=W(q. oz)—at, (10.14) where W(q, a)iscalled Hamilton ’scharacteristic function. Thephysical signiﬁ- cance ofWcanbeunderstood bywriting itstotaltimederivative dW_aw,_ dl _Bqiq” Comparing thisexpression totheresults ofsubstituting Eq.(10.14) intoEq.(10.7), itisclearthat 8W-=—, 10.15 P1 aqi ( ) andhence, ‘;—t'=pm (10.16) Thiscanbeintegrated togive W=/p,-Q; dt=/pi dq,-, (10.17) which isjusttheabbreviated action deﬁned byEq.(8.80). THEHARMONIC OSCILLATOR PROBLEM ASANEXAMPLE OFTHEHAMILTON-IACOBI METHOD Toillustrate theHamilton-Jacobi technique forsolving themotion ofmechanical systems, weshallwork outindetail thesimple problem ofaone-dimensional harmonic oscillator. TheHamiltonian is 1H=2_(p2 +mzcozqz) EE, (10.18)m co= (10.19)mwhere 10.2 TheHarmonic Oscillator Problem asanExample 435 kbeing theforce constant. Weobtain theHamilton-Jacobi equations forSby setting pequal to8S/8q andsubstituting intheHamiltonian; therequirement thatthenewHamiltonian vanishes becomes 1as2222as-- -= . 0.2 2m[(aq) +m (uq +at 0 (l0) Since theexplicit dependence ofSontispresent onlyinthelastterm, Eq.(10.14) canbeusedtoeliminate thetimefrom theHamilton-Jacobi equation (10.20) 1aw2Zn‘ +m2CD2q2:| =(1. Theintegration constant oristhustobeidentiﬁed withthetotal energy E.This canalsoberecognized directly from Eq.(10.14) andtherelation (cf.Eq.(10.3)) E5+H=0, which thenreduces to H=oz. Equation (10.21) canbeintegrated irmnediately to Z2 W=~/2m0zIdq‘ll-1"-92;-, (10.22) 22 s=~/2mafdq‘ll -1% -at. (10.23) While theintegration involved inEq.(10.23) isnotparticularly difﬁcult, there isnoreason tocarry itoutatthisstage, forwhatisdesired isnotSbutitspartial derivatives. Thesolution forqarises outofthetransformation equation (10.8):sothat ,8S m dq '6:80:=201 12_t’ mu) ,/1-$4- which canbeintegrated without trouble togive 1 _ 2t+;9’ =5arcs1nq‘lT—2:L. (10.24) 6 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables Equation (10.24) canbeimmediately “tumed inside out”tofumish qasa function oftandthetwoconstants ofintegration ozandf3=,8’w: q=,/£2 Sl1'l((1)l‘+B), (10.25)ma) which isthefamiliar solution foraharmonic oscillator. Formally, thesolution forthemomentum comes from thetransformation equation (10.7), which, using Eq.(10.22), canbewritten 8S 8Wp=-—=——=,/2ma —m2a)2 2. (10.26) aq aq q Inconjunction withthesolution forq,Eq.(10.25), thisbecomes p=,/2ma(1- sin2(a)t +5)). or p=\/2ma cos(wt +,5) (10.27) Ofcourse, thisresult checks withthesimple identiﬁcation ofpasmq. Tocomplete thestory, theconstants ozand,3must beconnected withtheinitial conditions qgandpgattimet=0.Bysquaring Eqs.(10.25) and(10.27), itis clearly seenthatozisgiven intenns ofqgandpgbytheequation 2moz =pg+mzwzqg. (10.28) Thesame result follows immediately ofcourse from theprevious identiﬁcation of aastheconserved totalenergy E.Finally, thephase constant ,8isrelated toqg andpgby tanp=mag. (10.29) Thechoice qg=0andhence ,8=0corresponds tostarting themotion withthe oscillator atitsequilibrium position q=0. Thus, Hamilton’s principle function isthegenerator ofacanonical transforma- tiontoanewcoordinate thatmeasures thephase angle oftheoscillation andtoa newcanonical momentum identiﬁed asthetotalenergy. Ifthesolution forqissubstituted intoEq.(10.23), Hamilton’s principal func- tioncanbewritten as s=201Icos2(wt +5)at-at=211f(cos2(wt +#1)-§)at.(10.30) 10.2 TheHarmonic Oscillator Problem asanExample 437 Now, theLagrangian is __1 2 222L-2m(p mwq) =oz(cos2(a)t +,8)—sin2(a)t +ﬁ)) =2ce(cos2(wt +,3)-1), sothatSisthetime integral oftheLagrangian, inagreement with thegeneral relation (10.13). Note thattheidentity could notbeproved untilafter thesolution totheproblem hadbeenobtained. Asanother illustration fortheHa1nilton—Jacobi method, itisinstructive tocon- siderthetwo-dimensional anisotropic harmonic oscillator. Ifweletmbethemass oftheoscillating body andkxandkybethespring constants inthex-andy- directions, respectively, theHamiltonian is 1E=%(p§ +pg+mzwixz +mzwgyz), /k /kcox: Ex and coy: Since thecoordinates andmomenta separate intotwodistinct sets,theprincipal function canbewritten asasumofthecharacteristic function foreachpair.As- suming thatwesolve they-functional dependency ﬁrst,thismeanswhere 50¢»y.<1.cry.r)=Fx(x.<1)+Fy(y.11))—<11. (10-31) andtheHa1nilton—Jacobi equation assumes theform 1 3W2 22 3W2 222+m mix +m wyy =01 (10.32) inanalogy withEq.(10.18). Since thevariables areseparated, they-part ofthe Eq.(10.32) must beequal toaconstant, which wecallozy,so 1aw21'2; +Emwgyz =ﬂy, (10.33) andwereplace they-term in(10.32) withozyfrom (10.33), yielding 21aw 15; +Emwfxz =ax, (10.34) where wewrite oz-ozy=01,,showing thesymmetry ofEqs.(10.33) and(10.34). Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables Eachequation hasasolution analogous toEqs.(10.25) and(10.27), so 201 _x= s1n(w,,t +ﬁx)mwx px=\/2ma,, cos(w,,t +Bx) 201 ,y=—% sm(wyt +By)Vmm), py=,/Zmay cos(a),t +By), where the,6;’sarephase constants andthetotalenergy isgiven by(10.35) E=a,c+ay=a. Asathird example ofHamilton-Jacobi theory, weagain consider thetwo- dimensional harmonic oscillator; onlywewillassume theoscillator isisotropic, so k,,=ky=k and w,,=cuy=w, andusepolar coordinates towrite x=rcos19 r=,/x2+y2 y=rsin9 6=tan_1X X (1036)px=mJE p,-=mf py=my pg=mrzé. TheHamiltonian nowwritten as 1 P2 2 E=_-g+%+MMr (mm2m r iscyclic intheangular coordinate 6.Theprinciple function canthenbewritten as S(rv 6!as(19) =W70! a) + a9) _at =W,(r,oz)+0019—at, (10.38) where, asweshow later,acyclic coordinate q,-always hasthecharacteristic func- tioncomponent Wq,=q,-oz,-. Thecanonical momentum pgassociated withthe cyclic coordinate, 6,iscalculated from thegenerating function _3F9_a p9— -‘ 9 hasitsexpected constant value. 10.2 TheHarmonic Oscillator Problem asanExample 439 When thispgissubstituted intoEqs.(10.37) and(10.38), W,(r,a)satisﬁes 1aw,’ 115 122ﬁ( ar) +ﬁ+2mwr -01. (10.39) Rather thansolving thisequation directly forW,,weshall write theCartesian coordinate solution forthese conditions as x=k sin(wt +)3) px=\/2moz cos(a)t +,8)Vmwz ,2a (10.35 ) y=,/izsinwt py=\/2mozcoswt,ma) andusethese togetthepolar counterparts, l201 /r=—2 sinzwt+sin2(a)t +°B), p,=mi,ma) and (10.40) 9: -1 =29'_ ta“[sin(a)t+ﬂ) P”W There aretwolimiting cases. Thelinear caseiswhen )9=0,forwhich I4a .r=Z2 sinwt, p,=\/2moz coswt,ma) and (10.41) It 9=—, =0. 4 P19 Themotion inanx-yplotwillbeanoscillation along adiagonal lineasshown inFig.l0.la. Theother limiting caseiswhen 5=rr/2, forwhich 201 r=r0= —,. p.=0ma) 0=wt, pg=mr§w.(10.42) Themotion inanx-yplotforthislimiting caseisacircle ofradius rgasisshown inFigure 10.lb. Forother values ofB(0<B<rt/2), theorbit incoordinate space isanellipse. Thecasefor)3=rt/4isshown inFig.l0.lc. Theplots shown inFig.10.1arefurther examples ofLissajous ﬁgures. 10.3 IChapter 10Hamilton-Jacobi Theory andAction-Angle Variables Y Y Y X X X (4)/i=0 (b>ﬁ=l‘ (oﬁ=-’-’ 2 4 FIGURE 10.1 Thetwolimiting cases (a)and(b)fortheharmonic oscillator andan intermediate example (c). THE HAMILTON-IACOBI EQUATION FOR HAMlLTON'S CHARACTERISTIC FUNCTION Itwaspossible tointegrate theHamilton-Jacobi equation forthesimple harmonic oscillator primarily because Scould beseparated intotwoparts, oneinvolving q only andtheother only time. Such aseparation ofvariables using Hamilton’s characteristic function W(q, oz)(Eq.(10.14))isalways possible whenever theold Hamiltonian does notinvolve timeexplicitly. Thisprovides uswiththerestricted Hamilton-Jacobi equation H(Q5, =(11, Bq. which nolonger involves thetime. Oneoftheconstants ofintegration, namely 111,isthusequal totheconstant value ofH.(Normally Hwillbetheenergy, but remember thatthisneed notalways bethecase, cf.Section 8.2.) Thetime-independent function, Hamilton’s characteristic function W,appears here merely asapartofthegenerating function Swhen Hisconstant. Itcan alsobeshown thatWseparately generates itsowncontact transfonnation with properties quite different from thatgenerated byS.Letusconsider acanonical transformation inwhich thenewmomenta areallconstants ofthemotion oz,,and where 0:1inparticular istheconstant ofmotion H.Ifthegenerating function for thistransformation bedenoted byW(q, P),thentheequations oftransformation are 3W 0W 3W -=i, -=i =i. (10.44) pl 8q,- Qt 3P," 301; While these equations resemble Eqs. (10.7) and(10.8) respectively forHamil- ton’sprincipal function S,thecondition nowdetermining WisthatHisthenew canonical momentum 0:1: H(qi.Pi) =v11- 10.3 Hamilton’s Characteristic Function 441 Using Eqs.(10.44), thisrequirement becomes thepartial differential equation: 3W Hq1.—— =41.341 which isseentobeidentical withEq.(10.43). Since Wdoesnotinvolve thetime, thenewandoldHamiltonians areequal, anditfollows thatK=111. Hamilton’s characteristic function Wthusgenerates acanonical transforma- tioninwhich allthenewcoordinates arecyclic. Itwasnoted intheintroduction tothischapter thatwhen Hisaconstant ofthemotion, atransformation ofthis nature ineffect solves themechanical problem involved, fortheintegration ofthe newequations ofmotion isthentrivial. Thecanonical equations forP,-,infact, merely repeat thestatement thatthemomenta conjugate tothecyclic coordinates areallconstant: - 3K P=-—— =0, P-='. 10.45 z aQi 1at ( ) Because thenewHamiltonian depends upon only oneofthemomenta oz,-,the equations ofmotion forQ;are .8K-zizl, ‘=1, Qt 301; I =0,i;é1, withtheimmediate solutions 3W (10.46) Q1= (1.-5% 1741- Theonlycoordinate thatisnotsimply aconstant ofthemotion isQ1,which is equal tothetimeplusaconstant. Wehave hereanother instance oftheconjugate relationship between thetimeasacoordinate andtheHamiltonian asitsconjugate momentum. Thedependence ofWontheoldcoordinates q,-isdetermined bythepar- tialdifferential equation (lO.43), which, likeEq.(10.3), isalsoreferred toasthe Hamilton—Jacobi equation. There willnowbenconstants ofintegration inacom- plete solution, butagain oneofthem must bemerely anadditive constant. The n—1remaining independent constants. 0:2,...,an,together with011maythenbe taken asthenewconstant canonical momenta. When evaluated attgtheﬁrsthalf ofEqs.(10.44) serve torelate thenconstants oz;withtheinitial values ofq,-and p,-.Finally, Eqs.(10.45) and(10.46) canbesolved fortheqiasafunction of01,-, 5;,andthetimet,thuscompleting thesolution oftheproblem. Itwillbenoted Chapter 10Hamilton—Jacobi Theory andAction-Angle Variables that(n—1)oftheEqs.(10.46) donotinvolve thetimeatall.Oneoftheq;’scan bechosen asanindependent variable, andtheremaining coordinates canthenbe expressed intenns ofitbysolving onlythese time-independent equations. Weare thusleddirectly totheorbit equations ofthemotion. Incentral force motion, for example, thistechnique would furnish rasafunction of9,without theneed for separately ﬁnding rand0asfunctions oftime. Itisnotalways necessary totakea1andtheconstants ofintegration inWas thenewconstant canonical momenta. Occasionally itisdesirable rather touse some particular setofnindependent functions ofthea,-’sasthetransformed mo- menta. Designating these constants byy,-thecharacteristic function Wcanthen beexpressed interms ofq;andy,-astheindependent variables. TheHamiltonian willingeneral depend upon more thanoneofthey,-’sandtheequations ofmotion forQ;become - 8K Q1—TM—v1- where the11,-’sarefunctions ofyi.Inthiscase,allthenewcoordinates arelinear functions oftime: Qi=1)iZ‘+ ﬂi. (10.47) Thefonn ofWcannot befound apriori without obtaining acomplete integral of theHarnilton—Jacobi equation. Theprocedures involved insolving amechanical problem byeither Hamilton’s principal orcharacteristic function may nowby summarized inthefollowing tabular form: Thetwomethods ofsolution areapplicable when theHamiltonian isanygeneral function ofq,p,t: isconserved: H(q, p,t). H(q, p)=constant. Weseekcanonical transformations tonewvariables suchthat allthecoordinates andmomenta allthemomenta P;areconstants. Q,',P,-areconstants ofthemotion. Tomeet these requirements itissufﬁcient todemand thatthenewHamiltonian shallvanish identically: shall becyclic inallthecoordi- K=0. nates: K=H(Pi) =061. Under these conditions, thenewequations ofmotion become . 8K . 8K-zizo, -ziz -,Q1 aPi Q1 api vl '._3K_ -.__K_ P_ -0, P- -0, 'aQ.- 'an 10.3 Hamilton’s Characteristic Function 443 withtheimmediate solutions Qi=.51, Q1=vii+151 ﬁ=W, ﬁ=n which satisfy thestipulated requirements. Thegenerating function producing thedesired transformation isHamilton’s Principal Function: Characteristic Function: S(q.PJ), W(q, P), satisfying theHamilton-Jacobi partial differential equation: 6S 8S 8WH ,—, —=0. H ,-— —=0. lq311t)+at l (Q341) al Acomplete solution totheequation contains nnontrivial constants ofintegra- n—1nontrivial constants ofin- tion(11,...,an. tegration, which together withon fonn asetofnindependent con- stants a1,...,an. Thenewconstant momenta, Pi=y,-,canbechosen asanynindependent func- tionsofthenconstants ofintegration: H=Mmh~w%L I ﬁ=wWb~w%% sothatthecomplete solutions totheHamilton-Jacobi equation maybeconsidered asfunctions ofthenewmomenta: $=5(qi,Vi,1‘)- l W= W(qi,)/i)- Inparticular, they,-’smaybechosen tobetheai’sthemselves. One-half ofthe transformations equations, __8S __8W P1 _' aqia P1— aqis arefulﬁlled automatically, sincetheyhavebeenusedinconstructing theHarnilton- Jacobi equation. Theother half, BS 8W Q1=—=5.. ]Q1~=—=v.—0»,~>:+12.-.371' 3)/i canbesolved forq;interms oftandthe2nconstants )3),yi.Thesolution tothe problem isthencompleted byevaluating these2nconstants interms oftheinitial values, (q,-g,p,-g), ofthecoordinates andmomenta. 444 10.4 IChapter 10Hamilton-Jacobi Theory andAction-Angle Variables When theHamiltonian doesnotinvolve timeexplicitly, bothmethods aresuit- able,andthegenerating functions arethenrelated toeachother according tothe fonnula S(q, P.t)=W(q, P)—alt. SEPARATION OFVARIABLES INTHE HAMILTON-IACOBI EQUATION Itmight appear fromthepreceding section thatlittlepractical advantage hasbeen gained through theintroduction oftheHamilton-Jacobi procedure. Instead of solving the2nordinary differential equations thatmake upthecanonical equa- tionsofmotion, wenowmustsolvethepartial differential Hamilton-Jacobi equa- tion, andpartial differential equations canbenotoriously complicated tosolve. Under certain conditions, however, itispossible toseparate thevariables inthe Hamilton-Jacobi equation, andthesolution canthenalways bereduced toquadra- tures. Inpractice, theHamilton-Jacobi technique becomes auseful computational toolonlywhen suchaseparation canbeeffected. Acoordinate q1-issaidtobeseparable intheHamilton-Jacobi equation when (say) Ha.milton’s principal ftmction canbesplitintotwoadditive parts, oneof which depends onlyonthecoordinate qjandtheother isentirely independent of q1-.Thus, ifq1istaken asaseparable coordinate, thentheHamiltonian must be suchthatonecanwrite S(q1.---.qn; <11.---.11"; t)=S1(q1: <11.---.04.; 1) +$'(q2,---.41»; <11.---.11”; I).(10-43) andtheHamilton-Jacobi equation canbesplitintotwoequations—one separately forS1andtheother forS’.Similarly theHamilton-Jacobi equation isdescribed as completely separable (orsimply, separable) ifallthecoordinates intheproblem areseparable. Asolution forHamilton’s principal function oftheform s=ZS.-(q.-1 a1.....a,.; o (10.49) willthensplittheHamilton-Jacobi equation intonequations ofthetype 8S~ BS-H,-(qj; ?:_;a1,...,a,,;t)+T3=0. (10.50) IftheHamiltonian does notexplicitly depend upon thetime, then, foreach S,-we have $1(q,~; 111.---.11”;I)=W1-(qj; <11.---.11»;I)—arr. (10-51) which provide nrestricted Hamilton-Jacobi equations, 10.5 I10.5 lgnorable Coordinates andtheKepler Problem 445 8W-H,-(q,-; ——'; a1,...,a,,) =01,-. (10.52) 341' (Nosummation inEqs.(10.50) to(10.52)!) Thefunctions H;inEqs.(10.50) and(10.52) mayormaynotbeHamiltonians, andthe01,-maybeanenergy, anangular momentum squared, orsome other quan- titydepending onthenature ofq,-.Weshallshow thisbyexample intheKepler problem inthenextsection. Theconstants 01,-arereferred tonowastheseparation constants. Each ofthe Eqs.(10.52) involves onlyoneofthecoordinates q,-andthecorresponding partial derivative ofW;withrespect toq,-.They aretherefore asetofordinary differential equations ofaparticularly simple form. Since theequations areonlyofﬁrstorder, itisalways possible toreduce them toquadratures; wehave onlytosolve forthe partial derivative ofW;withrespect toq,-andthenintegrate overq,-.Inpractice, each H;willonly contain oneoratmost afewoftheoz’s.There willalsobe cases where asubset ofrvariables canbeseparated inthisfashion, leaving n—r variables, which willnotseparate. Weshallalsoexamine thiseventuality inthe nextsection. Itispossible toﬁndexamples inwhich theHamilton-Jacobi equation canbe solved without separating thetimevariable (cf.Exercise 8).Nonetheless, almost alluseful applications oftheHamilton—Jacobi method involve Hamiltonians not explicitly dependent upontime,forwhich tistherefore aseparable variable. The subsequent discussion onseparability isthusrestricted tosuch systems where H isaconstant ofmotion, andHamilton’s characteristic function Wwillbeused exclusively. IGNORABLE COORDINATES AND THE KEPLER PROBLEM Wecaneasily show thatanycyclic orignorable coordinate isseparable. Suppose thatthecyclic coordinate isq];theconjugate momentum plisaconstant, sayy. TheHamilton-Jacobi equation forWisthen 3W 3WH ,..., ;;i,...,i = . 10.53 (112 qnVaqz aqn) <11 ( ) Ifwetryaseparated solution ofthefonn W=W1(q1. <1)+W’(q2. -.-,q,.;11). (10-54) thenitisobvious thatEq.(10.53) involves only theseparate function W’,while W1isthesolution oftheequation P1/8W1 1: :4 3411(10.55) 446 Chapter 10Hamilton—]ac0bi Theory andAction-Angle Variables Theconstant yisthustheseparation constant, andtheobvious solution forW1 (towithin atrivial additive constant) is W1=W11, andWisgiven by W=W’+yql. (10.56) There isanobvious resemblance between Eq.(10.56) andtheform Sassumes when Hisnotanexplicit function oftime, Eq.(10.43). Indeed, both equations canbeconsidered asarising under similar circumstances. Wehaveseenthattmay beconsidered insome sense asageneralized coordinate with—Hasitscanonical momentum (cf.Eq.(8.58)). IfHisconserved, thentmaybetreated asacyclic coordinate. IfSofthencoordinates arenoncyclic (thatis,theyappear explicitly inthe Hamiltonian), thentheHamiltonian isofthefonn H(q1, ...,q,;111, ...,an;t). Thecharacteristic function canthenbewritten as 5' ll W(q1. ....411;111.-.-.11”)=ZW1(q1; 011.---.11»)+Zarm. (10-56’) i=1 z=s+1 andthere aresHamilton-Jacobi equations tobesolved: 8Hq1;l’-I-;a2,...,a,, =a1. (10.57)8q1 Since these areordinary ﬁrst-order differential equations intheindependent vari- ableql,theycanbeimmediately reduced toquadratures, andthecomplete solu- tions forWcanbeobtained. Ingeneral, acoordinate q1-canbeseparated ifqjandtheconjugate momentum pjcanbesegregated intheHamiltonian intosome function f(qJ-,pI-)thatdoes notcontain anyoftheother variables. Ifwethenseekatrialsolution ofthefonn W=Wj(q,-.11) +W'(q1-.11), where q,-represents thesetofallq’sexcept q1-,thentheHa1nilton—Jacobi equation appears as aw’ aw,-qi, 3? =. 10.5 ”(--((4))~- <*2Inprinciple, atleast, Eq.(10.58) canbeinverted soastosolve forf: 8W- 3W’ qr 3411 10.5 lgnorable Coordinates andtheKepler Problem 447 Theargument used previously inconnection with Eq.(10.51) holds here in slightly varied guise; fisnotafunction ofanyoftheq’sexcept q1-;gonthe other hand isindependent ofq1-.Hence, Eq.(10.59) canholdonly ifboth sides areequal tothesame constant, independent ofallq’s: aw,- f(qr74,-)e8W’g(qh :(Yj, 1 andtheseparation ofthevariable hasbeen accomplished. Notethattheseparability oftheHamilton-Jacobi equation depends notonly onthephysical problem involved butalsoonthechoice ofthesystem ofgeneral- izedcoordinates employed. Thus, theone-body central forceproblem isseparable inpolar coordinates, butnotinCartesian coordinates. Forsome problems, itisnot possible tocompletely separate theHami1ton—J acobi equation, thefamous three- body problem being oneillustration. Ontheother hand, inmany ofthebasic prob- lems ofmechanics andatomic physics, separation ispossible inmore thanoneset ofcoordinates. Ingeneral, itisfeasible tosolve theHamilton-Jacobi equation in closed fonnonlywhen thevariables arecompletely separable. Considerable inge- nuity hastherefore been devoted toﬁnding theseparable systems ofcoordinates appropriate toeachproblem. Nosimple criterion canbegiven toindicate what coordinate systems leadto separable Hamilton-Jacobi equations foranyparticular problem. Inthecaseof orthogonal coordinate systems, theso-called Staeckel conditions haveproved use- ful.Theyprovide necessary andsufﬁcient conditions forseparability under certain circumstances. Aproof ofthesufﬁciency oftheconditions andreferences willbe found inAppendix Dofthesecond edition ofthistext. TheStaeckel conditions fortheseparation oftheHamilton-Jacobi equations are: 1.TheHamiltonian isconserved. 2.TheLagrangian isnomore than aquadratic function ofthegeneralized velocities, sotheHamiltonian takes thefonn: H=%(p-a)T"‘(p -a)+V(q). (8.27) 3.Thevector ahaselements a,~thatarefunctions onlyofthecorresponding coordinate, thatisa,-=a,-(q,-). 4.thepotential function canbewritten asasumofthefonn V(q)= (10.61) 48 Chapter 10Hamilton—Jacobi Theory andAction-Angle Variables 5.Consider thematrix ¢'1,withaninverse ¢whose elements are _ l . .6;j¢iJ-1= (nosummation onz) (10.62) I where 3W1—at)=25u<¢kj)/j withyaconstant unspeciﬁed vector. Ifthediagonal elements ofboth¢ and¢"ldepend only upon theassociated coordinate, thatis,¢_1,-,- and ¢,-iareconstants orafunction ofq;only, thenprovided 1-4aretrue, the Hamiltonian—Jacobi equations separate. Since wehave assumed thatthegeneralized coordinates q,-form anorthogonal coordinate system, thematrix T(introduced inSection 8.1)isdiagonal. Itfollows thattheinverse matrix T‘1isalsodiagonal and,ifwearedealing withaparticle inanexternal force ﬁeld,thediagonal elements are: 1 1¢,T,T1=5;=—, (nosummation) (10.63) ii m sotheﬁfthStackel condition issatisﬁed. LftheStaeckel conditions aresatisﬁed, thenHamilton’s characteristic function iscompletely separable: W(q)=ZW.-(ql),i withtheW,-satisfying equations oftheform 6W- 2(Eli —dz)=—2V1(q1)+2¢ijVj. (10-64)l where yjareconstants ofintegration (andthere issummation only overthein- dexj). While these conditions appear mysterious andcomplicated, their application usually isfairly straightforward. Asanillustration ofsome oftheideas developed here about separability, theHamilton-Jacobi equation foraparticle moving in acentral force willbediscussed inpolar coordinates. Theproblem willthenbe generalized toarbitrary potential laws, tofurnish anapplication oftheStaeckel conditions. Letusﬁrstconsider thecentral force problem interms ofthepolar coordinates (r,1//)intheplane oftheorbit. Themotion theninvolves onlytwodegrees of freedom andtheHamiltonian hasthefonn 12PiH=27p,+r_2+V(r), (10.65) 10.5 lgnorable Coordinates andtheKepler Problem 449 which iscyclic in1/r.Consequently, Hamilton’s characteristic function appears as W=W1(7')+a,;,¢, (10.66) where oz,/,istheconstant angular momentum ppconjugate to1//.TheHamilton- Jacobi equation thenbecomes aW2012+7'2?+2mV(r) =2moz1, (10.67) where a1istheconstant identiﬁed physically asthetotalenergy ofthesystem. Solving Eq.(10.66) forthepartial derivative ofW1weobtain 3W1 (xi W =‘l2m(f¥1 -V)—7;, ~i W=/dr 2m(a —V)—7+0: 1//. (10.68) 1 r ip With thisform forthecharacteristic function, thetransformation equations (10.46) appear as 8W t+ﬂl : =/ mdr , 3&1 \l2m(oz1 —V)— aw d52=_=-f——"‘-"’-L-— +(11. (10696)811,), r2‘l2m(0z1 —V)— Equation (lO.69a) furnishes rasafunction oftandagreees withthecorrespond- ingsolution, Eq.(3.18), found inChapter 3,witha1and01¢written explicitly asE andl,respectively. Ithasbeenremarked previously thattheremaining transforma- tionequations forQ,~,hereonlyEq.(10.69b), should provide theorbitequation. Ifthevariable ofintegration inEq.(lO.69b) ischanged tou=1/r,theequation reduces to I du /%;(oz1 -v)-ul which agrees withEq.(3.37) previously found fortheorbit, identifying Was9 and/32as90.sothatWis (lO.69a) \~I<§N and *..|.;=.. 1/I=/32— 4 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables Asafurther example ofseparation ofvariables, weshallexamine thesame central force problem, butinspherical polar coordinates, thatis,ignoring our apriori knowledge thattheorbitliesinaplane. Theappropriate Hamiltonian has beenshown tobe(cf.Eq.(8.29)): H—i 2+5%+ pf’ +V(r) (1070)_2m pr F2 r2sin26 . i Ifthevariables inthecorresponding Hamilton-Jacobi equation areseparable, then Hamilton’s characteristic function musthavethefonn W=Wr(r) +We(9) +W¢(¢)- (10-71) Thecoordinate ¢iscyclic intheHamiltonian andhence W¢ =d¢¢ (10.72) where 01¢isaconstant ofintegration. Interms ofthisform forW,theHamilton- Jacobi equation reduces to aw,21aw,211$2 =2E, 10.7 (Br) +r2 39) +Sin26 +mV(r) m (3) where wehaveexplicitly identiﬁed theconstant Hamiltonian withthetotalen- ergyE.Note thatalldependence on9,andon0alone, hasbeen segregated into theexpression within thesquare brackets. TheHamilton-Jacobi equation then conforms totheappearance ofEq.(10.58), andfollowing theargument given there weseethatthequantity inthesquare brackets must beaconstant: 3W0 2 “ii 2-—— i =. 10.74(80) +8164) °“’ () Finally thedependence ofWonrisgiven bytheremainder oftheHamilton- Jacobi equation: 2 2 +%=2m(E-V(r)). (10.75) Thevariables intheHamilton-Jacobi equation arethuscompletely separated. Equations (10.74) and(10.75) maybeeasily reduced toquadratures providing atleast aformal solution forW9(0)andW,(r),respectively. Notethattheconstants ofintegration 01¢,(19,a1allhavedirectly recognizable physical meanings. Thequantity 01¢isofcourse theconstant value oftheangular momentum about thepolar axis(cf.Eq.(10.44)): aw04,=p¢= (10.76) 10.5 ignorable Coordinates andtheKepler Problem 451 Toidentify 0:9weuseEq.(10.44) torewrite Eq.(10.74) as 2 Pg 2 7P9+ =(19, SlIl6 sothattheHamiltonian, Eq.(10.70) appears as 1 2 (X3 /H=2— p,+—2 +V(r). (10.70)m r Comparison withEq.(10.65) fortheHamiltonian asexpressed intenns ofpolar coordinates intheplane oftheorbitshows that(Y9isthesame asP1/,,themagni- tudeofthetotalangular momentum: 0:9=pd,E1. (10.77) Lastly, a1isofcourse thetotalenergy E.Indeed, thethree differential equations forthecomponent parts ofWcanbelooked onasstatements ofconservation the- orems. Equation (10.75) saysthez-component oftheangular momentum vector, L,isconserved. while Eq.(10.74) states theconservation ofthemagnitude, I, oftheangular momentum. AndEq.(10.75) isaform oftheenergy conservation theorem. Inthissimple example, some ofthepower andelegance oftheHamilton- Jacobi method begins tobeapparent. Afewshort steps sufﬁce toobtain thede- pendence ofrontandtheorbitequation, Eqs.(10.69a andb),results derived earlier onlywithconsiderable labor. Theconserved quantities ofthecentral force problem alsoappear automatically. Separation ofvariables forthepurely central force problem canalsobeperformed inother coordinate systems, forexample, parabolic coordinates, andtheconserved quantities appear there informs appro- priate totheparticular coordinates. Finally, wecanemploy theStaeckel conditions toﬁndthemost general form of ascalar potential Vforasingle particle forwhich theHamilton—Jacobi equation isseparable inspherical polar coordinates. Thematrix ¢oftheStaeckel condi- tions depends only onthecoordinate system andnotonthepotential. Since the Hamilton-Jacobi equation isseparable inspherical polar coordinates foratleast onepotential, thatis,thecentral force potential, itfollows thatthematrix ¢does exist. Thespeciﬁc fonnof¢isnotneeded toanswer ourquestion. Further, sincea byhypothesis iszero,allweneeddoisapply Eq.(10.62) toﬁndthemostgeneral separable fonn ofV.From thekinetic energy (Eq.8.28’), thediagonal elements ofTare Tr,=m, T99=mr2, T¢¢,=mr2sin26. ByEq.(10.62) itfollows thatthedesired potential musthavetheform V(q) =V,(r) +1/672?) + (10.78) 452 10.6 IChapter 10Hamilton-Jacobi Theory andAction—Angle Variables Itiseasytoverify directly thatwiththispotential theHamilton-Jacobi equation isstillcompletely separable inspherical polarcoordinates. ACTION-ANGLE VARIABLES IN SYSTEMS OFONE DEGREE OFFREEDOM Ofespecial importance inmany branches ofphysics aresystems inwhich the motion isperiodic. Very often weareinterested notsomuch inthedetails ofthe orbit asinthefrequencies ofthemotion. Anelegant andpowerful method ofhan- dling suchsystems isprovided byavariation oftheHamilton-Jacobi procedure. Inthistechnique, theintegration constants 01,-appearing directly inthesolution of theHami1ton—J acobi equation arenotthemselves chosen tobethenewmomenta. Instead, weusesuitably deﬁned constants J,-.which formasetofnindependent functions ofthe01,-’s,andwhich areknown astheaction variables. Forsimplicity, weshall ﬁrstconsider inthissection systems ofonedegree of freedom. Itisassumed thesystem isconservative sothattheHamiltonian canbe written as H(q,p)=0:1. Solving forthemomentum, wehave that P=P(q.u1). (10-79) which canbelooked onastheequation oftheorbittraced outbythesystem point inthetwo-dimensional phase space, p,qwhen theHamiltonian hasthe constant value 011.What ismeant bytheterm “periodic motion” isdetermined by thecharacteristics ofthephase space orbit. Twotypes ofperiodic motion maybe distinguished: 1.Intheﬁrsttype, theorbit isclosed, asshown inFig.10.2(a), andthesystem point retraces itssteps periodically. Both qandparethenperiodic functions ofthetimewiththesame frequency. Periodic motion ofthisnature willbe found when theinitial position liesbetween twozeros ofthekinetic energy. Itisoften designated bytheastronomical name libration, although toa physicist itismore likely tocalltomind thecommon oscillatory systems, suchastheone-dimensional harmonic oscillator. 2.Inthesecond typeofperiodic motion, theorbit inphase space issuchthatp issome periodic function ofq,withperiod q(),asillustrated inFig.10.2(b). Equivalently, thiskind ofmotion implies thatwhen aisincreased byq(), theconﬁguration ofthesystem remains essentially unchanged. Themost familiar example isthatofarigid body constrained torotate about agiven axis, withqastheangle ofrotation. Increasing qby22:thenproduces no essential change inthestate ofthesystem. Indeed, theposition coordinate inthistypeofperiodicity isinvariably anangle ofrotation, andthemotion 10.6 Action-angle Variables inSystems ofOne Degree ofFreedom 453 P P (a)Libration (b)Rotation FIGURE 10.2 Orbit ofthesystem point inphase space forperiodic motion ofone- dimensional systems. willbereferred tosimply asrotation, incontrast tolibration. Thevalues of qarenolonger bounded butcanincrease indeﬁnitely. Itmayserve toclarify these ideas tonotethatboth types ofperiodicity may occur inthesame physical system. Theclassic example isthesimple pendulum where qistheangle ofdeﬂection 9.Ifthelength ofthependulum islandthe potential energy istaken aszeroatthepoint ofsuspension, thentheconstant energy ofthesystem isgiven by 2 E=5%-mglcos0. (10.80) Solving Eq.(10.64) forP9,theequation ofthepathofthesystem point inphase space is pg=:l:,/2ml2(E +mglcos6). (10.81) IfEislessthanmgl, thenphysical motion ofthesystem canonlyoccur for[9| lessthanabound, 9’,deﬁned bytheequation E cos6’=—:.mgl Under these conditions, thependulum oscillates between -9’and+9’,which isa periodic motion ofthelibration type. Thesystem point thentraverses some such pathinphase space asthecurve 1ofFig.10.3. However, ifE>mgl, allvalues of6correspond tophysical motion and9canincrease without limit toproduce a periodic motion oftherotation type. What happens physically inthiscaseisthat thependulum hassomuch energy thatitcanswing through thevertical positionq_l (I 44 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables C FIGURE 10.3 Phase space orbits forthesimple pendulum. 9=rtandtherefore continues rotating. Curve 3inFig.10.3corresponds tothe rotation motion ofthependulum. Thelimiting casewhen E=mglisillustrated bycurves 2and2’inFig.10.3. Atthisenergy, thependulum arrives at9=rr,the vertical position, withzerokinetic energy, thatis,pg=0.Itistheninunstable equilibrium andcould inprinciple remain there indeﬁnitely. However, ifthere istheslightest perturbation, itcould continue itsmotion either along curve 2or switch tocurve 2’—it could falldown either way.Thepoint 9=rr,pg=0 isasaddle point oftheHamiltonian function H=E(p9, 9)andthere aretwo paths ofconstant Einphase space thatintersect atthesaddle point. Wehavehere aninstance ofwhat iscalled abiﬁrrcation, aphenomenon thatwillbediscussed extensively inthenextchapter. (SeealsoSection 6.6.) Foreither typeofperiodic motion, wecanintroduce anewvariable Jdesigned toreplace 011asthetransformed (constant) momentum. Theso-called action vari- ableJisdeﬁned as(cf.Eq.(8.80)) J=%pdq, (10.82) where theintegration istobecarried overacomplete period oflibration orof rotation, asthecasemaybe.(The designation asaction variable stems from the resemblance ofEq.(10.82) totheabbreviated action ofSection 8.6.Note thatJ always hasthedimensions ofanangular momentum.) From Eq.(10.79), itfollows thatJisalways some function of0:1alone: 0112 H=H(J). (10.83) Hence, Hamilton’s characteristic function canbewritten as W=W(q, J). (10.84) 10.6 Action-angle Variables inSystems ofOne Degree ofFreedom 455 Thegeneralized coordinate conjugate toJ,known astheangle variable w,is deﬁned bythetransformation equation: aw=_. 10.85waJ () Correspondingly, theequation ofmotion forwis ._011(1)_w-T] _-v(J), (10.86) where visaconstant function ofJonly. Equation (10.86) hastheimmediate solution w=vt+,5, (10.87) sothatwisalinear function oftime, exactly asinEq.(10.47). Sofartheaction-angle variables appear asnomore thanaparticular setofthe general class oftransformed coordinates towhich theHamilton-Jacobi equation leads. Equation (10.85) could besolved forqasafunction ofwandJ,which, in combination withEq.(10.87), would givethedesired solution forqasaftmction oftime. Butwhen employed inthisfashion thevariables have nosigniﬁcant ad- vantage overanyother setofcoordinates generated byW.Their particular merit risesrather fromthephysical interpretation thatcanbegiven tov.Consider the change inwasqgoes through acomplete cycle oflibration orrotation, asgiven by Aw=§aldq. (10.88) Bq ByEq.(10.85), thiscanalsobewritten azwA=——d. 10.89w_¢8q9J ‘I () Because Jisaconstant, thederivative withrespect toJcanbetaken outside the integral sign: daw dA =— ———d =— d=1, 10.90 wdjylaq qdjylpq () where thelaststepfollows from thedeﬁnition forJ,Eq.(10.82). Equation (10.90) states thatwchanges byunity asqgoes through acomplete period. ButfromEq.(10.87), itfollows thatif-ristheperiod foracomplete cycle ofq,then Aw=1=v1..'. 4 Chapter 10 Hamilton-Jacobi Theory andAction-Angle Variables Hence, theconstant vcanbeidentiﬁed asthereciprocal oftheperiod, 1v=—, (10.91)1' andistherefore thefrequency associated withtheperiodic motion ofq.Theuse ofaction-angle variables thusprovides apowerful technique forobtaining the frequency ofperiodic motion without ﬁnding acomplete solution tothemotion of thesystem. Ifitisknown apriori thatasystem ofonedegree offreedom ispe- riodic according tothedeﬁnitions given above, thenthefrequency canbefound onceHisdetemrined asafunction ofJ.Thederivative ofHwithrespect toJ, byEq.(10.86), thendirectly gives thefrequency vofthemotion. Thedesigna- tionofwasanangle variable becomes obvious from theidentiﬁcation ofvin Eq.(10.87) asafrequency. Since Jhasthedimensions ofanangular momentum, thecoordinate wconjugate toitisanangle.* Asanillustration oftheapplication ofaction-angle variables toﬁndfrequen- cies,letusagain consider thefamiliar linear harmonic oscillator problem. From Eqs. (10.26) andthedeﬁning equation (10.82), theconstant action variable Jis given by J=%pdq =¢“/2mcz —mzwzqzdq, (10.92) where oristheconstant totalenergy and0:2=k/m.Thesubstitution (10.25) l201 _q= Q S1119 2a 27: J=—Icos29d9, (10.93)w0reduces theintegral to where thelimits aresuchastocorrespond toacomplete cycle inq.Thisintegrates to 2rrorJ=i(1) or,solving foror, ozEH= (10.94)2:1 Thefrequency ofoscillation istherefore *Forsome applications theaction variable ISdeﬁned intheliterature ofcelestial mechanics as(2rr)‘1 times thevalue given inEq.(10.82). ByEq.(10.90), thecorresponding angle variable is2ntimes our deﬁnition andinplace ofvwehavew,theangular frequency. However, weshall stickthroughout to thefamiliar deﬁnitions usedinphysics, asgiven above. 10.7 I10.7 Action-Angle Variables forCompletely Separable Systems 457 3H 0) 1 k W-"=5;= <‘°-95> which isthecustomary formula forthefrequency ofalinear harmonic oscillator. Although itisentirely unnecessary forobtaining thefrequencies, itisnevertheless instructive (anduseful forfuture applications) towrite thesolutions, Eqs.(10.25) and(10.27), interms ofJandw.Itwillberecognized ﬁrstthatthecombination (cut+,8)isbyEqs.(10.95) and(10.87) thesame as2rrw, withtheconstant ofintegration suitably redeﬁned. Hence, thesolutions forq,Eq.(10.25), andp, Eq.(10.27), takeontheform q=‘lL sin2rrw, (10.96)rrmw lJp=ii cos2rrw. (10.97) NotethatEqs.(10.96) and(10.97) canalsobelooked onasthetransformation equations from the(w,J)setofcanonical variables tothe(q,p)canonical set. ACTION-ANGLE VARIABLES FOR COMPLETELY SEPARABLE SYSTEMS* Action-angle variables canalsobeintroduced forcertain types ofmotion ofsys- temswithmany degrees offreedom, providing there exists oneormore setsof coordinates inwhich theHarnilton—Jacobi equation iscompletely separable. As before, onlyconservative systems willbeconsidered, sothatHa1nilton’s charac- teristic function willbeused. Complete separability means thattheequations of canonical transformation havetheform 8W- ~;,..., , (10.98)liq, which provides eachpiasafunction oftheq;andthenintegration constants 0:]-1 Pi=Pr'(qi§ 9Yl,---»€Yn)- (10-99) Equation (10.99) isthecounterpart ofEq.(10.79), which applied tosystems of onedegree offreedom. Itwillberecognized thatEq.(10.99) hererepresents theorbitequation oftheprojection ofthesystem point onthe(pi,qi)plane in phase space. Wecandeﬁne action-angle variables forthesystem when theorbit equations forallofthe(q,-,pi)pairs describe either closed orbits (libration, asin Fig.10.2(a)) orperiodic functions ofq,-(rotation, asinFig.10.2(b)). Note thatthischaracterization ofthemotion does notmean thateach q,-and p,-willnecessarily beperiodic functions ofthetime, thatis,thattheyrepeat their *Unless otherwise stated, thesummation convention willnotbeusedinthissection. 458 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables values atﬁxedtimeintervals. Even when eachoftheseparated (q,-,p,-)setsarein- deedperiodic inthissense, theoverall system motion neednotbeperiodic. Thus, inathree-dimensional harmonic oscillator thefrequencies ofmotion along the threeCartesian axesmayallbedifferent. Insuchanexample, itisclearthecom- pletemotion oftheparticle maynotbeperiodic. Iftheseparate frequencies are notrational fractions ofeach other, theparticle willnottraverse aclosed curve in space butwilldescribe anopen “Lissajous ﬁgure.” Such motion willbedescribed asmultiply periodic. Itistheadvantage oftheaction-angle variables thatthey leadtoanevaluation ofallthefrequencies involved inmultiply periodic motion without requiring acomplete solution ofthemotion. Inanalogy toEq.(10.82), theaction variables J,-aredeﬁned interms ofline integrals overcomplete periods oftheorbitinthe(q,-,p,-)plane: J,=§p,' dqi. (10.100) Ifoneoftheseparation coordinates iscyclic, itsconjugate momentum isconstant. Thecorresponding orbitintheqi,piplane ofphase space isthenahorizontal straight line,which would notappear tobeinthenature ofaperiodic motion. Actually themotion canbeconsidered asalimiting caseoftherotation typeof periodicity, inwhich q,-maybeassigned anyarbitrary period. Since thecoordinate inarotation periodicity isinvariably anangle, suchacyclic q,-always hasanatural period of2n.Accordingly, theintegral inthedeﬁnition oftheaction variable corresponding toacyclic angle coordinate istobeevaluated from 0to2rr,and hence J,-=21),), (10.101) forallcyclic variables. ByEq.(10.98), J,-canalsobewritten as JZf gxl.1*"aall) qr Since q,-isheremerely avariable ofintegration, eachaction variable J,-isa function only ofthenconstants ofintegration appearing inthesolution ofthe Hamilton-Jacobi equation. Further, itfollows from theindependence ofthesep- aratevariable pairs(q,-,p,-)thattheJ,-’sformnindependent functions oftheor,-’s andhence aresuitable foruseasasetofnewconstant momenta. Expressing the or;‘sasfunctions oftheaction variables, thecharacteristic function Wcanbewrit- tenintheform W=W(q1.--..q-.; J1.....J..> =ZW,<q,-; J1.....J,.>.I while theHamiltonian appears asafunction oftheJ;’sonly: H=0:1: H(J1,...,J,,). (10.l03) 10.7 Action-Angle Variables forCompletely Separable Systems 459 Asinthesystem ofonedegree offreedom, wecandeﬁne conjugate angle variables w,-bytheequations oftransformation thathereappear as 3W n3Wj(qj; J1,...,./n) -=—= -M. 1.14 w,Mi all (00) Noteingeneral w;could beafunction ofseveral oralloftheq,-;thatis,w,-= w;(q,-,...,q,,;J,-,...,Jn).Thew,-’ssatisfy equations ofmotion given by , 8HJ,..., Jw,-=(il-ll =v,~(J1,...,J,,). (10105)3]," Because thevfsareconstants, functions oftheaction variables only, theangle variables arealllinear functions oftime ‘LU;=v,~t-l-,5)‘. (10106) Notethatingeneral theseparate w,-’sincrease intimeatdifferent rates. Theconstants v;canbeidentiﬁed withthefrequencies ofthemultiply peri- odicmotion, buttheargument todemonstrate therelation ismore subtle thanfor periodic systems ofonedegree offreedom. Thetransformation equations tothe (w,J)setofvariables implies thateachqj(andpJ-)isafunction oftheconstants J;andthevariables w,-.What wewant toﬁndiswhat sortofmathematical func- tiontheq’sareofthew’s.Todothis,weexamine thechange inaparticular w; when eachofthevariables q_,-istaken through anintegral number, mJ-,ofcycles oflibration orrotation. Incarrying outthispurely mathematical procedure, we areclearly notfollowing themotion ofthesystem intime. Itisasiftheﬂowof timewere suspended andeach oftheq’swere moved, manually asitwere, inde- pendently through anumber ofcycles oftheirmotion. Ineffect, wearedealing withanalogues ofthevirtual displacements ofChapter 1,andaccordingly thein- ﬁnitesimal change inw,-astheqJ-’sarechanged inﬁnitesimally willbedenoted by8w,-andisgiven by aw; 32W 8w-= Zdq-= i——dq-,I Hqj J 2 3],‘8q,- J where usehasbeenmade ofEq.(l0.l04). Thederivative withrespect toq,-van- ishes except fortheWjconstituent ofW,sothatbyEq.(10.98) 6w,-reduces to a8w,-=5]-Zp,-(q,-,J)dq,-. (10107) '1 Equation (10.107) represents 6w,-asthesumofindependent contributions each involving theq1-motion. Thetotalchange inw,-asaresult ofthespeciﬁed ma- 4 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables neuver istherefore 8Aw;=ZHy§p,(q,~,J)dq,-. (10.108) 1',,,, thedifferential operator withrespect toJ;canbekeptoutside theintegral signs because throughout thecyclic motion ofq,-alltheJ’sareofcourse constant. Be- loweach integral sign, thesymbol m_,-indicates theintegration isovermjcycles ofq1-.Buteachoftheintegrals is,bythedeﬁnition oftheaction variables, exactly m1-JJ-.Since theJ’sareindependent, itfollows that Aw; =m,-. (10.109) Further, notethatifanyqjdoesnotgothrough acomplete number ofcycles, then intheintegration overq1-there willbearemainder ofanintegral overafraction ofacycle andAw; willnothave anintegral value. Ifthesetsofw’sandm’sare treated asvectors wandm,respectively, Eq.(10.109) canbewritten as Aw=m. (l0.l09’) Suppose, ﬁrst,thattheseparable motions areallofthelibration typesothat each qJ-,aswellasp_,-,returns toitsinitial value oncompletion ofacomplete cycle. Theresult described byEq.(l0.l09’) could nowbeexpressed somewhat asfollows: 1|(thevector ofq’sandp’s)issuchafunction ofwthatachange An=0corresponds toachange Aw=m,avector ofinteger values. Since the number ofcycles inthechosen motions ofqjarearbitrary, mcanbetaken aszero except form,-=1,andallthecomponents of1)remain unchanged orretum to their original values. Hence, inthemost general casethecomponents of1|must beperiodic functions ofeach w,-with period unity; thatis,theq’sandp’sare multiply periodic functions ofthew’swithunitperiods. Such amultiply periodic function canalways berepresented byamultiple Foruier expansion, which forqk, say,would appear as O0 X X ‘1'<= Z Z Z“i1i....1,."’2m("w'+’m+”w’+"'+’"'”"). (libration11=—<>9 .iz=—99 .i»=—99 (10.110) where thej’sareninteger indices running from—ootooo.Bytreating thesetof j’salsoasavector inthesamen-dimensional space withw,theexpansion canbe written morecompactly as qk=Zaj(k)a2"’~l'“’, (libration). (10110) 1 Ifwesimilarly writeEq.(l0.l09’) asavector equation, w=vt+B, (10.106') 10.7 Action-Angle Variables forCompletely Separable Systems 461 thenthetimedependence ofqkappears inthefonn qk(t)=ZaJ$"’e2"1"<"+B>, (libration). (10111) .i Note thatingeneral qk(t)isnotaperiodic function oft.Unless thevarious v,-’s arecommensurate (thatis,rational multiples ofeachother), qkwillnotrepeat its values atregular intervals oftime. Considered asafunction oft,qkisdesignated asaquasi-periodic function. Finally itshould beremembered thatthecoefﬁcients aj-k)canbefound bythestandard procedure forFourier coefﬁcients; thatis,they aregiven bythemultiple integral overtheunitcellinwspace: l l 85$"): [0 [Oqk(w)e_2”’J"'(dw). (10112) Here (dw) stands forthevolume element inthen-dimensional space ofthew,’s. When themotion isinthenature ofarotation, theninacomplete cycle ofthe separated variable pair(qk.pk)thecoordinate qkdoes notretum toitsoriginal value, butinstead increases bythevalue ofitsperiod qok.Sucharotation coordi- nateistherefore notitselfevenmultiply periodic. However, during thecycle we have seenthatwkincreases byunity. Hence, thefunction qk—wkqok doesreturn toitsinitial value and,likethelibrational coordinates, isamultiply periodic func- tionofallthew’swith unitperiods. Wecantherefore expand thefunction ina multiple Fourier‘ series analogous toEq.(10.110) qk-wkqok=Zaj(k)e2’"j'w, (rotation) (10113) 1 01' qk=q()k(vkt +ﬁr)+Zaj(k)e2”ij'(v'+B), (rotation). (10.114) J Thus, itisalways possible toderive amultiply periodic function from arotation coordinate, which canthenbehandled exactly likealibration coordinate. Tosim- plifythefurther discussion, weshalltherefore conﬁne ourselves primarily tothe libration typeofmotion. Theseparable momentum coordinates, pk,arebythenature oftheassumed motion alsomultiply periodic functions ofthew’sandcanbeexpanded inamul- tipleFourier series similar toEq.(10.110).Itfollows thenthatanyfunction ofthe several variable pairs (qk,pk)willalsobemultiply periodic functions ofthew’s andcanbewritten intheform f(q,p)=2bj€2”ij'w =Zbje2""i'<”'+1*>. (10115) .i i Forexample, where theCartesian coordinate ofparticles inthesystem arenot themselves theseparation coordinates, theycanstillbewritten asfunctions of timeinthefashion ofEq.(10.1 15). Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables While Eqs.(10.110)and(10.l1l) represent themostgeneral typeofmotion consistent withtheassumed nature oftheproblem, notallsystems willexhibit thisfullgenerality. Inparticular, formostproblems simple enough tobeusedas illustrations oftheapplication ofaction-angle variables, Eq.(l0.l04) simpliﬁes t0 3w- wi=—‘<ql; J1,...,J,,> (10116)3J,‘ andeach separation coordinate q,isafunction only ofitscorresponding wk. When thishappens, qkisthenaperiodic function ofwk(andtherefore oftime), andthemultiple Fourier series reduces toasingle Fourier series: qk=Za§k)e2nijwk :2a£_k)e27'rij(v1;t+l5k)_ (10117) 1 J Inthelanguage ofChapter 6,insuchproblems theqk’sareineffect thenormal coordinates ofthesystem. However, evenwhen themotion intheq’scanbeso simpliﬁed, itfrequently happens thatfunctions ofalltheq’s,suchasCartesian co- ordinates, remain multiply periodic functions ofthew’sandmustberepresented asinEq.(10.1 15).Ifthevarious frequencies vkareincommensurate, thensuch functions arenotperiodic functions oftime. Themotion ofatwo-dimensional anisotropic harmonic oscillator provides aconvenient andfamiliar example of these considerations. Suppose thatinaparticular setofCartesian coordinates theHamiltonian is given by H=i1;[(Pi +47z2m2v§x2) +(pg,+4n'2m2v§y2)]. These Cartesian coordinates aretherefore suitable separation variables, andeach willexhibit simple harmonic motion with frequencies vxandvy,respectively. Thus, thesolutions forxandyareparticularly simple forms ofthesingle Fourier expansions ofEq.(l0.117). Suppose nowthatthecoordinates arerotated 45° about thezaxis;thecomponents ofthemotion along thenewx’,y’axeswillbe n—~§|r—~l\-7xi=‘[950 COS2770-’xt +ﬁx)+Y0C03277-'(vyt +l3y)], y’=-\7i[y0 cos2rr(vyt +ﬂy)—x0cos2rr(vxt +/3x)]. (10.118) Ifvx/vyisarational number, these twoexpressions willbecommensurate. corre- sponding toclosed Lissajous ﬁgures ofthetypeshown inFig.10.4. Butifv,and vyareincommensurable, theLissajous ﬁgure never exactly retraces itsstepsand Eqs.(10.1 18)provide simple examples ofmultiply periodic series expansions of thefonn (10.1 17). Even when qkisamultiply periodic function ofallthew’s,weintuitively feel there mustbeaspecial relationship between qkanditscorresponding u-1,(and xi xl10.7 Action-Angle Variables forCompletely Separable Systems 463 y’ >1’ .p_1-1,_<1;(a)Bx=ﬁy= ~:-;- .§|>-FIGURE 10.4 Lissajous ﬁgures forEq. (l0.l18). (a)ﬁx=ﬂy= 7'1?=%(b)ﬁx=711. 5), ZO1 gt; Z %' therefore vk).After all,theargument culminating inEq.(10.l09) saysthatwhen qkalone goes through itscomplete cycle, wkincreases byunity, while theother w’sretum totheir initial values. Itwasonlyin1961 thatJ.Vinti succeeded in expressing thisintuitive feeling inaprecise andrigorous statement.* Suppose thatthetimeinterval Tcontains mcomplete cycles ofqkplusafrac- tionofacycle. Ingeneral, thetimes required foreach successive cycle willbe different, since qkwillnotbeaperiodic function oft.Then Vinti showed, onthe basis ofatheorem innumber theory, thatasTincreases indeﬁnitely, =vk. (10.119) Themean frequency ofthemotion ofqkistherefore always given byvk,even when theentire motion ismore complicated than aperiodic function with fre- quency vk. Barring comrnensurability ofallthefrequencies, amultiply periodic function canalways beformed from thegenerating function W.Thedeﬁning equation forJ,-,Eq.(10.102), ineffect states thatwhen q;goes through acomplete cycle; thatis,when w,~changes byunity, thecharacteristic function increases byJ,-.It follows thatthefunction W’=W-Zwﬂk (10.120)k remains unchanged when each wkisincreased byunity, alltheother angle vari- ables remaining constant. Equation (10.l20) therefore represents amultiply peri- odicfunction thatcanbeexpanded interms ofthew,-(orofthefrequencies v,-) byaseries oftheformofEq.(10.115). Since thetransformation equations forthe *J.Vinti, J.Res.Nat.BunStandards, 65B, 131(1961). 64 Chapter 1OHamilton-Jacobi Theory andAction-Angle Variables angle variables are wk=w 3./k, itwillberecognized thatEq.(10.120) deﬁnes aLegendre transformation from theq,Jbasis totheq.wbasis. Indeed, comparison withEq.(9.15) incombina- tionwithEq.(9.12) shows thatifW(q, J)isagenerating function ofthef0I'lI1 F2(q. P),then W’(q,w)isthecorresponding generating function ofthetype F1(q, Q),transforming inbothcases from the(q,p)variables tothe(w,J)vari- ables. While W’thusgenerates thesame transformation asW,itisofcourse not asolution oftheHamilton Jacobi equation. Ithasbeen emphasized thatthesystem conﬁguration ismultiply periodic only ifthefrequencies v,-arenotrational fractions ofeach other. Otherwise, thecon- ﬁguration repeats after asufﬁciently long time andwould therefore besimply periodic. Theformal condition forthecomrnensurability oftwofrequencies v, andvjisthattheysatisfy therelation j,-v,-=jJv1(nosum) where j,-andjjare nonzero positive integers. Forcomplete commensurability, allpairs offrequencies must satisfy relations oftheform j,'v,-=jkvk. (nosum) (10.121) where thej,-andjkarenonzero positive integers. When wecanexpress anyv;asarational fraction ofanyoftheother frequen- cies, thesystem issaidtobecompletely commensurate. Ifonlym+lofthen frequencies satisfy Eq.(10.l21), thesystem issaidtobem-fold commensurate. Forexample, consider thesetofseven frequencies v1=3MHz, U2=5MHz, v3=7MHz, v4=2x/5MHz, v5=3~/5 MHz, v5=\/-I-5‘MHz, v7=~/7MHz. Theﬁrstthree v1,U2,andv3aretriply commensurate, thenexttwov4andv5are doubly commensurate. There isaninteresting connection between comrnensurability andthecoordi- nates inwhich theHamilton—Jacobi equation isseparable. Itcanbeshown thatthe pathofthesystem point foranoncommensurate system completely ﬁllsalimited region ofbothconﬁguration andphase space. This canbeseenintheLissajous ﬁgures ofincommensurate frequencies. Suppose theproblem issuchthatthemotion inanyoneoftheseparation coor- dinates issimply periodic andhastherefore been shown tobeindependent ofthe motion oftheother coordinates. Hence, thepathofthesystem point asawhole must belimited bythesurfaces ofconstant q,-andpithatmark thebounds ofthe oscillatory motion oftheseparation variables. (The argument iseasily extended to rotation bylimiting allangles totheregion 0to27:.)These surfaces therefore de- ﬁnethevolume inspace thatisdensely ﬁlled bythesystem point orbit. Itfollows thattheseparation ofvariables innoncommensurate systems must beunique: the Hamilton-Jacobi equation cannot beseparated intwodifferent coordinate sys- tems (aside from trivial variations such aschange ofscale). Thepossibility of separating themotion inmore thanonesetofcoordinates thusnormally provides evidence thatthesystem iscommensurate. 10.7 Action-Angle Variables forCompletely Separable Systems 465 Thesimplest example ofbeing commensurate isdegeneracy which occurs when twoormore ofthefrequencies areequal. Iftwooftheforce constants inathree-dimensional harmonic oscillator areequal, thenthecorresponding fre- quencies areidentical andthesystem issingly degenerate. Inanisotropic linear oscillator, theforce constants arethesame along alldirections, allfrequencies are equal, andthesystem iscompletely degenerate. Whenever thissimple degeneracy ispresent, thefundamental frequencies are nolonger independent, andtheperiodic motion ofthesystem canbedescribed bylessthanthefullcomplement ofnfrequencies. Indeed, themconditions of degeneracy canbeused toreduce thenumber offrequencies ton—m+1.The reduction ofthefrequencies maybemost elegantly performed bymeans ofapoint transformation oftheaction-angle variables. Themdegeneracy conditions maybe written where jk;arepositive ornegative integers II Zjkiv,-=0, k=1,...,m. (10122) i=l Consider nowapoint transformation from (w,J)to(w’,J')deﬁned bythe generating function (cf.Eq.(9.26) Where thesummation convention isused): 3'11_[‘1=R4§ F2= ,1jk,w,-+ZJ,§w,.. (10123)== l<=m+1 Thetransformed coordinates are fl w;‘=Z:jki, k=1,...,m, i=1 =wk, k=m+l,...,n. (lO.l24) Correspondingly, thenewfrequencies are n I);<=li);<=Zjk,"l),'=0 k=1,...,m, l=1 =vk k=m-l—1,...,n. (10.l25) Thus inthetransformed coordinates, mofthefrequencies arezero, andweareleft withasetofn—mindependent frequencies plusthezerofrequency. Itisobvious thattheneww,’€mayalsobetermed asangle variables inthesense thatthesystem conﬁguration ismultiply periodic inthewf,coordinates with thefundamental period unity. Thecorresponding constant action variables aregiven asthesolution ofthenequations oftransformation m fl J,~=ZJ,§j,.,-+ ZJ,§a,.,-. (10.126) k=1 k=m+l 466 10.8 IChapter 10Hamilton-Jacobi Theory andAction-Angle Variables Thezerofrequencies correspond toconstant factors intheFourier expansion. These areofcourse alsopresent intheoriginal Fourier series interms ofthe v’s,Eq.(10.1l0), occurring whenever theindices j,-aresuch thatdegeneracy conditions aresatisﬁed. Since ,,/_?£l_aJ-ii’ theHamiltonian must beindependent oftheaction variables J,-’whose corre- sponding frequencies vanish. Inacompletely degenerate system, theHamiltonian cantherefore bemade todepend upon onlyoneoftheaction variables. Note thatHamilton’s characteristic function Walsoserves asthegenerating function forthetransformation from the(q,p)settothe(w’,J’)set.Since theJ’ quantities arenindependent constants, theoriginal constants ofintegration may beexpressed interms oftheJ’set,andWgiven asW(q, J’).Inthisform, itisa generating function toanewsetofcanonical variables forwhich theJ’quantities arethecanonical momenta. Butbyvirtue ofthepoint transformation generated bytheF2ofEq.(l0.l23), weknow thatw’isconjugate toJ’.Hence, itfollows thatthenewcoordinates generated byW(q, J’)must betheangle variable w’set, withequations oftransformation given by 8W (Foramore formal proof ofEq.(10.127)based onthealgebraic structure of Eq.(l0.l23), seeDerivation 3.) Theproblem ofthebound motion ofaparticle inaninverse-square lawcentral force illustrates many ofthephenomena involved indegeneracy. Adiscussion of thisproblem alsoaffords anopportunity toshow howtheaction-angle technique is applied tospeciﬁc systems, andtoindicate theconnections withBohr’s quantum mechanics andwithcelestial mechanics. Accordingly, thenextsection isdevoted toadetailed treatment oftheKepler problem interms ofaction-angle variables. THE KEPLER PROBLEM INACTION-ANGLE VARIABlES* Toexhibit alloftheproperties ofthesolution, weshall examine themotion in three dimensional space, rather thanmake useofourapriori knowledge thatthe orbit liesinaplane. Interms ofspherical polar coordinates, theKepler problem becomes aspecial caseofthegeneral treatment given above inSection 10.5for central force motion inspace. Equations (10.70) through (10.77) canbetaken overhereimmediately, replacing V(r) wherever itoccurs byitsspeciﬁc form kV(r) =——. (l0.l28)r *Thesummation convention willberesumed fromhereon. 10.8 TheKepler Problem inAction-angle Variables 467 Since thepotential V(r) depends onlyupon oneofthethree coordinates, itfol- lows thattheHamilton-Jacobi equation iscompletely separable inspherical polar coordinates. Weshall conﬁne ourdiscussion tothebound case, thatis,E<0. Hence, themotion ineach ofthecoordinates willbeperiodic—libration inrand 6,androtation in45.Theconditions fortheapplication ofaction-angle variables arethussatisﬁed, andwecanproceed toconstruct theaction variables onthebasis ofthedeﬁning equation (10.102). From Eq.(10.72), itfollows that 6WJ¢=1; 8—¢-d¢=fay) d¢. (10.129a) Similarly, onthebasis ofEq.(10.74), J9isgiven by aw l 11$J9=¢%d6 =y§ 113- ;12—6d6. (10.129b) Finally theintegral forJ,from Eq.(10.75), is W 2k1J,=5iLdr =y§2mE+L-ﬁdr. (10.129¢)Br r r2 Theﬁrstintegral istrivial; <15goes through 21:radians inacomplete revolution andtherefore J¢=21ra¢ =27rp¢. (10.l30) Thisresult could have been predicted beforehand, for¢isacyclic coordinate inH,andEq.(10.130) ismerely aspecial caseofEq.(l0.l01) fortheaction variables corresponding tocyclic coordinates. Integration ofEq.(l0.129b) can beperformed invarious ways; aprocedure involving only elementary rules of integration willbesketched here. Ifthepolar angle ofthetotalangular momentum vector isdenoted byi,sothat cosi=E, (10131)<16 thenEq.(10.l29b) canalsobewritten as J9=a9f\/1—-coszicsc26d0. (l0.132) Thecomplete circuital pathofintegration isfor6going from alimit -00to+190 andback again, where sin60=cosi, or60=(Jr/2) —i.Hence, thecircuital integral canbewritten as4times theintegral over from Oto60,orafter some manipulation, 90 J9=4119/ csc91/sinzi—cosz0d6. 0 68 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables Thesubstitution cos6=sinisin(Zr transforms theintegral to It/2 2 d J9=4a9sin2i f (10133)0 l—s1n lS1l'l iﬂ Finally, withthesubstitution u=tan1//, theintegral becomes °° du (1+u2)(l +uzcosz i) °° l coszi= iii . 1.144%,‘) du<l+u2 1+u2cos2i) (03)J9=40:9sinzif 0 Thislastfonn involves onlywell-known integrals, andtheﬁnalresult* is J9=21ra9(l— cosi) =2rr(oz9 —0:4,). (l0.l35) Thelastintegral (Eq.(10.l29c)), forJr,cannowbewritten as 2mk (J9+J¢)1 After performing theintegration, thisequation canbesolved fortheenergy EE Hinterms ofthethree action variables J¢,J9,Jr.Note thatJ¢andJ9canoccur inEonlyinthecombination J9+J¢,andhence thecorresponding frequencies v¢andv9must beequal, indicating adegeneracy. Thisresult hasnotinvolved the inverse-square lawnature ofthecentral force; anymotion produced byacentral force isatleast singly degenerate. Thedegeneracy isofcourse aconsequence ofthefactthatthemotion isconﬁned toaplane normal totheconstant angular momentum vector L.Motion inthisplane implies that9and¢arerelated to each other such thatas45goes through acomplete 21rperiod, 6varies through a complete cycle between thelimits (rr/2) :l:i.Hence, thefrequencies in6and¢ arenecessarily equal. Theintegral involved inEq.(l0.136) canbeevaluated byelementary means, buttheintegration ismost elegantly andquickly performed using thecomplex *Inevaluating theintegral ofthesecond termintheﬁnalintegrand ofEq.(l0.l34), ithasbeenassumed thatcosi ispositive. Thisisalways possible, since there isnopreferred direction forthezaxisinthe problem anditmaybechosen atwill.Ifcosi were negative, thesignof11¢inEq.(10.l35) would be positive. Forchanges inthesubsequent formulas, seeExercise 23. 10.8 TheKepler Problem inAction-angle Variables 469 contour integration method ofresidues. Forthebeneﬁt ofthose familiar withthis technique, weshall outline thesteps involved inintegrating Eq.(l0.136). Bound motion canoccur only when Eisnegative (cf.Section 3.3), andsince theinte- grand isequal top,—mi,thelimits ofthemotion aredeﬁned bytheroots r1and T2oftheexpression inthesquare rootsign. Ifr1istheinner bound, asinFig.3.6, acomplete cycle ofrinvolves going from r1tor2andthenback again tor1.On theoutward halfofthejoumey, from r1tor2,p,ispositive andwemust take thepositive square root. However, ontheretum triptor1,p,isnegative andthe square rootmust likewise benegative. Theintegration thusinvolves bothbranches ofadouble-valued function, withr1and7'2asthebranch points. Consequently, thecomplex plane canberepresented asoneofthesheets ofaRiemann surface, slitalong therealaxisfrom r1toV2(asindicated inFig.10.5). Since thepath ofintegration encloses thelinebetween thebranch points r1 and7'2,themethod ofresidues cannot beapplied directly. However, wemayalso consider thepathasenclosing alltherestofthecomplex plane, thedirection of integration nowbeing inthereverse (clockwise) direction. Theintegrand issingle- valued inthisregion, andthere isnownobartotheapplication ofthemethod of residues. Only twosingular points arepresent, namely, theorigin andinﬁnity, and theintegration pathcanbedistorted intotwoclockwise circles enclosing these twopoints. Now, thesigninfront ofthesquare rootintheintegrand must beneg- ative fortheregion along therealaxisbelow r1,ascanbeseenbyexamining the behavior ofthefunction intheneighborhood ofr1.Iftheintegrand isrepresented as l 2B C—A-l———-——2.r r R()=—~/-—-C. Above T2,thesignofthesquare rootontherealaxisisfound tobepositive, andtheresidue isobtained bythestandard technique ofchanging thevariable of integration toz=r'1:theresidue attheorigin is —‘¢i2\/A+2Bz—Cz2 dz. (lO.137) Z Negative ---- PositiveO-——————— ——i square root r] ++++ r2 square root FIGURE 10.5 Thecomplex rplane intheneighborhood oftherealaxis; showing the paths ofintegration occurring intheevaluation ofJ. 70 Chapter 1OHamilton-Jacobi Theory andAction-Angle Variables Expansion about z=0nowfurnishes theresidue R_ B O0 1 H ' Thetotalintegral is—2rri times thesumoftheresidues: J,=Zrri(V-C + . (l0.l38) or,upon substituting thecoefﬁcients A,B,andC: 2J,=—(J9+J9)+rrk,l:%. (10.139) Equation (10.139) supplies thefunctional dependence ofHupon theaction variables; forsolving forE,wehave 27z'2mk2 r 6 45 Note that,aspredicted, J9andJ4,occur onlyinthecombination J9+J¢.More than that, allthree oftheaction variables appear only intheform J,+J9+ J¢.Hence. allofthefrequencies areequal; themotion iscompletely degenerate. This result could alsohave been predicted beforehand, forweknow thatwith aninverse-square lawofforce theorbit isclosed fornegative energies. With a closed orbit, themotion issimply periodic andtherefore, inthiscase, completely degenerate. Ifthecentral force contained anr"3term, suchasisprovided byﬁrst- order relativistic corrections, thentheorbit isnolonger closed butisintheform ofaprecessing ellipse. Oneofthedegeneracies willberemoved inthiscase, but themotion isstillsingly degenerate, since v9=v¢forallcentral forces. Theone frequency forthemotion hereisgiven by arr an an 41k2v= = = = 7’m3. (10141) Ifweevaluate thesumoftheJ’sinterms oftheenergy from Eq.(10.l40) the period oftheorbit is 1=rrk%. (1o.142) This formula fortheperiod agrees with Kepler’s third law,Eq.(3.71), ifitis remembered thatthesemimajor axisaisequal to—k/2E. Thedegenerate frequencies may beeliminated bycanonical transformation toanewsetofaction-angle variables, following theprocedure outlined inthe previous section. Expressing thedegeneracy conditions as v¢—v9=0, v9-v,=O, 10.8 TheKepler Problem inAction-angle Variables 471 theappropriate generating function is F=(w¢—wo)J1+(we —wr)J2 +wrJ3- (10-143) Thenewangle variables are w1=w¢—w9 ‘(D2 2- ‘ U);-, w3=w,, (l0.l44) and,asplanned, twoofthenewfrequencies, v1andv2,arezero. Wecanobtain thenewaction variables from thetransfomration equations J¢=J|, h=h—h b=h—h which yields therelations J1=J45, J;=J9+J9, (10.l45) J3=J¢+J9+J,. Interms ofthese transformed variables theHamiltonian appears as 2 2 H=-2'-‘Ii, (10146)J3 aform involving onlythataction variable forwhich thecorresponding frequency isdifferent from zero. Ifwearewilling touse,from thestart, ourapriori knowledge thatthemotion forthebound Kepler problem isaparticular closed orbit inaplane, thentheinte- grals forJ9andJ,canbeevaluated veryquickly andsimply. FortheJ9integral, wecanapply thefollowing procedure. Itwillberecalled thatwhen thedeﬁning equations forthegeneralized coordinates donotinvolve timeexplicitly, then(cf. Eq.(8.20) andthematerial following (8.20)) Piéi=Zlzéiér =2T- Knowing thatthemotion isconﬁned toaplane, wecanexpress thekinetic energy Teither inspherical polar coordinates orintheplane polar coordinates (r,1/r).It follows, then, that 21"=11.»+P09+91¢=9.»+pi. (10141) Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables where p(El)isthemagnitude ofthetotalangular momentum. Hence, thedeﬁ- nition forJ9canalsobewritten as J9Eyl99d6=(£941) -y§p¢d¢. (10148) Because thefrequencies for6and¢areequal, both¢and1,0varyby2rras6goes through acomplete cycle oflibration, andtheintegrals deﬁning J9reduce to J0=2TF(P -19¢)=2Tl'(0le —<1¢)~ inagreement withEq.(10.135). Theintegral forJ,,Eq.(10.136), wasevaluated inorder toobtain HEEin terms ofthethree action variables. Ifweusethefactthattheclosed elliptical orbit inthebound Kepler problem issuch thatthefrequency forristhesame asthat for6and¢,thenthefunctional dependence ofHonJcanalsobeobtained from Eq.(10.l47). Ineffect thenweareevaluating J,inadifferent way. Thevirial theorem forthebound orbits intheKepler problem saysthat(cf.Eq.(3.30)) V=—2T, where thebardenotes anaverage overasingle complete period ofthemotion. It follows that HE13'='1_'+T/' =-T. (10.149) Integrating Eq.(10.I47)withrespect totimeoveracomplete period ofmotion we have 2TT=J,-l—J9-l—J¢=J3, (10.l50) 3 where v3isthefrequency ofthemotion, thatis,thereciprocal oftheperiod. Combining Eqs.(10.149)and(l0.150) leads totherelation 2v31arr-_=-=_-, 10.151)J3HHdJ3 ( ' where usehasbeen made ofEq.(10.105). Equation (10.151) isineffect adiffer- ential equation forthefunctional behavior ofHonJ3.Integration oftheequation immediately leads tothesolution H=2, (10.l52) 1;where Disaconstant thatcannot involve anyoftheJ’s,andmust therefore de- pend onlyupon mandk.Hence, wecanevaluate Dbyconsidering theelementary caseofacircular orbit, ofradius rg,forwhich J,=0andJ3=2rrp. Thetotal energy ishere 10.8 TheKepler Problem inAction-angle Variables 473 kH=—2— (10.153) ro (ascanmost immediately been seen from thevirial theorem). Further, thecon- dition forcircularity, Eq.(3.40), canbewritten fortheinverse-square force law as k P2 J3 3=m=T.» “‘“5“’ 0 0 0 Eliminating r9between Eqs.(l0.l53) and(l0.l54) leads to 22k2 H=_l’;’_. (10155)J3 Thisresult hasbeen derived onlyforcircular orbits. ButEq.(l0.152) saysitmust alsobecorrect forallbound orbits oftheKepler problem, andindeed itisidentical withEq.(lO.l46). Thus, iftheexistence ofasingle period forallcoordinates is taken asknown beforehand, itispossible toobtain H(J)without direct evaluation ofthecircuital integrals. Inanyproblem with three degrees offreedom, there must ofcourse besix constants ofmotion. Ithaspreviously been pointed outthatintheKepler problem ﬁveofthese arealgebraic functions ofthecoordinates andmomenta anddescribe thenature oftheorbit inspace, andonly thelastrefers totheposition ofthe particle intheorbitatagiven time(cf.Sections 3.7to3.9).Itiseasytoseethat ﬁveparameters areneeded tocompletely specify, say,theelliptical orbit ofthe bound Kepler problem inspace. Since themotion isinaplane, twoconstants are needed todescribe theorientation ofthatplane inspace. Oneconstant isrequired togivethescale oftheellipse, forexample, thesemimajor axisa,andtheother theshape oftheellipse, say.through theeccentricity e.Finally, theﬁfthparameter mustspecify theorientation oftheellipse relative tosome arbitrary direction in theorbital plane. Theclassical astronomical elements oftheorbit provide theorbital parameters ahnost directly intheform given above. Two oftheangles appearing inthese elements haveunfamiliar buttime-honored names. Their deﬁnitions, andfunc- tions asorbital parameters, canbestbeseenfrom adiagram, such asisgiven in Fig.10.6. Here xyzdeﬁnes thechosen setofaxes ﬁxed inspace, andtheunit vector ncharacterizes thenormal totheorbital plane. Theintersection between thexyplane andtheorbital plane iscalled thelineofnodes. There aretwopoints onthelineofnodes atwhich theelliptical orbit intersects thexyplane; thepoint atwhich theparticle enters from below intotheupper hemisphere (orgoesfrom the“southem” tothe“northem” hemispheres) isknown astheascending node. In Fig.10.6, theportion oftheorbit inthesouthem hemisphere isshown, forclarity, asadashed line.Thedot-dashed lineONisaportion ofthelineofnodes contain- ingtheascending node. Wecanmeasure thedirection ofONinthexyplane by theangle x0N, which iscustomarily denoted byS2,andisknown asthelongitude 474 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables Z C/n Z 1 /’If / // B 0y60 st\\ .NX A»"” \ FIGURE 10.6 Angular elements oftheorbit inthebound Kepler problem. oftheascending node. Finally, ifCdenotes thepoint ofperiapsis intheorbit, thentheangle N0Cintheorbital plane isdenoted bytoandiscalled theargu- ment oftheperihelion.* Themore familiar angle i,introduced inEq.(10.13 1),is initsastronomical usage known astheinclination oftheorbit. Oneusual setof astronomical elements therefore consists ofthesixconstants i,S2,a.e,w,T, where thelastone,T,isthetimeofpassage through theperiapsis point. Ofthe remaining ﬁve,theﬁrsttwodeﬁne theorientation oftheorbital plane inspace, while a,e,andcodirectly specify thescale, shape, andorientation oftheelliptic orbit, respectively. Theaction-angle variable treatment oftheKepler problem alsoleads toﬁve algebraic constants ofthemotion. Three ofthem areobvious asthethree constant action variables, J1,J2,andJ3.Theremaining twoaretheangle variables wl andwg,which areconstants, because theircorresponding frequencies arezero. It must therefore bepossible toexpress theﬁveconstants J1,J2,J3,wl,andwgin terms oftheclassical orbital elements i,S2,a,e,andw,andviceversa. Some of these interrelations areimmediately obvious. From Eqs.(10.145)and(10.135) it follows that J2=2rra9 E2rrl, (l0.l56) andhence, byEq.(10.l31), JT‘=cosi. (10.15?)2 Asiswellknown, thesemimajor axisaisafunction onlyofthetotalenergy E (cf.Eq.(3.61)) andtherefore, byEq.(10.l46), aisgiven directly interms ofJ3: *This terminology appears tobecommonly used even fororbits thatarenotaround thesun.The proper termfororbits about starsisperiastra; forEarth-orbiting satellites, thistennis perigee. 10.8 TheKepler Problem inAction-angle Variables 475 —k—J52 10158)“T2E—4J'r2mk' (' Interms ofJ2,Eq.(3.62) fortheeccentricities canbewritten as /1% e=1-—-,41r2mka 2 9=‘/1- . (10.159) Itremains onlytorelate theangle variables wlandw2totheclassic orbital elements. Obviously, theymust involve S2andw.Infact,itcanbeshown thatfor suitable choice ofadditive constants ofintegration theyareindeed proportional to Qandco,respectively. Thiswillbedemonstrated forw1;theidentiﬁcation ofw2 willbeleftasanexercise. Theequation oftransformation deﬁning w1is,byEq.(l0.l27), _a_“1w]-8J1.OI‘ Itcanbeseenfromtheseparated formofW,Eq.(10.71), thatWcanbewritten asthesumofindeﬁnite integrals: W:/p¢d¢+/p9d6+/prdr. (l0.l60) Aswehave seenfrom thediscussion onJ,,theradial momenttun prdoesnot involve J1,butonly J3(through E)andthecombination J9+J9,=J2.Only the ﬁrsttwointegrals aretherefore involved inthederivative withrespect toJ1.By Eq.(l0.l30). Jp¢=6,,= (10.161) andbyEq.(10.74), withthehelpofEqs.(l0.156) and(l0.16l), l <12 1l J2_ 2_._L -_ 2____1_ S1116 S1Il6 Itturns outthatinorder torelate wltotheascending node, itisnecessary to choose thenegative signofthesquare root.* Theangular variable w1istherefore determined by *Note thatwhen theparticle passes through theascending node(cf.Fig.10.6)6isdecreasing and thecorresponding momentum isnegative. Incalculating J9,itwasnotnecessary toworry about the choice ofsignbecause ingoing through acomplete cyclebothsignsareencountered. 4 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables J d6 245+21I , Tr 7: sinz6,/J5‘ —J12csc26w1= OI‘ 2rrw -¢+cosi/ d6 1 sin26~/1 —cos2icsc26 __¢+/ coticsc26d6 \/1—cot2icot26' Byachange ofvariable tou,deﬁned through sinu =coticot6, (l0.l63) theintegration canbeperformed trivially, andtheexpression forwrreduces to 2rrw1=¢—u. (l0.l64) Theangle coordinate ¢istheazimuthal angle oftheprojection onthexyplane measured relative tothexaxis. Clearly, from Eq.(l0.l63) uisafunction ofthe polar angle 6oftheparticle. Butwhat isitsgeometrical signiﬁcance? Wecan seewhat uisbyreference toNapier’s rules* asapplied tothespherical triangle deﬁned bythelineofnodes, theradius vector, andtheprojection oftheradius vector onthexyplane. However, itmaybemore satisfying toindulge inalittle trigonometric manipulation andderive therelation abinitio. InFig.10.7, theline ONisthelineofnodes. ORisthelineoftheradius vector atsome time, andthe dotted lineOPistheprojection oftheradius vector onthexyplane. Theangle thatOPmakes withthexaxisistheazimuth angle ¢.Wecontend thatuisthe angle OPmakes withthelineofnodes. Toprove this,imagine aplane normal both tothexyplane andtothelineofnodes, which intersects theradius vector atunitdistance OBfrom theorigin O.Thepoints ofintersection A,B,andCof thisplane, withthethree lines from theorigin, deﬁne with theorigin fourright triangles. Since OBhasunitlength, itfollows thatBC=cos6 andtherefore AC=cos6coti.Ontheother hand, OC=sin6andtherefore itisalsotruethat AC=sin6sinu.Hence. sinu=coticot6, which isidentical withEq.(l0.l63) andproves thestipulated identiﬁcation oftheangle u.Figure 10.7shows clearly thatthedifference between ¢andumust beS2,sothat 2rrw1= S2. (l0.l65) Inasimilar fashion, wecanidentify thephysical nature oftheconstant w2.Of theintegrals making upW,Eq.(10.160), thetwoover6andrcontain J2and *Foranexplanation ofNapier’s rulesforspherical triangles, seehandbooks suchastheHandbook of Mathematical Tables (Chemical Rubber Publishing Co.)orHandbook ofApplied Mathematics (Van Nostrand-Reinhold). 10.8 TheKepler Problem inAction-angle Variables 477 Z B R I 509O =1_Z'X<‘-;j//\\y’\// JA‘< "0 N FIGURE 10.7 Diagram illustrating angles appearing inaction-angle treatment ofthe Kepler problem. aretherefore involved inﬁnding w2.After differentiation withrespect toJ2,the integral over6canbeperformed bythesame typeoftrigonometric substitution as employed forw|.Thecorresponding integral overrcanbecarried outinanumber ofways, most directly byusing theorbit equation forrinterms ofthepolar coordinate angle intheorbital plane. Bysuitable choice ofthearbitrary lower limit ofintegration, itcanthusbefound that211'w2isthedifference between two angles intheorbital plane, oneofwhich istheangle oftheradius vector relative tothelineofnodes andtheother isthesame angle butrelative tothelineofthe periapsis. Inother words, 2rrw2istheargument oftheperihelion: 2rtw2 =co. (10.l66) Detailed derivation islefttooneoftheexercises. Themethod ofaction-angle variables iscertainly notthequickest waytosolve theKepler problem, andthepractical usefulness ofthesetofvariables isnotob- vious. However, their value haslongbeen demonstrated incelestial mechanics, where theyappear under theguise oftheDelaunay variables.* Aswillbeseenin Section 12.2, theyprovide thenatural orbital elements thatcanbeusedinpertur- bation theory, todescribe themodiﬁcations ofthenominal Kepler orbits produced bysmall deviations oftheforce from theinverse-square law.Many ofthebasic studies onpossible perturbations ofsatellite orbits were carried outinterms of theaction-angle variables. *Ascustomarily deﬁned, theDelaunay variables differ from the(J,,w,)setbymultiplicative con- stants. 478 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables DERIVATIONS 1. 2 3. 4.Foraconservative system show thatbysolving anappropriate partial differential equation wecanconstruct acanonical transformation suchthatthenewHamiltonian isafunction ofthenewcoordinates only. (Donotusetheexchange transfonnation, F1.)Show howaformal solution tothemotion ofthesystem isgiven interms ofthe newcoordinates andmomenta. Inthetext, theHamilton-Jacobi equation forSwasobtained byseeking acon- tacttransformation from thecanonical coordinates (q,p)totheconstants (oz,ﬁ). Conversely, ifS(q,-,oz,-,t)isanycomplete solution oftheHamilton-Jacobi equa- tion(l0.3), show thatthesetofvariables (Q,-,p,-)deﬁned byEqs.(10.7) and(10.8) arecanonical variables, thatis,thattheysatisfy Hamilton’s equations. Intheaction-angle formalism, thearguments ofHamilton’s characteristic function are theoriginal coordinates qkandtheaction variables Jk.Inthecaseofdegeneracy. a subsequent canonical transformation ismade tonewvariables (wg,Ji')from (wk,Jk), inorder toreplace thedegeneracies byzerofrequencies. Byconsidering each Jka function oftheJ‘-'quantities asdeﬁned byEq.(l0.126), show thatitremains truethat aw ,T’l,=w‘-. Theso-called Poincare elements oftheKepler orbits canbewritten as w1+I02+I03, J¢, J J.;rcos27r(w2+w1), ;rs1n2zr(wg +w1), J J -0cos21rw1, —6sin2nw1.21 71' Show thattheyfonn acanonical setofcoordinates, withthenewcoordinates forming theleft-hand column, theirconjugate momenta being given ontheright-hand side. EXERCISES 5.Show thatthefunction 6.S=$012 +0:2)cotwt —mwqa cscwt isasolution oftheHamilton-Jacobi forHamilton’s principal function forthelinear harmonic oscillator with l H=T(p2 +mzwzqz).m Show thatthisfunction generates acorrect solution tothemotion oftheharmonic oscillator. Acharged particle isconstrained tomove inaplane under theinﬂuence ofacentral force potential (nonelectromagnetic) V=%kr2, andaconstant magnetic ﬁeld B Exercises 479 7. 8. 9. 10. 11. 12.perpendicular totheplane, sothat A=%Bxr SetuptheHamilton-Jacobi equation forHamilton’s characteristic function inplane polar coordinates. Separate theequation andreduce ittoquadratures. Discuss the motion ifthecanonical momentum pgiszeroattimet=0. (a)Asingle particle moves inspace under aconservative potential. Setupthe Harnilton—Jacobi equation inellipsoidal coordinates u,v,¢deﬁned interms of theusual cylindrical coordinates r,z,42bytheequations r=asinhvsinu, z=acoshvcosu. Forwhat forms ofV(u, v,¢)istheequation separable? (b)Usetheresults ofpart(a)toreduce toquadratures theproblem ofapoint particle ofmass mmoving inthegravitational ﬁeldoftwounequal mass points ﬁxed on thezaxisadistance 2aapart. Suppose thepotential inaproblem ofonedegree offreedom islinearly dependent upontime,suchthattheHamiltonian hastheform 2PH=——-A, 2m mtx where Aisaconstant. Solve thedynamical problem bymeans ofHamilton’s principal function, under theinitial conditions: =0,x=0,p=mvo. Setuptheplane Kepler problem interms ofthegeneralized coordinates u=r+x, v=r—x. Obtain theHamilton-Jacobi equation interms ofthese coordinates, andreduce itto quadratures (atleast). Oneendofauniform rodoflength 2lamdmassmrestsagainst asmooth horizontal ﬂoor andtheother against asmooth vertical surface. Assuming thattherodiscon- strained tomove under gravity withitsends always incontact withthesurfaces, use theHamilton-Jacobi equations toreduce thesolution oftheproblem toquadratures. Aparticle isconstrained tomove onaroller coaster, theequation ofwhose curve is z=Acoszﬁ.A There istheusual constant downward force ofgravity. Discuss thesystem trajectories inphase space under allpossible initial conditions, describing thephase space orbits inasmuch detail asyoucan,paying special attention toturning points andtransitions between different types ofmotion. Aparticle ofmass mmoves inaplane inasquare wellpotential: V(r)=—Vg O<r<r0, =O r>r0. 4 Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables (a)Under what initial conditions canthemethod ofaction-angle variables beapplied? (b)Assuming these conditions hold, usethemethod ofaction-angle variables toﬁnd thefrequencies ofthemotion. Aparticle moves inperiodic motion inonedimension under theinﬂuence ofapoten- tialV(x) =F|x|,where Fisaconstant. Using action-angle variables, ﬁndtheperiod ofthemotion asafunction oftheparticle’s energy. Aparticle ofmass mmoves inonedimension under apotential V=—k/|x|. For energies thatarenegative, themotion isbounded andoscillatory. Bythemethod of action-angle variables, ﬁndanexpression fortheperiod ofmotion asafunction ofthe particle‘s energy. Aparticle ofmass mmoves inonedimension subject tothepotential G . Obtain anintegral expression forHamilton’s characteristic function. Under what con- ditions canaction-angle variables beused? Assuming these aremet.ﬁndthefrequency ofoscillation bytheaction-angle method. (Theintegral forJcanbeevaluated byma- nipulating theintegrand sothatthesquare rootappears inthedenominator.) Check yourresult inthelimit ofoscillations ofsmall amplitude. Aparticle ofmass misconstrained tomove onacurve inthevertical plane deﬁned bytheparametric equations y=l(l—cos2¢), x=l(2¢+sin2¢). There istheusual constant gravitational force acting inthevertical ydirection. By themethod ofaction-angle variables. ﬁndthefrequency ofoscillation forallinitial conditions suchthatthemaximum of¢islessthanorequal to7!/4. Solve theproblem ofthemotion ofapoint projectile inavertical plane. using the Hamilton-Jacobi method. Findboththeequation ofthetrajectory andthedependence ofthecoordinates ontime, assuming theprojectile isﬁredoffattimet=0from the origin withthevelocity v0,making anangle ozwiththehorizontal. Forthesystem described inExercise 12ofChapter 6,ﬁndalinear point transformation tovariables inwhich theHamilton-Jacobi equation isseparable. Byuseoftheaction- angle variables, ﬁndtheeigenfrequencies ofthesystem. Athree-dimensional harmonic oscillator hastheforce constant klinthex-andy- directions andk3inthez-direction. Using cylindrical coordinates (with theaxisof thecylinder inthezdirection), describe themotion interms ofthecorresponding action-angle variables, showing howthefrequencies canbeobtained. Transform to the“proper” action-angle variables toeliminate degenerate frequencies. Find thefrequencies ofathree-dimensional harmonic oscillator with unequal force constants using themethod ofaction-angle variables. Obtain thesolution foreach Cartesian coordinate andconjugate momentum asfunctions oftheaction-angle vari- ables. Exercises 481 21.(a) 22. 23. 24 25 26 27 28.Intheharmonic oscillator ofExercise 20,allow allthefrequencies tobecome equal (isotropic oscillator) sothatthemotion iscompletely degenerate. Transform tothe“proper” action-angle variables, expressing theenergy interms ofonlyone oftheaction variables. (b)Solve theproblem oftheisotropic oscillator inaction-angle variables using spher- icalpolar coordinates. Transform again toproper action-angle variables andcom- parewith theresult ofpart(a).Arethetwosetsofproper variables thesame? What aretheirphysical signiﬁcances? Thisproblem illustrates thefeasibility of separating adegenerate motion inmore thanonesetofcoordinates. Thenonde- generate oscillator canbeseparated only inCartesian coordinates, notinpolar coordinates. Themotion ofadegenerate plane harmonic oscillator canbeseparated inanyCarte- siancoordinate system. Obtain therelations between thetwosetsofaction-angle vari- ables corresponding totwoCartesian systems ofaxesmaking anangle 0witheach other. Notethatthetransformation between thetwosetsisnottheorthogonal trans- formation oftherotation. (a)Evaluate theJ9integral intheKepler problem bythemethod ofcomplex con- tourintegration. Togettheintegral intoauseful form, itissuggested thatthe substitution cos9=xsinimight bemade. (b)Verify theintegration procedure used forJ9inthetext,carrying outtheﬁnal integrations inEq.(10.134). (c)Follow theconsequences oftheinclination being greater than90°,thatis,cosi negative. Inparticular. what arethechanges inEq.(10.135), inthecanonical transformations tozerofrequencies andtherefore inEqs.(10.145)? Canyouwrite these equations insuch aform thattheyarevalid whether cosi ispositive or negative? Justify youranswer. Evaluate theintegral forJ,intheKepler problem byelementary means. Thisincludes using tables ofintegrals, butifso,explicit anddetailed references should begiven to thetables used. Show, butthemethod outlined inthetext(oranyother), that2n1.02isw,theargument oftheperiapsis, inthethree-dimensional Kepler problem. Setuptheproblem oftheheavy symmetrical top.withonepoint ﬁxed, inthe Hamilton-Jacobi method, andobtain theformal solution tothemotion asgiven byEq.(5.63). Describe thephenomenon ofsmall radial oscillations about steady circular motion in acentral force potential asaone-dimensional problem intheaction-angle formalism. With asuitable Taylor series expansion ofthepotential, ﬁndtheperiod ofthesmall oscillations. Express themotion interms ofJanditsconjugate angle variable. Setuptheproblem oftherelativistic Kepler motion inaction-angle variables, using theHamiltonian intheform given byEq.(8.54). Show inparticular thatthetotal energy (including restmass) isgiven by Chapter 10Hamilton-Jacobi Theory andAction-Angle Variables E_ 1 mc2 I+ 4,,2k2 ' [(J’—J’)c+ J’2¢2-4191812 3 2 2 Note thatthedegeneracy hasbeen partly lifted, because theorbit isnolonger closed, butisstillconﬁned toaplane. Inthelimit ascapproaches inﬁnity, show thatthis reduces toEq.(lO.l46). CHAPTER Classical Chaos Wehaveintheprevious chapters devoted most ofourattention tointegrable prob- lems, thatis,problems inwhich theequations ofmotion canbeintegrated to provide solutions inclosed form. Forexample, inSections 3.7and3.8wefound exact solutions forthetwo-body, inverse-square force lawproblem byintegrations oftheequations ofmotion. Formany physical situations exact solutions cannot befound. Inthenextchapter weshall examine problems withpotentials thatcan bebroken intoamainintegrable partandaweaker additional partthatrenders the problem nonintegrable, butthatcanbetaken intoaccount byapplying classical perturbation theory. Aweak interaction termmight, forexample, couple together twoequations ofmotion sothevariables arenolonger separable. Thepresent chapter deals withsome situations involving perturbations andlackofintegrabil- itythatcannot beconveniently handled byclassical perturbation theory. lftheinteraction termisnolonger “small” inthesense ofclassical perturbation theory (cf.Section 12.1), thesolutions maybecome complex anddiffer consider- ablyfromthose oftheuncoupled equations. Insome cases newsolutions appear thatcannot begenerated fromtheuncoupled equations. These solutions areoften wellbehaved inthesense thatasmall change intheinitial conditions brings about onlyasmall change inthemotion. When thisisthecase,thesolutions arereferred toasregular ornormal. There alsoexist cases inwhich themotion evolves inen- tirely different ways evenfornearly identical starting circumstances. Solutions ofthistypearereferred toaschaotic. Itisimportant topoint outthatthischaos stillinvolves deterministic solutions todeterministic equations. They arecalled chaotic because, although deterministic, theyarenotpredictable because theyare highly sensitive toinitial conditions. Ifweconsider twobounded solutions in thenonchaotic regime thatstartnearby within asmall region ofphase space, the phase space region covered bythesolutions atalatertimewillstillberelatively small andcompact asexpected fromLiouville’s theorem (cf.Section 9.9).Inthe chaotic regime, thesector ofphase space covered bythese solutions willcontinu- allydisperse inoneormore directions withthepassage oftime. Chaos isatypeofmotion thatliesbetween theregular detemrinistic tra- jectories arising from solutions ofintegrable equations andastateofnoise or unpredictable stochastic behavior characterized bycomplete randomness. Chaos exhibits extensive randomness tempered bysome regularity. Chaotic trajectories arise from themotion ofnonlinear systems, which isnonperiodic, butstillsome- what predictable. Speciﬁc solutions change exponentially inresponse tosmall 483 484 11.1 IChapter llClassical Chaos changes intheinitial conditions. Inthischapter weshallexamine some ofthe properties ofthischaotic motion, andgiveexamples ofit. This chapter isonly anintroduction tothesubject ofchaos; itpresents the general principles thatunderlie chaotic motion. Webegin withadiscussion ofpe- riodic motion ingeneral, andwediscuss ways totransform ittocircular motions inphase space. Then weaddperturbations thatdisturb theregular motion, and examine theKolrnogorov-Arnold—Moser (KAM) theorem, which provides con- ditions forthebreakdown ofregularity. Weintroduce theLiapunov exponent as aquantitative measure ofchaos through dispersion inphase space anduseitto summarize some predictions concerning thestability ofthesolar system. Therole played byattractors innonchaotic motion isexplained, aswellasthecharacter- istics ofthestrange attractor involved inchaos. Ournexttaskistoshow howto conveniently display theregularities andirregularities ofmotion withtheaidof Poincaré sections. Wethenexamine themotions ofindependent oscillators and, using theHénon—Heiles Hamiltonian asanexample, weintroduce theeffect ofa perturbation interaction anddemonstrate thatorbits thatareinitially regular will, when subject toacontinual increase inthemagnitude oftheperturbing coupling potential, gradually transform toastateofchaos. Thelogistic equation istreated indetail andusedtoexplain bifurcations andinvariants, including auniversal constant associated withchaos. Some briefcomments aremade onnonintegral dimensionality andfractals before closing. PERIODIC MOTION InChapter 3,wediscussed bounded motion withanemphasis onmotion inwhich theorbits areclosed; thatis,thetrajectory repeats itself every period. Thesim- pleharmonic oscillator andtheKepler problem areexamples ofclosed periodic motion. Inthelatter casethere aretwoperiodicities, theradial coordinate rvaries fromitsminimum value r1atperihelion toitsmaximum r3ataphelion andthen backtoperihelion during thetimethattheangular motion goesfrom 0=Oto 6=2n.Hence, theperiods fortheradial andtheangular motions arethesame. These periods exemplify twotypes ofmotion thataredegenerate. Weknow from Section 3.2thattherateofchange, 9,depends upontheradial distance r eh)=L, (3.8)mr2 andtherateofchange ofrisacomplicated analytical closed-fonn expression. Theangular speed v9=rt?depends upon theangle 0inthemanner sketched in Fig.3.17. InChapter 3.weshowed howtointegrate theequations ofmotion to obtain thepolar coordinate equation fortheorbit a(1—e2)=-——, 3.64rl+ecost9 () 11.1 Periodic Motion 485 where theorigin oftheangular coordinate, 0=O,ischosen atperihelion. Fig- ures3.16and3.17present phase space plotsinthev,versus randv9versus 0 planes, respectively, forKepler orbits withthesame energy anddifferent eccen- tricities. InSection 10.6, wefound thataconvenient waytorepresent periodic motion is tocarryoutavariant oftheHamilton-Jacobi procedure andtransform theHamil- tonian toaction-angle variables. Thenewmomentum, called theaction variable J=fpdq isaconstant ofthemotion, andthenewconjugate coordinate w depends linearly upon thetime: w=wt+19.Weareinterested inaHamilto- nian'H(q1, q2,...,qn;pl,pg,...,p,,;r)ofaconservative system containing several variables p,-,q,-,which exhibits bounded motion. IfthisHamiltonian His transfomied toanewsetofcanonical variables P,-,Q;inwhich alloftheQ,’sare cyclic, thatis,H=7-((P1, P2,....P,,;t),thenHamilton’s equations (8.18) can bereadily integrated toprovide thesolution Qr(t)=w(l)=wil+l9i P10) =Pi(0)—0lr. (ll-1) where the2nconstants ofintegration 18,-and01,-areinvariants ofthemotion. When canonical transfonnations exist thatprovide thistypeofsolution, thentheHamil- tonian issaidtobeintegrable. Thissolution issimilar totheaction-angle variables discussed inChapter 10.Forthemotion toremain bounded, thatis,conﬁned to aﬁnite region ofphase space, thecoordinates w(t), which aregrowing linearly withthetime, mustbearguments ofbounded ftmctions, andinmany cases, they willbearguments ofperiodic functions, asisthecasewiththeradial variable rof Eq.(3.64) quoted above. InSections 10.2and10.7, weshowed thattheHamiltonian ofaharmonic os- cillator canundergo acanonical transfonnation toconjugate coordinates andmo- menta with thetime dependencies ofEqs. (11.1). Itfollows thataHamiltonian withthecoordinates Q,-(r) andP,-(2) canbetransfonned tothatofaharmonic oscillator instandard fonn, withthecoordinates qlf,pf.Forthecasen=2,this gives '2 '2p 1 Ip 1 IH=ﬁll+imiwiqlz +i+Emir/»§q22. (11-2) which corresponds toasystem oftwouncoupled harmonic oscillators with a Hamiltonian thatequals thetotalenergy 'H='H1+'H3=E1, (11.3) where wehave, inaction variable notation (cf.(10.94)) J J7-l1= ‘—""=E1 and 7-£2=La”=E2. (11.4)2rr 21: 486 Chapter l1Classical Chaos Tovisualize themotion, wecanexpress each individual oscillator innormalized coordinates I Pi§ and q,-=>q;(%mw,?)‘/2. (11.5)I Each part'H,-ofHamiltonian (11.3) corresponds totheequation ofacircle inits p,-.q,-plane ofphase space p%+q?=E.-- <11-6) Figure 11.1illustrates these circles bypresenting constant total energy ET= E1+E2plots inthepl,qlplane forE1<E3(small circle), E1~E2(medium- sizecircle) andE1>E2(large circle). This representation ofanoscillator byunifonn circular motion provides us withaneasywaytopicture themotion associated withthedouble oscilla- tor(11.2), where forconvenience weselect C02>>(01.Consider themovement ofthelow-frequency oscillator anproceeding along acircle oflarge radius in thepl,qlplane andthenplotthetrajectory ofthehigh-frequency oscillator (02 along asmall circle inapg,qgplane drawn perpendicular tothecircle ofmland centered onitscircumference, asshown inFig.11.2forthecase(02>>wl.The jointmotion inthetotalphase space isaspiraling ofthesystem point along the surface ofatorus, asillustrated intheﬁgure. Ifthefrequency mgisamultiple of (v1,meaning thattheirratio isaninteger 2=n, (11.7)wl P1 E,>E2 E,~E2 E1<E2 '11 FIGURE 11.1 Circular orbits inthepl,qlphase space forthree values oftheenergy ratio E1/E2oftwouncoupled harmonic oscillators plotted forthesame totalenergy E1= E1+E2. 11.2 I11.2 Perturbations andtheKo|mogorov—Arnold—Moser Theorem 487 P2 ‘I1 P l i-> v=mlrl ‘I1 FIGURE 11.2 Circular motions ofalow-frequency (C01)harmonic oscillator inthehor- izontal p1,q1 plane andofahigh-frequency (0)2>ml)harmonic oscillator intheuni- formly moving pg,qgvertical plane. Theoscillators areuncoupled, andtheresultant spi- raling motion ofthesecond oscillator generates atorus, asshown. thenthetrajectory willclose onitself andrepeat thesame pattem every period 1|=2n/an. More generally, ifthefrequencies arecommensurate, meaning that ninthisEq.(11.7) isarational number like thentheorbit willstillbeclosed, butitwilltraceoutmore thanonepatharound thep1,q1 circle before closing onitself. If,however, thefrequencies areincommensurate, meaning thatnin Eq.(11.7) isanirrational number, thenthetrajectory willnever close, butwill gradually cover thesurface ofthetorus, without everpassing through exactly the samepoint twice. Eventually, however, itwillpassarbitrarily closetoevery point onthestuface. Thisiscalled adense periodic orbit. Such anorbit isbounded and conﬁned toasurface, butitisnotclosed. Thisapproach canbegeneralized tomore thantwooscillators. Ifthere are three such oscillators with thefrequencies col,(4)2,and(03,thenthemotion will beconﬁned toathree-dimensional surface called a3-torus inthesix-dimensional pl,pg,p3,qj,q2,q3phase space. ForNoscillators, therewillbeanN-torus ina 2N-dimensional phase space. Itisnoteasytovisualize theN-toriforN>2. PERTURBATIONS AND THE KOLMOGOROV- ARNOLD-MOSER THEOREM Intherealworld wecanoften express thedynamics ofasystem interms ofanin- tegrable Hamiltonian perturbed byasmall interaction thatmakes itnonintegrable. Anexample isthemotion ofEarth inaKeplerian orbit around theSunprimarily perturbed bythepresence oftheplanets Mars andJupiter. Thisinteraction isso weak thatthere isverylittledisturbance ofEarth’s orbit. Weak interactions ofthis typearemost conveniently treated withtheaidofcanonical perturbation theory, which isexplained indetail inChapter 12.Thefollowing outline ofthemethod Chapter 11Classical Chaos discussed inSection 12.2issufﬁcient fortheconsideration ofchaos. References aregiven totheequations inChapter 12butreading thechapter isnotnecessary tofollow thearguments, sowehaveplaced thischapter ﬁrst. Weassume aHamiltonian Hinvolving adominant interaction arising froman integrable Hamiltonian H0forwhich thesolution isknown, plusanadditional interaction arising from asmall perturbation term AH H=H()+AH. (11.8) Itisconvenient tousethegenerating function S(q,P,t)=F2(q, P,t)intro- duced inSection 9.1totransform thedominant Hamiltonian tennHQfrom the phase space coordinates p,qtonewcoordinates P,Qofatransfonned Hamilto- nianK0(Q, P),thatisidentically zero,aswasillustrated intheHamilton-Jacobi approach ofChapter 10.Hamilton’s equations (10.1) forK0=0provide new coordinates andmomenta, Q0andP0,which areconstants ofthemotion. The sametransformation carried outforthetotalHamiltonian, H=H0+AH0, pro- vides atransformed Hamiltonian AK0, which canbeused toobtain ﬁrst-order corrections P1,Q1tothetimederivatives ofthecoordinates andmomenta via Hamilton’s equations (cf.Equation (12.4)) 3 - 3 - -5;AKo(P, Q)=Q1 EAKMP, Q)=—P1- (11-9) After differentiation, QandParereplaced inAK0bytheirunperturbed forms, thatis,byq=Q0andp=P0.These expressions (11.9) canbeintegrated overtimetogivetheﬁrst-order detennination ofQ=Q1andP=P1.The procedure provides uswithanewgenerating function S(Q1,P1,r),andhence a newperturbed Hamiltonian AK1,which canbeiterated togivethenexthigher- order terms Q2andP2,andsoon.Further cycles ofperturbation areobtained by iteration withtheaidofthefollowing relations (cf.Eq.(12.6)) withnosummation intended: 3 - 8 . §iAK1(Pi, Qt)=Qi+1, 3—Q—!_AKi(Pi. Qt)=—Pi+l- (11-10) Thus, wehaveasystematic canonical iteration technique forobtaining better and better approximations tothesolution when theperturbation AHispresent. This method canbecontinued tohigher order, asdiscussed inChapter 12. Wehaveseenthatperturbation theory provides uswithasolution when AHis small relative toH0,butthequestion arises astowhether theperturbed solution is stable, andwhether ornottheorbits willremain close totheunperturbed onesover longperiods oftime.Large perturbations canclearly disturb theregular motion. A theorem known astheKolmogorov—Amold—Moser (KAM) theorem provides the conditions forthebreakdown ofregularity. Thistheorem tellsusthat Ifthebounded motion ofanintegrable Hamiltonian HQisdisturbed byasmall perturbation, AH, thatmakes thetotal Hamiltonian, H=Hr)+AH, noninte- grable andiftwoconditions aresatisﬁed: 11.3 I11.3Attractors 489 (a)theperturbation AHissmall, and (b)thefrequencies anofH0areincommensurate, thenthemotion remains conﬁned toanN-torus, except foranegligible setof initial conditions thatresult inameandering trajectory ontheenergy surface. Thus, theperturbed orbits willbestable, only slightly altered inshape, andlo- calized inthesame region astheunperturbed ones. Another waytosaythisis toobserve thatforaperturbation oftheHamiltonian thatissufﬁciently small, most quasi-periodic orbits willonlyexperience minimal changes. Themethod of proof forthistheorem wasoriginally suggested byKolmogorov in1954, andthe proofs themselves, approached from different viewpoints, were worked outinde- pendently byArnold andbyMoser adecade later.Agreat dealofmathematical sophistication isneeded fortheproof, andreferences canbeconsulted forde- tails.* Forexample, thesecond condition (b)ofthetheorem ismathematically more complex thansimple incommensurability. Thecaveat “except foranegligible setofinitial conditions” introduces thepos- sibility ofinitial conditions forwhich thetheorem doesnothold. Thisisanalogous tothecaseofadifferential equation with well-behaved solutions overanentire domain except foroneormore singular points where thesolutions blow upto inﬁnity. Theexceptions aresofewthattheyhaveverylittleeffect onapplications. Chaos canoccur when KAM doesnothold. ATTRACTORS Theprevious section wasconcerned with anintegrable Hamiltonian HQbeing disturbed byasmall perturbation AH. Wefound thatstable orbits ofHQpersist asslightly modiﬁed butstillstable orbits ofthetotalHamiltonian, H=H0+AH. Another casetoconsider isthatofasystem inwhich theinitial conditions startthe motion onatrajectory thatdoesnotlieonastable pathbutthatevolves toward a particular ﬁxedpoint inphase space ortoward astable orbitinphase space called alimit cycle. Aﬁxed point ofthistypeaswellasalimit cycle areexamples of attractors. Ingeneral, anattractor isasetofpoints inphase space towhich thesolution ofanequation evolves longaftertransients have diedout.Itmight beapoint with dimension dg=O,atrajectory orlimitcycle orbit(cf.Fig.11.1)withdimension dA=1,orperhaps atoroidal surface ortoruswithdimension d,4=2.Forareg- ularattractor, theattractor dimension, dA,isaninteger thatislessthantheoverall dimensions ofthephase space. Inhigher dimensions, theattractors canbeN- dimensional tori,where d,4=2forthetorus generated bytheorbit inFig.11.2. There alsoexist somewhat bizarre types ofattractors called strange attractors, *See, forexample. H.Bai-Lin, Chaos, Singapore: World Science, 1984; E.A.Jackson, Perspectives ofNonlinear Dynamics, Cambridge, England: Cambridge University Press, 1989; L.E.Reich], The Transition toChaos, Berlin: Springer-Verlag, 1992. 0 Chapter 11Classical Chaos associated withchaos, which tendtobewidely dispersed rather thanlocalized in phase space. Inaddition, theyhavefractal dimensions—-in other words, dimen- sions thatarefractions orirrational numbers rather thanwhole numbers. These properties, aswellasthetennfractal dimension, arecounter-intuitive. Weshall clarify themeanings ofstrange attractors andfractal dimensions laterinthechap- ter. Anexample ofaﬁxed-point attractor istheequilibrium position ofapendu- lumatrest.Ifthependulum isoscillating while subject totheaction ofaweak frictional dragforce, thensuccessive oscillations willdecrease inamplitude until thependulum ﬁnally comes toastopatitsequilibrium position. Wesaythatthe motion isdrawn totheattractor. Ifthedragforce isaperturbation onthemain Hamiltonian, thenthemotion isunderdamped andthependulum undergoes many oscillations before stopping attheattractor point. Ifthedamping termexceeds the mainHamiltonian term, thenthemotion isoverdamped andthependulum fallsto restwithout undergoing anyoscillations. Either way,themotion ofthependulum ﬁndsitswaytotheattractor. Being apoint, itisclearthatthedimensionality of thisattractor iszero;d,(=0. Anexample ofalimitcycle typeofattractor isprovided bythevanderPol equation, dz dmT;—6(l—x2)?:+mw3x=FC0S amt, (11.11) which hasbeen employed todescribe oscillations inmechanical andelectrical systems, aswellascardiac rhythms. Ifwesete=0,thenwehaveadriven simple harmonic oscillator witharesonant frequency tooandadriving frequency (DD.If (ODisclose tocoo,thenthemotion repeats itself atthefrequency (ODoftheapplied force. IfF=0,thenthemotion willbesimple hamronic attheresonant frequency coo.Ifthe small damping terma(l—x2)dx/dt isincluded intheequation, thenthe motion willbedrawn toward thelimit cycle, which inthiscaseisacircle ofunit radius. Ifx2>1,thedamping ispositive andthemotion spirals inward toward thelimit cycle, while forx2<1,thedamping isnegative andthemotion spirals outward toward thelimit cycle. Both cases areshown inFig.l1.3a. Theﬁnal stateofmotion haslong-term stability since thedamping vanishes forx=1,and thesystem pointmoves along thecircular path,which byitsnature hasdimension dA=1.Ifeislargeenough, thedamping tennbecomes comparable inmagnitude totheother terms intheequation ofmotion, andthedamping stilldraws the trajectories toward thelimit cycle, butthecycle itself becomes distorted from a circular shape, asshown inFig.11.3b. Thedistortion inshape doesnotchange the dimension ofthepath, which remains d,4=1.Inaddition, thestrong damping causes thepreviously simple harmonic oscillations x=S1l10)()l‘ todecrease in frequency andbecome distorted, asshown inFig.ll.3c. Forverylarge damping, theshape approximates asquare wave. 1 =1)X 11.4 I11.4 Chaotic Trajectories andLiapunov Exponents 491 11': x2>l x2<l /X X time (b) (C) FIGURE 11.3 Limit cycles (darkened curves) ofthevanderPolequation inthe22,x phase space showing (a)circular motion forasmall damping coefﬁcient 6,and(b)distorted curve forlarge damping. Approaches tothelimit cycles viaorbits outside andinside them areshown. Part(c)sketches thedistorted sinewave obtained forthecaseofappreciable damping (large 6). CHAOTIC TRAIECTORIES AND LIAPUNOV EXPONENTS Theorbits thatwehavediscussed thusfarhavebeenwellbehaved, andconﬁned toarelatively small region ofphase space. Examples aretheellipses oftheKepler problem, thecircles ofthesimple harmonic oscillator, andthelimit cycle ofthe vanderPolequation (11.11).Under certain conditions, trajectories, called chaotic trajectories, willbeencountered inwhich themotion wanders around anextensive andperhaps irregularly shaped region ofphase space inamanner thatappears toberandom, butthatinfactistempered byconstraints. This path orregion where themeandering takesplace isanexample ofastrange attractor. Itiscalled strange because ofits(fractal) geometry andchaotic because ofitsdynamics.* Thechaotic trajectory roams hereandthere, back andforth through thisstrange attractor region seeming toﬁllthespace, butwithout everactually passing through thesame point twice. Inshort, chaotic motion hasafﬁnities withergotic motion (cf.Section 9.8), with characteristics between regular detemrinistic trajectories andtotally random roaming. Themotion involved inchaos hastheproperties ofmixing, dense quasi- periodic orbits, andsensitivity toinitial conditions. Theproperties areasfollows. Mixing means thatifwechoose twoarbitrarily small butnonzero regions, I1and I2,ofthedomain ofthemotion andwefollow anorbitthatpasses through region I1,thenitwilleventually passthrough region I2.Theorbits arequasi-periodic *See A.B.Cambel, Applied Chaos Theory, NewYork: Academic Press. 1993, p.70. 2 Chapter 11Classical Chaos inthesense thattheyrepeatedly andirregularly passthrough thewhole range ofthedomain without everclosing onthemselves, andwithout anyparticular timeperiod associated withsuccessive transits. They aredense because theypass through orarbitrarily closetoevery point ofthedomain, aproperty thatconfonns withtheergotic hypothesis (cf.Section 9.8).Achaotic orbitthatvisits andrevisits (thatis,mixes with) allregions oftheavailable phase space isidentiﬁed withwhat iscalled astrange attractor. Itsassociation isnotwithalocalized attractor such asaﬁxedpoint oralimitcycle, butrather withaveryextended region ofphase space, hence thedesignation strange. Theproperty ofergodicity, which involves covering allaccessible regions ofadomain, isshared byincommensurate non- chaotic orbits withrespect toanordinary attractor (forexample, atorus), andby chaotic orbits withrespect toastrange attractor. Sensitivity toinitial conditions means thatasmall change intheinitial con- ditions canresult inalarge change inposition andvelocity many transits orit- erations later. Forexample, asmall change canconvert aparabolic orbitofthe Kepler problem toeither aweakly bound elliptic orbitortoahyperbolic orbitthat extends toinﬁnity. IntheHénon—Heiles Hamiltonian, (cf.Section 11.6), asmall increase intheenergy caninduce theonset ofchaos withtheLiapunov exponent (deﬁned below) giving thetimescale forthisbreakdown oforder. TheKAM theorem oftheprevious section isvalid forsmall perturbations. As theperturbation increases, theeffect onthemotion ofthesystem becomes more andmore pronounced. Iftheperturbation becomes sufﬁciently large, thebehavior maybecome chaotic. Then successively calculated orbits move farther andfar- theraway fromeachother. Even iftheﬁrstfeworbits ofachaotic sequence lie relatively close totheoriginal one,eachiteration involves agreater recession than theprevious one,sotheextent towhich theymove apart canincrease exponen- tially withthenumber ofiterations. Anexample isaspaceship inanEarth orbit. Asmall rocket boost willmove ittoanearby orbit whereas astrong boost could throw itoutoforbit, heading forouter space. Another common example ofhow linear andchaotic motions differ when periodicity isnotpresent isturbulence in water. While there isstreamline ﬂow,twonearby points inthewater stayclose together astheymove along; aftertheonset ofturbulence thesametwopoints, on average, keepmoving farther andfarther apart. Aquantitative measure ofthisexponential divergence isacoefﬁcient, 2.,called aLiapunov exponent, (sometimes spelled Lyapunov orLjapunov). Inthechaotic region ofmany systems, iftwoorbits areseparated bythesmall distance soatthe timet=0,thenatalatertimettheir separation isgiven by 3(1)~toe“. (11.12) IfA>0themotion ischaotic, andtheLiapunov exponent Aquantiﬁes theaverage growth ofaninﬁnitesimally small deviation ofaregular orbit arising from aper- turbation. Itsetsatimescale1:~1/Aforthegrowth ofdivergences brought about bysufﬁciently large perturbations. Thechaos becomes appreciable fort>>r when thetrajectory winds itswayaround theextensive. butbounded, phase space 11.4 Chaotic Trajectories andLiapunov Exponents 493 ofthestrange attractor. Eventually theseparation s(t)becomes comparable tothe dimensions oftheaccessible coordinate space soitcannolonger increase further, andfrom thatpoint ontheseparations s(t)varyrandomly intime. Ifthesystem evolves byaniterative process rather thanbyatemporal process thenEq.(11.12) assumes theform s(n)~s()e”f‘, (11.13) where nisthenumber ofiterations, andtheexponent Aisnowdimensionless. Moreover, thisdivergence oforbits isnotreversible. Inachaotic region itisim- possible toreconstruct thedistant pasthistory ofasystem fromitspresent state. Thismeans thatcurrent trajectories cannolonger beprojected back todetennine theinitial conﬁguration. IftheLiapunov exponent isnegative itmeasures therateatwhich asystem point approaches aregular attractor. Inother words, inthenonchaotic region A< 0andthedistance s(t)from anattractor attimetisgiven bytheexpression s(t)~s()e'lM' (11.14) where soistheinitial distance attimet=0.Foraniterative process wehave the analogous expression s(n)~s()e_"|}‘l (11.15) forthedistance s(n)afterniterations. Anegative exponent characterizes therate atwhich theorbit spirals intothecircle onFig.ll.3a. Inthepreviously consid- ereddamped pendulum case thetime constant 1'ofthedamping process isthe reciprocal oftheassociated negative Liapunov exponent, t~1/|)t|. Asanexample, consider theelliptic orbit ofaplanet inthesolar system that isperturbed bythegravitational interaction withanother planet. Theperturbation isnonlinear, anditisalsosmall since thegravitational interactions ofthetwo planets withthemuch larger Sunaredominant. Wemight expect thattheKAM theorem would predict thatanyperturbed orbitisstable, butthisisnotcorrect fortworeasons. First, many natural frequencies inthesolarsystem correspond toresonances involving individual planets andasteroids. Second, many ofthe objects inthesolar system areasteroids, andperturbations resulting from their presence nolonger remain small. Both ofthese effects leadtochaotic results. Some ofthischaos simply means thatwecannot make exact predictions about thefuture. Other effects mayleadtotheeventual ejection ofoneormore bodies from bound orbits, apossibility thatwasmentioned inSection 3.12onthethree- bodyproblem. When weconsider natural frequencies, itisnotonlytheorbital periods thatare important. Therotation, obliquity (axial tilt),rotational plane, orbital plane, and eccentricity provide some oftheother frequencies thatmayinteract insurprising ways. Themassive planets oftheouter solar system have apparently settled into quasi-periodic orbits ofmarginal stability. Marginal stability means thattheir or- 494 11.5 IChapter 11Classical Chaos bitalmotion isstable onatimescale comparable withtheageofthesolar system. Other orbital parameters occasionally change. Theobliquity ofEarth’s axisisap- parently stabilized bythepresence oftheMoon. Both Venus andEarth interact inabounded chaotic fashion withlittle change intheir periods. Mercury, Mars, Pluto, andmany asteroids mayundergo much more chaotic motion. Calculations, projecting motions forthenext100Gyr,show thatthere isaﬁnite probability thatMercury willbeejected orcollide withVenus some timeduring thenext3.5Gyr.Using theapproximation r~l/|)»| with1:=3.5Gyrprovides aLiapunov exponent A~3x10_'° peryear asthetime scale forplanetary chaos. Theeccentricity oftheorbit ofMars could increase to0.2,while itsaxial tiltcanvaryby60°,perhaps sufﬁcient torelease water onthesurface through the possible melting ofitsicecaps. Pluto alsohaschaotic motions, buttheyseem tobebounded. Thus, chaos hasbeen amechanism forthereorganization ofthe planetary bodies since theformation ofthesolar system. Motions inboth theouter (>2.8AU)andinner (<2.5AU) asteroid belts arechaotic. Theouter beltchaos isdominated byJupiter andtheJupiter-Saturn- asteroid interactions, while theinner beltchaos involves Mars andMars~Jupiter— asteroid resonances. These interactions provide asteady impetus forMars cross- ingasteroids. Once established along suchapath, theLiapunov exponent ismuch larger, leading tochanges inorbit. Wemust notethatthese conclusions arebased upon theresults ofnumerical calculations. Every effort hasbeen made toensure thatcurrent limits ofnumerical accuracy, aswellastheinclusion orexclusion ofmembers ofthesolar family, do notaffect theconclusions. Although there isevidence ofpastchaos inthesolar system, wemust remember thatourfuture predictions arebased upon ourmodel ofthesolar system, notthesystem itself. Stability could bebetter orworse than themodel predicts, butthechaos itself isdeﬁnitely present. POINCARE MAPS InSection 11.1, wediscussed theperiodic motion ofuncoupled oscillators. When twoone-dimensional oscillators become coupled byadding atermsuchasx2yto theHamiltonian, thenthemotion becomes rather complex inthefour-dimensional pxxpy yphase space, anditisnolonger feasible tofollow thetrajectories. Itis more convenient tosample themotion atregular intervals andusetheresulting information todeduce some ofitsgeneral characteristics. Aconvenient wayto sample themotion istomapitonacross section ofphase space. When thetotalenergy, ET,ofadouble oscillator isﬁxed, thedimensionality ofthespace islowered byone,andthemotion isconﬁned toathree-dimensional region inthisphase space called anenergy hypersurface. Some authors refer to itasa“three-dimensional energy surface.” Toavoid thecomplications oftracing outorbits wandering around thisthree-dimensional region, itismore advanta- geous tostudy atwo-dimensional slice orsection through thehypersurface. The slice iscalled aPoincare’ section. Wecalculate thepositions ofpoints where or- 11.5 Poincaré Maps 495 bitspassthrough thesection. Aconvenient choice forthissection iseither the pxxorthepyyplane. Since theequations ofmotion areknown viaHamilton’s equations (8.18), thepositions where successive orbits pass through thistwo- dimensional section canbecalculated. Forbounded motion, such sequences of points mapoutclosed curves. Thepaths onthesection deﬁned bythese points constitute what iscalled aPoincaré map. Asanexample ofthedetemrination ofaPoincaré map, consider theKepler problem thatwassolved inSection 3.7forthecaseofnegative energies. Wenow reexamine thisproblem using Cartesian coordinates x,px=mi.yandpy=my, taking intoaccount aperturbation thatcauses theelliptical orbit toprecess inthe xy(that is,inther,(9)coordinate space plane, asshown inFig.l1.4.Theenergy Eisconserved withthevalue E=%mJt2+111192-k(x2+y2)-1/2. (11.16) Onthisﬁgure weimagine avertical plane located attheposition y=0,withthe vertical ordinate pxaxisandthehorizontal abscissa xaxisshown inFig.11.5. To calculate aPoincare maponthispxxcross section, westartthemotion (t=O)at theperihelion point AofFig.11.4withtheinitial values x=r1,y=0,at=O, andthevelocity component yamaximum value determined byEq.(11.16). The polar coordinates forthisstarting point arer=r1and6=0.Theequations of J’ v9=ré l 'e\,A’ C’ q AC ./’/ FIGURE 11.4 Precessing elliptic orbits oftheKeplerian problem sketched inCartesian coordinate space. Theﬁgure shows thevector velocity vtangent totheorbit atapoint (r,6),together withitsradial (r)andangular (rél)components. Points A,B,andCalong thexaxisnearperihelion denote successive penetrations oforbits through thei,xPoincaré section ofFig.11.5located along thexaxiswhere y=0. 496 11.6 IChapter 11Classical Chaos mic=px A C I III I —-a—r2 —r1 0r1 r2 a x FIGURE 11.5 ApxxPoincaré section fortheKepler problem withthesolid curve on theright tracing outtheorbit generated bypoints A,B,C,...oftheprecessing ellipse of Fig.11.4. Points A’,B’,C’arenotshown butarelocated atnegative values ofx. motion areused tocalculate successive points thattrace outtheorbit. Every time theorbit passes through thepxxsection, apoint ismarked onitindicating the value ofpx.Since theorbit isﬁxed fortheunperturbed Kepler problem, theorbit willalways passthrough thesame twopoints onthesection, point Agoing from back tofront andA’going from front toback, with px=Oforbothpoints, as indicated inFig.11.5. Poincaré maps generally only show points going through thesection inonedirection, which does notinclude point A’,sothisPoincaré mapconsists ofonly onepoint A.When theperturbation istaken intoaccount, perhaps arising from theattractive forces ofother planets onEarth asittravels around theSun, thentheorbit canprecess intime, inthefashion ofFig.11.4. Successive orbits passthrough thexaxisatdifferent orbital distances indicated bypoints A,B,C,...onFig.11.4. These points mapontothepxxsection atthe positions indicated inFig.11.5, andtrace outthesolid curve called thePoincare mapontheright sideoftheﬁgure. Theamount ofprecession thattakes place for eachcycle hasbeen greatly exaggerated onthese ﬁgures. Wehaveseenthatinafour-dimensional phase space aPoincaré section isa two-dimensional slicethrough athree-dimensional constant-energy hypersurface. More generally, aPoincare section isa2N-2 dimensional slice through a2N- 1dimensional constant energy hypersurface ina2Ndimensional phase space. Although theconcept ofaPoincaré section isdeﬁned forthese higher dimensions, itsmain usefulness isfortheN=2casewhere itprovides atwo-dimensional representation oftheorbits, which iseasytovisualize. ForN>2,itisnotnearly aseasytovisualize theorbits. HENON-HEILES HAMILTONIAN Over three decades ago.M.Hénon andC.Heiles were investigating themotion of starsabout thegalactic center. Twoconstants ofthemotion arethevector angular 11.6 Hénon-Heiles Hamiltonian 497 momentum (Zandthescalar energy E.Theobserved motions ofstarsneartheSun suggested thatoneadditional constraint might, under certain conditions, restrict thepossible motions. Under other energy conditions, however, themotion isnot restricted, soonly thetwostandard constants theangular momentum llandthe energy Eareavailable. Rather thansolve thisproblem withthetheactual potential ofthegalaxy, which isrelatively unmanageable, Hénon andHeiles restricted the motion tothexyplane, asintheKepler problem, andstudied arelatively simple analytic potential V(x, y)thatillustrates thegeneral features oftheproblem.* Thispotential, called theHénon—Heiles potential, provides twocubic perturbation terms, which couple together twostandard harmonic oscillators, corresponding to theHamiltonian, H=5i+Ll+lk(x2+y2)+>I x2y—-1-y3 (1117)2m 2m 2 3 ’ ' where thecoefﬁcient Aissmall sothelastterm serves asaperturbation. These cubic terms prevent theequations ofmotion frombeing integrated inclosed form. When thisHamiltonian isexpressed inpolar coordinates x=rcos6,y=rsin6 theperturbation potential exhibits threefold symmetry, 11--’£+Ll+1r<r2+1xr3s1n so (1118)_2m 2mr2 2 3 ' ' Tosimplify theircomputer calculations, Hénon andHeiles setpx=miand py=my,expressed theHamiltonian innormalized form using dimensionless units, andsetitequal toadimensionless energy E,withit=1, E:%x2+%y2+%x2+%y2+x2y—%y3. (11.19) Theequations ofmotion, which maybeobtained from either Lagrange’s equa- tions orHamilton’s equations, 55=—x—2xy 1'=—y-x2+y’. (11.20) arecoupled together andnonlinear, sothere isnosolution inclosed form. We canseefrom theform ofthedimensionless potential energy expressed inpolar coordinates, V(r,0)=1%+-1-r3sin30. (11.21) thatforaparticular value ofV,theradial coordinate rattains itsmaximum value forsin36=-1(that is,for6=90°,210°, 330°), anditattains itsminimum *M.Hénon, Numerical Exploration ofHamiltonian Systems, Course 2inChaotic Behavior ofDeter- ministic Systems, atthe1981LesHouches Ecole D’Eté dePhysique Théoretique, Session 36,G.Iooss, R.H.G.Helleman, andR.Stora (eds.), NewYork: North Holland, 1983. 4 Chapter 11Classical Chaos L 1 o.s- 6l_ 31 0.4- E1 y _ h _ ()_ “ 2‘l0"2 -0.4- l_I IIIIIIII -0.8 -0.4 0 0.4 0.8 X FIGURE 11.6 Hénon-Heiles equipotentials labeled with their dimensionless energies Eplotted onthey,xplane. Closed curves forenergies E5éreduce toanequilateral triangle forthelimit E=%.Open curves outside thetriangle (notshown) exist forhigher energies. Adapted from M.Hénon (1983), Fig.19. value forsin36=+1(that is,for6=30°,150°, 270°). Figure 11.6presents equipotential curves (thatis,curves ofconstant V)drawn forseveral values ofthe energy E.Forthelimit E<<%,notrepresented intheﬁgure, thecubic perturba- tionterms xzy—31-y3arenegligible relative tothequadratic harmonic oscillator potential terms, %(x2 +yz),andthecurves closely approximate circles centered atx=y=O.When thecubic terms areappreciable forE<%,theequipotentials form closed curves asshown ,andforE=é,thecurve becomes anequilateral triangle with rmax/rmin =2.Forenergies exceeding é,theequipotentials (not shown) liebeyond theequilateral triangle, areopen, anddiverge toinﬁnity. Thus, themagnitude oftheenergy determines whether ornotthecubic terms constitute aperturbation orserve asmain potential terms. When theenergy isﬁxed atavalue E<é,thesumoftheterms intheHamil- tonian must beequal toE,which means thatthekinetic andpotential terms both satisfy theinequalities V(x,y) 5E 112+1125E, (11.22) because thepotential ispositive deﬁnite. Theﬁrstinequality tellsusthatanytra- jectory started inside theclosed equipotential curve V(x, y)=Emust remain entirely within thatline,thesecond inequality setslimits totheallowed kinetic energy, andtheoverall effect istorestrict themotion toaﬁnite region infour- dimensional phase space. Tohelpusvisualize what ishappening, weexamine Poincaré sections inthey,yplane located atx=0.Theaccessible region insuch 11.6 Hénon-Heiles Hamiltonian 499 asection lieswithin thelimits setbyletting x=Oand1=0inEq.(11.19), 112+-gyz-119:5. (11.23) Themaximum velocity yoccurs aty=0,andtheextrema ofthecoordinate y arefound bysolving thecubic equation (11.23) withysetequal tozero. Togiveanexample ofthecalculation ofaPoincare map onay,ysection located attheposition x=0inphase space, wefollow Hénon andHeiles and select theenergy E=éandthevalues y1=-0.08, y=0.02asstarting points forthecalculation. Theinitial velocity, x1,isﬁxed byEq.(11.19) withx=0 . . 1/21,=(215-yf -y,2+§yf') , (11.24) where wesetx1=0since thestarting point isonthesection. Anumerical calcu- lation provides thesequence ofpoints (1)2,yg),(133,y3),(y4,y4),...,which are labeled 2,3,4,...onthePoincare mapofFig.11.7. Theﬁrsteight points lieon aclosed curve, asshown, thenextninepoints retrace thissame curve, asdothe subsequent points 18,19,20,....Anumber oftrajectories thatwere calculated byHénon-Heiles forthesame energy anddifferent starting points aredisplayed inFig.ll.8(a). Note thatFig.11.7provides anenlargement ofthelarge ovalcurve ontheright sideofFig.1l.8(a). Theoutermost curve ofthelatter ﬁgure marks theboundary oftheaccessible region deﬁned bysolutions toEq.(11.23). ForE=é,the velocity yreaches itsmaximum valuey=:l:(2E)1/2 =0.40sattheposition y=0,andthecoordinate yattains itsextremal values atthevelocity y=0given bythetworoots tothecubic equation (11.23), y=§. -1(~/5-1) (11.25) 167 1425 0.1- 2 511 _ 26 2,2 y 0- 9 "1. ‘3 01 1 41019 2]zs21] 12 -o.2- 2°3 I I I I I I 0 0.1 0.2 0.3 0.4 0.5 Y FIGURE 11.7 Poincare section intheyyplane showing thesuccessive points 1,2,3,... ofaHénon—Heiles orbit fortheenergy E= Thisparticular curve alsoappears onthe right sideofFig.11.8a.From M.Hénon (1983), Fig.20. Chapter 11Classical Chaos _ I I I 1 I 1 1 l 1 I y EIOOOJ3 04- - // 0: / - -03 .. -04 _ - -04 -0: -0’: -0'1 1’:0:1 oi: 0'3 0'4 0'5 oley (H) FIGURE 11.8(a) Poincare maps inthey,yplane showing several Hénon—Heiles orbits: (a)E=éwithregular orbits. asindicated inFig.ll.8(a). Thethird rootofthecubic equation +%(~/3 +1) violates condition (11.22), soitisnotacceptable. Theﬁgure shows thatthere are fourregions withoval-shaped orbits, which (ifcalculated forsmaller andsmaller circumferences) would shrink tofourﬁxed points called elliptic ﬁxed points. Sep- arating andbounding these regions ofelliptic typeclosed orbits isasingle con- tinuous curve thatcrosses itself three times atwhat arecalled hyperbolic points. Ahorizontal linedrawn fory=0isalineofmirror symmetry withthecurves above thislinebeing mirror images ofthose below it.Thissymmetry results from theHamiltonian being invariant under thetransformation y—>—y,butnotin- variant under thetransformation y—>—ybecause oddpowers ofyinEq.(11.19) produce asymmetry inthey-direction. Iftheenergy isincreased toE=§andthecalculations arerepeated, anun- expected result isobtained. Theregions where theovalorbits were found for thelower energy E=éstillproduce closed trajectories with ﬁxed points at their centers; however, intheregions between these closed trajectories, there is nocontinuous curve andthepoints there appear tohave noregularity, asshown inFig.ll.8(b). Ifwefollow theorder inwhich these scattered points appear, we ﬁndthat,instead offollowing aregular curve, theyjump around inamore orless random fashion fromonepartofthePoincare section toanother. Allofthescat- tered points onFig.1l.8(b) arose from thesame single chaotic trajectory, andthe chaotic region where theyappear ontheﬁgures constitutes across section ofa strange attractor. Inother words, theyalloriginate from asingle orbit meandering through thestrange attractor region ofphase space andrepeatedly penetrating the Poincare section randomly throughout thechaotic region ofthissection. Raising theenergy stillfurther tothecritical value E=écauses thestrange attractor 11.6 Hénon-Heiles Hamiltonian 501 9 1 l . - I I 1 A I 0 E-012500 04- - . o3- _ ~_ - : __. -s__'~. 0Q- 'n - O '.._" - I /i .- / '0‘ \ / II I I I ~~ I I r I I -04 -03 -OZ -OI 0 OI OZ 03 O4 O5 06 y (bl Z5_E=Ol6657 Z _ , ..//..~ ~cs - _ OZ - O, .. . __ _.~.1-:' .'-_-A. _""_:“ i'-\ . 0 -.--C) ""€-."‘ "-."-._-_--..:.'-'- .»-~.-,,.__.___ ‘_ -°| ,,‘ - -oz _ 0 1'1 ~ -93 . . - .. _-04 ".\'i -'. ' -05 ‘- '' "- - -0'5-014-0'3-dz-(in 6,111 0'2:1‘!G3o’50's0'1"'6'b isy (9) FIGURE ll.8(b&c) (b)E=%withregions ofregular motion andregions ofchaos, and (c)E=%withchaos dominant. Theorbit ontheright sideof(a)isplotted inFig.11.7on anenlarged scale. From M.Hénon (1983), Figs. 21,22,and23. toﬁllmost oftheavailable phase space, andthishastheeffect ofextending the chaotic region toinclude almost theentire accessible areaofFig.1l.8(c). Anin- dexoftheextent ofthechaos isthefraction oftheaccessible region where the calculated points lieonregular trajectories. Figure 11.9shows howtherelative areaoftheregular region declines astheenergy increases. Theonset ofchaos occurs nearE=5,beyond which theregion ofregularity decreases linearly with theenergy until complete chaos setsinatabout E=%.Calculations forthis Chapter 11Classical Chaos 1.0 0.9— 0.8- ' 0.7— - 0.6— Relativearea9U1 94:- 0.3- 0.2- 0.1- _ I | I u 1 1 | 1 | 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Energy FIGURE 11.9 Fraction oftheavailable Hénon-Heiles Hamiltonian phase space occu- piedbyregular (nonchaotic) orbits plotted asafunction oftheenergy E.From M.Hénon (1983), Fig.24. ﬁgure athigher energies arenotmeaningful because theequipotential lines no longer close onthemselves. andtheaccessible areabecomes inﬁnite. Chaos canalsobeviewed asabreakdown ofintegrability. Thetrajectories of Figs. 11.7and1l.8(a) forE=1-12canbeobtained byintegrating theequations ofmotion forparticular initial conditions; theresults obtained bycarrying outthe integrations areunique andreproducible, andthepathfollowed bytheposition point ispredictable. Atthehigher energy E=é,theequations areintegrable forsome initial conditions, butproduce points randomly located inthechaotic region forother initial conditions, inaccordance withFig.ll.8(b). ForE=é, integrability breaks down overvirtually theentire accessible region ofphase space depicted inFig.ll.8(c). Another interesting feature ofchaos istheappearance ofwhat arecalled is- lands. Forverysmall coupling, such asforenergies intherange E~10_3, the )3versus ysection consists ofclosed orbits slightly perturbed from being circular. Themuch larger perturbation fortheenergy E=%produces foursetsofelliptic typeorbits, andtheincrease intheenergy toE=%results intheappearance of ﬁveislands ofintegrability along theborder ofthechaotic region ontheright side ofFig.ll.8(b). Figure 11.8(c) shows thatsuchislands persist even when almost complete chaos reigns. 11.6 Hénon—Heiles Hamiltonian 503 Inaddition totheabove features, thechaotic region canexhibit anhierarchy ofislands, andthese aremosteasily visualized byplotting constant energy orbits inxycoordinate space. ThiscanbedonefortheHénon-Heiles system, butitwill bemore instructive forustoplotthese coordinate space orbits anddisplay some features ofthehierarchy ofislands withtheaidofanother chaotic system called quadratic mapping, which arises from thesetofcoupled equations x,,+1 =x,,cosa —y,,sina +xisina y,,+1 =x,,sinu +Y»Cosoe —x3cos0!, (11.26) where thevariables lieintheranges -1<x<+1,—l<y<+1,andoz,which might becalled acontrol parameter, detennines theextent towhich thesolutions areregular orchaotic. These equations aresolved byaniteration technique similar Q . -' 1 1 1 1 r» ‘ '’’ - r ‘ ' \. ~-'’ ', .. ._. /; . I,‘ 1 I‘ '1 \ II ,,, __j._-;.' § ,_I ,.-1-_1'?':'.§)'__ ' _. 1.O I-l,‘ / Mn.“ __’,‘_'-"’, :_- Q1 ..‘ _ . .--...-_,........., .1 ,._. x,.-_/--. I' I0" .-"ii; :24 -.-|;- I.¢ '._._ --- _: 4 -_ -\-.-Q 2!; ' E. . /0 \ /' -‘ /‘ 1 I I, m / ~.\r'Y .._/Q._._. 1.‘ _._._.\_._ Z____ _ , 4 . .,.'.- .' -v- - -_ . -. /a. .... ‘J: I I..'. ,Iz N-J; I.. .......- ‘.... /_/ 2_- ?-_ ’// '\'§-Q‘-,.‘~,,.-~ ,-:'_-:"::.'. 0,. - ’ ."~.. ‘E,, J- Ir ’ -.___;-: .‘Z . I L ., .. / :'- - O 1 ,_; 1 I ‘1.0 -0.5 X0 0.5 1.0 (8) FIGURE ll.l0(a) (a)Trajectories incoordinate space forthequadratic mapping sys- tem(11.26) atanenergy neartheonset ofchaos. From M.Hénon (1983), Figs. 33and34. 4 Chapter 11Classical Chaos tothatdescribed forthelogistic equation atthebeginning ofSection 11.8.Trajec- tories calculated numerically forthecasecosoz=0.4areplotted inFig.11.lO(a). Weseefromtheﬁgure thatthissystem exhibits onemaincentrally located elliptic- typeregion, ﬁvehyperbolic points where trajectories appear tocross, ﬁveoutlying elliptic-type regions, andwhatappears tobeasomewhat irregular distribution of dotscalled islands. Themain trajectories canalsobereferred toaszero-order is- lands. When theareanearoneofthehyperbolic points (forexample, x=0.57, y=0.15) isenlarged byafactor of20,weseefromFig.11.10(b) thatthetra- jectories donotactually cross atahyperbolic point, butrather there areseveral series ofislands inthisregion, andsome hintofincipient chaos. Thewell-formed curves ontheleftareassociated withthemaincentral elliptic-type zero-order is- landregion, andthestructure attheupper right involves acontinuous curve that Q 3._rbi "-.U 05.‘-. +‘J§ OQ\*1=:-*-- >£P°~r'u-1‘ﬂa- .,f‘$-'v‘~ho-'-'-rd:"0§-. ya_,-__.-asx~03I/IIs‘,\‘QI-.-——I‘IQ0.‘.—T)(...~__ 1'.-"'9A»;‘K ' § 0'0 m | I ._ E ‘Q 3..k-4 .I‘ ’;1‘_.!,I -"‘k*:-/.-'\ ‘l| :a- ,9 t ... <2:-H‘ it ‘“"F: I .n-r'°"'ﬂ"‘.."‘ '0-. ‘.0 ‘e J~"' “""-. "-2;.1 '1 -.‘ .-,iv ‘Qn “J 0‘ _’ ’ "...- -._ --.-~__. .. 'I O ‘O ‘- I ‘ "wJn f ,1 ~__ s f _a ~.., 4-/ 4 v “~.J 0 ~_ 0? 1 ,2’ ‘ 0 I ,- ,- I IIu §l 4 4 1 1 .525 .550 X.S75 .600 (b) FIGURE ll.l0(b) (b)Enlargement oftherightmost hyperbolic point of(a)showing several orders of“islands.” 11.7 I11.7 Bifurcations, Harmonic Oscillator, Resonance 505 encircles theordered region ofFig.11.l0(a). Thelongdashed curves atthebottom ofFig.ll.10(b) arepartofanoutlying elliptic-type region ofzero-order islands, anddirectly above them areﬁrst-order islands, eachofwhich hasfoursecond- order islands nearby. Attheupper border oftheﬁgure areﬁrst-andsecond-order islands associated withthezero-order island outofviewabove theﬁgure. Ingen- eral,islands tendtobeorganized inaninﬁnite hierarchy, theyareself-similar, and therelatively fewislands atonelevelofenlargement areassociated withmany islands atthenextlower level. Indeed, Fig.11.10(b) shows islands withalarge range ofdiameters: ~1.0,0.3,0.01, 0.003, and0.0005. Theproperty of“islands” being replicated athigher andhigher levels ofmag- niﬁcation isaproperty characteristic ofentities called fractals. Thisself-similarity ismuch more regular inthecaseoffractals because highly magniﬁed regions can lookalmost identical toviews atmuch lower magniﬁcation. Thenonintegral di- mensionality associated withastrange attractor thatwasmentioned earlier inthe chapter isalsocharacteristic offractals. Wewillhave more tosayabout fractals laterinthechapter. BIFURCATIONS, DRIVEN-DAMPED HARMONIC OSCILLATOR, AND PARAMETRIC RESONANCE Theminimal requirements forasystem ofﬁrst-order equations toexhibit chaos isthattheybenonlinear andhaveatleastthreevariables. While many nonlinear equations inphysics aresecond order, itispossible toreduce asetofsecond-order nonlinear differential equations toalarger system ofﬁrst-order nonlinear differ- ential equations. Recall fromSection 8.1thatasetofNsecond-order Lagrange equations reduces toasetof2Nﬁrst-order Hamilton equations. Ourpresent topic deals withthenonlinear analogue ofthisbehavior. TheHénon-Heiles Hamiltonian satisﬁes these minimum criteria forchaotic motion. Thiscanbeseenbyrewriting itstwononlinear second-order equations of motion (11.20) asfourﬁrst-order equations, twoofwhich arenonlinear dx_ dv, _ 2 at_"" at‘xxy dy dvy 2 2—= —=— — , 11.27dt vy dt y+ x+y ( ) where therearenowfourgeneralized coordinates x,y,vx,andvy. Letusconsider, asanother example, thedriven, damped, harmonic oscillator thathasthefollowing equation ofmotion (cf.Eq.(6.90)): d26 1d6.72-+ E+sin6 =gcos(wDt), (11.28) where (ODisthedriving frequency which isindependent oftime, andtheangle andtimecoordinates havebeenrenorrnalized toabsorb theexcess constants. This Chapter 11Classical Chaos nonlinear second-order differential equation canbeconverted toasystem ofthree ﬁrst-order differential equations bywriting d d9—= 11.2dt w (9) —=——w— 1 dw 1 s'n0+ cos¢dt q g where ¢isthephase ofthedriving term. There arenowthreedependent variables, ¢(t),9(t),w(t), andoneindependent variable t.Since thethirdoftheseequations isnonlinear, weexpect thatparticular values oftheparameters q,g,and601)might produce chaotic motion. Onephysical waytojustify thisexpectation istonote thatthemotion ofthependulum should depend upon theinterplay between the “natural” frequency toandthedriving frequency (OD. Toobtain quantitative results, wechoose q=2andlettheamplitude gof theforcing function playtheroleofwhatiscalled acontrol parameter. Such a parameter isanindex thatdelineates regions ofnonnal andchaotic behavior. In Fig.11.11(a),weshowthew=9versus 9Poincaré section forthecontrol param- eterg=0.9.Weseefromthisﬁgure thatthemotion isregular, while Fig.11.1l(b) constructed forg=1.15displays chaotic motion, thatis,randomness inthedis- tribution ofpoints. Theperiodic nature ofthedifferential equations (11.29) pro- duces regions ofstability, andthenregions ofchaos asthecontrol parameter gis increased. Ifweexamine howthefrequency ofoscillation, w,depends upon thisforc- ingfunction amplitude gforaﬁxed choice ofphase, ¢,weﬁndthatthesystem undergoes anumber ofbifurcations inthemeasured frequency oftheoscilla- tor.Ateachbifurcation, thenumber ofallowed frequencies doubles. Aplotof thisisshown inFig.11.12. Thebifurcations areassociated withnormal ornon- chaotic behavior. Theﬁgure alsoshows shaded regions where theoscillator ex- hibits chaos. Fig.l1.12(b), which isafactor oftenenlargement ofaregion of(a), shows thatbifurcations andchaos have acomplex dependence upon thecontrol parameter. Figures ofthistype arecalled bifurcation diagrams orFeigenbaum plots. Acomparison ofthetwoFeigenbaum plots ofFig.11.12 makes itclear thatthissystem exhibits theproperty ofself-similarity whereby thebehavior of w(t)intheneighborhood ofonebifurcation resembles thatintheneighborhood ofother bifurcations, even though thescale orlinear dimensions aresomuch dif- ferent. Itisalsoevident thatthequantity gseems to“control” theextent towhich thesystem bifurcates anddisplays chaos. IntheHénon—Heiles system discussed intheprevious section, thecontrol parameter isthemagnitude oftheperturba- tionA7-L=x2y—%y3.Inthedimensionless unitsbeing usedthere, theeffective magnitude ofAHwassetbythechoice ofenergy. 11.7 3L 2- 1 :00 -1 _2- ‘$1’ 3L 21- 1 m0 -1 _2- _.3rBifurcations, Harmonic Oscillator, Resonance 507 -| l I I 1 1 I 1 1.1 -" I I Fj -r F" I I I 1 1 'l-3 -2 —1T 10 9 (8) 1’ l t’ t i J 'l-a -2 -1 9 (blt t i i 1 t 1 2 3 FIGURE 11.11 Phase space diagram ofanorbit ofthedriven, damped harmonic oscil- lator(a)inthenormal behavior region forq=2andcontrol parameter g=0.9,and(b)in thechaotic region forq=2andg=1.15. Reprinted withthepermission ofCambridge University Press. From G.L.Baker andJ.P.Gollub, Chaotic Dynamics, AnIntroduction, Cambridge, England: Cambridge University Press. 1990, Figs. 3-4a and3—4c. Chapter 11Classical Chaos I I I I I I t I I I I 3— 7 1? 1 .' ."'7:'!w ,2.. Q , -,1_.,‘.;._"§)’» §:§§’,'.| - *" ‘Q1--1% £§;;>&.=,~=.-.1‘- ‘A . '‘ 1'.‘ '1hf.‘ ._-,r1- -_ \ ‘- .. guru‘ ‘J’ }1“ to1— "'1' ., ~_~.',-.- - ' :V;-:\"Y::(t,‘ 1"-lI\.5'1=.';' I: \\ 1;:-~18 I"-\Hg.‘ -in-,~. -,-_;5.gI"-'.~¢£~4.';. ".'",..a5\ a11r3" 't\-5- ‘W -,.-1- 0- 11,- Q1.‘-.--.1 - - _'|. ~f ’.-'§’?:’):- 3;"‘2i..II,.. 611+"-I -1- - 1 1 1 1 1 1 1 1 1 1 1 10 11 1.2 1.3 1.4 1.5 (8) 91 1 1 1 1 1 1 1 1 I *1 3- - _ ,__ -~,§=:1,;=': ' 2I‘ _ §_ i @ _ _ -' \;\,.’ __, .. I".(I) ‘I- I i51‘1§':'*‘ ' _ _ _..__ .-_..-.. _.-, ,",i=:*.=I-." ——.,,.¢.-:~.. 3*...~:;. ., ‘\ g 1‘;~%i1-its0_ ' 4*‘ . ikﬁ§‘:i.""; '' 1 P3511:-J1‘ ' -1- - I I -I I I — I I I I I I 1.45 1.46 1.47 1.48 1.49 1.50 (bl g FIGURE 11.12 Feigenbaum plotofthedriven, damped harmonic oscillator showing regions ofregular andofchaotic behavior. Part(b)isanenlargement oftheregion on therightsideof(a).Reprinted withthepermission ofCambridge University Press. From G.L.Baker andJ.P.Gollub, Chaotic Dynamics, AnIntroduction, Cambridge, England: Cambridge University Press, 1990, Fig.4-22. Anexample ofaparametric harmonic oscillator typesystem thatcanbecome chaotic istheparametric oscillator, which satisﬁes theequation d2 d2 mTI;+G(t,1.')x =mT:+(mag+k(t)) x=0 (11.30) 11.8 I11.8 TheLogistic Equation 509 where G(t,r)istheparameter oftheoscillator, andthetennk(t)=k(t+r) isaperturbation periodic inthetime-t.Many functions k(t)produce what is called parametric resonance, andwegiveanexample ofone.Recall thatfora simple rigidrodpendulum oflength L,corresponding tok=0inEq.(11.30), theresonant frequency wo=(g/L)1/2andtheoscillations canbeperturbed by changing thelength ofthebob.Parametric resonance canbeinduced inasimple pendulum byshortening thelength Lbyasmall amount ALwhen themass isat itslowest point withthemaximum kinetic energy, andincreasing thelength by thesame amount ALatthetopofthemotion where themassisinstantaneously atrest,withthekinetic energy zeroandthepotential energy amaximum. More energy isadded atthebottom thanissubtracted atthetop,sothereisacontinual increase inenergy every cycle. Ingeneral, theevolution intimeofthesolution ofEq.(11.30) canbehighly sensitive tosmall changes intheinitial conditions andthenature ofk(t). Thisis acondition forchaos. THE LOGISTIC EQUATION Since thedriven-damped harmonic oscillator andtheparametric resonance os- cillator solutions canonlybecalculated withtheaidofsophisticated numerical techniques, weshall consider thedetailed analysis ofamuch simpler mathemat- icalequation called thelogistic equation orquadratic iterator, which lends itself toelementary calculations andexempliﬁes mostofthecharacteristics ofchaos. Itssolutions exhibit regularities aswellaschaotic behavior. Theproperties ofthis equation, using successive iterations, areeasytocarry outonasmall calculator, andthedescription ofchaos thatthecalculations provide hasmuch incormnon with many realistic physical situations. This ubiquitous equation describes be- havior invarious disciplines suchasphysics, engineering andeconomics. For example, inbiology itdescribes population dynamics, ortheriseanddecline ofpopulations interacting with each other through predator—prey relationships. Other simple functions withaquadratic term alsogivequalitative andquantita- tiveresults similar tothose ofthequadratic iterator. Thelogistic equation isdeﬁned bytheexpression x,,+1=ax,,(1—x,,), (11.31) where aisthecontrol parameter, withthevariable xisrestricted tothedomain 05x51. (11.32) Successive iterations ofthisequation areexpected tobring x,,+1closer andcloser toalimiting value, xoo,sothatfurther iterations produce noadditional change in x,,.This limiting value xooiscalled aﬁxed point, anditisobtained bysetting x,,+1 =x,,inthelogistic equation (11.31), which gives -1X00= (11.33) 5 Chapter 11Classical Chaos Since x,,islimited totherange given byEq.(11.32), thecontrol parameter must bepositive withthelimit 15a.Equation (11.33) doesnotsetanyupper limit on thecontrol parameter, andordinarily therange 15a54isstudied. Itisofinterest toknow theconditions fortheﬁxed point tobestable. For stability, avalue ofx,,neartheﬁxed point williterate toavalue x,,+1, which is closer toxxthanx,,was.Tocheck this,wecanselect avalue ofx,,thatisclose totheﬁxed-point value bywriting -1x,,=“-a-:l:6, (1l.34a) where 5<<1.Weshallshow inDerivation 4thatthisgives, toﬁrstorder in8 -1x,,+1= 3a—:l:8(2—a). (l1.34b) Forconvergence toxoo,werequire thecoefﬁcient (2—a)of6tohaveanabsolute value lessthan1,which means thatthisstable ﬁxedpoint hasthecondition 1<a<3. (11.35) Suchaﬁxedpointconstitutes anattractor sincevalues ofx,,areattracted toit;that is,theyiterate toward it.Weseefrom aleft-hand column ofTable 11.1thatfor thechoice a=2andtheinitial value xi)=0.3,lessthanhalfadozen iterations areneeded toreachtheﬁxedpoint35,,=§obtained fromEq.(11.33). Itisofinterest toﬁndoutwhat happens when weiterate thelogistic equation forcontrol parameters beyond thevalue a=3.Fora=3.2then, weﬁndthat aftertwodozen iterations thevalue ofx,,alternates between twoﬁnalvalues or attractors asfollows: x,,=0.51304 x,,+1=0.79946, (11.36) asshown inthecenter columns ofTable 11.1, andforthecontrol parameter a=3.5,adouble bifurcation corresponds toafourfold cycle involving thefour attractors x,,=O.5_0l xn+1 Z x,,+2=0.383 x,,+3=0.827. (11.37) There isaneightfold cycle fora=3.55, asixteenfold cycle fora=3.566, .... Figures 11.13(a) andFig.11.14 illustrate thebifurcations. These Feigenbaum di- agrams, which plotxooagainst a,show howthenumber ofvalues ofxoosucces- 11.8 TheLogistic Equation TABLE 11.1 Examples ofiterations ofthelogistic equation (11.31) before bifurcation511 withcontrol parameter a=2.0(leftside), afteronebifurcation withcontrol parameter a=3.2(center), andinthechaotic region (a>aoo)withcontrol parameter a=4.0 (right side). Inthenormal regions, values of|x,,=xoolaregiven, andinthechaotic region, values ofAx"=|x,,—x{,|aregiven fortwoiterations xnandxfl,which start close together. Normal, a=2.0 Normal, a=3.2 Chaotic, a=4.0 n xn |xr,—xoo| x104 xn |x,,—xoQ| ><104 xn xf, Ax"x10 00.3000 10.4200 20.4872 30.4997 40.5000 50.5000 60.5000 70.5000 80.5000 90.5000 100.5000 11 12 13 14 15 16 17 18 19 20 212000 800 128 3 0 OOOOOQ0.3000 0.6720 0.7053 0.6651 0.7128 0.6551 0.7230 0.6408 0.7365 0.6210 0.7531 0.5950 0.7711 0.5647 0.7866 0.5372 0.7960 0.5204 0.7987 0.5146 0.7993 0.51332130 1590 942 1521 867 1421 765 1278 630 1080 264 820 284 517 129 242 35 74 8 16 2 30.3000 0.8400 0.5376 0.9943 0.0225 0.0879 0.3208 0.8716 0.4476 0.9890 0.0434 0.1661 0.5542 0.0734 0.2720 0.7922 0.6586 0.8999 0.3619 0.9237 0.2819 0.80970.3001 0.8402 0.5372 0.9948 0.0220 0.0859 0.3143 0.8621 0.4755 0.9976 0.0096 0.0381 0.1465 0.4714 0.9967 0.0199 0.2877 0.8197 0.5911 0.9668 0.1282 0.4472LIILII-Pl\>'-‘ 20 65 95 279 86 338 1280 4077 4268 7247 7723 3709 802 2292 431 1537 3625 sively doubles: 1,2,4,8,...,forincreasing control parameter auntilthevalue aw=3.5699456 ... (11.38) called theFeigenbaum point isreached, beyond which thebehavior becomes chaotic. Forthechoice ofcontrol parameter a=4.0inthechaotic region be- yond aw,successive x,,-terms generate asequence ofwhat seems likerandom numbers. Ifwestartwithtwoveryclose initial values, such asx0=0.3000 and x6=0.3001, weseefrom theright-hand column ofTable 11.1thatafter 10or11 iterations x,,andxf,become widely separated fromeachother, andtheirdiffer- ence Ax"=|x,,—xf,|becomes comparable totheir values. Additional iterations produce seemly random values ofx,,andxf,. AFeigenbaum diagram hassome other interesting properties. When there- gionneareachbifurcation isenlarged, weﬁndsuccessive bifurcations thatare XChapter 11Classical Chaos 1.1 l.1 -M» . tma;- -aw~___W.»1¢-_~Jt‘L;_,-.-.,q.if ,.~:*~33-.1.‘EEa‘3.;...»-».._,4‘i=1 »'1:5.12%: 4%»:17.-.‘Fa '-..11.~ .g~.*',,;-. '»-> ~'t;,1*.:a§' ~'=» '113< +1’'5:2!‘ W"J.xi,.;»;--~‘;>\‘JI-1 =< 1-~1.'".... ~w»-.1"1 1-~'-=; Q7.11».-51);"__ .’=1_ .t:~;g=. "»‘-sari’, ;>%;;~:?;§»..,r -'-rt?‘....1 I.K»’1¢£=.'.‘»'1r¢‘»i~.. ~."far’ "‘~<;#"I-331, 1~'<}.‘ ..'5 '{V'?“[,5’ié§’4fz3 *3!“.14 '.1#33 ‘ea. ‘£3 R. 1%» -~43’,-. IX-ea .-ww,;,- »:. . .. 1“vi?! 1.11:; taa=.§= :i'~;’.'=""1,5:1:»-1..‘=-.-_. >3--:~ ~-.1/.1:.~ '-2* ‘4 ..1; ‘i:r§/§‘._ :>*~= 1:~:-‘w>;' :11»'x~:--.~.=..-“.'..3»."1'~§;_g? 1,?tis-: ii“;;'5f~r=..W‘;- $1"sai- ....:, 7,,.3% Jay ,~-_t,_;';, "6. 8'1B-ii--.. '1-.!~.-1.-..""<'iL".' mi ‘“‘"¥'13".1}‘ii»‘ﬁ_ri.%,.* .-, "1‘1»-1.; 1 1 | 1 X 1 1 -‘H0 3 4 (8) 1- 0.5- 7»0 -0.5- -1- 3.43.5 sis 3173.8 3:94io (b) FIGURE 11.13 Correlation between Feigenbaum plot(a)andtheLiapunov exponent A(b)ofthelogistic equation inthecontrol parameter range from a=3.4toa=4.0. Theﬁgures arealigned withcorresponding values ofatoshow howsharp minima inA correlate withbands ofnormal behavior embedded inthechaos. TheLiapunov exponent isnegative intherange a<awofnormal behavior, andpositive inthechaotic region a>aoo,except where regions ofnormal behavior appear inthechaos beyond a=aoo. From Peitgen etal.(1992), Fig.11-1 (upper ﬁgure) andR.Shaw, Z.Naturforsch, 36a,80 (1981) (lower ﬁgure). 11.8 TheLogistic Equation 513 .12.-’1 ,'1"‘'i:»‘-‘191? / ‘iv .,-:32\‘,.~:::,=j»_¢,1'f"5ﬁg} / 31; /’ ‘Bf:63.- 1:1;-.£128‘ / 0 1 4 0.2"/22. s?~“i"11‘ "1»»1’§.'-‘I.1'11‘ (b) \i'01i. L . ;. “‘ ’0.7287 3 3.5 3.678 "0.594I.‘.1; A‘Y, 9 "‘*15%.. ~1" -1. .4,» Y1? =31E. ,.‘,¢_~.,;_i"F,*9 >- je~:!a'.- . \ at" azafggl \ /5?~92:3¥:§:Z=?§ c ‘ti ‘i'§<.=1<..r.51~,i§~:; 21;; ~+'.-31:11ma ‘(Hﬁx; =.~=.:z*1‘:*3/‘hf ' 1:2.15'\1 -1-1 1-|-1 1 1 1 1 1 1 1-M04 0 3.45122 3.5 3.59383 30.46361c;-_.'j.¢=/ "“$5 , 1:55;}?-i-»<. 1zgi/R,3;f;_».f.\ .-3 4"if' i E~.» /H*'*11'i~;»}...t-1'y,,-3;-;fe'.\i»§;..3“E5/,7..~,-...-:;s,-t'"Jit '1f1.%f;:%; \0.5357 3.54416 3.56 3.57490 FIGURE 11.14 Feigenbaum diagram ofthelogistic equation overawide range (1-4) of control parameter a(a).Diagrams (b),(c),and(d)show successively greater enlargements ofregions nearbifurcations. Note thereversals inorder oftheordinate scales ontheright sideofsuccessive ﬁgures. From Peitgen etal.(1992), Fig.11.3. 4 Chapter 11Classical Chaos self-similar toeachother, butonsuccessively smaller scales. Thisisillustrated graphically inthesequence ofenlargements, Figs. 11.14(a—d). Theratio ofthe horizontal spacing between successive bifurcations converges toalimitcalled the Feigenbaum number 8 5=limT =4.6692016. (11.39)n_’°° an+l _an andtheratioofsuccessive vertical spacings alsoconverges toalimitoz: xn_xn-1oz=limi-—— =2.50290787.. .. (11.40)n—>o0 _xn+1 —xn TheFeigenbaum number 8isauniversal constant found withmany chaotic sys- tems, butthenumbers aandawdepend upon thespeciﬁc model, which inthe present caseisthelogistic equation. Another interesting property ofaFeigen- baum diagram isthepresence ofregions ofnonnality embedded inthechaos. Thisisevident inFig.ll.l3(a), andismore prominent intheexpanded diagrams ofFig.11.15, which display bifurcations forthree levels ofenlargements. Each enlargement displays more bifurcations andnewregions ofnormality within the chaos. Thefractal property ofself-similarity isevident. InSection 11.4, wediscussed howtherateofapproach toanonnal-state ﬁxed point ortorandomization inthechaotic region isdetermined bythevalue ofthe associated Liapunov exponent A.Thisexponent Afrom Eq.(11.13) isdimension- less,andwewrite forthenormal andchaotic regions, respectively, as |xn—xoo|=e")‘=e_”'M (normal region) (l1.41a) |x,,-x;,1=e"*=¢"l*' (chaotic region). (11.41b) Note thattheexponent nAiswritten as—n]A| forthenormal region because A isnegative there. Inthenonnal region x,,Qxooforlarge n,sothedifference [xn—xoolgoes tozero. Inthechaotic region, thedifference Ixn—x,’,|grows exponentially until itbecomes comparable totheoverall range ofvalues, namely O<x<1,which means exponential growth inseparation until perhaps lxn— x,’,|>0.2.Further iterations keep thisseparation xn—xflintheapproximate range 0.2<x<1.These behaviors areclear from thedataintheright-hand colunms ofTable 11.1. Figure 11.13 shows howtheLiapunov exponent depends upon the control parameter. Weseefrom theﬁgure thatAisnegative inthenormal range a<aw,andrisestozeroatbifurcation points, ascanbeseenbycomparing Figs. 11.13(a) and(b).Itispositive inthechaotic region where a>aoo,except where regions ofnonnal behavior thatappear white inFig.11.l3(a) areembedded inthechaos. Near control parameter a=3.83, weseethree successive minima of Aintheregion ofnegative values thatcorrespond totheperiod doublings visible inFig.ll.l3(a), andthatappear considerably enlarged inFig.11.15. Wemust remember when studying systems such asthelogistic equation that values ofx,,obtained from theiterative process ofEq.(11.31) donotcorrespond 11.8 TheLogistic Equation 515 .., .. 1..4 1. :3-;~.-'*,~:i.' <?;..:.4-4",,- -.»,1. -1» ".1-1 #75-;r=~i*'* ~4‘.-'<‘-.r- .1V < . "»"-"‘4?' ‘13w.~1‘k=fx~~ -.=‘;'1"_‘2_g’.~§5 pt-§ .<|. r.1.,av, ¥.€'t.r_;_:‘¢_._._,-_ _.-w- --g=1;‘~~"‘~,f{..1f—_,'-».*‘ L1~..~r.r'$. s~»s‘ii.';.“a'-1)f"~:<‘i:. .f,..\\.1-_*:“;_-»;*r."_,~?<,!',.,§’ .,' ;?t=rTﬂ5:f;§'- ,1.9"-5'?‘ 7 ''".‘1 .“‘P“;\i;%a<a,.. »~.1*’».»;.~.~.T.;-.i»*..; an ,.1r'+?i1;.:§‘..~ir;.;:5,;.. l;;;,;-.g.,;_.~€;§$§*,; Iggy 11:3 .-~*».;.+r~=‘§»@. as;II1.».a~<r.§.-,..- ._:.-..1.:rr‘:,-.<.~**~r=1;1- ~Q.M.’.ear.‘ . .._--;.e@:>M1,.-»_.% .\@\-M. z.\t‘a'~-1; ..* ..~.-w <9},1 '4' *:‘%‘4;“-;1;,’$ *‘~‘ ‘jI,.~.="'<=.;.i-.'’~ d7"i‘§~ 1»~25-ﬁx” "‘1‘,,”€1t.‘;n ,?_;-"4.5 aw4;, * Ie 1-mi‘ '‘“M I iI. asU‘t,._-p3;‘ :3 -_ .51? E;,§.W‘ ,3, ,,Ir:.. “1.-‘aw =.»-Q.-t,ﬁ;§_§ 4*'».§-'.*.a"£ .. ._ _..' ,....,, .1;, ..., ..¢-_; ..,.44., .11,.,~..,1!?’1A ".“';r=11"**'*J'> 1”: 1&5‘. ~ "1-1.1...;‘..:;.-§~.~.,»a< 3.1.3 ~_-,:-.».».:,=.»3..~.>;;»aa ~------------ ..-...." <...1 >,*,._‘%:-ad .,=.<,§L ‘Gaga. 1,r_-1 »,‘E¢;§ ,_:c.- , ‘ml_31. " 18 _d ‘ 4\3;._3;,‘€i4§.;_. "ﬁt .‘. . ‘i' .:i."‘4'!“”;-"$1" *1¢;6;*,_~.'_ '5 perm 6period 7_ —- ""‘~;,, =4-perwd5 ‘Ieriod3 "'~’*i-.*?*»0 (10., 3.8 4ﬁlial: W/3149" éi .e»:+’<$:=s 0.558 -....-ta,-i\ -.M,.,:~r-3~ .3-;*.»>-j'.~:.f. 43;§'.’gl’~»’l*Y'¥£:.'.-,‘r‘-‘f».=:;‘§.< =3;~1:1;-2$¢.~. ~ A,1‘-.*-*r;<§%::;=‘..~'~=~~<:1.-'-'‘»:-as..'F_ 1;"..";i;'?:'3\,,3:§5l pita";,.,; _»,~.->...;.,,v,=:§.§:.--=3.\\;kif-r’»$'."1*"’?1=§t‘.:>'.’s’>1.'~ ;~’£“4*.'-'1'.1~1:.»<:i~'C‘2.e?:r' *~3<f"i‘P§FF“4. .=»..»~»=.»/4’.‘ ~=.:zi;<"<:;4¥;.@- ".1:-~‘.:, _:‘E!.'“‘2'<‘-:/¢T~" ~ ..r=»;~.<g_,?{.~. ..¢~,\>.=,s¢-:~.-.6: viigg -.-@*~'*%.<=:1.~. 8"..erz2- isW~~ 11-3»1*‘:=:‘-PI::-.;‘.. , Ti‘/. 1'»'\:’:" ‘*'~=._-..;:; if 0.443 3.828 3.84 3.85 3.8570.508 / .;‘-ti‘fail ,- .-1,;:,3‘ /i".=;“-.;:l31.7.. \ ,-:i<:;'f'~:>- xi,E"{!'.'.“-_ 21. r a3'~',"“? y; .»’‘$1.1' 5.I °.~, ‘_,~.§'.* A 2111...~r‘ ~..1.¢ 8‘. _. ~, 2 ‘ V... _ \;;_ imi"1“§'L‘§ ‘i’:;l‘_‘:2‘. ~€1_,f~.‘l.‘-T."“Q-“.5. 1' 0.491-1 I1|IIIIII1IIIIIIIIII 3.8478 3.848 3.849 3.8498 FIGURE 11.15 Feigenbaum diagram ofthelogistic equation showing regions ofnormal behavior embedded inregions ofchaos. Three successive enlargement ﬁgures areshown, asindicated bytheirabscissa andordinate scales. From Peitgen etal.(1992), Fig.11.41. 5 11.9 IChapter 11Classical Chaos toaparticle moving inspace. Successive iterations merely illustrate some ofthe properties ofchaos. Wemust maintain aclear distinction between thechaos that results from simpliﬁed models suchasthatofHénon-Heiles, andtheactual mo- tionofrealstarsinthegalaxy. These simpliﬁed models display many ofthefea- tures thatarefound innumerical solutions thatmore closely approximate thereal world, buttheycannot make reasonable quantitative predictions about theonset ofchaos inrealphysical situations. Bywayofsummary, wehave seen thatwhere thelogistic equation behaves inanonnal manner, thesolutions occur atvalues ofxcalled attractors, stable ﬁxed points thatconstitute one-dimensional analogues oflimitcycles. Beyond this,successive bifurcations arefound. Inthechaotic region theequation gener- atesnumbers inarandom manner sothatifwestartwithavalue ofxinonesmall interval, theiteration willeventually produce anumber inanother previously des- ignated small interval, corresponding totheproperty ofmixing. Wealsosawthat inthechaotic region twopoints thatareinitially veryclose generate successive sequences thatdonotremain neareach other, corresponding totheproperty of sensitivity toinitial conditions. There arealsoregions oforder with attractors, period doublings, andnegative Liapunov exponents imbedded inthechaos. FRACTALS AND DIMENSIONALITY Thephenomenon of“islands” being replicated athigher andhigher levels ofmag- niﬁcation, asdescribed inSection 11.6, ischaracteristic ofmany chaotic systems, andalsoofentities called fractals. Afractal isanobject orsetwithnonintegral dimensions thatexhibits theproperty ofself-similarity. Forexample, consider a linesegment, remove itsmiddle third toproduce twolinesegments, remove the middle third ofthese latter linesegments toproduce atotal offour, andsoon, asindicated inFig.1l.16(a). Ifthisprocess ofremoving themiddle third ofsuc- cessively smaller linesegments iscontinued indeﬁnitely, weendupwithaseries ofdotswithcharacteristic spacings called aCantor set.TheCantor setatvarious stages initsgeneration isself-similar inthesense thatmagniﬁcations oftheset atlater stages ofgeneration have thesame appearance asthesetitself atearlier stages offormation. Thedimensionality oftheCantor setisalittlemore subtle to deduce because therecursion process ofitsgeneration continually increases the number andreduces thesizeoftheresidual “dots.” Before discussing thedimensionality oftheCantor setitwillbehelpful tosay afewwords about dimensionality dinordinary Cartesian orEuclidian space. In onedimension consider alinesegment oflength a0divided intoalargenumber ofequal subdivisions eachoflength a<<a0.Intwodimensions wehave asquare ofsidea0subdivided intomany equal subdivisions each ofsidea<<a0.Inthree dimensions thesame typetinysquares aremade ofacube ofsidea0.Ineachcase thetotalnumber ofsubdivisions, which wedenote byN(a),isgiven by ma)=tao/ar‘ 11.9 Fractals andDimensionality 517 (8) (bl FIGURE 11.16 Recursive procedure thatgenerates (a)theCantor set,and(b)theSier- pinski carpet shown after foursteps ofiteration. From R.J.Creswick, H.A.Farach, and C.P.Poole, Jr.,Introduction toRenormalization Groups inPhysics, New York: Wiley (1992), Figs. 1.1.1 and1.2.3. where thedimensionality d=1,2.3forthese three cases. Solving thisexpression forthedimensionality ofthespace weobtain _logN(a)d_$g(a0/a) (11.42) This formula forthedimension disintuitively obvious forsystematic subdivi- sions ofordinary Euclidian space inanynumber ofdimensions. Wewillalsoﬁnd itapplicable forwhat wemight callthepathological subdivisions ofspace that arecharacteristic offractals. Inthisapplication thedimensionality a’determined bytheapplication ofEq.(11.42) iscalled theHausdorff orfractal dimension dp. Chapter 11Classical Chaos Returning totheCantor set,itinvolves subdividing alineoriginally oflength a0,which isone-dimensional; thatis,itsEuclidean dimensionality dE—1.Even- tually wefeelbyintuition thatafter aninﬁnity ofsplittings thelines diminish topoints thathave adimensionality ofzero, andwesaythatthetopological di- mensionality oftheCantor setd7=0.Further consideration, however, leads us tothink thatthelimit isnever really reached, andthatanylarge butﬁnite num- berofsplittings stillleaves anenormous number ofinﬁnitesimal one-dimensional linesegments present. This suggests thatweneed another waytoassign dimen- sionality. Thiscanbedone bynoting thatatthenthlevel ofsubdivision theline segments areoflength a=a0/3", andthenumber ofthem N(a)is2".Thus, we have a=3_"a0 N(a) =2". (11.43) Thefractal dimension orHausdorﬁ dimension dpisdeﬁned bytheexpression _logN(a)dp_i1Og(a0/a). (11.44) Thisdeﬁnition ischosen tobeconsistent withtheresults ofEq.(l1.42). Inserting Eqs.(11.43) intoEq.(11.44) togetfortheCantor set 12dp=K=0.6309. (11.45)log3 Inthefollowing discussion, weshall usedgfortheinitial Euclidean dimen- sion. d7-,fortheﬁnal limiting Euclidean (called topological) dimension, and dpforthecounterintuitive non-integer dimension characteristic offractals and strange attractors. Thefractal dimension dpisalways between thetwolimiting values d1anddE, dT<dp<d5, (11.46) andweseethatthisrelation issatisﬁed fortheCantor set 0<0.6309 <1. (11.47) Itwillbeinstructive todetermine thefractal dimensions ofaninitially two- dimensional (dE=2)self-similar ﬁgure called theSierpinski carpet oflinear dimension a0andareaA0=a3illustrated inFig.11.l6(b). Tostart, asquare is divided intoninesquares oflength a=a0/3andareaA=a2=(a0/3)2, andthe middle square removed. Then each oftheremaining eight squares isdivided into ninesmaller squares, andthemiddle oneofeach isremoved. Theﬁgure shows thefourth stepinthisiteration process. Atthenthlevel ofsubdivision, thesquares areoflength a=a03'" andthenumber ofthem N(a)is8".Thus, wehave 11.9 Fractals andDimensionality 519 a=a()3_” N(a) =8”. (11.48) Theappropriate limit isasetofedges ofsquares delineating intersecting jagged ﬁlamentary lines, which become progressively thirmer andthinner withsuccessive iterations, appearing toapproach d1=1.Thefractal dimension dpisagain given byEq.(11.4-4), 1s.1,=l;_3=1.s92s, (11.49) andEq.(11.46) issatisﬁed bytheSierpinski carpet, asexpected, 1<1.8928 <2. (11.50) Inthegeneral caseofafractal object inadg-dimensional Euclidean space, we deﬁne thefractal dimensionality dp,alsoreferred toasthecapacity dimension, byacovering oftheregion occupied bytheobject bydE-dimensional spheres in accordance withtheexpression _logN(r) where ristheradius ofthedE-dimensional spheres. This deﬁnition isclearly independent ofthevalue ofr0.Ifd5=2,thesphere isa2-sphere orcircle of radius r,andif(15=1,the“sphere” isal-sphere orlinesegment oflength 2r. InthecaseoftheCantor settheobject being covered bylinesegments orone dimensional spheres ofradius r=a/2isthemultitude ofresidual linesegments aftermany subdivisions. IntheSierpinski carpet casethecovering isbycircles of radius r=a/4/2,where acircle ofradius r=ao/4/2 covers theinitial square before anysubdivisions. Fractal dimensions havebeenevaluated formany chaotic systems.* Forexample, thelogistic equation wasquoted ashaving astrange at- tractor dimension of0.538, which isbetween thetopological dimension d1=0 corresponding totheindividual points xnandtheEuclidean dimension d5=l corresponding totherange ofxgiven byEq.(11.32). Thedriven-damped pen- dulum withtheequation ofmotion (11.25) exists intwo-dimensional (x,y)Eu- clidean space, andhasone-dimensional orbits ofthetypeshown inFig.11.11(a). Itsfractal dimensionality determined from Liapunov exponents ranges from 1.2 to1.4forvarious damping factors, which isbetween thevalues ofdT=1and d5=2thatwejustmentioned. Wesawintheprevious section thatchaotic systems exhibit atype ofself- similarity, butlessregular thaninthecaseofsystematically constructed fractals such asthose inFig.11.16. This does, however, suggest thatchaotic systems could have afractal-type nature, andthatnonintegral dimensionality might bea *See A.B.Cambel, Applied Chaos Theory, NewYork: Academic Press, 1993, p.70;G.LBaker and J.P.Gollub, Chaotic Dynamics, AnIntroduction, Cambridge, England: Cambridge University Press, 1990. 520 Chapter 11Classical Chaos characteristic ofchaos. Thissuggestion iscorrect. Thefractal dimensionality dp ofastrange attractor canbecalculated from theLiapunov exponents associated withitsexpansion inphase space. Toillustrate this,weconsider theparticular case ofastrange attractor intwo-dimensional conﬁguration space, which evolves in timebycontinuously expanding inonedirection andcontinuously contracting in itsorthogonal direction insuchamanner thatitsareaA(t)continuously decreases inmagnitude with thepassage oftime. Thispermits ittocontinuously elongate andmeander throughout theavailable regions ofphase space. Westartwith a square zone inthex,yplane ofachaotic region withtheinitial dimension a0in thex-andy-directions andthecorresponding initial area, A0=ag,asshown in Fig.ll.l7a. This means thatdg=2.Iftheareaevolves intime bycontracting inthex-direction withthenegative Liapunov exponent 11andexpanding inthe y-direction withthepositive Liapunov exponent A2itgetscontinuously thinner andevolves toward alineoftopological dimension dT=1.Interms ofthese Liapunov exponents, thex-andy-dimensions oftheareahavetherespective time T_ a>(z) =aoeh’ ~—~ -Ia0,,-|i.|/ "0 _v_:ao |_,i ax(z) =a0e’|'"'|' (3) (b) FIGURE 11.17 Role oftheLiapunov exponents Al<0andA2>O,subject tothecon- dition IA1|>A2,intheevolution ofaninitially square area(a)inphase space thatexpands along onecoordinate direction andcontracts along theother withthepassage oftime(b). 11.9 Fractals andDimensionality 521 dependencies from Eq.(11.12), ax(t)=a0e_p“|' ay(t) =aQe)‘2', (11.52) andtheareaA(t)evolves intimeas A(t)=A0e<*1-W)’, (11.53) where A0=ag.Since A1isnegative andA2ispositive, itisnecessary tohave IA]l>A2sothatthearea(11.50) willcontinually decrease withtime. Thefeature ofacontinuous decrease inthefractal areaA(t)ofEq.(11.50) isanalogous tothe continuous decrease inoverall length ofthelinesegments intheCantor set,and ofthecontinuous decrease inthenetremaining areaintheSierpinski carpet case, astheiterations progress tothelimit n=oo. Ifweconsider theevolved elongated areaA(t) ascontaining anumber N(t) ofsmall squares ofindividual areaAA(t) =af,asindicated inFig.ll.l7b then wehave AA(t)=age-2'*1", (11.54) where Misnegative, and A 2(12-I111)! NU)=L) =f‘_0_‘1_i =e(>\2+|A1|>r_ (1155)AA“) agg_2|)~1lT Byanalogy withEq.(11.44), thestrange attractor dimension dF,isgiven by _ logN(t) _ 2 dF_1°8(¢10/¢1x(F)) ~1+|7~1|’ (11.56) which hasanonintegral orfractal value. Forthepresent case, A2<|A1|, so Eq.(11.46) issatisﬁed with 1<df-‘<2.Thus, astrange attractor isrelated toafractal inthesense thatitsdimension is“strange”; thatis,itisnotaninteger. There isafundamental difference between thetimeevolution andthespace- ﬁlling effect ofregular trajectories andchaotic trajectories. WesawinSec- tion11.1howtheorbits ofincommensurate oscillators can“ﬁll” thespace ofa torus byranging over theentire domain. However, technically speaking, these regular orbits donotoccupy anyoftheareaofthetoroidal surface because they areone-dimensional curves without anywidth, meaning thattheactual areataken upbythem iszero. Chaotic orbits alsorange overtheir entire domain ofphase space, buttheydosobyoccupying areainthisspace. What isstrange isthatthe more thechaotic orbits “ﬁll” phase space, thesmaller theareathattheyactually occupy (cfEq.(11.53)). This makes itappropriate torefer tothedomain over which thechaotic orbits roam asastrange attractor. Theonset ofchaos maybe looked upon astheincrease inthedimension ofaregular orbit from itstopolog- icalvalue d1=1toitsfractal value 1<dp<2asitbegins tooccupy space inanareaofEuclidean dimension dE=2.Thefractal dimension maybelooked Chapter llClassical Chaos upon asanindex ofhowmuch space isoccupied bythefractal orbit. Weshould ofcourse continue tobearinmind thefactthatthemain difference between the space-“ﬁlling” aspects ofregular andchaotic orbits isthatintheregular incom- mensurate casethespace is“ﬁlled” inasystematic manner bythepredetennined spiraling motion around thetorus, while inthechaotic casetheorbit “ﬁlls” space inarandom, meandering, manner. This nonintuitive manner inwhich chaotic orbits inasense spread outmore andmore, andinanother sense become more attenuated, astheydevelop intime isveryanalogous tothebehavior offractals. Wesawabove howtheCantor set andtheSierpinski carpet illustrated inFig.11.16 both become more disperse andmore attenuated astheygothrough successive iterations, always remaining ﬁnite throughout theprocess. There isananalogue oftheSierpinski carpet in three-dimensional Euclidian space called aSierpinski sponge, which evolves in ananalogous manner through aniterative process, dispersing through space while losing volume inaccordance wiﬂa afractal dimension. Chaotic trajectories are indeed closely related tofractals. Inourtreatment ofthequantitative aspects ofchaos, wehave placed more emphasis onthefractal property ofnonintegral dimensionality thanwehave on itsproperty ofself-similarity. Intheapplications offractals outside thedomain ofclassical mechanics, theemphasis isoften more ontheself-similarity aspect. Many books display beautiful pictures ofprecisely drawn ﬁgures thatillustrate self-similarity down toinﬁnite levels ofsubdivision, suchastheSierpinski carpet sketched inFig.11.16.There arealsoexamples from nature, suchasthedendritic growth ofthebranches ofatree,inwhich theself-similarity ismore approximate andirregular. DERIVATIONS 1.Show thatthesystem y,,+1 =1-J/Y3with -1<y<land 0<y52can be transformed tothelogistic equation (11.31) bythesubstitution y=cx+d.Findy, c,anddinterms ofthecontrol parameter aofthelogistic equation. 2.Show thattheHénon—Heiles Hamiltonian (11.17) canbewritten inpolar coordinates as H=+51%+ékrz+gmsin36. Thisform explicitly exhibits thethreefold symmetry. 3.Show thatfortheenergy E=é,thebounding equipotential (V(r, 0)=%)forthe dimensionless Hénon—Heiles potential V(r,0)=;1_,r2+31;?sin30. forms anequilateral triangle inthex,yplane (cf.Fig.11.6). 4.Show thatEq.(1l.34b) follows from inserting Eq.(11.34a) intoEq.(11.31). Inaddi- tion.showthatthestability range given inthetext(cf.Eq.(11.35)) alsofollows. Exercises 523 EXERCISES Most ofthefollowing exercises arebestcompleted using apersonal computer abletorunprograms such asMapleTM, Mathematicam, orMaximaTM. Inthese exercises thenotation dz/dt=2isused. 5.Find theﬁrstthree bifurcations forthesystem y,,+1 =l—byg,where —l<y<1 and0<b52. 6.Inanattempt topredict weather pattems, Edward N.Lorenz developed amodel in 1969 with thefollowing three coupled equations (Lorenz model) inx(t), y(t), and z(t): d d d7:=0(y—x), I:=rx—y—xz. ?:=xy—bz, where 0,r,andbarepositive constants andx,y,andzarereal.Lorenz chose, for physical reasons, 0=10andb=g,andtheparameter risincreased from 0.Let x(0) =2,y(0)=S,and1(0)=5.Investigate thebehavior for (a)r=O, 10,and20, 05: <20 (b)r=28,O5t<20,where chaotic behavior setsinfortw7 Inbothcases, investigate thetrajectories byusing either three-dimensional plotsof thecoordinates x(t), y(t), z(t)fordifferent time steps or,ifyour numeric programs donotgenerate suchplots, plotx(t)versus t. 7.Asystem ofequations simpler thantheLorenz equations ofExcrcisc 6wcrc proposed byO.E.Rossler in1976, withonlyonenonlinear system coupling term. Thissystem hadnophysical intent except toshowchaos. dx dy dz E‘-x+a_),v E—b+Z(x_c)1 witha,b,andcpositive constants andx(t), y(t), andz(t)real. (a)Takea=b=0.2,andinitial conditions x(O) =-1,y(O)=z(O)=O.Investigate theeffects ofchanging caround c=5.7,holding aandbﬁxed. (b)Takea=b=0.2,c=5.7andinvestigate theeffects ofchanging theinitial conditions starting withx(0)=-1,y(0)=z(0)=0. 8.Thegeneral forced damped oscillator equation studied byF.Dufﬁng in1918(Dujﬁng oscillator) canbewritten as 2 375+2y %+ozx+/3x3 =Fcoswt (a)Take oz=1,)3=0.2,y=0,F=4.0,[dx/dt],=Q =0andchoose aset ofvalues ofx,=0 anda>toshow thattheamplitude (absolute magnitude ofthe maximum x)ofthesteady-state oscillation shows hysteresis. Thisisbestdone by plotting thebehavior forincreasing wuntil there isajump intheamplitude and thencontinuing theplotforslowly decreasing wfrom avalue slightly larger than where thejump occurred. (b)Having solved part(a),pickavalue ofa)intherange ofthejump andslightly varyFtodetennine howtheamplitude varies foraﬁxedcoasFischanged. Chapter 11Classical Chaos 9Study thevanderPolequation (11.11), d2x 2dx 2 m-E —s(l —x)E +mwOx=FcoswDt (a)Fortheinitial conditions nearx=0.5anddx/dt =0forthevalues ofs=0, 0.1,0.2,and0.3.Plotx(t)asafunction oftimetodetermine empirically therate atwhich theorbit approaches theattractor atx=1. (b)Repeat fortheinitial conditions x=1.5,dx/dt =0. Construct thePoincaré section xpfortheparticular Dufﬁng oscillator d2x dx 3F+0.7 I+x =0.75cost, where p=it=%§,withinitial conditions x(0) =%f-(0) =0. Construct thePoincaré section, asinExercise 10,fortheinverted Dufﬁng oscillator, d2x dx 3—X+-X =FCOSY, forvalues ofFintherange 0.24to0.35. Thisoscillator issaidtobeinverted because thecoefﬁcient ofthelinear tennisnegative. Thediffusion equation is814/8t =r)V2u where u(x,t)isthedensity andr;isthe diffusion constant. Themodel ofdiffusion byWitten andSadler canbeapproximated fornumerical integration intwodimensions byconsidering atwo-dimensional square lattice anddeﬁning thesizeofacluster astheminimum radius thatincludes allofits particles. Mathematically perform thefollowing: (a)Place aparticle atthecenter ofa25x25lattice ofspacing a. (b)Place aparticle atarandom position away from thecenter butnotadjacent to thecenter andallow thisparticle torandomly move onelocation atatimeuntilit either leaves thelattice orbecomes adjacent totheoriginal particle. Forthelatter eventuality, draw acircle centered onthecenter ofthecluster thatjustincludes these twoparticles. Callthisradius Rmin. After completing thisstepRmin =a/2. (c)Repeat thisprocess byadding additional particles atrandom, increasing Rmin if necessary toinclude alladjacent particles. (d)After areasonable number ofparticles, N,areaggregated, calculate thefractal dimension, D,bytherule D=1lnRmin Construct aPoincare section fortheHénon—Heiles potential. Itissuggested thatyou make theplotintheyyplane sothatyoucancompare yourresults withFigs. 11.7 and11.8. Choose anenergy, E,andinitial conditions, x=0andat=0,andinitial conditions onyandytosatisfy theenergy condition andﬁndtheboundary curve. Relax thecondition onatandchoose conditions onX,y,and>3thatsatisfy theenergy condition forx=0.Integrate theequations ofmotion toﬁndthecrossings. (a)Choose E=115,yo=0.01, 5,0=0.02, andxo=0.Usetheenergy equation to determine fro.Integrate theequations toﬁndthevalues oftwhere x(t)»'='»0saving Exercises 525 thevalues t,x(t) ~0,k(t), y(t), yo). Find theﬁrst27crossings andcompare withFig.11.7. (b)Repeat thisprocess forE=-,1;andplotthechaotic behavior. 14.Construct theentries inTable 11.1fora=3.55anda=3.60. 15.Refer toFigs. 11.13 and 11.15 forthelogistic equation. Find thevalues ofthethree cycleattractors embedded intheregion ofchaos. Usethecontrol parameter a=3.83. Also ﬁndthevalues ofthenexthigher cycle obtained foralarger control parameter in thissame embedded region ofnormality. 16.Show thatinthecontrol parameter range between theﬁrstandsecond bifurcations of thelogistic equation thetwoﬁnalvalues oftheattractors, xnandx,,+1 suchasthose given byEq.(11.36) satisfy thecubic equation a3x2(2 -x)-@2111+1)x+(.12-1)=0. CHAPTER Canonical Perturbation Theory 12.1 IINTRODUCTION 526Almost alloftheproblems inclassical mechanics discussed inChapters 1-10, whether inthetextorintheexercises, have hadexact solutions. Nevertheless, itshould beclear from Chapter llonchaos thatthegreat majority ofproblems inclassical mechanics cannot besolved exactly. Wehave found solutions forthe two-body Kepler problem, butwiththeexception ofafewspecial cases theclas- sical moti_on ofthree-point bodies acted upon onlybytheir mutual gravitational forces hasproved intractable (seeSection 3.12). Even fortwobodies thesolutions areimplicit; noclosed explicit formula canbefound forthecoordinates asafunc- tionoftime(cf.Section 3.8).There isthusconsiderable incentive fordeveloping approximate methods ofsolution. Itoften happens, fortunately, thatinaphysical problem thatcannot besolved directly theHamiltonian differs only slightly from theHamiltonian foraprob- lemthatcanbesolved rigorously. Themore complicated problem isthen said tobeaperturbation ofthesoluble problem, andthedifference between thetwo Hamiltonians iscalled theperturbation Hamiltonian. Perturbation theory consists oftechniques forobtaining approximate solutions based onthesmallness ofthe perturbation Hamiltonian andontheassumed smallness ofthechanges intheso- lutions. Weknow from thediscussion inChapter llthateven when thechange in theHamiltonian issmall, theeventual effect oftheperturbation onthemotion can belarge. Thissuggests thatanyperturbation solution must becarefully analyzed tobesurethatitisphysically correct. Thedevelopment ofperturbation theory goes back totheearliest days ofce- lestial mechanics. Newton realized, forexample, thatmost oftheoscillations in theMoon’s motion were theresult ofsmall changes intheattraction totheSun astheMoon revolves about Earth. Hisinitial attempts atalunar theory including these effects corresponded roughly toaform ofperturbation theory. Many of thesubsequent developments intheformal structure ofclassical mechanics, such asHan1ilton’s canonical theory, stemmed inlarge measure from thedesire to perfect perturbation techniques incelestial mechanics. Theneed forpredicting highly accurate orbits forspace vehicles andtheenonnously increased capacity fornumerical computations have spurred further improvements inperturbation theory. 12.2 I12.2 Time-dependent Perturbation Theory 527 Classical perturbation theory canbedivided into twoapproaches: time- dependent andtime-independent perturbations. Theterminology ischosen with aneyetoperturbation theory asdeveloped forquantum mechanics, andindeed there aremany points ofanalogy between theclassical perturbation techniques andtheirquantum counterparts. Generally speaking, classical perturbation theory isconsiderably more complicated thanthecorresponding quantum mechanical version. Weshall treattime-dependent perturbation ﬁrstasbeing theeasier form tounderstand. While perturbation theory canbedeveloped forallversions of classical mechanics, itissimplest tousetheHamilton-Jacobi formulation. TIME-DEPEN DENT PERTU RBATION THEORY LetH0(q,p,t)represent theHamiltonian forthesoluble, unperturbed problem. Weimagine thesolution hasbeen obtained through Hamilton’s principal function S(q,oz,t),which generates acanonical transformation inwhich thenewHamilto- nian, K0,fortheunperturbed problem isidentically zero. Thetransformed canon- icalvariables, (Ct,13),arethenallconstant intheunperturbed situation. Now let usconsider theperturbed problem forwhich wewrite theHamiltonian as(cf. Eq.(11.8)) H(q.P.I)=Ho(q-P.t)+AH(q. p.t)- (12-1) Ashasbeen emphasized before, thecanonical property ofagiven coordinate transformation isindependent oftheparticular form oftheHamiltonian. There- fore, thetransformation (P.q)—>(<1./3) generated byS(q,oz,t)remains acanonical transformation fortheperturbed problem. Only nowthenewHamiltonian willnotvanish andthetransformed variables maynotbeconstant. Fortheperturbed problem, thetransformed Hamil- tonian willbe K(0z,/3,t)=H0+AH+%€=AH(oz, 5,t). (12.2) Hence, theequations ofmotion satisﬁed bythetransformed variables arenow éi=_8AH(a, ,8,t)! ﬂi:3AH(<x,ﬁ, t)‘ (12.3) (9)3; 30!,‘ Equations (12.3) arerigorous; noapproximation hasyetbeen made. Ifthesetof 2nequations canbesolved foroz,and)8;asfunctions oftime, thentheequations oftransformation between (p,q)and(oz,)8)giveq1-andpjasfunctions oftime, thatis,solve theproblem. However, theexact solution ofEqs.(12.3) isusually nolessdifﬁcult toobtain thanfortheoriginal equations ofmotion. Theuseof Chapter 12Canonical Perturbation Theory Eqs. (12.3) asanaltemative approach totherigorous solution istherefore not particularly fruitful. Intheperturbation technique, however, advantage istaken ofthefactthatAH issmall. Thequantities (ct,)9),while nolonger constant, therefore donotchange rapidly, atleastcompared totheexplicit dependence ofAHontime.Aﬁrst-order approximation tothetimevariation of(oz,,8)isobtained byreplacing orand)3on theright-hand sideofEqs.(12.3) bytheir constant unperturbed values: dh_=_3AH(a,)3,t) vBh_:8AH(a,ﬂ,t) '(124) 3/31 0 3011' 0 Here a|,-and51,stand fortheﬁrst-order perturbation solutions for01,-andBi, respectively, andthevertical lines withsubscript 0indicate thatafterdifferentia- tionozand19aretobereplaced bytheir unperturbed forms; thatis,theconstants (060,ﬁg).Equations (12.4) canbeplaced inmatrix form bydesignating ‘yasthe column matrix oftheﬂandorcanonical variables, sothat 3 r.=|i;5“”—’) . <12.s>'1’ 0 where Iisthematrix given byEq.(8.38a). Equations (12.4) cannowbeintegrated directly toyield theor;andBlasfunctions oftime. Through thetransformation equations. wethenobtain (q,p)asfunctions oftimetoﬁrstorder inthepertur- bation. Clearly, thesecond-order perturbation isobtained byusing theﬁrst-order dependence oforand)3ontime intheright-hand sides ofEqs. (12.4), andso on.Ingeneral, thenth-order perturbation solution isobtained byintegrating the equations (inmatrix form) for'y,,given by ._aAH(1/.t)1',.—1 67 . (12.6) n—1 Asatrivial example ofthese procedures, letusconsider astheunperturbed system theforce-free motion inonedimension ofaparticle ofmass m.Theun- perturbed Hamiltonian is 2PH=——.O2m Themomentum pisclearly conserved; callitsconstant value or.Forthissystem theHamilton-Jacobi equation is 1as2as Because thesystem isconservative andxiscyclic, weknow immediately thatthe solution forHamilton’s principal function is 12.2 "lime-dependent Perturbation Theory 529 2 s=ax-E. (12.8)2m Thetransformed momentum isa;thetransformed constant coordinate is Q§ﬂ=Zi=x—5'C! "1 01' x=1'+5, (12.9)"1 theexpected solution fortheforce-free motion. While Eq.(12.9) isobvious apri- ori,thisformal derivation viatheHamilton-Jacobi equation atleast shows thatoz and19,sodeﬁned, form acanonical set. Now suppose theperturbation Hamiltonian is 22 AH= (12.10) where coissome constant. ThetotalHamiltonian is 1H=H0+AH=5—(p2 +m2a>2x2). (12.11)I1’! Wearethusconsidering theharmonic oscillator potential asaperturbation on force-free motion! Interms ofthea,;3variables, theperturbation Hamiltonian, byEq.(12.9), is 2 2 AH=2<1’+)6). (12.12)2 m Intheperturbed system, theequations ofmotion foroz,)3are(cf.Eqs.(12.3)) 2atdz=—ma) (—+5), (l2.13a)m 3:(02! +,5). (121311) Note that 6+£61=0. (12.14) Arigorous solution ofEqs.(12.13) canbeobtained bytaking thetimederivative ofEq.(12.l3a): bi=——a>2oz —mwz —it) =—a>2oz. (12.15)m Chapter 12Canonical Perturbation Theory Thus, aintheperturbed system rigorously hasasimple harmonic variation with time. From Eqs. (12.l3a) and(12.9), itfollows x=-61/(mwz), andhence the solution forxisalsosimple harmonic motion. Considered asrigorous equations ofmotion, Eqs. (12.13) therefore leadproperly totheconrect andwell-known solution. Butnowletustreatmwz (Ek,theforce constant) asasmall parameter and seekperturbation solutions. Theﬁrst-order perturbation isobtained byreplacing ozand,3ontheright bytheir unperturbed values anandB0.Forsimplicity, we shall takex=0initially, sothatB0=0;theinitial value ofpisthen0:0.The ﬁrst-order equations ofmotion arethen _ 2,2 <1,=-090101, 5,=C\€()—-wT1—, (12.16) withimmediate solutions 22 23 011=010——Wdot ,/31=Low I. (12.17) 2 3m Solutions forxandptoﬁrstorder arethen 33 x=51+,s1= 3<0»-Q), (l2.l8a)m ma) 6 and 22 p=011=0z()(1— . (12.18b) Substituting Eqs.(12.17) forozand)8ontheright-hand sideofEqs.(12.13), the second-order equations ofmotion become w2t3 dz2=—oz()a>2 IT , _ 2 2,4 )2=223;:_26.), (12.19) withsolutions aJ20t()t2 a>2a()t4 “Z=“°-_2_+—24—’ otgwz :3w2t5=_- ---_ . 12.2'6’ m(3 30 (0) Thecorresponding second-order solutions forxandpare 12.2 Time-dependent Perturbation Theory 531 ao I@323+a>2t5x: — ———- i- ,ma)"’3! 51 22 44cot cot= -—— -i . 2.2 p<¥0(1 2,+4‘) <11) Bynowwehave enough toseewhere thenth-order solution isgoing. Thequan- tities intheparentheses inEqs.(12.21) aretheﬁrstthree tenns intheexpansion ofthesineandcosine, respectively. Inthelimit ofinﬁnite order ofperturbation, clearly <10 .x—>——sma>t, p—>aocoswt, mw which arethestandard solutions consistent withtheinitial conditions. Theconstant transformed variables (oz,/3)incorporate information onthepa- rameters oftheunperturbed orbit. Thus, iftheKepler problem inthree dimensions describes theunperturbed system, thenasuitable setof(oz,fl)aretheDelaunay variables, thatis,theconstant action variables J1andtheconstant terms inthecor- responding angle variables wi.Wehave seen inSection 10.8thattheDelaunay variables aresimply related totheorbital parameters—semimajor axis,eccentric- ity,inclination, andsoon.Theeffect oftheperturbation istocause these parame- terstovarywithtime. Iftheperturbation issmall, thevariation oftheparameters within oneperiod oftheunperturbed motion willalsobesmall. Time-dependent perturbation theory thusimplies apicture inwhich theperturbed system moves during small intervals oftimeinanorbit ofthesame functional form astheunper- turbed system, anorbit whose parameters however willbechanging intime. The unperturbed orbit along which thesystem ismomentarily traveling issometimes described asthe“osculating orbit.” Inposition andtangent direction, itmatches instantaneously thetruetrajectory. Asdetermined byaperturbation treatment, theparameters oftheosculating orbit may vary with time intwoways. There may beaperiodic variation, in which aparameter comes back toaninitial value inatime interval thattoﬁrst order isusually theperiod oftheunperturbed motion. Orthere mayremain a netincrement inthevalue oftheparameter attheendofeach successive orbital period—and theperturbed parameters aresaidtoexhibit secular change. Peri- odiceffects ofperturbation donotchange theaverage parameters oftheorbit; onthewhole, thetrajectory remains looking much liketheunperturbed orbit. A secular change, nomatter howsmall perorbital period, means thateventually, after many periods, theinstantaneous perturbed parameters maybequite differ- entfrom theirunperturbed values. Therefore, themajor interest inaperturbation calculation willoften beinthesecular terms only, andtheperiodic effects maybe eliminated early inthegame byaveraging theperturbation overtheunperturbed period. Effectively, thisiswhat wasdone inSection 5.8when theperturbing Chapter 12Canonical Perturbation Theory gravitational potential oftheoblate Earth wasaveraged overthesatellite period (cf.Eq.(5.90)).* Often wewould liketodetermine thetime dependence oftheorbital “con- stants”—for example, eccentricity, orinclination—directly, rather thanthrough theintermediary ofthecanonical set(av,,6).This canbedone easily through the Poisson bracket formalism. Letc,-beanysetof2nindependent functions ofthe (or,,8)constants oftheunperturbed system: c,-=c,-(oz, B). (12.22) Oneormore ofthec;maybethedesired orbital parameters. Then intheperturbed system thetimedependence ofthec,-quantities isdetermined bytheequations of motion é,=[c,-,K]=[c,-,AH]. (12.23) ButAH(oz,/3,t)mayequally well, bytheinverse ofEqs.(12.22), beconsidered afunction ofthec’sandt,sothat(cf.Eq(9.68)) 8-BAH 8-BAH6 - rc.~.AH1E °’1 =“I "’31] 81) 81) Bcj 61] _[C‘ c_]GAH _ HJac; . Hence, , 3AHCi=[Ci,Cj] J Aswith Eqs.(12.3), Eqs. (12.24) arerigorous equations ofmotion forthec,-’s. They become ﬁrst-order perturbation equations when theright-hand sides, in- cluding thePoisson brackets, areevaluated fortheunperturbed motion. Ingeneral thenth-order perturbation isobtained when theright-hand sides areevaluated in terms ofthe(n—1)storder ofperturbation. Equations (12.24) thuscorrespond, ingeneralized form, toEqs.(12.6). *The circumstances areoften bemore complicated thanasdescribed inthisparagraph. Forexample, theperiodic variation oforbital parameters canexhibit more thanoneperiod. Thiswould obviously occur when theperturbing potential hasitsownintrinsic periodicity, forexample, thevarying pertur- bation oftheSun’s gravity onEarth-Moon orbit asEarth revolves around theSun.Multiply periodic behavior canalsoappear through interactions between perturbations. 'l11us, theperiodic perturbation ofsatellite parameters canshow bothshort andlongperiods, anditisnecessary toaverage overboth kinds ofperiods toﬁndthesecular perturbation effects. Sometimes thedividing linebetween periodic andsecular perturbations becomes abitvague. What mayappear asasecular perturbation inﬁrstorder willattimes oncloser examination tumouttobeaperiodic perturbation withaverylongperiod, as wediscovered inSection 1l.lwiththeharmonic oscillator perturbation calculation. Depending onthe purpose ofthecalculation, itmaystillbeadvisable totreatitasasecular perturbation term. Nonethe- less,thedistinction between periodic andsecular terms remains useful andnormally straightforward, especially inﬁrst-order perturbation theory. 12.3 I12.3 Illustrations oflime-dependent Perturbation Theory 533 Aversion ofEqs.(12.24) expressed inLagrange brackets (cf.Eq.(9.79)) is often found intheliterature ofcelestial mechanics. Multiply theequation forc,-, bytheLagrange bracket {ck,c,-}andsumoveri: , 8AH {C/<,¢z}¢r={C/<.Cr}{¢t.Cjl T-C1 Bythetheorem expressed inEq.(9.83), thisreduces to 8AH _—-3? ={Cj, C;}C,'. (12.25) J Historically, theperturbation equations ofcelestial mechanics areexpressed in terms ofthedisturbing ﬁmction R,deﬁned as—AH, sothatEqs.(12.25) appear as 8R _E ={Cj, C;‘}C|'. Equations (12.24) or(12.25) arefrequently denoted astheLagrange perturbation equations. ILLUSTRATIONS OFTIME-DEPENDENT PERTURBATION THEORY A.Period oftheplane pendulum withﬁnite amplitude. Inthelimit ofsmall oscil- lations aplane pendulum behaves likeaharmonic oscillator andisisochronous; thatis,thefrequency isindependent oftheamplitude. Astheamplitude increases, however, thecorrect potential energy deviates from theharmonic oscillator form, andthefrequency shows asmall dependence ontheamplitude. Thesmall differ- ence between thepotential energy andtheharmonic oscillator limit canbecon- sidered astheperturbation Hamiltonian, andtheshiftinfrequency derived from thetimevariation oftheperturbed phase angle. TheHamiltonian foraplane pendulum, consisting ofamasspointmattheend ofaweightless rodoflength I,is Z H=—”—2+mgl(l-cose). (12.26)2ml where, forsimplicity, themomentum conjugate to0isdenoted byp.Expanding thecos6terminaTaylor series, theHamiltonian canbewritten as _p2mgl92 e204 127 H_2ml2+ 2112+360 ' (Z) Thesmall amplitude limit consists ofdropping allbuttheﬁrsttermintheparen- theses. Wecangetanideaofthemagnitude ofthecorrection terms byintroducing 4 Chapter 12Canonical Perturbation Theory artiﬁcially aparameter 2150,2=_- (12.28)mgl andtherelated parameter 02 E)(=_1=i 6 3mgl Theseries intheparentheses (cf.(12.27)) thenlooks like 2.0211021__ _ __ _..._ 2<91)+10(91) Now, theratio 6/91risestotheorder ofunity atthemaximum amplitude. Indeed, 91isthemaximum amplitude ofoscillation when E,andtherefore theamplitude, issmall. Hence, therateofconvergence oftheexpansion isdetermined bythe magnitude ofA. Ifonlyonecorrection tennisretained, ﬁrst-order perturbation introduces terms oftheorder Ainthemotion. Second-order perturbation withthesame perturbation Hamiltonian introduces 2.2tenns. Thus, toobtain modiﬁcations ofthemotion consistently correct toA2,wewould have tocompute second-order perturbation ontheAtermintheHamiltonian, andﬁrst-order perturbation ontheA2termin theHamiltonian. Weshall herecontent ourselves with aconsistent treatment to order A;thatis,retain onlytheﬁrstcorrection tenn intheHamiltonian andcarry outaﬁrst-order perturbation solution. Theunperturbed Hamiltonian derived from Eq.(12.27) canbeputinthefonn ofaharmonic oscillator bywriting itas(cf.Eq.(10.18)) 1H=E(E+1202262), (12.29) where I=mlz,themoment ofinertia ofthependulum, and l Asuitable setofcanonical variables corresponding toavanishing Kfortheun- perturbed system aretheaction variable Jandthephase angle ﬂintheangle variable: w=vt+5, 1»= (12.31) Theeffect oftheperturbation istocause both Jand,5tovary with time. The equations oftransfonnation relating pand9toJand,6,respectively, havealready been given inEqs.(10.96) and(10.97), which heretaketheform 12.3 Illustrations oflime-dependent Perturbation Theory 535 6= '7l'i(‘)Sll’l27l'(1)t-l-5), (12.32) /IJp=Twcos2a(vt +,B). Intheunperturbed system Jand,6areconstant andEqs.(12.32) constitute the complete solutions forthemotion. Buttheequations remain valid fortheper- turbed case, only Jand,5have timedependencies tobedetermined. Theunperturbed Hamiltonian isH0=Jv,buttheperturbation Hamiltonian takes theform3 1 J2 .AH=-%e“ =-@5712 S1114211(1):+)3). (12.33) Theﬁrst-order timedependence of)3andJaretobeobtained from .aAH .8AH=i, J=-—-, 12.34'6aJ an () where ontheright-hand sideofeachequation theunperturbed solutions forJand )6aretobeused; thatis,Jand,6areconsidered constant. Thus, . J _ 5=- S1114 2J'l'(l)l + Equation (12.35) saysthattoﬁrstorder, varies overthecycle oftheunperturbed oscillation. Butthere isanetvalue for when averaged overacomplete cycle, fortheaverage ofsin4is-3-.Hence, exhibits asecular perturbation ataconstant rategiven by - J Viewed overtimes longcompared totheunperturbed period, ,5hasatimedepen- dence )3_)3:+50. (12.37) Suchavariation, when inserted inEq.(12.32), saysthat,onaverage, theﬁrst-order solution isstillsimple-harmonic withafrequency v’=v+ Now, intheunperturbed motion 2E ElJ=L =27rw—, (0 5’ Chapter 12Canonical Perturbation Theory sothat)3,Eq.(12.36), becomes T E 02p=_%nﬂ =_‘1_é. (12.38) Theﬁrst-order fractional change inthefrequency ataﬁnite amplitude 91isthere- fore A 02-U3=g=-é, (12.39) awell-known result thatcanalsobeobtained byapproximating theelliptic- function representation ofthemotion. From Eqs.(12.33) and(12.34), itisseenthattoﬁrstorder thetimevariation of Jis . J2 _J=W s1n32rr(vt +,6)cos2rr(vt +,5). Theaverage ofsin3¢cos¢overeven ahalfperiod of¢iszero; hence, Jshows nosecular perturbation. Wewould expect thisresult physically, asJisameasure oftheamplitude oftheoscillations (cf.Eqs.(12.32)), andtheperturbation would notbesuchastocause theamplitude togrow ordecay withtime. B.Acentral force perturbation ofthebound Kepler problem. InExercise 21, Chapter 3,itwasshown rigorously thatifapotential witha1/r2formisadded totheCoulomb potential, theorbit inthebound problem isanellipse inarotat- ingcoordinate system. Ineffect, theellipse rotates, andtheperiapsis appears to precess. Here wewillﬁndtheprecession ratebyﬁrst-order perturbation theory, considering asomewhat more general fonn fortheperturbing potential. Suppose thetotalpotential is khv=-;-,7, (12.40) where nisaninteger greater thanorequal to+2.Theconstant hwillbeassumed tobesuchthatthesecond tennisasmall perturbation ontheﬁrstfortherange of rconsidered. Theperturbation Hamiltonian isthus AH=-L n32. (12.41)rn Intheunperturbed problem theangular position oftheperiapsis intheplane ofthe orbit isgiven bytheconstant to=21:wg(cf.Eq.(10.166)). With theperturbation, tohasatimedependence detennined by aAH aAH'=2___=—, 12.42“’”an at () 12.3 Illustrations oflime-dependent Perturbation Theory 537 using therelation J2=2rrl(Eq.(10.156)). First-order perturbation results are obtained byevaluating AH, andthederivative, interms oftheunperturbed mo- tion.Further, theinstantaneous change incoisrarely ofinterest. Inmost situations where theperturbation formalism isofvalue, cbissosmall thechange inwisdif- ﬁcultorimpossible toperceive within asingle orbital period, anditissufﬁcient to measure onlythesecular change incuaftermany orbits. Therefore, what iswanted iscbaveraged overatimeinterval 1',theperiod oftheunperturbed orbit: - 1 TAH (bi —‘/‘ L dt. T 0 Thederivative canbetaken outside theintegral sign, since risafunction ofJ3 only(Eq.(10.142) combined withEq.(10.146)), whereas thederivative iswith respect toI=J2/2rr. Hence, _a1I 8AH'=--AHd =__. 12.43“’at<1/0 t) al () Butthetimeaverage oftheperturbation Hamiltonian ishere W I AH=-a(l) =if Q. (12.44)r" 1'0r" Byusing theconservation ofangular momentum intheform ldt=mrzd1//,the integral canbeconverted intooneover(0: _ mh2"a(l mh mk "-2 2" ,,_2 ,=_F (IT) A[1+ecos(r//——1//')] <11//,(12.45) where rhasbeen expressed interms oflbthrough theorbit equation. Eq.(3.56) (with (/1used inplace of0).Ingeneral, onlyterms involving even powers ofthe eccentricity ewillgivenonvanishing contributions totheintegral. Thederivative with respect tolalsoinvolves eanditspowers, since, byEq.(10.159), eisa function onlyofJ2andJ3. Twospecial cases areofparticular interest. Oneoccurs when n=2,mentioned brieﬂy atthestartofthisillustration. Theaverage perturbation Hamiltonian isthen simply ﬁrm,l1: andthesecular precession rateis Chapter 12Canonical Perturbation Theory 5=Lmh, (12.46)Z21: which agrees withExercise 21ofChapter 3. Theother caseofinterest isforn=3(a1/r3perturbation potential), forwhich Eq.(l2.45’) reduces to _ 2rrm2hkAH=-iI31: and T6Zakw=—-7%. (12.47) What makes thischoice ofnofparticular signiﬁcance isthatgeneral relativity theory predicts acorrection toNewtonian motion thatcanbeconstrued asan F3potential. Theso-called Schwarzschild spherically symmetric solution ofthe Einstein ﬁeldequations corresponds forweak ﬁelds toanadditional Hamiltonian termintheKepler problem oftheform ofEq.(12.41), withn=3and /<12h=Z, (12.48) sothatEq.(12.47) becomes T6/<2(0%. (12.49) Toapply Eq.(12.49) tothesecular precession ratefortheprecession ofabody revolving around theSun, kissetequal toGMmandEq.(3.63), valid forthe unperturbed ellipse, isused 12=m/(a(l-e2). (12.50) Equation (12.49) canthenbeputintheform _ 6rr R'=M _ 12.51 (U r(1—e2) (a)’ ( ) where Ristheso-called gravitational radius oftheSunis GMR=7 =1.4766 km. (12.52) Fortheplanet Mercury, 1:=0.2409 sidereal years, e=0.2056, anda= 5.790 ><107km;Eq.(12.51) then predicts aprecession oftheperihelion of Mercury arising from general relativity atanaverage rateof ab=42.98”/century. 12.3 Illustrations oflime-dependent Perturbation Theory 539 Theobserved secular precession oftheperihelion ofMercury isover 100times larger thanthisvalue, namely 5599.74 :1:0.41”/century. Most ofthisisduetothe precession oftheequinoxes, oftheremainder, about 531.54’ ’/century arises from perturbations oftheorbitofMercury byother planets. Only afterthese twosetsof effects aresubtracted from theobserved precession doesthesmall general relativ- ityeffect ofapproximately 43"/century become visible. Thecurrently accepted observational value isstated tobe43.l” :l:0.5”/century; thedeviation from the theoretical prediction isnotconsidered signiﬁcant. Onepoint remains tobemade. Intheapplication torelativistic effects, the constant h,Eq.(12.48), isafunction ofthevalue ofl.Itmight beasked therefore thatinﬁnding rb,whydoesn’t thederivative withrespect tolactalsoonh?The keyhereisthathisnotfunctionally dependent onlasacanonical momentum, Equation (12.48) saysonlyhowthevalue oftheconstant hisdetermined intenns ofthevalue oftheorbit parameter l.Inother words, theperturbation potential is afunction ofthedynamical variables onlythrough r;itisnottobeconstrued as velocity dependent. C.Precession oftheequinoxes andofsatellite orbits. Thefamily ofproblems to beconsidered herewasdiscussed previously inSection 5.8,which bears thesame title. Wewish todescribe therelative motion oftwobodies interacting through theirgravitational attraction, oneaspherically symmetric orpoint body, theother being slightly oblate witharesultant gravitational quadrupole moment. Theeffect oftheslight oblate shape ofEarth isphysically thatthetorques exerted bythe SunandMoon ontheequatorial bulge cause Earth’s rotation axistoprecess very slowly. Reciprocally, theeffect onanobject orbiting around Earth, such asthe Moon oranartiﬁcial satellite, istocause theplane oftheorbittoprecess about theﬁgure axisofEarth. Thesmall magnitude ofthegravitational quadrupole term, manifested bytheveryslowrateofprecession, suggests thataperturbation treat- ment should beanextremely good approximation. Weshall actually examine here onlythecaseoftheperturbation ofasatellite’s orbit; thereciprocal phenomenon oftheprecession oftheequinoxes proceeds verysimilarly (though withdifferent notation) from thesame perturbation Hamiltonian, andwillbeleftfortheexer- cises. Since theemphasis herewillbeonapoint satellite moving about amuch more massive Earth, thenotation ofSection (5.8) willbereversed here andmused todenote themass ofthesatellite while Mstands forEarth’s mass. Thetotal potential acting onthesatellite, byEq.(5.88), isthen kk(I3-11)V=—;+H—’7P2(V), (12-53) where k=GMm, P2(y) isthesecond-order Legendre polynomial, andyisthe cosine oftheangle 0between theradius vector tothesatellite andEarth’s ﬁgure axis.Fortheperturbation Hamiltonian, wetherefore have I3—11 2 Chapter 12Canonical Perturbation Theory Thepolar angle 6canbeexpressed interms oftheinclination angle oftheorbit, i,andtheangle oftheradius vector intheorbital plane relative totheperiapsis, 1//,(theso-called trueanomaly) bytherelation* cos6 =sinisin(1// +to), (12.55) where coistheargument oftheperiapsis. Asmall amount ofmanipulation enables ustorewrite theangular dependence ofAHas 3cos29 -—1=(%—%coszi)—gsin2icos2(1// +a>). (12.56) Now, because ofthesmall sizeoftheperturbation, thechief interest isinthe cumulative effects ofthesecular portion. Thus, theprecession oftheorbital plane shows upasasecular change inQ,theangle ofthelineofnodes (orlongitude of theascending node). Bythesame argument used intheprevious illustration we canobtain thesecular effects byaveraging AHprior totaking derivatives: i_ 1r m 21: AHE—f AHdt=—f r2AHd1//T0 ‘L’10 ZkZ I__I 27! ="%(M?3?-ll A(1+ec0S1//)(3cos26 -l)d(//. (12.57) Thetenn incos2(1//+co)inEq.(12.56) gives zerocontribution totheintegral because itisorthogonal, intheinterval ofintegration, tobothIandcos1//.Hence theaveraged perturbation Hamiltonian is i 7Tm2k2(I3 —I1) 2 A =4-? 1—3 '. 12.58 H 2Ml3T ( cosz) ( ) Inview ofEqs.(10.157) and(10.165) linking S2andiwiththeaction-angle vari- ables, theﬁrst-order perturbation value forQistobefound from ._ aAH 1aAH§Z=2 '=2 —i=—i— W‘ 7'an l8cosi 01' 5__3rrm2k2(I3 —I1)cosi 7 M141 ' Finally, using Eq.(12.50), theaverage fractional change inQperunperturbed revolution is *Equation (12.55) canbeobtained inmany ways, forexample, bymatrix rotation oftheplane of theorbitintothexyplane. Itisgiven, mostsimply perhaps, bysome old-fashioned trigonometric reasoning based onFig.10.7.AsOB=1,BC=cos6,butAB=sin(r// +0))andtherefore BCis alsosinisin(r// +co). 12.4 I12.4 lime-independent Perturbation Theory 541 Q1: 3I3-I1cosi=—— , 12.59221 2Maz (1—e2)2 ( ) which istheappropriate generalization ofEq.(5.96) toanelliptic satellite orbit. Once theaverage perturbation Hamiltonian isknown, theeffect ofthepertur- bation onother average parameters oftheorbit canbefound. Thus, thesecular precession oftheperiapsis intheplane oftheorbit isimmediately given by T2n__ 2”SE 8EQ): w = i =-4 2 an at Thecanonical variable J2occurs inAHasgiven byEq.(12.58) intwofonns: in theI3term inthedenominator andintheterm containing cosi =J1/J2. Upon carrying outthederivative, itisfound that T31-1 _g=Z (5cos2z -1). (12.60) Themaximum value of5isthusabout thesame asthatofQ,butthedependence upon iisquite different. Atcritical inclinations of63°26’ and1l6°34’, thepre- cession oftheperiapsis vanishes (atleast toﬁrstorder) andchanges signabove andbelow these points. Itisclear that,toﬁrstorder, there isnosecular change ineither aore,since Edoes notcontain theconstant parts ofanyoftheangle variables. Theshape andsizeoftheosculating ellipse, when averaged overthe orbital period, thusdoesnotchange withtime. Itmaybenoted from thelasttwoillustrations thatthegeneral relativity cor- rection andthegravitational quadrupole ﬁeldbothgiverisetoaprecession ofthe periapsis ofanorbiting body. Theformer isbelieved tobethemore dominant factor contributing totheobserved precession oftheperihelion ofMercury, since themeasured quadrupole component oftheSun’s massistoosmall. TIME-INDEPENDENT PERTURBATION THEORY Consider conservative periodic separable systems ofarbitrary number ofdegrees offreedom withaperturbation parameter e.Fortheunperturbed problem, weas- sume asetofaction-angle variables (Joi,wo,-)suchthattheunperturbed Hamil- tonian, H0,isafunction onlyoftheaction variables J0,-,andcorrespondingly, the w(),-arethenlinear functions oftime. Inthenotation ofEq.(10.1 10’),therelation between, say,qkandthew(),-canbewritten compactly as qt=ZA,F"’<J6)e2’"'~‘""°. (12.61).i where j,W0,andJ0aren-dimensional vectors oftheinteger indices, angle vari- ables, andaction variables, respectively. Chapter 12Canonical Perturbation Theory Intheperturbed system, (W0,J0)remain avalid canonical setofvariables. When expressed interms oftheset(W0,J0),theperturbed Hamiltonian canbe expanded inpowers ofasmall perturbation parameter e: H<w0,J0.6)=H<><J<>>+6H1(Wo, Jo)+e’H2<wo, Jo)+---.(12.62) Weseekacanonical transformation from (W0,J0)toanewset(W,J),such that theJareallconstants andthewtherefore linear functions oftime. Inthisset,H isafunction onlyofJ(and6)and,initsfunctional form withrespect toJ,willbe written as <1(J,e)=a0(J)+6011(1)+e2a2(J) +---. (12.63) Toobtain theperturbed frequencies through agiven order ine,itsufﬁces toﬁnd theappropriate functions 010,0:1,...,forthenthevector representing thefrequen- ciesis 8 8v=v0+6€‘1.]i+e2ai‘I2+---. (12.64) Thegenerator ofthecanonical transformation from (W0,J0) to(W,J)is Y(W0,J,6),withacorresponding expansion ine: Y(W0,J,6)=W0-J+eY1(Wo, J)+62Y2(W0, J)+---. (12.65) WeseektoﬁndYasthesolution oftheappropriate Hamilton-Jacobi equation: H(W0, aaTf0, 6)=a(J,e). (12.66) Asbefore, theterms inatoagiven order inearefound byexpanding bothsides in powers ofeandcollecting coefﬁcients ofthesame order onbothsides. Weshall illustrate theprocess forasecond-order calculation, where theHamilton-Jacobi equation reduces to H0 +6H1 (W0. +€2H2 (W0, =0l0(J)+e0z1(J)+'='20l2(J)-W0 W0 W0 (12.67) Each ofthetenns ontheleftarefunctions ofethrough thederivative ofY: 3Y BY BYJ0=—=J+€_‘+€2-2-. (12.68)8W0 6W0 8W0 Weagain expand theterms H;inaTaylor series around J0=J,retaining terms oforder 62inH0andoforder einH,-,withJ0replaced directly byJinH2.The expansions forH0andH1,inmatrix notation, arethen 12.4 Time-independent Perturbation Theory 543 3}’ 3Y1 23Y2 3H0 H°(aw0> 'H°(J)+(6570+6 576)61 1anBZH0(an--__ _- 2.+2(63W0) BJBJ 6BW0) (169) BY BYBH H1(»-10.55) =H1<w@,;n +6 <12-10> Collecting powers ofeinEq.(12.67) thenleads tothefollowing expressions for theﬁrstthree terms inoz: 010=H0(J), (12.71?!) arCV1=voﬁ +H1(W0, J), (12.7lb) BYB2=voﬁ+2<w0.J), (lam) where anam IBY1B2H0BY1<1>,= , - . 2.2(w°‘DH2(w° J)+8W0BJ+2BW0BJBJBW0 (172) Again, theequation oftransfonnation linking WandW0isgiven by BY BY1 2BY2=—= —— —— 12.73 W aJ w0+e 61+6 aJ+ ( ) Inorder forthe(q,p)settobeperiodic inbothW0andWwithperiod 1,allofthe Y1,terms must beperiodic functions ofW0,thatis,oftheform Y1<(Wo.J)=ZB§"’<J>e2"'“'"". <12-74>J Hence, allderivatives ofYkwith respect toW0have noconstant term, andthe ﬁrsttenns ontheright ofEqs.(12.7lb,c) donotcontribute totheJdependence. Equations (12.71) cantherefore alsobewritten as a0(J) =H0(J) (l2.75a) w1(J) =Hi(Wo,J), (12-75b) 0l2(J) =¢2(W0, J). (12-75¢) where thebardenotes anaverage overtheperiods ofallW0.Wecanconveniently express allofEqs.(12.75) inacommon fonnat by a,'(J) =<l>,-(W0, J), (12.75’) 44 Chapter 12Canonical Perturbation Theory where 0=H0and<1>1=H1.Inaddition, Eqs.(12.71) havecounterparts peri- odicinW0withzeromean: vogi =6,-<1>,-. (12.76)3W0 Notethatinsecond-order perturbation theterms inY1donotnecessarily vanish inthemean. Itistruethatthederivatives ofY1themselves have zeromean, but theyaremultiplied byother functions thatwillbeperiodic inW0,andthere is noguarantee thattheaverage oftheproduct vanishes. Hence, toﬁndthesecond- order correction tothefrequencies, weneed toknow theﬁrst-order canonical transformation. (Analogously inquantum mechanics, asecond-order eigenvalue involves ﬁrst-order corrections ofthewave function.) Inprinciple, thecoefﬁcients B9)deﬁning Y1through Eq.(12.74) canbefound directly from Eq.(12.76) for i=1.Subtraction oftheaverage means thatH1—H1canbeexpanded ina Fourier series analogous toEqs.(12.61) or(12.74) butwithout anyconstant term: H1-F,=Zc1(J)e2"‘i""@. (12.77) #0 Using thederivative ofY1inEq.(12.76) withrespect tooneoftheW0,sayw0k, willbring down afactor 211'ijk.Hence, thematrix product ontheleft-hand sideof Eq.(12.76) canbewritten voQ=ZBj<”<J>2.,1u -»0>@1m»w._ (12.18)aw° #0 From Eqs.(12.76) and(12.77), thecoefﬁcients intheseries forY1canbeobtained as 1 C‘(J) .B,‘>(J)= 1¢0. (12.79) Itistruetheconstant tenns inY1arenotd61ZCl‘I1'1iI'l€d inthisway,butitisonlythe derivatives ofY1thatenter intotheexpressions foranandthese donotinvolve the constant terms (cf.Eqs(12.71)). While wehave carried outtheprocedure indetail onlyforsecond-order per- turbation, itiseasytoseethatthegeneral formofthehigher-order calculations must besimilar; onlythedetails ofthealgebra willbemore complex. Fortheith order perturbation, wewillagain beabletowrite 011inthefonn BY a1(J)=v<>a—'+<1>.-(wt).J). (12.?1d)W0 Theﬁrstterm ontheright willcome from theﬁrst-derivative tenn intheTaylor expansion ofH(J0)about J0=J,where allterms inthedifference J0—Jare keptthrough order e‘.OnlyinthistermwillY;appear; hence, <l>,-cancontain only 12.4 Time-independent Perturbation Theory 545 thegenerators Ykfororder lessthani.Byvirtue ofthearguments already used forﬁrst~ andsecond-order perturbations, theﬁrsttermontheright intheprevious equation (12.7ld) haszeromean when averaged overcomplete cycles inW0,and hence, Eqs.(12.75) and(12.76) arevalid inallorders. Ofcourse, fori>2,<l>1 becomes increasingly more complicated thanEq.(12.72), butitalways contains onlysuchfunctions ashavealready beenfound inlower order calculations. Thus, stepbystep, wecould inprinciple work uptoanyorder perturbation. There arepractical problems insuch aseries ofcalculations ofcourse, butthe most serious andobvious conceptual difﬁculty occurs iftheunperturbed system isdegenerate. Asweseefrom Eq.(10.122), theexistence ofadegeneracy means there willbeatleastonevector ofindices jsuchthatj-v0=0.Thecorresponding coefﬁcient BjmintheFourier series forY1willtherefore, byEq.(12.79), blow up. Indeed, something similar takes place even when theunperturbed system isnot degenerate. Even ifthefrequencies arenotexactly equal, aswegotohigher and higher values oftheinteger indices inj,eventually there willbefound avector j forwhich j-v0isverysmall even ifnotzero, andthecorresponding coefﬁcients Bbecome verylarge (theso-called problem of“small divisors”).* Thiscrudely qualitative observation isthebasis oftheelegant proof byPoincare attheendof thelastcentury thattheFourier series forY1,andtherefore forthemotion, are onlysemiconvergent. Nonetheless, theseries canbetruncated atsome reasonable values oftheindices andstillgiveextremely precise results, atleastfortimes that arenottoolong. Weshall discuss laterwhat canbedone inthepresence ofdegeneracy, butat thispoint itmaybewelltoillustrate asecond-order calculation with aspeciﬁc example ofasystem withonedegree offreedom. Consider aone-dimensional anharmonic oscillator, thatis,onewithaq3term inthepotential energy. TheHamiltonian canbewritten as 12 222( qH=— +m a) l+e— , (12.80) 2m[P oq qo where a>0istheunperturbed angular frequency: /ka)0=2n'v0=21r —~,m q0isareference amplitude thatcanbeleftunspeciﬁed forthemoment, andeisa small dimensionless parameter. Taken asanexpansion inpowers of6,Hconsists ofthetenns 1H0=$012 +m2a>%q2), (12.8la) *Similar phenomena, itwillberecalled, arefound inquantum mechanics, where degeneracy means thatthere areseveral states withthesame energy E.Denominators oftheform E,—Ejwillthen vanish. orbecome small evenifthereisnoexact degeneracy. 546 Chapter 12Canonical Perturbation Theory mw2 3 H1=2:1”, (12.81b)Q0 and (l2.81c) H1=O, i32. (12.81d) Using theunperturbed action-angle variables (J0,w0)ascanonical variables the nonvanishing parts ofHcan,byEqs.(10.96) and(10.97). bewritten as H0=./0110 02.8221) and 2 3/2 H1=%L) S111321111111. (12.82b)2q0 1rma>0 Therecipes ofEqs.(l2.75a,b) thengiveasthelowest twoterms ina(J) 010(1) =Hvo; 111(1) =0- Toobtain thesecond-order terma2(J), wenotethatsince H0islinear inJ,and H2vanishes, then<l>2(cf.Eq.(12.72)) reduces to 8Y1,,=_1£11_31.00 3] Butthevanishing ofH1means thatEq.(12.76) fori=1hasthesimple form 3Y1 H1 8w0 v0 Combining these tworesults leads to 1BH12=___. 12.83(P2 211,BJ () Now from Eq.(12.82b), J3 H12(w0, J)=-19-5 8111621111111,2rr2mq0 leading to 312 _6¢2('LU(), =-4? S111 2rrw(1. 71'mqo Since theavera eofsin6overone eriod isE,012J)issiml 8 P 43 P 12.4 Time-independent Perturbation Theory 547 1512 andtosecond order inetheperturbed frequency is 801 215]U=§:U()—€ . Itisconvenient touseforq0themaximum amplitude theoscillator would have forthegiven energy initsunperturbed form, sothattolowest order 2 9 or,since E=Jw0/(2n'), Jmqg=_. (12.87)Irwo Interms ofthisreference amplitude, Eq.(12.86) isequivalent tosaying thatthe second-order fractional shiftinthefrequency issimply AU 2 —— =—— . 12. v0 166 (88) Mention hasalready been made ofthedifﬁculties thatappear inperturbation theory arising outoftheexistence ofdegeneracy, forexample, thevanishing (or nearvanishing) ofj-v0inthedenominators ofEq.(12.79). Treatment ofdegen- eracies inclassical perturbation theory ismuch more complicated thaninquantum mechanics. Themathematics thathasbeen brought tobearontheproblem isboth subtle andcomplicated, andafullexposition would beoutofplace here. Only some brief andintroductory remarks canbemade atthispoint. Wespeak ofexact (or“proper”) degeneracy, asinSection 10.7, when theun- perturbed frequencies 110aresuchthatthere areoneormore setsofintegersj for which j-v0=0.Ashasbeen pointed outinSection 10.7, wecanthentransfonn toanewsetofvariables (J0,w0)forwhich thedegeneracies appear aszerofre- quencies andtheremaining nonzero unperturbed frequencies arenotdegenerate. Theeffect oftheperturbation istoliftthedegeneracy sothatthecorresponding frequencies arenotexactly zerobuthave small values. Inconsequence, there ap- pearinthesolution terms thathave small frequencies, thatis,longperiods. The corresponding angle variables areknown as“slow” variables. incontrast tothe angle variables with nondegenerate frequencies, which aretherefore called the “fast” variables. Long-period terms mayappear assecular terms overrestricted timeintervals; forexample, sin211'vtcanbetaken asalinear function oftsolong asvt<<1. When there isexact degeneracy, atransformation isﬁrstmade tothe(w0,J0) set.Theunperturbed Hamiltonian willbeafunction only ofthenondegenerate 4 Chapter 12Canonical Perturbation Theory J0variables; inallother respects Eq.(12.82) stillrepresents thecomplete Hamil- tonian. Wenowcarry through thecanonical transformation oftheperturbation calculation, butonlyforthenonperturbed variables, leaving thedegenerate vari- ables unchanged. ThenewHamiltonian, Eq.(12.62). nowhastheform ¢1(J.Jf1.W{1.6) =t1o(J) +6w1(J.J{,.w(1) +620120. J0.W11)+-~-- Here W6stands forthem(degenerate) variables thatintheunperturbed problem have zerovalues andJ6fortheirconjugate momenta. Thetransformed nondegen- erate momenta arerepresented byJ.Theresult ofthecanonical transformation isthustoeliminate the“fast” variables, buttoleave interms with the“slow” variables. Note thatsince oriscyclic inw,thetransformed Jmomenta aretrue constants ofthemotion, anda(J,J6,W6,e)canbeconsidered asaHamiltonian ofasystem withmdegrees offreedom. Further, since a0(J) isaconstant, inde- pendent oftheremaining variables, itdoesn’t matter fortheequations ofmotion of(J6,W6)andcanbedropped from oi.Thus, theneweffective Hamiltonian is nowoforder e;ineffect, the“unperturbed Hamiltonian” isea1(J,J6,W6),andin thisunperturbed problem w6nolonger consists ofzerovalues. Ifthere isonlyone degeneracy condition, theeffective problem isofonlyonedegree offreedom and isinprinciple immediately integrable. With more degeneracy conditions, wecan seekasecond canonical transformation toeliminate the“slow” variable terms just aswasdone forthe“fast” variables. Inpractice, theprocedure obviously becomes quitecomplicated. Ithasalready been pointed out,inconnection withEq.(12.79), thateven with nondegenerate frequencies, small values ofthedivisor j-v0willinevitably oc- curastheindices jbecome larger andlarger. Thisphenomenon isreferred toas resonance, implying thattheamplitude ofsome particular term intheFourier expansions becomes verylarge. Itwould seem therefore thattheproblems ofde- generacy willalways bewithus,nomatter what theunperturbed frequencies are! Thesituation isnotallasbadasthat,inpartbecause ofthenature oftheperturba- tionHamiltonians encountered inpractice. From Eq.(12.79), itwillbenoted that what counts isnotsomuch thevalue ofj-v0astheratio C1 .i'v0’ where C1istheFourier series expansion oftheperturbation Hamiltonian H1,cf. Eq.(12.77). Itturns outthatincelestial mechanics, atleast, most perturbation Hamiltonians have what iscalled theD’Alembert characteristic. While thefonnal mathematical deﬁnition oftheproperty iscomplicated, what itsays, roughly, is thatwhen thevalues oftheintegers inthejindices arelarger thantheexponent of eintheHamiltonian, themagnitudes ofC1fallrapidly (generally exponentially) withincreasing values oftheindices. Theratios inEq.(12.79) thendonotbecome toolarge, andtheexpansion process actually canbeproved toconverge when the frequencies v0areincommensurate. 12.5 I12.5 Adiabatic Invariants 549 Resonant behavior inthepresence oftheD’Alembert characteristic, orgener- allywhen C1/(j-110)<O(61/2), isdescribed asashallow resonance. Inprinciple, atleast, shallow resonances maynotupset theperturbation expansion process and canbetolerated without introducing newmethods. There aresituations where the ratio C1/(j-v0)becomes large, atleast larger thanorder 51/2, andthese arere- ferred toasdeep resonances. Special methods have tobedevised tohandle deep resonances, such astheso-called Bohlin expansion inpowers of61/2rather than inpowers ofe. ADIABATIC INVARIANTS AttheﬁrstSolvay Conference in1911, which grappled withtheproblems ofin- troducing quantum notions intophysics, adeceptively simple problem inclassical mechanics wasraised. Consider abobonastring oscillating asaplane pendulum, withthestring passing through asmall holeintheceiling. Now imagine thatthe string iseither pulled uporletdown slowly, soslowly thatthere islittlechange in thelength ofthependulum during oneperiod ofoscillation. What happens tothe frequency ofoscillation during thisprocess? Note thattheenergy ofthependu- lumisnotconserved, forwork isdone onthesystem (orextracted from it)asthe length ofthestring isaltered. Byelementary means itwasdemonstrated thatfor veryslow change oftheratio E/vwould beconstant. Itwillberecognized that thisratioisprecisely theaction variable J.Theadiabatic invariance oftheaction variables under slowchange ofparameters wasaverysatisfying property tophysi- cistsdeveloping quantum mechanics. Forsimplicity, weshall examine onlyperi- odicsystems Withonedegree offreedom, although theextension tomany degrees offreedom nomially isnotdifﬁcult intheabsence ofdegeneracy. Weconsider a system thatinitially hasnodependence onthetime, andthatinvolves aparameter a.Implicit inthemethod isapicture ofthesystem asinitially conservative witha constant. Time dependence ofaisthen“switched on,”andavaries slowly overa longtime, eventually reaching aconstant value. When aisconstant, themotion is periodic, andtheslowchange intheparameter doesnotaltertheperiodic nature of themotion. Although thechanges inthemotion aresmall inanyoneperiod, over alonginterval oftimetheproperties ofthemotion canaccumulate large quanti- tative changes. Theswitching onofthetimedependence isthusinthenature ofa small perturbation, andwearelooking forsecular changes inthemotion. When theparameter aisconstant, thesystem willbedescribed byaction-angle variables (J0,w0)such thattheHamiltonian isH=H(J0,a).Itwillbeuseful toconsider these variables asderived from anoriginal canonical set(q,p)via anF1generating function W*(q, w0,a).Theusual Hamilton-Jacobi equation of course leads toanF2generating function oftheform W(q, J0,a),butthese two generating functions arenormally connected byaLegendre transformation (cf. Eq.(9.19)): W*(q. wo.11)=W(q.J(1. ¢1)—J0w0- (12-39) Chapter 12Canonical Perturbation Theory When aisallowed tovarywithtime,(w0,J0)ofcourse remain asvalidcanonical variables, butthegenerating function isnowanexplicit function oftimethrough thetimedependence ofa.Hence, theappropriate Hamiltonian forthe(w0,J)set isnow i)W* K0002 J01 a) = a) +Y 3 * =H(J0,11)+at (12.90)a Since J0isnolonger aconstant andw0does notvarylinearly withtime, the second term intheHamiltonian isaperturbation. Thetime dependence ofJ0is governed bytheequation ofmotion .ax aBW*J=-_=-'— _-, 12.91°8w0 ”aw0( 8a) () where ofcourse thederivative inparenthesis isexpressed, asisK,interms of J0,w0,anda.Inthespirit ofaﬁrst-order perturbation theory, welook fora secular term, theaverage ofJ0overtheperiod oftheunperturbed motion forthe appropriate a.Since avaries slowly, acanbetaken asconstant during thistime interval, andtheaverage canbewritten as . 1 B BW* J=—— '— i d0 r,/;aBw0(8a)t a 8 6W* ,2_=-- __ . 2. {La (aa)dt+O(a,a) (192)W0 Itwillberemembered from Eq.(10.17) thatWisgiven bytheindeﬁnite integral W:/pdq. Inoneperiod ofw0,thegenerating function, W,therefore increases byJ0.At thesame time, J0w0 alsoincreases byJ0.since w0increases byunity. Hence, by Eq.(12.89), W*isaperiodic function ofw0,andbothitandthederivative with respect toacanbeexpressed asaFourier series: a* .-“L=ZA1,(J0, a)@2’"'""°. (12.93)3a k Theaverage, T0,therefore hastheform 1'0=_£IX:2rrikA1<(J0,a)e2"""“’° at+0(a2,a).rTheo 12.5 Adiabatic Invariants 551 Since theintegrand hasnoconstant term, theintegral vanishes, 71]=0+0(a2,a), (12.94) andj0hasnosecular variation toﬁrstorder ina,proving thedesired property of adiabatic invariance. Letusseehowthisderivation would work indetail fortheproblem ofthe harmonic oscillator: 1H= +m2w2q2), m where a)may beanexplicit function oftime. Theequations ofthecanonical transfonnation from the(q,p)settothe(J0,w0)setaregiven byEqs. (10.21) and(10.97), which canbewritten soastofacilitate theevaluation ofW*: BW*J0=rrmwqz csc2221-100 =—a——,waw, ° (12.95) p=mwq cot2n'w0 = Towithin constant (andtherefore irrelevant) terms, W*isfound byintegration of Eqs.(12.95) tobe 2 W*(q, w0,co)=% cot2rrw0. (12.96) Thederivative withrespect towis BW* 2W =% cot2rrw0, or,using Eq.(10.96) asafunction ofw0,J0,and(0, * i =i sin4rrw0. (12.97)Ba) 4zra> Thus, J0isgiven bytheone-terrn Fourier expansion J0=-311,cos41111111, (12.98) which, aspredicted, hasnoconstant term.S0far,Eq.(12.98) isrigorous. Similarly thengorous connection between w0andtimeisdetemiined bythew0equation ofmotion 1b—aK—aH+(ba aW* —°’+°3 '4 1299°—0J0—0J0 010 01» T211 ».nws‘“ "“’°' ('2 Chapter 12Canonical Perturbation Theory Inorder tocalculate anaverage of10overaperiod, including atleast theﬁrst correction term, webegin tomake approximations. Firstweshallassume thatover aparticular period oftheperturbed motion theratio 3Ee (12100)(0 isaconstant, andonesuchthatst51.Equation (12.100) corresponds toavaria- tion a>=a>0e" %w0(1+et), (12.10l) where tismeasured from thestartoftheperiod interval, atwhich timew(0)=(00. Equation (12.99) nowlooks like we=3+5-311141111111. (12.99')2n’ 42': Thezeroth-order solution is 2z1:w60) =a)0t, where theconstant term hasbeen setzerobysuitable choice oftheinitial phase. Toﬁrstorder ine,Eq.(l2.99’) becomes _ (1+et) e_1116"=£2? +Es1n2a>0t, (12.102) withthesolution 1- 12rrw61) =(110:+5@012+-i-Ml . (12103)2 2a>0 Correspondingly theequation forJ0correct tosecond order in6canbewritten as dlnJ0_ 6608 2t+€( t2+l—cos2w0t)] dt _ mo (D0 2020 ' Expanding thecosine, treating theterm ineasasmall quantity toﬁrstorder, the derivative reduces to lJ 1- 2E =-6cos2w0t+62(»0t2+—-mi sin2w0t.dt 2:00 Toﬁndthesecular behavior, thisequation canbeaveraged overtheperiod ofthe motion asitisatt=0,thatis,overaninterval r=211:/a>0. Intheaveraging, almost allterms ontheright drop out,except theﬁrstinside theparentheses, involving t2.Theﬁnalresult is 12.5 Adiabatic Invariants 553 1 2 2 %=%=%, (12104) where 8=er,thatis,fractional change inwovertheperiod r.Correspondingly, thefractional secular change inJovertheperiod is 2 AT’= (12.105) Asexpected from themore general considerations, thesecular change intheac- tionvariable hasnoterm inﬁrstorder ine.Only byretaining quantities ofthe order 62=(cb/(0)2 doweﬁndanynonvanishing long-term change inJ. Theadiabatic invariance oftheaction variables hasproven tobeespecially useful inapplications involving themotion ofcharged particles inelectromag- netic ﬁelds. Oneofthesimplest instances, andonewithimportant practical con- sequences, concems themotion ofelectrons inauniform (ornearly uniform) constant magnetic ﬁeld. Asiswellknown, thecharged particle insuchasituation circles around themagnetic ﬁeldlines. Atthemost basic level, thiscanbeshown from Newton’s equations ofmotion. TheLorentz force inaconstant magnetic ﬁeldBis(vxqB); hence, theequation ofmotion, Eq.(1.4), is dv qB—= ——. 12.1 dt vxm (06) Equation (12.106)saysthevelocity vector vrotates, without change ofmagnitude, about thedirection ofthemagnetic ﬁeld. withanangular frequency Bwt=-‘17. (12107) Thefrequency, called thecyclotron frequency, hasavalue twice theLarmor fre- quency ofEq.(5.104) (cf.Eq.(7.154)). Anequivalent derivation canbeformulated intenns ofLagrangian mechanics. Itwasshown, inSection 5.9,thattheLagrangian inthiscasecanbewritten as 2 L=%+M-B, (12108) where Mismagnetic moment ofthemoving particle deﬁned interms ofitsangu- larmomentum Lby LM=%;. (12.109) (Cf.Eq.(5.108).) Incylindrical coordinates withthezaxisalong thedirection of B,thecomponent ofMparallel toBis 2. M,=9'76, (12.110) Chapter 12Canonical Perturbation Theory andtheLagrangian is L=%(r2+r2é2+z2)+ %Br2é§. (12.111) Since 6iscyclic intheLagrangian, thecorresponding canonical momentum pg, .B211.,=mr26+qTr, (12112) isaconstant ofthemotion. Further, theradial equation ofmotion is mi‘—r0(mt9 +qB)=0. (12118) Asteady-motion solution toEqs.(12.1 12)and(12.ll3) corresponds torand0 constant, with6having thecyclotron value - B 0=wcE-‘*7, (12114) inagreement with Eq.(12.lO7). Inthiscase, pg=—(qBr2/2) andtheaction variable corresponding to6is J9=pip9d9=—rrqBr2. (12.115) By(12.1 10),wecanwrite qrz=2%we (asMzisequal toMforthismotion), andtherefore J0canalsobewritten as 2rrMB 2J9=-? =EM. (12116)we q Theadiabatic invariance theorem implies thatunder sufﬁciently slow variation of themagnetic ﬁeld J9remains constant. Equation (12.1 16)saysthatthemagnetic moment issimilarly invariant adiabatically. Analtemative statement, onthebasis ofEq.(12.1 15),isthatBtimes thearearrrzoftheorbit (that is,thenumber of lines offorce threading through theorbit) remains constant. Anadiabatic variation ofBmight arise ifthemagnetic ﬁeldconﬁguration re- mained static butwasslightly nonuniform. Ifthentheparticle hadasmall zcom- ponent ofvelocity, theresultant driftwould move theparticle slowly intoregions ofdifferent Bvalues. From Eqs.(12.1 14),(12.1 15),and(12.1 16),itfollows sim- plythatthekinetic energy ofmotion around thelines ofBis 26'2 T1,,=% =MB. (12.117) Exercises 555 Suppose acharged particle drifts inthedirection ofincreasing B;byEq.(12.1 17), thekinetic energy ofrotation increases. Asthetotalkinetic energy isconserved, thekinetic energy oflongitudinal driftmiz/2 along thelines offorce must de- crease. Eventually, thedrift velocity 2goes tozero andthemotion reverses in direction. Ifitcanbearranged thatBeventually increases intheother direction, thecharged particle willremain conﬁned, drifting back andforth between thetwo ends—the principle oftheso-called mirror conﬁnement. Themirror principle is used tocontain hotplasmas forthermonuclear energy generation. Thecomplete story isofcourse more complicated, butthesigniﬁcance oftheadiabatic invari- anceofMisclearly demonstrated. Wehave seen thatalmost allphenomena ofsmall oscillations about steady- state orsteady motion canbedescribed intenns ofharmonic oscillators. Incon- sequence, there isagood dealofpractical interest inquestions oftheinvariance of Jforaharmonic oscillator under slow, andnotsoslow, variations ofaparameter. Thestudy ofoscillations incharged particle accelerators, forexample, hasledto anumber ofnewinsights. Ithasbeen possible tosketch hereonlythehighlights ofthesubject ofadia- batic invariants. Theramiﬁcations oftheﬁeldgointomany areas ofclassical and quantum physics andofmathematics. EXERCISES 1.Bythemethod oftime-dependent perturbation theory, carrythesolution forthelinear harmonic oscillator (inwhich thepotential isconsidered aperturbation onthefree particle motion) outthrough third-order terms, assuming theinitial condition B0=0. Find expressions forbothxandpasfunctions oftimeandshow thattheyagree with thecorresponding terms intheexpansion oftheusual harmonic solutions. 2.Amass point mhangs atoneendofavertically hung Hook’s-law spring offorce constant k.Theother endofthespring isoscillated upanddown according toz1= acosw1t.Bytreating aasasmall quantity, obtain aﬁrst-order solution tothemo- tionofmintime, using timedependent perturbation theory. What happens as(1)1 approaches theunperturbed frequency C00? 3.(a)Alinear harmonic oscillator offorce constant khasitsmass suddenly increased byafractional amount e.Useﬁrst-order time-independent perturbation theory, to ﬁndtheresultant shiftinthefrequency oftheoscillator toﬁrstorder ine.Compare yourresults withtheexact solution anddiscuss. (b)Repeat part(a),fortheeffect ofincreasing kbyafractional amount e. 4.Carry outaconsistent second-order perturbation calculation (using whichever method youchoose) ofthecorrection tothefrequency ofaplane pendulum astheresult ofa ﬁnite amplitude ofoscillation. Allterms oforder A2should beretained intheHamil- tonian andintheperturbation treatment. 5.Amass particle isconstrained tomove inahorizontal straight lineandisattached to theendsoftwoidealsprings ofequal force constants, asshown inthediagram. The Chapter 12Canonical Perturbation Theory unstretched length ofeachspring isb5a.Useperturbation theory toﬁrst-order to ﬁndthelowest order correction tothefrequency ofoscillation forﬁnite amplitude of oscillation. What happens asaapproaches binmagnitude? '/ Ita m a k A (a)Show thattolowest order incorrection tenns therelativistic (butnoncovariant) Hamiltonian fortheone-dimensional harmonic oscillator hasthefonn 2m 8711302. (b)Useﬁrstorder perturbation theory tocalculate thelowest-order relativistic correc- tiontothefrequency oftheharmonic oscillator. Express yoturesult asafractional change inthefrequency. Aplane isotropic harmonic oscillator isperturbed byachange intheHamiltonian of theform EH1=brfpi where bisaconstant. Usetime-independent perturbation theory toﬁrstorder ﬁndthe shiftinthefrequencies. Amodel oftheatomic Stark effect canbemade bytaking theKepler elliptic orbit in aplane andperturbing itbyapotential AV=—Kx. Useperturbation theory toﬁrst order todetermine whathappens tothefrequencies ofmotion. Thismodel canalso beusedasaﬁrstapproximation totheeffect ofthelightpressure ofsolar radiation on theorbitofanEarth satellite. Byconsidering theworkdonetoalteradiabatically thelength lofaplane pendulum, prove byelementary means theadiabatic invariance ofJfortheplane pendulum in thelimitofvanishing amplitude. Consider thesystem described inExercise 13ofChapter 10.Suppose theparameter Fisslowly varied from aninitial value. What happens totheenergy oftheparticle? Theamplitude ofoscillation? Theperiod? Exercises 557 I m a '11-_---iii -,-r’ ll.Aplane pendulum ofsmall amplitude isconstrained tomove onaninclined plane, as shown intheaccompanying ﬁgure. How doesitsamplitude change when theinclina- tionangle aoftheplane ischanged slowly? CHAPTER 13.1I 558Introduction totheLagrangian andHamiltonian Formulations forContinuous Systems andFields Alltheformulations ofmechanics discussed thusfarhavebeen devised fortreat- ingsystems withaﬁnite oratmost adenumerably inﬁnite number ofdegrees of freedom. There aresome mechanical problems, however, thatinvolve continuous systems, as,forexample, theproblem ofavibrating elastic solid. Here eachpoint ofthecontinuous solid partakes intheoscillations, andthecomplete motion can only bedescribed byspecifying theposition coordinates ofallpoints. Itisnot difﬁcult tomodify theprevious formulations ofmechanics soastohandle such problems. Theconcepts ofﬁeld theory canbedeveloped byapproximating the continuous system with adiscrete system, solving thatproblem, andtaking the continuous limit. THE TRANSITION FROM ADISCRETE TOACONTINUOUS SYSTEM Weshall apply thisprocedure toaninﬁnitely long elastic rodthatcanundergo small longitudinal vibrations, thatis,oscillatory displacements oftheparticles of therodparallel totheaxisoftherod.Asystem composed ofdiscrete particles that approximates thecontinuous rodisaninﬁnite chain ofequal mass points spaced adistance aapart andconnected byunifonn massless springs having force con- stants k(cf.Fig.13.1). Itwillbeassumed thatthemass points canmove only along thelength ofthechain. Thediscrete system willberecognized asanexten- sionofthelinear polyatomic molecule discussed inSection 6.4.Wecantherefore 4 G In equilibrium If___"- II______- II"“'“'_A__n Displaced ¥F U5UUU“‘*F6UT§;°u§;Mum "1-I "1 '7i+I FIGURE 13.1 Adiscrete system ofequal mass points connected bysprings, asanap- proximation toacontinuous elastic rod. 13.1 TheTransition fromaDiscrete toaContinuous System 559 obtain theequations describing themotion bythecustomary techniques forsmall oscillations. Denoting thedisplacement oftheithparticle from itsequilibrium position by27,-,thekinetic energy is 1 .T=5Zmnf, (13.1)i where misthemass ofeach particle. Thecorresponding potential energy isthe sumofthepotential energies ofeach spring astheresult ofbeing stretched or compressed from itsequilibrium length (cf.Section 6.4): 1 v=5ikm.-+1 -11.->2. (13.2) Combining Eqs.(13.1) and(13.2), theLagrangian forthesystem is 1 .L=T-v=5inn”?-/<(m+1—m')2], (13.3) which canalsobewritten as 1 , ,'—'2 1.=5;a[iZ-1;,-2-ka ]=‘;aL,-, (13.4) where aistheequilibrium separation between thepoints (cf.Fig.13.1). There- sulting Lagrange equations ofmotion forthecoordinates 17,-are 35,--ka +ka =0. (13.5)a (.1 G Theparticular form ofLinEq.(13.4), andofthecorresponding equations of motion, hasbeen chosen forconvenience ingoing tothelimit ofacontinuous rod asaapproaches zero. Itisclear thatm/areduces toit,themass perunitlength of thecontinuous system, butthelimiting value ofkamaynotbesoobvious. Foran elastic rodobeying Hooke’s law,itwillberemembered thattheextension ofthe rodperunitlength isdirectly proportional totheforce ortension exerted onthe rod,arelation thatcanbewritten as F=Y& where §istheelongation perunitlength andYisYoung’s modulus. Now the extension ofalength aofadiscrete system, perunitlength, willbe§=(r),-+1 — r),)/a. Theforce necessary tostretch thespring bythisamount is F=MmH—mJ=M(ﬁ%§1). Chapter 13Formulations forContinuous Systems andFields sothatkamust correspond totheYoung’s modulus ofthecontinuous rod.In going from thediscrete tothecontinuous case, theinteger index iidentifying the particular mass point becomes thecontinuous position coordinate x;instead of thevariable r;,-wehave n(x). Further, thequantity m-+1— m=n(x+01)—n(x) G d occuning inL,-obviously approaches thelimit dn dx' asa,playing theroleofdx,approaches zero. Finally, thesummation overadis- crete number ofparticles becomes anintegral overx,thelength oftherod,and theLagrangian (13.4) appears as 2 L= [M2_Y(g) :|dx. (13.6) Inthelimit asagoes tozero, thelasttwoterms intheequation ofmotion (13.5) become amll(“—”>»<d-1 a—>0 a dxX dxx_a which clearly deﬁnes asecond derivative ofr).Hence, theequation ofmotion for thecontinuous elastic rodis dzn dzn thefamiliar wave equation inonedimension withthepropagation velocity v=\/Z. (13.8) M Equation (13.8) isthewell-known formula forthevelocity oflongitudinal elastic waves. This simple example issufﬁcient toillustrate thesalient features ofthetran- sition from adiscrete toacontinuous system. Themost important facttograsp istheroleplayed bytheposition coordinate x.Itisnotageneralized coordi- nate; itserves merely asacontinuous index replacing thediscrete i.Justaseach value oficorresponds toadifferent oneofthegeneralized coordinates, r),-,of thesystem. sohere foreach value ofxthere isageneralized coordinate n(x). Since 17depends alsoupon thecontinuous variable t,weshould perhaps write more accurately 17(x,t),indicating thatx,liket,canbeconsidered asaparameter entering intotheLagrangian. Ifthecontinuous system were three-dimensional, 13.2 I13.2 TheLagrangian Formulation forContinuous Systems 561 rather thanone-dimensional ashere, thegeneralized coordinates would bedistin— guished bythree continuous indices x.y,z,andwould bewritten as17(x,y,z,t). Note thatthequantities x,y,z,andtarecompletely independent ofeach other, andappear onlyasexplicit variables inr).Derivatives ofr)withrespect toanyof them cantherefore always bewritten astotalderivatives without anyambiguity. Equation (13.6) alsoshows thattheLagrangian appears asanintegral over the continuous index x;inthecorresponding three-dimensional casetheLagrangian would have theform L=/-f/Ldxdydz, (13.9) where Lisknown astheLagrangian density. Forthelongitudinal vibrations of thecontinuous rodtheLagrangian density is 1 d172 d172L=— — —Y— , 13.12l"(dr) (nix) (0) corresponding tothecontinuous limit ofthequantity L,-,appearing inEq.(13.4). ItistheLagrangian density, rather thantheLagrangian itself, thatwillbeused to describe themotion ofthesystem. THE LAGRANGIAN FORMULATION FOR CONTINUOUS SYSTEMS Itwillbenoted from Eq.(13.9) thatLlfortheelastic rod,besides being afunction of15E317/61, alsoinvolves aspatial derivative of11,namely, Hr)/8x; xandtthus playasimilar roleasparameters oftheLagrangian density. Ifthere were local forces present inaddition tothenearest neighbor interactions, thenLwould bea function of17itself aswellasofthespatial gradient of17.Ofcourse, inthegeneral case Lmight well beanexplicit function ofxandtalso. SotheLagrangian density foranyone—dimensional continuous system would appear asafunction of theform 4d£=Ll(17,£,?;l,x,t). (13.11) Thetotal Lagrangian, following Eq.(13.10), isthentheintegral ofLover the range ofxdeﬁning thesystem, andHamilton’s principle, Eq.(2.2), inthelimit of thecontinuous system appears as 2 81=8! /[ldxdt=O. (13.12)1 IfHamilton’s principle forthecontinuous system istohave anyusefulness, it must bepossible toderive thecontinuous limit oftheequation ofmotion, forex- Chapter 13Formulations forContinuous Systems andFields ample, Eq.(13.7), directly byvariation ofthedouble integral ofLinEq.(13.12). Wecancarry outthisvariation bymethods thatdiffer only slightly from those usedinChapter 2foradiscrete system. Thevariation isonlyon17anditsderiva- tives; theparameters xandtarenotaffected bythevariation either directly orin theranges ofintegration. Justasthevariation ofr7istaken tobezeroattheend points t1andI2,sothevariation of17atthelimits x1andx2oftheintegration inx isalsotobezero. AsinSection 2.2,asuitable varied pathofintegration inthe17 space canbeobtained, forexample, bychoosing 17from aone-parameter family ofpossible 17functions: 17(x,t;or) =r7(x,t;0)+0zg“(x,t). (13.13) Here r)(x,t:0)stands forthecorrect function thatwillsatisfy Hamilton’s princi- ple,andZisanywell-behaved function thatvanishes attheendpoints intandx. IfIisconsidered asafunction ofor,tobeanextremum for17(x,t:0)thederivative ofIwithrespect toorvanishes ata=0.Bystraightforward differentiation, d1 '1*2 8.6817 3Z13d17 8118dr)—= dd .13.14da /,-Ift, xtli3173<x+3%;Z8oz(dt)+3%3a1(dx) ( ) Because thevariation of17,thatis,01;,vanishes attheendpoints, integration by parts inxandtyields therelations '23C 3dr) tzd 3C 317—-- -d=— -_-11,_/,13%3a <dt) t [1dz(3%) 801t and "Z3L8 d17 _ ‘Id 8L 817d ma.17“"-‘:1;,25*-xl dx XI dx Hamilton’s principle cantherefore bewritten as zv/‘2\/Xzdxdt 811_d 35 _d 351 (817) :0, (13.15) 7-‘ XI dt dx 301 0 andbythesamearguments asinSection 2.2thearbitrary nature ofthevaried path implies thevanishing oftheexpression inthebrackets: .1 ad3::+ ad‘:_L=0. (13.16)dt33!} dx3% 817 TheEuler-Lagrange equations (13.16) (cf.Eq.(2.18)) istheappropriate form of theequation ofmotion asderived from Hamilton’s principle, Eq.(13.12). 13.2 TheLagrangian Formulation forContinuous Systems 563 Asystem ofndiscrete degrees offreedom willhave nLagrange equations of motion; forthecontinuous system withaninﬁnite number ofdegrees offreedom weseemtoobtain onlyoneLagrange equation! Itmustberemembered, however, thattheequation ofmotion for17isadifferential equation involving thetimeonly, andinthatsense Eq.(13.15) furnishes aseparate equation ofmotion foreach value ofx.Thecontinuous nature oftheindices xappears inthatEq.(13.15) isa partial differential equation inthetwovariables xandt,yielding r7asr7(x,t). Forthespeciﬁc instance oflongitudinal vibrations inanelastic rod,itisseen from theform oftheLagrangian density, Eq.(13.10), that £_/Ld_1; 8£_ Yd.) ac_0 3% dz’ 3% dx’ 817 ' Thus, asdesired, Eq.(13.16), reduces properly totheequation ofmotion, Eq.(13.7). TheLagrangian formulation developed hereforone-dimensional continuous systems needs obviously tobeextended totwo-andthree-dimensional situations, forexample, ageneral elastic solid. Further, instead ofoneﬁeldquantity 17there maybeseveral; forexample, displacement from anequilibrium position would bedescribed byaspatial vector 17withthreecomponents. There isnodifﬁculty incarrying outthemathematical steps forthemore general situation inclose parallelism totheone-component one-dimensional case. However, theformulas become lengthy andcumbersome ifwritten inthesame manner, especially in view ofthetwotiersofderivatives. Considerable gain innotational simplicity canbeachieved bynoticing thattime tandthespatial coordinates x,y,zplay thesame typeofmathematical roleinHamilton’s principle. Theﬁeldquantities arefunctions ofthecoordinates ofboth time andspace thataretobetreated as independent variables. Novariation oftheﬁeldquantities occurs atthelimits of integration inHamilton’s principle overbothtimeandspace. Itismathematically convenient tothink interms ofafour-dimensional space withcoordinates xo=ct,x1=x,x2=y,x3=z.Nophysical signiﬁcance is implied forthisspace. Thecinx0isthespeed oflight used onlytoconvert the units ofx0tothesame asthose usedforxi.Theentire tensor formalism developed inChapter 7applies. Themetric tensor gwillhaveaEuclidean metric withthe Galilean transformation group astheallowed coordinate transfonnations onthe space components ofthemetric tensor restricted bygig=go;=0.ARoman letter superscript refers onlytothethree coordinates ofthephysical space, aGreek letter superscript orsubscript refers toallfourcoordinates. Useofthesummation convention with respect torepeated indices willberesumed fortherestofthe chapter. Thevarious components oftheﬁeldquantities willbesymbolized bya subscript p,which maycover amultitude offomrs. Attimes, itwillstand fora single index having two,three, four, ormore values. Oritmaystand formultiple indices. Thus, iftheﬁeldquantity isaspatial tensor ofsecond rank, thenpreally refers totwosubscript indices. Finally, aderivative oftheﬁeld quantities with respect toanyoneofthefourcoordinates xl‘willbedenoted bythesubscript v Chapter 13Formulations forContinuous Systems andFields separated frompbyacormna. Where there isonlyoneﬁeldquantity theindex doesnotappear. Examples are _dm>. _dv. 11211"~~=m~ "Fm "3-1”Onlythederivatives oftheﬁeldquantities willbesymbolized inthismanner. Inthisnotation, themost general form oftheLagrangian density tobeconsid- eredhereiswritten as L=L§(177,,17,,_,,x”). (13.18) ThetotalLagrangian isthenanintegral overthree-space: L=fL(dxi), (13.19) butitrarely occurs explicitly. Hamilton’s principle appears asanintegral overa region in4-space: 51=afcw“) =0, (13.20) where thevariation ofthe17Pvanishes atthebounding surface Softheregion of integration. Thederivation ofthecorresponding Euler—Lagrange equations ofmo- tionproceeds symbolically asbefore. Weconsider aone-parameter setofvaried functions thatreduce to17,,(x")astheparameter 01goes tozero. Aspreviously, a possible suitable setcanbeconstructed, forexample, byadding to177,theproduct 01§,,, where {P(x")areconvenient arbitrary functions vanishing onthebounding surface. Thevanishing ofthevariation ofIisequivalent tosetting thederivative ofIwithrespect to01equal tozero:* dl=/i .7. (dx#)_ doz 817p801 817707,, 801 Integration bypartsyields dl 8L d 8£ 817,,_= _ dIL da _/[a177, dx”(817,,,,,):l 801( x) dana+f(dx“)F (13.21)p,V Thesecond integral vanishes inthelimit asagoes tozero, ascanbeseen in various ways. Wecanexamine ittennbyterm: carrying outtheintegration forthe particular x”ofeach derivative term, which thenvanishes because thederivative withrespect to01iszeroattheendpoints. Ortheintegral canbetransformed by *Unless otherwise noted, thesummation convention willbeusedintheremainder ofthischapter, for alltypes ofsubscript-superscript pairs. 13.2 TheLagrangian Formulation forContinuous Systems 565 afour-dimensional divergence theorem intoanintegral overthesurface bounding theregion ofintegration in4-space. Thesurface integral again vanishes because thevariation of17,,inthevicinity ofthecorrect ﬁeld functions iszero onthe surface. Equation (13.21) inthelimit asorgoes tozerotherefore reduces to dl 85 d 8L <->-1~<>1(1)1 da0 817,, dx” 817,,,,, 801 0 Again, thearbitrary nature ofthevariation ofeach17,,means thatEq.(13.22) is satisﬁed onlywhen eachofthesquare brackets vanishes: d 8L 8£— 1 -—=0. 13.23) dx“(817%)) 817,, ( Equations (13.23) represent asetofpartial differential equations fortheﬁeld quantities, with asmany equations asthere aredifferent values ofp.Itmaybe worth repeating thatsince thespace coordinates xiareindices fortheﬁeldquan- tities, each ofEqs.(13.23) ineffect corresponds toanentire setofLagrange dif- ferential equations ofmotion inthediscrete case. Foraone-dimensional continuous system, where vtakes ononlythevalues 0and1,Eq.(13.23) expands tothesame form asEq.(13.16). Thecompactness ofthenotation isevident even insosimple anexample. Although wehave used covariant notation, theuseofafour-dimensional space forsymbolic convenience innowayrequires covariant behavior (inthephysicist’s sense oftheword) ofany ofthequantities inthatspace. Fordiscrete systems, theLagrangian isuncertain toatotaltimederivative of anarbitrary function ofthegeneralized coordinates andtime. With continuous systems, thecorresponding statement isthatLisuncertain toany“4-divergence,” thatis,toatermoftheform dFv(Tlp,X")dxv (13.24) where theF,areanyfour(differentiable) functions oftheﬁeldquantities 17,,and thecoordinates xl‘.That such aterm makes nocontribution tothevariation of theaction integral isobvious. Application ofthedivergence theorem in4-space converts thevolume integral intoanintegral overthebounding surface where the variation ofF,iszero. Insymbols, therelevant variation canbewritten d I4 8f(dx“)% =sfF,,(17,,,x“)do“ =0, (13.25) where dc”represents thecomponents ofanelement ofsurface (inEuclidean 4- space) oriented along thedirection oftheoutward normal. TheLagrangian formulation foracontinuous setofgeneralized coordinates hasbeendeveloped inorder totreatcontinuous mechanical systems suchasan 13.3 IChapter 13Formulations forContinuous Systems andFields elastic solidinlongitudinal oscillation, oragasvibrating insuchamanner asto setupacoustic waves. Ashasbeenimplied, theformulation mayalsobeused, even intheabsence ofamechanical system, todescribe theequations governing aﬁeld. Mathematically, aﬁeldisnomore thanasetofoneormore independent functions ofspace andtime, andthegeneralized coordinates ﬁtthisdeﬁnition. There isnorequirement thattheﬁeldberelated tosome underlying mechanical system. Inthusbreaking theconnection between theLagrangian ﬁeld descrip- tionandpurely mechanical motion, wearemerely recapitulating thehistory of physics. Forexample, theelectromagnetic ﬁeld waslongthought ofinterms of theelastic vibrations ofamysterious ether. Onlyinrecent times wasitgenerally realized thattheether hadnoother rolethanbeing thesubject oftheverb“to undulate.” Werecognize equally wellthatthevariational procedures developed herealsostand independent ofthenotion ofacontinuous mechanical system, and thattheyserve tofurnish theequations describing anyspacetime ﬁeld. Hamilton’s principle thenbecomes ineffect aconvenient andcompact description oftheﬁeld, onethatupon expansion leads totheﬁeldequations. Inaddition toimplying theﬁeldequations, theLagrangian density hasmore to tellusabout thephysical nature oftheﬁeld.Aswithsystems ofadiscrete number ofdegrees offreedom, thestructure oftheLagrangian alsocontains information onconserved properties ofthesystem. Onesuch setofconservation theorems is discussed inthenextsection.* THE STRESS-ENERGY TENSOR AND CONSERVATION THEOREMS Ananalog totheconservation ofJacobi’s integral inpoint mechanics found in Section 2.6,canbederived here, andinmuch thesame manner. Allwehave to remember isthatthetreatment oftime must beextended inparallel fashion to thexisince theyareallindependent parameters inLI.Thus, instead ofthetime derivative ofL,weseektoevaluate thetotalderivative of£withrespect tox“: (IL: 31: 3C all Z; =$1711.71 +%'np.uv + (13-26) Bytheequations ofmotion, Eq.(13.23), thisbecomes (with aslight change in notation), dﬁ d 8£ 8£ d17,,,,, 3C —— =i ——- T]p_]_l, +——iv -l"—-dxll dx“ 817,,_,, 317,,,,, dx 8x/‘ d 8L BL *Amore general attack ontheconservation properties inherent intheLagrangian willbefound in Section 13.7onNoether’s theorem. 13.3 TheStress-energy Tensor andConservation Theorems 567 Combining totalderivatives, thiscanbewritten d 8£ 811F ['8-;;:T],;,,_7, -Ldﬂv] ="-—axT. Letussuppose, now, that1Cdoes notdepend explicitly upon xi‘.Thisusually means that£represents afreeﬁeld, thatis,contains noexternal driving sources orsinks thatinteract with theﬁeld atexplicit space points andwith given time dependence. Ineffect, thismeans nointeraction between theﬁeld andpoint particles moving inspace andtime through theﬁeld. Under thiscondition, Eq.(13.28) takes ontheform ofasetofdivergence conditions, UdT T:=T,,"_,,=0 (13.29) onaquantity withtheformofa4-tensor ofthesecond rank: T,,"=5-2lc—17,,,,, -L8,)’. (13.30) 7lp.v Thatthese equations haveonlytheform oftensor equations in4-space isem- phasized because asyetthe4-space hasnotransformation properties-—space and timearestilldistinct—and there isnotransformation requirement onT,,".How- ever,thespace portions ofthese quantities dobehave likevectors andtensors in ordinary space; thatis,T77arethecomponents ofathree-dimensional tensor ofthe second rank. Before considering thepossible transformations, wewilldetennine thephysical meaning ofT,,". Thesimilarity between T,,"andJacobi’s integral, Eq.(2.54), isobvious. It becomes especially clear forthecomponent T00: acT°=—' —£. 13.31 0 aﬁp7lp ( ) Inmechanical systems, theLagrangian density oftenhastheformL=T—V,the difference between akinetic energy density andapotential energy density. This isthecase, forexample, with theLagrangian densities fortheelastic rod,with thekinetic energy density having thefonn ofone-half themass density times a square ofthedisplacement velocity: T=iwlp'11»- Bythesame arguments asused indiscrete mechanics, T00canthenbeidentiﬁed asatotalenergy density. Thecorresponding identiﬁcation tagstobeplaced ontheother elements ofT,," canbesuggested bywriting thesetofEqs.(13.29) as dT,,° dT,,1'_ -=0, 13.32dt +dx ( ) Chapter 13Formulations forContinuous Systems andFields 01' dT° dTl dT°T,f’,,=T':+fc'f=T';+V-T,,=0 (13.33) where T,,,whose components areT,,",areasetof4-space vectors. Ineither fonn, Eqs.(13.32) or(13.33) appear asequations ofcontinuity, which istosaythatthe timerateofchange ofsome density plusthedivergence ofsome corresponding ﬂuxorcurrent density vanishes. Intum, theequations ofcontinuity imply the conservation ofsome integral quantities providing theﬁeldvolume isﬁnite; that is,theﬁeldcanbecontained within avolmne beyond which theﬁeldquantities arezero,deﬁned, insuchacase,integral quantities R,,by R,,=fT,,°dV, (13.34) where thevolume integral extends beyond theregion containing theﬁeld. Then, byEqs.(13.33), dRT“:/‘V-T,,dV=fT,,-dA=O. (13.35) Itisbecause ofthese conservation theorems, derived from Eq.(13.29), thatthe fourarrays T,,",it=0,1.2,3areknown asconserved currents, inanalogy with theconservation equations forelectromagnetic current. Weshould therefore expect Totoplaytheroleofthecomponents ofanenergy current density. That thisisreasonable canbeseenagain from considerations of thelongitudinal vibration ﬁeld inanelastic rod.Imagine theroddivided byan imaginary cutatpoint x(cf.Fig.13.2). From theconsiderations thatledtothe Lagrangian, Eq.(13.6), theforce exerted bythepartoftherodontheright to extend thepartthatistotheleftofthecutis dY_”. (13.36)dx Tension, —Yg—;l Force, Y3—;7 44> ><----------i p1 ><+-_________.___._._Q"=3(X+dx) n FIGURE 13.2 Diagram illustrating calculation ofenergy current density inelastic rod. 13.3 TheStress-energy Tensor andConservation Theorems 569 Hence, there isatension atxintheleft-hand portion ofequal magnitude butof opposite direction. Further, theleft-hand portion isbeing stretched byanamount thatatxisr7,andtherateatwhich thisextension changes intimeis17.Hence, the rateofwork being done bythetension atthecutis _d-Mi. (13.31) which isthustherateatwhich energy isbeing transferred totherightperunittime. Comparison shows thatthisisexactly T01fortheappropriate Lagrangian density ofEq.(13.10). IfT00isanenergy density thenthequantity, R0,ofEq.(13.34) canbeidentiﬁed asthetotalenergy intheﬁeld. Thefourth component ofthecon- servation equation (l3.35) therefore saysthatthetotalﬁeldenergy isconserved if Toivanishes onthebounding surface, thatis,ifthesystem doesnotradiate energy totheoutside. Physical meaning fortheT70components canbesuggested similarly byturning oncemore tothevibrations oftheelastic rod.Iftheparticles intherodmove by thesame amount allalong therod,themotion willbethatofarigid body, that is,nooscillatory disturbances. Thenetchange ofmass inalength dxoftherod asaresult ofthemotion would clearly bezero, since asmuch mass moves past x+dxaspastx.There would stillbeanetmomentum density 7.117forthiscase ofrigid-body motion. When wave motion takes place, anetmass change inthe length dxexists, amounting atanygiven timeto(cf.Fig.13.2) d7.t[17(x)— 17(x+111)]=-7id—:dx. (13.38) Theadditional momentum intheinterval resulting from thewave motion isthere- fore .d11-/Lflgadx. Thus, anadditional momentum density, above andbeyond thatofthesteady-state motion, canbeidentiﬁed asthewave orﬁeldmomentum density: —/.11‘;S-2. (13.39)X Thisquantity isjust—T1° fortheLagrangian density given byEq.(13.10). Thus, weareledtoidentify —T7° asthecomponents ofﬁeld momentum density and —R,-.asthetotal(linear) momentum oftheﬁeld, atleastinthisfour-dimensional convention. Theequations ofcontinuity, Eqs.(13.33), thensuggest that—T,must represent thevector ﬂuxdensity fortheithcomponent oftheﬁeldmomentum density. We ascribe avector property toTibecause there canbe,forexample, aﬂowinthe y-direction ofthex-component ofthemomentum density, asmeasured by—T,,>’ . Analternative interpretation ofTHcomes from considering thedisplacement ﬁeld Chapter 13Formulations forContinuous Systems andFields ofanelastic solid. Itiswellknown thatinsuchasolid there arealsoshear forces (besides thecompression forces normal toasurface) along asurface element. The entire assemblage offorces canbedescribed bysaying thattheforce dFacting onanelement ofareadAisexpressed interms ofastress tensor Tsuchthat dF=T-dA. (13.40) Hence, thenetforce, sayinthex-direction, onarectangular volume element dxdydzhasacontribution from theforces onthesurfaces inyzplanes given by (cf.Fig.13.3)(where 1indicates thexcomponent, 2they,etc.) dT1[T|'(x+dx)-T1l(x)] dydz=T;dxdydz. (13.41) butthere isalsoacontribution from thesurfaces inthexzplane; dT2[T12(y+dy)-T12(y)] dxdz=T;dxdydz, (13.42) andsimilarly from thexyplanes. Newton’s equations ofmotion herecorrespond tosaying thatthetimerateofchange ofthemomentum density inthexdirection, —T1°, isequal tothex-component oftheforce onaunitvolume element: dT1° dT11 dT12 dT13- = , 13.43cdt dx+dy+dz ( ) which isprecisely thex-component ofEq.(13.33). Forthisparticular ﬁeld T}-/l canbeidentiﬁed astheelements ofthethree-dimensional stress tensor, hence the origin ofthename “stress-energy tensor" forT,,,". J’ dz T?(z+dz) °——-—-a I .__1=// Ti<1).// my)cly 157/ / / / dx 2/ x FIGURE 13.3 Force inxdirection onavolume element dxdydzofanelastic solid. 13.3 TheStress-energy Tensor andConservation Theorems 571 Byconsiderations ofacontinuous mechanical system, wehavethusbeen able toattach physical identiﬁcations, orassociations, toeachofthecomponents of thestress-energy tensor. Thus, thecomponents are T00 ﬁeldenergy density divided byc, T0,withcomponents T01 ﬁeldenergy current density, —T;° ﬁeldmomentum density, ithcomponent, —Ti,withcomponents T,-O current density fortheithcomponent oftheﬁeld momentum density, T,-7 three-dimensional stress tensor where, aswesawdiscussed following Eqs.(13.33) and(13.35), T0andT7fonn 4-space vectors eachofwhich isconserved andthusidentiﬁed, inanalogy with thecharge-current vector ofelectromagnetic theory asa“4-current”. Allsuch conserved objects arecalled currents inﬁeldtheory. Inalmost allcases thethree-dimensional tensor Tissymmetric. This isnot only physically desirable, butalmost necessarily acharacteristic forthespatial portion ofthestress-energy tensor. Itmust beremembered thatalthough theexample ofmechanical systems gave birth totheprocedures andnomenclature, theformalism canbeapplied toany ﬁeld irrespective ofitsnature ororigin. Aclassical theory ofﬁelds canbecon- structed notonlyforvibrations ofanelastic solid, butalsofortheelectromagnetic ﬁeld, forthe“ﬁeld” oftheSchrodinger wave function, orfortherelativistic ﬁeld describing a“scalar” meson, among others. Weshall examine some ofthese ex- amples inmore detail lateron. Recalling theidentiﬁcation ofR,-,theconservation equations, Eq.(13.35), say thatforaclosed noninteracting system thetotal linear momentum oftheﬁeldis conserved. Wewould expect noless.Butthere should beacorresponding con- servation theorem forthetotal angular momentum oftheﬁeld. Itissimple to construct aquantity thatshould actasanangular momentum density. Since angu- larmomentum isanaxial vector. Weexpect thatthecomponents oftheangular momentum density aretheelements ofanantisymmetric tensor ofthesecond rank. Asuitable form forthistensor is MU=-(x"T1'° -x1'T*'°), (13.44) withthetotalangular momentum oftheﬁeldgiven by M”=fM”dV. (13.45) Inasmuch astandxiarecompletely independent variables, thetime rateof change ofM'7is ij _jO _i0 2;.=-7 dv, (.3...) 7 13.4 IChapter 13Formulations forContinuous Systems andFields or,from thecontinuity conditions, Eqs.(13.32), dM"f -dT1"< -dT”‘'-ET =-‘I (Ila? —.XJ'd?) Integration byparts converts thisexpression to ,1ii .. .. .. .. %-=-f%(x'T1'< —x1T'k)dV +f(T'1 -T/')dV. (13.43)X Theﬁrstintegral ontheright isintheform ofavolume integral ofadivergence. Itistherefore equal toanintegral overthebounding surface, which vanishes fora closed nonradiating system. Finally, ifTil=T/ll,thesecond integral isalsozero. Thus, thetotalangular momentum oftheﬁeldisconserved ifTissymmetric. Ifthestress tensor isnotsymmetric, wecanoften make useoftheambiguity indeﬁning thestress tensor torestore thissymmetry. JustasfortheLagrangian, theform ofthestress-energy tensor, Eq.(13.30), waschosen tosatisfy diver- gence conditions (cf.Eq.(13.29)). Therefore T,,"isindeterminate byanyfunc- tionwhose 4-divergence vanishes. Usually itispossible toﬁndsuchaquantity to “symmetrize” thestress-energy tensor. HAMILTONIAN FORMULATION Itispossible toobtain aHamiltonian formulation forsystems withacontinuous setofcoordinates much aswasdone inChapter 8fordiscrete systems. Toindicate themethod ofapproach. wereturn brieﬂy tothelinear chain ofmass points dis- cussed inSection 13.1. Conjugate toeachﬁeldcomponent, 177,there isacanonical momentum at atp,=—,=d-_i. (13.49)3'71‘ 3111 TheHamiltonian forthesystem istherefore . 3L1.HE -"—L= i '—L, P17)! ﬂaﬁiﬂi 01‘ 8L-H=d<—_'z7, -L,). (13.50)all! Itwillberemembered thatinthelimit ofthecontinuous rod,when agoestozero, L,-—>Candthesummation inEq.(13.50) becomes anintegral: 85.H=~/dx (8—ﬁn—£). (13.51) 13.4 Hamiltonian Formulation 573 Theindividual canonical momenta p,~,asgiven byEq.(13.49), vanish inthecon- tinuous limit, butwecandeﬁne amomentum density, rr,thatremains ﬁnite: - aLim£'- E71.’= (13.52)a—->0 £1 31] Equation (13.51) isintheformofaspace integral overaHamiltonian density, ‘H, deﬁned by H=1117——L. (13.53) While aHamiltonian formulation canthusbeintroduced inastraightforward manner forclassical ﬁelds, notethattheprocedure singles outthetimevariable forspecial treatment. Itistherefore incontrast tothedevelopment wehavegiven fortheLagrangian formulation where theindependent variables oftimeandspace werehandled symmetrically. Forthisreason theHamiltonian approach, atleastas introduced here, lends itself lesseasily toincorporation inarelativistically covari- antdescription ofﬁelds. TheHamiltonian wayoflooking atﬁelds hastherefore notproved asuseful astheLagrangian method, andarather briefdescription should sufﬁce here. Theobvious route forgeneralizing toathree-dimensional ﬁelddescribed by ﬁeldquantities 17,,istodeﬁne, analogously toEq.(13.52), thecanonical momen- tumdensities 1r”(x”) = (13.54)817,, Thequantities 17,,(xi,t),rr/’(xi, t)together deﬁne theinﬁnite-dimensional phase space describing theclassical ﬁeldanditstimedevelopment. Aconservation the- orem canbefound forrr,,thatisroughly similar tothatforthecanonical momen- tumindiscrete systems. Ifagiven ﬁeld quantity 17,,iscyclic inthesense thatLI does notcontain 17,,explicitly (asinthecaseofEq.(13.10)), thentheLagrange ﬁeldequation looks likeanexistence statement foraconserved current: da__L =0, (13.55)dx#817,,_,, 01' drrp d8L——- ++——— =O. (13.56) Itfollows thatif17/’iscyclic, there isanintegral conseryed quantity r1/1=fdv11/’(x", 1). Theobvious generalization ofEq.(13.53) foraHamiltonian density is H<11P.11..,1.z1,..x"> =1131,.—1:. (13.51) Chapter 13Formulations forContinuous Systems andFields where itisassumed thatfunctional dependence upon 1),,canbeeliminated by inversion ofthedeﬁning equations (13.49). From thisdeﬁnition itfollows that 871 , A81); 8L81); _ w-"P"5;;/¥"w5;;;-"P, <13-58> byEq.(13.51). Theotherhalfofthecanonical ﬁeldequation ismorecumbersome. When expressed intenns ofthecanonical variables, Hisafunction of17,,through theexplicit dependence ofL,andthrough 1)p.Hence, a a‘ aa‘ a aH=n*”‘- "*- L=- 5. (13.59)877p an,» aman,» an,» am» Using theLagrange equations, thiscanbewritten d 392=-L(i)=_,-,1__i(1.), (MO) Because oftheappearance ofL,westilldonothave auseful form. Byanexactly parallel derivation, however, weﬁndthat anznlam_apan,_aL=_6L. (13.61) 911,».-' 8%,: 8111arm 9%,: an,”- Hence, wecanwrite asthesecond halfofthecanonical equations ﬁ-i. =_¢rP. (13.62)817,, dxl 827p_,- Equations (13.58) and(13.62) canbeputinanotation more closely approaching Hamilton’s equations foradiscrete system byintroducing thenotion ofaﬁmc- tional derivative deﬁned as 8 8 d 3—=———.——. 13.6381/1 Btlr dx'31/1,; ( ) Since Hisnotafunction ofJr‘;-,Eqs.(13.58) and(13.62) canbewritten as 8 5ﬁp=l, fr"=—l. (13.64)67!/’ 827,, Note thatinthesame symbolism theLagrange equations, Eqs.(13.23), takethe form d8L 8L—___ -—=0. (13.65) dt(8%) 827,, 13.4 Hamiltonian Formulation 575 About theonlyadvantage ofthefunctional derivative, however, isthatofthe resultant similarity with discrete system. Itsuppresses, ontheother hand, the parallel treatment oftimeandspace variables. There isawaytotreatclassical ﬁelds thatprovides almost alloftheHamilto- nianformulation ofdiscrete mechanics. Themainideabehind thistreatment isto replace thecontinuous space variable orindex byadenumerable discrete index. Wecanseehowtodothisbyreferring again tothelongitudinal oscillations of theelastic rod.Letussuppose therodisofﬁnite length L=x2—x1.There- quirement that17vanish attheextremities isaboundary condition thatcould be achieved physically byplacing therodbetween twoperfectly rigidwalls. Then theamplitude ofoscillation canberepresented byaFourier series: 17(x)=it),sin (13.66) n=0 Instead ofthecontinuous index x,wehavethediscrete index n.Weareallowed to usethisrepresentation forallxonlywhen n(x)isawell-behaved function, which mostphysical ﬁeldquantities are. Forsimplicity inillustrating howthescheme maybecarried out,itwillbeas- sumed thatonlyonerealﬁeldquantity, 17,canbeexpanded inathree-dimensional Fourier series oftheform 1 ~.1,(r,1:)=T/EZqk(t)e'k ". (13.67)k Herekisawave vector thatcantakeononlydiscrete magnitudes anddirections, suchthatonlyanintegral (orsometimes, half-integral) number ofwavelengths ﬁt intoagiven linear dimension. Wesaythatkhasadiscrete spectrum. Thescalar index kstands forsome ordering ofthesetofinteger indices used todenumer- atethediscrete values ofk,andVisthevolume ofthesystem, appearing ina normalization factor. Because 17isreal,wemusthaveqz‘=q_k. Theorthogonality oftheexponentials overthevolume canbestated asthe relation 1.,VIe‘(""‘)"dV =a,,,,,. (13.68) Ineffect, theallowed values ofkarethose forwhich thecondition (13.68) is satisﬁed (ascanbeseenbylooking attheone-dimensional Fourier series). It follows thatthecoefﬁcients ofexpansion, qk(2),aregiven by 1 _-_ qk(t) =Y/T/2-‘/e ‘k'17(r, t)dV. (13.69) Insimilar fashion, thecanonical momentum density canbeexpanded as 57 Chapter 13Formulations forContinuous Systems andFields 1 _-k,7r(r,1)=W72.;pk(t)e *', (13.70) again withpf:=p_k.Correspondingly, theexpansion coefﬁcients, pk(t),areto befound from l .P/<(I) =W‘/e"‘ 'rr(r,t)dV. (13.71) Inasense wehave almost come fullcircle. Webegan thischapter withadis- crete system employing adenumerable number ofgeneralized coordinates. By then going tothelimit ofacontinuous setofvariables, wewere abletotreat continuous systems. Finally, wehave introduced adescription ofthecontinuous system interms ofadenumerable, discrete setofcoordinates thatobeythesame typeofmechanics asthediscrete system westarted with. Because oftheformal correspondence withthevariables ofdiscrete systems, theqkandpkquantities aretheobvious candidates forquantization when wegofrom classical toquantum ﬁeldtheory. Indeed, theqkcorrespond towhat arespoken ofasthe“occupation numbers” fortheﬁeld. Wecould describe theﬁeldinterms ofdiscrete coordinates because theﬁnite sizeofthesystem, andtheboundary conditions, permitted adiscrete Fourier ex- pansion. Equivalently, wecansaythattheexpansion ismade overadiscrete spec- trum ofplane waves. Since thewave vector kisinquantum mechanics directly proportional tothemomentum oftheparticle associated withtheplane wave, the expansions used hereareoften spoken ofasthemomentum representation. We TABLE 13.1 Comparison ofMinkowski 4-dimensional spacetime andsymplectic structure (after Misner, Thome &Wheeler, Gravitation. SanFrancisco: Freeman, 1973) Hamiltonian Minkowski spacetime Comparison item symplectic structure metric structure Canonical coordinates ql,q2,pl,p2 ct,x,y,z 662=C2dt®dt —dX®dx Canonical structure 6)=dpl/\dql+dpg/\dqz —dy®dy—dz®dz Nature of“metric” antisymmetric symmetric Name for“metric” canonically (ordynamically) structure conjugate coordinates Lorentz coordinates Field equations VG=0satisﬁed automatically R,,,,g),5 =0:ﬂatspacetime 4-dimensional manifold phase space spacetime Coordinate free description VG=0 Riemann =0 13.5 I13.5 Relativistic FieldTheory 577 neednotberestricted toplane wave expansions. Adenumerable setofcoordi- nates canbefound whenever theﬁeldfunctions canbeexpanded interms ofa discrete setoforthonormal eigenfunctions. Oneﬁnalcomment. TheHamiltonian orsymplectic structure canbeexpressed intensor notation. Table 13.1compares themetric structure of4—dimensional Minkowski spacetime with thesymplectic structure ofaHamiltonian with co- ordinates ql,q2,p1,andpg. RELATIVISTIC FIELD THEORY WesawinChapter 7thatthereisconsiderable difﬁculty inconstructing relativis- tically covariant Lagrangian andHamiltonian descriptions ofparticle mechanics. Partoftheproblem canbetraced totheseparate rolesplayed byspace andtime coordinates. Forpoint particles, thespace coordinates aremechanical variables while timeisamonotonic parameter. Butinclassical ﬁeldtheory there isanat- uralsimilarity inhandling space andtimecoordinates. They areallparameters, together deﬁning apoint inthespacetime continuum atwhich theﬁeldvariables aretobedetennined. While thefour-dimensional spacetime system hasbeen used sofaronlyforreasons ofnotational simplicity, theeasyandnatural wayitﬁtsinto theformulation suggests thatarelativistically covariant description isquite fea- sible forclassical ﬁelds. Indeed, onlyrelatively minor tinkering hastobedone totheformulation already presented sothatitcanhandle relativistic ﬁelds ina marmer thatismanifestly Lorentz covariant. Three points require speciﬁc attention: (1)thenature (andmetric) ofthefour- dimensional space used; (2)theLorentz transformation properties oftheﬁeld quantities, Lagrangian densities, andrelated functions; and(3)thecovariant de- scription ofthelimits ofintegration. Thesimple Cartesian, 4-space withcoordi- nates t,x,y,zthatwehave implicitly used sofarinthischapter isnotconve- nientforexhibiting Lorentz invariance. Wewillusethenotation andconventions adopted inChapter 7aswellastheresults ofthatchapter. Accordingly, theGreek letter indices willstillrunfrom Oto3,withx0=ct.Note thattheLagrange equations (13.23) areunaffected bythischange. Indeed, theterm d 8L dx" 817),’, remains unaltered byascale change ofanyofthex",andtheother terminthe Lagrange equation doesnotinvolve thecoordinates atall.Further, thechange in space doesnotaffect theformulation ofHamilton’s principle inEq.(13.20), since itonlyintroduces amultiplicative constant. Allofthequantities related totheﬁeldandassociated equations must nowhave some deﬁnite Lorentz covariant properties. Theﬁeld quantities must therefore consist of4-tensors ofsome given rank—scalar, 4-vector, andsoon.Inprinciple, 17pneednotberestricted toanyoneofthesecategories butmaystand forasetof such, forexample, twoscalars. TheLagrangian andHamiltonian densities must 5 Chapter 13Formulations forContinuous Systems andFields alsobecovariant. InHamilton’s principle, thevolume element (dx") of4-space is invariant under Lorentz transfonnation. Since weusually think oftheaction Iasa scalar, thismeans thattheLagrangian density (andtherefore H)should bescalars. Thatistosay,theymustbefunctions oftheﬁeldquantities (possibly along with external covariant quantities) insuchmanner astoform scalars under Lorentz transformations. Itthenfollows thatthestress-energy tensor T,,,,,asdeﬁned by Eq.(13.30) isautomatically a4-tensor ofthesecond rank. Thechange inthe4- space however means thatthecomponents ofT,”maybealtered invalue. Intensor notation, thestress-energy tensor, T,isalinear, symmetric “func- tional” withslotsfortwovectors. Ithasthefollowing properties: 1.Ifweinsert the4-velocity uoftheobserver intooneoftheslots andleave theother slotempty, theoutput is dT(u,...)=T(...,u)=—(density of4-momentum, (13.72) Theright-hand sideisthenegative ofthe4-momentum perunitthree- dimensional volume asmeasured intheobserver’s frame attheevent where Tismeasured. Incomponent notation, 0! divT°’,gu°’ =Tﬂ°’u5 =- (13.73) 2.Ifweinsert the4-velocity uoftheobserver intooneoftheslots andan arbitrary unitvector nintotheother slot,theoutput is T(u, n)=T(n, u)=—(n- (13.74) Theright-hand sideisthenegative ofthecomponent ofthe4-momentum density along thendirection. Incomponent notation du Tqguanﬂ =Tﬂuuﬂn“ =—n#FI)‘7. (13.75) 3.Ifweinsert the4-velocity oftheobserver intobothslots, theoutput is T(u, u)=(mass energy perunitvolume) (13.76) asmeasured intheframe with4-velocity u. Incomponent notation, dp“T,,,5u°’u'5 =Tﬂ,,,u'3u°’ =MW (13-75') 4.Ifwepickaframe andinsert twospacelike basis vectors e,-andekinthat frame, theoutput is 13.5 Relativistic FieldTheory 579 Tik=T/<1=T(@i,e1<) =T(@/t, er) =i-component offorce acting from sidexk—8tosidexk+8across aunitsurface areaperpendicular todirection ek =k-component offorce acting from sidexi—8tosidexi+8across aunitsurface areaperpendicular todirection e,- (13.77) Forexample, ifweassume theLorentz transformations apply andconsider a perfect ﬂuidmoving witha4-velocity u,which mayvaryinspacetime, wecan describe theﬂuidinterms ofitsmass density, p,andanisotropic pressure, p,both intherestframe oftheﬂuidelement. Thestress-energy tensor isgiven by T=(,o+p)u®u+pg (13.78) orincomponent form Tug=(p+p)u,,,u,9 +pgap. (13.79) Insert the4-velocity intooneslotgiving T°’,3uﬂ =[(p+p)u°’u)5 +p8°’;;]u'5 =pu“. (13.80) Intherestframe oftheﬂuid, thisbecomes T°,.,u/5=pc (13.81) and . d' T‘5145=%=momentum density =0, (13.82) where thelastequality follows from thechoice oftherestframe. Finally Tik=T(@1,er)=P5ik- (13-33) TheLagrangian density isofcourse uncertain toamultiplicative constant fac- tor.Itiscustomary tochoose thefactor such thatT()()(oritssyrmnetrized fonn) directly represents theenergy density intheﬁeld. Inthechosen 4-space thequan- titiesR,,,Eq.(13.34), arenowdeﬁned as 12,,=IT,,°dV. (13.84) Letusconsider arelated setP”deﬁned as 1P“=ER”. (13.85) Itfollows then, from Eqs.(13.72) to(13.76) andtheinterpretation given above for T,-0,thatP‘represents thecomponents ofthetotallinear momentum oftheﬁeld, 5 Chapter 13Formulations forContinuous Systems andFields andP0isE/c,where Eisthetotalenergy intheﬁeld.Thissuggests thatwecan interpret P“asthe4-momentum oftheﬁeld.However, westillhavetoshow that R“andP“transform like4-vectors under aLorentz transfonnation. Toprove this property, weshallexamine whatismeant byanintegration overthree-space ina covariant formulation andindeed howtheintegration limits aretobetreated in general. Theﬁrstinstance where thecovariance ofthelimits ofintegration may be questioned isinHamilton’s principle. InEq.(13.20), theintegral appears man- ifestly covariant, butthelimits ofintegration derived from Eq.(13.12) arenot. Thespatial integration isoversome ﬁxed volume inthree-space followed byan integration overtime between t1andI2.Butanintegration over Vforﬁxed tis notacovariant concept, forsimultaneity (“constant time”) isnotpreserved under Lorentz transformation. Asuitable covariant description istosaytheintegration is conducted overahypersurface ofthree dimensions thatisspacelike. Byaspace- likesurface, wemean oneinwhich all4-vectors lying initarespacelike. The vectors normal tosuch asurface aretimelike. Now, anyvector connecting two points onasurface ofconstant timeiscertainly spacelike, foritsx0-component vanishes. Hence, asurface atconstant timeisaparticular example ofaspacelike surface. Butsuchasurface retains itscharacter inallLorentz frames, because the spacelike ortimelike quality ofavector isnotaffected bytheLorentz transfor- mation. Inasimilar fashion, what isinoneframe anintegration overtataﬁxed point canbedescribed covariantly asanintegration overatimelike surface. With asystem ofonedimension (inphysical space), theintegration inHamilton’s prin- cipleasgiven inEq.(13.12) isovertherectangle shown inFig.13.4.ALorentz transformation isarotation inMinkowski space, andthesides oftherectangle willnotbeparallel totheaxesinthetransformed space. Butwecandescribe theintegration inallLorentz frames asbeing overaregion in4-space contained between twospacelike hypersurfaces andbounded byintersecting timelike sur- faces. \ \l"°\ '2 \ // \\ /// \/,/j - //1' __z, " \\ x3 \ ,l Z] \ Z2 \ \ FIGURE 13.4 Regions ofintegration inHamilton’s principle forasystem extending in onlyonespace dimension. 13.5 Relativistic FieldTheory 581 Theappropriate covariant description ofintegral quantities suchasP“isthen given as P”=éIT",,dS", (13.86)S where theintegration isoveraregion onaspacelike hypersurface forwhich the 1-fonn elements ofsurface, inthedirection ofthesurface normal, aredS”(agra- dient). AsTl”isa4-tensor ofthesecond rank, itisobvious thatP”sodeﬁned isa4-vector. Butnowwecanshow thatthecomponents ofP“given by(13.86) reduce toavolume integral inordinary three-space, providing itisdivergence- less,thatis,satisﬁes Eq.(13.29). Imagine aregion in4-space deﬁned bythree surfaces: S1andS2thatarespacelike, andS3thatistimelike (cf.Fig.13.5). By afour-dimensional divergence theorem, avolume integral ofadivergence canbe replaced byasurface integral: .1WL(dx4) = T”"(18,, (13.87)U V4dx s1+s2+s, where dx4istheinvariant 4-volume, \/Ecdtdx dydz.Theintegration over S3 corresponds toanintegration overtatconstant r.Byallowing thevoltune to expand sufﬁciently, theintegral overthissurface willinvolve routside thesystem, where allﬁeldquantities vanish. Because oftheassumed divergenceless property ofTl”, theintegral ontheleft-hand sidealsovanishes. Therefore, ifthenormals tothespacelike surfaces aretaken inthesame sense, v/‘T”'vdS,,=./l T“"dS,,. (13.88) S1 S1 IfS1isanyarbitrary spacelike surface, andS2isaparticular surface forwhich xo,ort,isconstant, thenbyEq.(13.88), IT”"dS,, =fT“°dV. (13.89)S1 I S3 )' x FIGURE 13.5 Schematic integration volume in4-space Chapter 13Formulations forContinuous Systems andFields The4-vector transformation property oftheleft-hand sideisobvious; hence, theright-hand side, i.e.,R“according toEq.(13.84), alsotransforms asa4- vector. Further, ifbothS1andS2aresurfaces atconstant t,sayt1andt2,respec- tively, thenEq.(13.88) isequivalent to Rl'L(l‘1) =R”'(t2), (13.90) which isthusthecovariant wayproving thatR“isconserved intime. With some care, therefore, theconserved integral quantities canstillbeused within theframework ofarelativistic theory ofclassical ﬁelds. Weshall notal- ways carry through thedetailed correspondence butwillletitsufﬁce inmost instances thatthevolume integration refers toaparticular Lorentz frame inwhich thespacelike hypersurface isaregion inthree-space atconstant t.Fortheangu- larmomentum density, notethatthecovariant analog ofMij, Eq.(13.44), isa 4-tensor ofthirdrank: M""*=%(x'*r"* -x"T/M), (13.91) which isantisymmetric in/4andv.Thecorresponding global orintegral quantity is 11/1""=I/1/1""*dS;,, (13.92) where theintegration isoveraspacelike hypersurface. IftheLorentz frame is chosen suchthatthesurface isoneatconstant t,then Ml”->fM“"°dV, (13.93) which corresponds totheprevious deﬁnition. Therestoftheargument onthe conservation ofM'7forsymmetrical stress-energy tensors thencanbecarried out asbefore byconsidering thisparticular Lorentz frame. Allofthisfollows from Chapter 7. Asconstructed intheprevious section, theHamiltonian formulation sharply distinguishes between thetimecoordinate andthespace coordinates. Thisisnot tosaythatitisnecessarily nonrelativistic, merely thattheformulation isnotman- ifestly covariant. Wemustimagine theHamiltonian framework asconstructed in terms ofthetimeasseenbyeachparticular observer. Providing theﬁeldquantities andderived functions have suitable transfonnation properties, thisconstruction foreachLorentz frame isnotinviolation ofspecial relativity. Onefurther point needs tobemade here. Byallowing 17,,tostand forasetof covariant ﬁeldquantities, weallow forthepossibility thatthesystem consists of twoormore ﬁelds thatinteract witheachother. Thecomplete Lagrangian den- sitymayconsist ofasumofLagrangian densities representing thefreeﬁelds plus terms thatdescribe theinteractions between theﬁelds. Itwillberemembered that 13.6 I13.6 Examples ofRelativistic Field Theories 583 oneofthedifﬁculties ofrelativistic point mechanics wastheproblem ofconsid- ering interactions between particles thatnecessarily implied action-at-a-distance. However, interactions between ﬁelds canbeatapoint and,therefore, consistent with special relativity. Wecanoften gofurther andtreattheinteraction between aﬁeldandaparticle atagiven point inspacetime. There isthusthepossibility ofconsidering relativistically asystem consisting ofacontinuous ﬁeld, adiscrete particle, andtheinteraction between them. How thiscanbedone inaspeciﬁc case willbeshown inthenextsection, which provides illustrations ofrelativistic ﬁeld theories. EXAMPLES OFRELATIVISTIC FIELD THEORIIES Weshall consider three examples, ofincreasing complexity. A.Complex scalar ﬁeld. Anycomplex ﬁeldwillbedescribed bytwoindependent parts, which canbeexpressed either astherealandimaginary partoftheﬁeldor asthecomplex ﬁelditself anditscomplex conjugate. Weshallfollow thelatter al- temative. Accordingly, theLagrangian density andassociated functions willhere begiven intenns oftwoindependent ﬁeldvariables, ¢and¢*,eachofwhich are 4-sca1ars.* Forthisparticular example, wechoose theLagrangian density 11=c’¢.t¢*'* —/»5c’¢¢* (13.94) where 11.1)isaconstant and¢,)(=83%,qb,‘=g’\"%% asgiven inEq.(13.17). Notice, thatasrequired, Lisaworld scalar. Expressed interms ofspace andtime variables, Liswritten as(where (I2=84>/61) £3=</343*-¢2v¢-v¢*-n3¢2¢¢*. (13.95) Toobtain theﬁeldequation forwhich 77,,=¢*,notethat ———— = ,,——=— . 13.96 84),,’ VC¢ a¢* /1105 ¢ ( ) Hence, theLagrange-Euler ﬁeldequation is ¢.i"+tr5¢ =0. (13.97) or,inequivalent form, 2 ¥(%% +71.54»=0 (13.98) *Asshallbeseeninthenextsection, complex ﬁelds leadnaturally toanassociated charge andcurrent density, andthisisthemain reason fortheirintroduction inphysical theories. Chapter 13Formulations forContinuous Systems andFields and 2142¢ 2_—V ¢+ 'l'I/l10¢—0. Interms oftheD’Alembertian (cf.Section 7.5), theﬁeld equation canalsobe written covariantly as (E12+/13)¢=(V2+11%)¢=0- (13.99) Similarly, from thesymmetry ofL,theﬁeldequation obtained when 77,,=42*is (132+;t3)¢*=(V2+n3)¢*=0. (13.100) Thisbasic ﬁeldequation satisﬁed bybothqband42*isknown astheKlein—Gordon equation and.asgiven here, represents therelativistic analog oftheSchrodinger equation foracharged zero-spin particle ofrestmass energy 11.0. Thestress-energy tensor deﬁned byEq.(13.30) hascomponents T...=c’¢...¢*..+c’¢*.,.<1».+@2(¢.r¢*.*'l'”'%¢¢*)gpLU (13-.101) andisclearly symmetrical. AstheLagrangian density describes afreeﬁeld, with- outinteractions withtheoutside world, lldoesnotcontain xexplicitly andthe conservation theorem (13.29) holds forT,1,,,ascanbeveriﬁed directly. Toin- troduce theHamiltonian formulation, wemust distinguish between thetimeand space coordinates insome particular Lorentz frame. Theconjugate momenta, ac- cording toEq.(13.54), arethen(cf.Eq.(13.95)) 811 . 811 . Itfollows thattheHamiltonian density (which hasthesame magnitude asT011) takes theform HE7rq5+rr*(8* -11, =1111*+<.~2v¢-v¢*+n§c2¢¢*. (13.103) Forthemoment, allthatweshall dohereisillustrate thetransformation tothe momentum representation. Theexpansions (13.67) and(13.70) canbeintroduced intotheHamiltonian density. Since theﬁeld isnotreal, wedonothave that qz‘=q_k. Ineffect, qkandqj:nowstand fortwoindependent setsofdiscrete coordinates, onerepresenting (15andtheother ¢*.ThetotalHamiltonian isasum ofvolume integrals overthethree terms inEq.(13.103). Asatypical example, let usconsider 2 115/¢¢*dV=%Zfqrq7§@“"'“""'dv. (13104)k,k’ 13.6 Examples ofRelativistic Field Theories 585 which byEq.(13.68) reduces to /1~§qkqi‘- Theonlyother term requiring anyspecial noteatallisthatinvolving thediver- gences, which introduce afactor (ik)-(-ik’) intheintegrand. Theﬁnalform for Hcanbewritten as H=pip;+wiqrqr. (13105) where wkisrelated tokthrough thedispersion relation 8,2=@208+113). (13.106) Each term ofthesummation inEq.(13.105) isintheform ofaharmonic oscil- lator ofunitmass withfrequency wk.This canbeseenexplicitly byevaluating Hamilton’s equations ofmotion. Inthemomentum orplane wave representations, theﬁelds ¢and¢*arethusreplaced bydiscrete systems ofharmonic oscillators, much inthesame manner thatthesound ﬁeldinasolid islooked onasacollection of“phonons.” Thediscrete spectrum of“vibrations” ofourscalar charged ﬁeld isgiven byEq.(13.106). Quantization oftheﬁeld (that is,theso-called second quantization) isdone most simply viathemomentum representation. Ineffect, the motion ofeach harmonic oscillator isquantized aswould bedone foranactual harmonic oscillator. Butthissubject certainly liesoutside ourprovince. B.TheSine-Gordon equation andassociated ﬁeld. Ifthescalar ﬁeldintheprevi- ousexample were taken asreal(that is,¢*=(0)andtoexist inonlyonespatial dimension, thentheobvious corresponding Lagrangian density along themodel ofEq.(13.95) would be ‘ 2 L=c2_2 _ _,,,,§¢2]. (13107) (The factor of%isintroduced forconvenience; itclearly doesnotaffect theform oftheequations ofmotion.) Theassociated ﬁeldequation (cf.Eq.(13.16)) 82¢ 1a2¢_ 2 istheone-dimensional Klein—Gordon equation. Note thatitislinear intheﬁeld ¢(x,I)- Wecanlookupon theLagrangian density ofEq.(l3.l07) asasmall-ﬁeld ap- proximation toaLagrangian density ofthefonn ‘ 2 E=é - ]_115820 -66845), (13.109) Chapter 13Formulations forContinuous Systems andFields which hasthecorresponding ﬁeldequation 2 2 37‘:-Ci2%t§ =;r§sin¢. (13.110) Inevitably, ifperhaps frivolously, Eq.(13.110) hascome tobeknown asthesine- Gordon equation. IftheKlein—Gordon equation, Eq.(13.99), isreminiscent ofthe harmonic oscillator, thenthe“potential” termintheLagrangian equation (13.109) recalls thepotential term ofthelinear pendulum. Indeed, Eq.(13.1 10)hasalso been called, perhaps more appropriately, thependulum equation. Inthisone-dimensional world, thestress-energy tensor hasonlyfourcompo- nents. Asxandtagain donotappear explicitly inL1,theelements ofthetensor satisfy conservation equations, which areheretwoinnumber. Details willbeleft totheexercises, butofparticular interest istheenergy density T01): 2 T01)=1[(132+8(33) ]+;r.§c2(1- 6654»), (13.111)2 8x which isofcourse thesame inmagnitude astheHamiltonian density 12234’2 22'H=5 rt+c 5 +/4.00 (1—cos¢), (l3.l12) where theconjugate momentum is rr(x,1)=<1. (13.113) Themomentum representation fortheKlein-Gordon ﬁeldasthesumoverhar- monic oscillators means thatintheone-dimensional casetheﬁeldcanbebuiltup asasuperposition ofplane waves oftheform qk(t)e”" =A11(/<)e"<’"-‘"'~'>, (13114) where kandwkarerelated bythedispersion relation, Eq.(13.106). Fortheﬁeld obeying thesine-Gordon equation, itismuch moredifﬁcult toapply amomentum representation, because ofthepresence ofthecos¢ tenn in‘H.Butwecanstill solve thesine-Gordon equation bysomething resembling atraveling wave. A solution forqbinEq.(l3.1l0) thathastheform ofadisturbance traveling witha speed v,butotherwise keeping itsshape, must beafunction onlyof1:=t—x/v. Inthatcase, Eq.(13.110) reduces to 2 ﬂ—Asin¢=0, (l3.ll5)d1'2 where Mzczvz A= (13.116) 13.6 Examples ofRelativistic FieldTheories 587 Interms ofthevariable 1:,theequation ofmotion isindeed thatforasimple pendulum ofﬁnite amplitude. Forvery small amplitude, weknow that¢isa simple harmonic motion inrwithorgiven byEq.(13.106) forawave number k=co/v. independent oftheamplitude. With ﬁnite amplitude, wealsoknow from ourstudy ofthependulum, thatwhile ¢willstillbeperiodic, thefrequency cowillalsodepend upon theamplitude. That istosay,thedispersion relation will beamplitude dependent. Thisisacharacteristic ofcourse ofnonlinear equations, ofwhich thesine-Gordon equation isoneexample. TheKlein-Gordon equation islinear, butthedispersion equation, Eq.(13.l06), issaidtobenonlinear; thatis, wkisnotalinear function ofk.Itbecomes linear onlywhen /.111—>0,reducing theKlein—Gordon equation theusual linear wave equation. Wecanthusdescribe thesine-Gordon equation asbeing nonlinear, withanon- linear amplitude-dependent dispersion relation. Further examination reveals that itcanhave solutions withproperties shared byonlyafewother nonlinear equa- tions. These solutions aretraveling wave disturbances thatcaninteract witheach other—pass through each other—and emerge withunchanged shape except per- hapsforaphase shift. Such solutions arealsofound, forexample, forthenonlinear Korteweg—deVries equation, a¢ a¢ 83¢ where orandvareconstants. These solitary waves thatpreserve their shape even through interactions have been tenned “solitons” andhave found many applica- tions throughout physics, from elementary particles through solid-state physics. Thependulum sine-Gordon equation, forexample, hasbeenusedtodescribe fam- iliesofelementary particles, anditalsoshows upinconnection withthetheory of theJosephson junction. C.TheElectromagnetic Field.* Theformalism andﬁeldequations fortheelectro- magnetic ﬁeldwere developed inSection 7.5.Itremains toexpress these ideas in terms oftheLagrangian formalism. Ifthecomponents A”oftheelectromagnetic potential aretreated astheﬁeldquantities, thenasuitable Lagrangian density for theelectromagnetic ﬁeldis KP L=-ﬁll +jkA*. (13.118) Toobtain theEuler-Lagrange equations, wenotethat 35_--_3_‘3___%_8.l”.’_8A#Th“ aA,,_., T2aA,,,., *Part ofthedifﬁculty inhandling theelectromagnetic ﬁeldarises fromthefactthatthecomponents A“ arenotentirely independent; tobeunique, theymustbeconnected through somegauge condition, such asEq.(7.66). However, itwillbesufﬁcient forourpresent purposes ifwetreatthegauge condition as a“weak” constraint. Chapter 13Formulations forContinuous Systems andFields Now, from thedeﬁning equations (7.71), thederivative ofFkpvanishes except whenk =/.i,p=vand). =v,p=/.1. Hence, ar: F,,,,F,,,-—-=—-_=F, 13.119aA,,,, 2 2 “" ( ) andtheEuler-Lagrange equations are dF"" /10.W_/5,11 =0. (13120) Finally, ithasalready been noted thatLforanelectromagnetic ﬁeldconsists ofafree-ﬁeld Lagrangian density plusaterm describing theinteraction ofacon- tinuous charge andcurrent density with theﬁeld. Itistempting toseehowfar wecangotoward introducing ﬁeld-particle interactions, bylocalizing thecharge toapoint. Thisismosteasily donebyconsidering thephysical situation insome particular Lorentz frame, thatis,asseenbyaparticular observer. Manifest covari- anceisthereby abandoned, buttheresult stillconfomrs tospecial relativity, asit derives from aclearly relativistic theory. Thecurrent density isameasure ofthe motion ofthecharges, andinanygiven system jisdeﬁned interms ofthecharge density pbytherelation j(r,t)=p(r,t)v(r, t). Here visthevelocity “ﬁeld” ofthecontinuous charge distribution. Thelocaliza- tioncanbecarried outthrough theuseofthewell-known Dirac 8-function. In three-dimensional form, the8-function hastheproperty thatiff(r)isanyfunc- tionofspace, then [dVf(r)6(r—s(r)) =f(s), (13.12l) where s(t)isthespatial position, say,ofaparticle attimet(solongassisinside thevolume ofintegration). Thus, thespatial charge andcurrent density cone- sponding toaparticle ofcharge qatpoint sis p=q8(r—s) (13.122) and j=q8(r —s)v(r). (l3.123) Ifwewrite LlofEq.(13.1 18)asthesumofafree-ﬁeld termL11)andaninteraction term, theLagrangian asseeninthegiven Lorentz frame is L=/dVL()—/dVp¢+/dVA-j=/dVL()—q¢+qA-v. (l3.l24) 13.7 I13.7 Noether’s Theorem 589 Theinteraction terms inEq.(13.124)areexactly thesame asthose inEq.(7.141) fortheLagrangian ofasingle particle inanelectromagnetic ﬁeld. Thissuggests thatasingle Lagrangian canbeformed forthecomplete system ofparticle and ﬁeldthat,analogous toEq.(7.141), would looklike L=—mc2,/l—,62—q¢+qv-A+fdVL(). (l3.l25) Considered asafunction oftheﬁeldtensor orpotentials, thisLagrangian implies theﬁeldequations; considered asafunction oftheparticle coordinates, Lleads to theparticle equations ofmotion. Themechanical descriptions ofthecontinuous ﬁeldandthediscrete particle have ineffect been putunder onewing, expressed inacommon formalism! Animportant branch ofmodern physics isconcemed with theconstruction ofﬁelds torepresent various types ofelementary particles. Ofcourse, allsuch theories arequantum-mechanical, butmany features ofquantum ﬁeld theories willhave concomitant ornearly corresponding classical analogs. There islittle apriori physical guidance intheconstruction ofpossible Lagrangian densities andinteraction terms forthevarious particles. Some constraint ontheform of these functions comes from covariance limitations. Forexample, theterms in[L must becombinations ofﬁeld andother quantities insuch amanner astopro- duce a4-scalar. Usually, Bisalsorestricted totheﬁeld quantities ortheir ﬁrst derivatives, although Lagrangian densities withhigher derivatives have alsobeen explored. Additional requirements ontheform oftheterms arealsoprovided, or suggested, byconservation andinvariance properties, implicit intheLagrangians. These properties gobeyond theconservation conditions contained inthest;ress- energy tensor andareusually tobefound bytheapplication ofapowerful pro- cedure known asNoether’s theorem, which forms thesubject ofthenextandlast section. NOETHER' STHEOREM Arecurring theme throughout thistexthasbeen thatsyrmnetry properties ofthe Lagrangian (orHamiltonian) imply theexistence ofconserved quantities. Thus, iftheLagrangian does notcontain explicitly aparticular coordinate ofdisplace- ment, then thecorresponding canonical momentum isconserved. Theabsence ofexplicit dependence onthecoordinate means theLagrangian isunaffected by atransformation thatalters thevalue ofthatcoordinate; itissaidtobeinvari- ant,orsymmetric, under thegiven transformation. Similarly, invariance ofthe Lagrangian under timedisplacement implies conservation ofenergy. Theformal description oftheconnection between invariance orsymmetry properties andcon- served quantities iscontained inNoether’s theorem. Itisinthe4-space ofclassi- calﬁeldtheory thatthetheorem attains itsmostsophisticated andfertile fonn. For thatreason, explicit discussion ofthetheorem hasbeen reserved forthetreatment ofﬁelds, although adiscrete-system version canalsobederived. Chapter 13Formulations forContinuous Systems andFields Symmetry under coordinate transformation refers totheeffects ofaninﬁnites- imaltransfonnation oftheform x“—>x"‘=x”+8x”, (13.l26) where theinﬁnitesimal change 8x” may beafunction ofalltheother x". Noether’s theorem alsoconsiders theeffect ofatransformation intheﬁeldquan- tities themselves, which maybedescribed by 77,,(x”) —>77;,(x"‘) =r7,,(x") +8r7,,(x”). (l3.l27) Here 8r7p(x/‘) measures theeffect ofboth thechanges inx”andin77,,andmay beafunction ofalltheother ﬁeldquantities 77k.Note thatthechange oneofthe ﬁeldvariables ataparticular point inx”space isadifferent quantity 877,,: n;,(x")=17p(x")+§n,.(x'*). (13128) Thedescription ofthetransformations interms ofinﬁnitesimal changes fromthe untransformed quantities indicates wearedealing onlywithcontinuous transfor- mations. Thus, symmetry under inversion inthree dimensions (parity symmetry) isnotoneofthesymmetries forwhich Noether’s theorem canbeapplied. Asa consequence ofthetransfonnations ofboththecoordinates andtheﬁeldquanti- tiestheLagrangian appears, ingeneral, asadifferent function ofboth theﬁeld variables andthespacetime coordinates: l3(r1,>(x").17p..)(x“).x“) ->c'(n;.(x’“).1);,,.(x'“).x'“). (13129) Theversion ofNoether’s theorem thatweshall present hereisnotthemost general form possible, butissuch astofacilitate thederivation without signiﬁ- cantly restricting thescope ofthetheorem ortheusefulness oftheconclusions. Three conditions willbeassumed tohold. Theﬁrsttwoare 1.The 4-space isﬂat;thatis,either itisEuclidean, orintheform of Eq.(7.171), R°‘7;,,, =0. 2.TheLagrangian density displays thesame functional form interms ofthe transformed quantities asitdoes oftheoriginal quantities, thatis, 17(1);(x"‘),11;,_,(x'“), X’/J’)=c()1;,(x'“), )1;,,,(x"‘), 11'“). (13.130) This type ofcondition hasnotpreviously entered ourdiscussions ofcon- served quantities, mainly because ithasbeen automatically satisﬁed under the transformations considered. When cyclic coordinates aretransformed bydis- placement, thefunctional dependence oftheLagrangian onthevariables is unaltered bytheimplied shift inorigin. Butinourpresent extended types oftransformation, itbecomes asymmetry property thatneeds study. Thus, thefree-ﬁeld version oftheLagrangian density fortheelectromagnetic ﬁeld, 13.7 Noether’s Theorem 591 Eq.(l3.ll8), retains itsfunctional form when A"issubject toagauge trans- formation, while other forms may not.Note also thatEq.(l3.l30) ensures thattheequations ofmotion have thesame fonn whether expressed interms oftheoldorthenew variables (form invariance). The condition ofform- invariance isnotthemost general circumstance under which thisistrue; the original andtransformed Lagrangian densities mayalsodiffer bya4-divergence without modifying theequations ofmotion. Indeed, itispossible tocarry out thederivation ofNoether’s theorem with such anextended version offonn- invariance because thevolume integral ofthe4-divergence term vanishes. But forsimplicity weshall restrict ourselves toEq.(13.130). Thethird condition 1S 3.Themagnitude oftheaction integral isinvariant under thetransformation, thatistosay,(cf.Hamilton’s principle Eq.(2.1)) 1'=/QI<dx"‘>11’(n;, <x'“>.v;,v<x'“>. x”‘) =L(dx4)L(17p(x"),17,,_,,(x"),x“), (13131) where dx4istheinvariant volume element isequal toJlg]dxodxldxzdx3 and~/lg]=,/|det(g)| isthesquare rootabsolute value ofthedetenninant Ofg. Again, Eq.(13.131) represents anextension of,andincludes, ourprevious symmetry properties suchascyclic coordinates. TheLagrangian does notchange numerically under translation ofacyclic coordinate, nordoesthevalue oftheac- tionintegral. Equation (l3.131) willbecalled thecondition ofscale-invariance. Oursecond andthirdconditions thusrepresent generalizations ofthesymmetry or invariance conditions thatledtotheexistence ofconserved quantities fordiscrete systems. Combining Eqs.(13.130)and(13.l31)gives therequirement L2’L1(r;;,(x"‘), n;,,,,(x'”), x'”)dx'4 —/s;,C(17,,(x“),17,,,,,(x"‘),x")dx4 =O. (l3.l32) Intheﬁrstintegral, x’”nowrepresents merely adummy variable ofintegration andcantherefore berelabeled x“.Butofcourse there remains achange inthe domain ofintegration, sothecondition becomes /S‘?£(17;(x”'),17;,’v(x"), x“)dx4—/QLI(n,,(x”'), 17,,_,,(x”'), x“)dx4=0. (13.l33) Thesequence oftransformations ofspace andofintegration region isillustrated inFig.13.6foraspace oftwodimensions. Equation (13.l33) saysthatifin theaction integral over (x”) space wereplace theoriginal ﬁeldvariables bythe b+6b b b /‘+5 (f(x)+8f(x))dx—/i f(x)dx=/‘ 8f(x)dx+f(x)8xChapter 13Formulations forContinuous Systems andFields X2 x'2 X2 Q7 Q! S2 xl xv] V xl FIGURE 13.6 Schematic illustration ofthetransformation oftheinvariant action inte- gral. transformed quantities, andtransform theregion ofintegration, thentheaction integral remains unaltered. Under theinﬁnitesimal transformations ofEqs.(l3.126) and(l3.l27), theﬁrst- order difference between theintegrals inEq.(l3.l33) thusconsists oftwoparts, onebeing anintegral overQandtheother anintegral overthedifference volume Q’—Q.Anexample inone-dimension willshow howtheterms aretobefonned. Consider thedifference oftwointegrals: I1-l-db I7 b L5<r<x>+8f<x>>dx—/ f<x>dx= f<v<x>dx b-I-517 +£ <f<x>+<tf<x>>dx a+¢Sa -f(foo+8f<x>>dx. (l3.l34) Toﬁrstorder insmall quantities, thelasttwoterms ontheright canbewritten as b+8b a+6a ii, f(x)dx—f f(x)dx =8bf(b)—8af(a). Tothisapproximation. Eq.(13.134) becomes (13135) of 1’ d=f[mo+5<@xf<x>>] dx. (l3.l36) Themultidimensional analog ofEq.(13.135)thensaysthattheinvariance con- dition ofEq.(13.133)takes theform 13.7 Noether’s Theorem 593 f£<n'.x'“>dx"‘— f2<n.x">dx‘= f[!J(n',x“)—£(n,x“)]dx4Q’ Q Q +I£(n)8x“dS,, =0. (13.13?)S Here, L(17,x”)isshorthand forthefullfunctional dependence, Sisthethree- dimensional surface oftheregion S2(corresponding totheendpoints aandb intheone-dimensional case), and8x”isineffect thedifference vector between points onSandcorresponding points onthetransformed surface S’(cf.Fig.13.7). Corresponding toEq.(l3.l36), thelastintegral canbetransformed bythefour- dimensional divergence theorem, sofortheinvariance condition wehave d0=Lax‘ {[L(1;’, x”)-£01,x")]+E(L(1;, x)8x")]. (13138) Now, byEq.(l3.128), thedifference terminthesquare brackets canbewritten to ﬁrstorder as , ac_ ac_ll(n§,(x""). n,,,U(x“). X”)—£(n(x"). np,»(x“), x“)=577-5m»+5- 877p,v-P Pr" (13139) Theimportant property ofthe5change isthatitisachange of17ataﬁxed point inx“space (unlike the8variation, Eq.(l3.127)). Hence, itcommutes with the spatial differentiation operator; thatis,theorder ofthequantities - d8 d—an dx" canbeinterchanged. Symbolically, ac- ac(131;' _— _____iL(17,x”) —LI(17,x“) -anp617,, +amp,” dxﬂ, (l3.l40) 2x ,~\ S, \\\ s \\ bx\\ \____ ,//\-‘\\-Z{Q~ ~11.»- xl FIGURE 13.7 Theintegration regions intwodimensions involved inthetransformation oftheaction integral. 4 Chapter 13Formulations forContinuous Systems andFields or,using theLagrange ﬁeldequations, d 3L_',“- ," =—- -——-8 . 13.14 £(nX)£01x)dx,(MW '7/J) (1) Hence, theinvariance condition, Eq.(13.138), appears as a’ 8L_(d“— [—8 +118 "}=0, (l3.l42)fx)dx" 8np,,, up x which isaconserved current equation (cf.arguments onpg.571). Itishelpful however todevelop thecondition further byspecifying theform oftheinﬁnitesimal transformation interms ofRinﬁnitesimal parameters 6,,r= l,2,...,R,suchthatthechange inx"and17,,islinear inthe6,: 8x”=e,X}’, 817,,=e,\II,,,. (13.143) Thefunctions XXand\I/nomaydepend upon theother coordinates andﬁeldvari- ables, respectively. Ifthetransformation symmetry relates tothecoordinates only, andcorresponds toadisplacement ofasingle coordinate x",thenthese functions aresimply X318?’ qlrp Z 0- Thus, thetransformations contained inthefonn ofEq.(l3.143) constitute afar more extensive testforsymmetries thanwehaveusedthusfar.From Eqs.(13.127) and(13.128),itfollows thattoﬁrstorder 827and817arerelated by - 3am,=am,+(“if8x”. (13145) Hence, 51),,=6,01/,,, -17,,,X5). (13146) Substituting Eqs.(l3.143) and(l3.146) intheinvariance condition, Eq.(l3.128), wehave 4 an an— -—— —L8" X“---tr d‘=0. 13.147 /Grdxv [(anp'v 7lp,a U) r am)” rp] x ( ) Since thee,parameters arearbitrary, there exist inanalogy withEq.(13.142), rconserved currents with differential conservation theorems: (integral ofdiver- gence =0) d 3L: 3L: i —— —L5v X”-—iii =0. 13.148 dxv [(anp’V77p,o 0) r am)!” no] ( ) 13.7 Noether’s Theorem 595 Equations (13.148) form themain conclusion ofNoether’s theorem, which thussaysthatifthesystem (ortheLagrangian density) hassymmetry prop- erties suchthatconditions (2)and(3)above holdfortransformations ofthe typeofEqs.(l3.143), thenthere existrconserved quantities. Theconservation ofthestress-energy tensor iseasily recovered asaspecial caseofEq.(13.l42). IfLdoes notcontain anyofthex”,thenit,andtherefore theaction integral, willbeinvariant under transformations such asEq.(13.l44), where Atakes onallthevalues /4,.Equation (13.l48) thenreduces to d ac dac ,E [(%'ﬂp_U — =w (?’)")7]p,IL — , which isidentical withEqs.(13.29) withT,”given byEq.(13.30). Alarge number ofother symmetries arecovered bytransformations ofthe form ofEq.(l3.142). Oneofthemost interesting isafamily oftransformations oftheﬁeld variables only, called gauge transformations oftheﬁrstkind,* such that 8x=O, 817,,=ecpnp (nosummation onp), (l3.l50) where thecpareconstants. IftheLagrangian density, andtherefore theaction in- tegral, isinvariant under thistransformation, thenthere isaconservation equation oftheform d®"E67=0, (13.151) where 3L @”= —-— . 13.152 cpan/"V 77p ( ) Equation (13.151) isintheform ofanequation ofcontinuity with (~)"intherole ofacurrent density j“.Hence, invariance under agauge transformation oftheﬁrst kind leads totheidentiﬁcation ofaconserved current thatwould beappropriate foranelectric charge andcurrent density tobeassociated withtheﬁeld. Asanillustration, letusconsider theﬁrstexample ofSection 13.6,thecomplex scalar ﬁeld. Atransformation ofthetype ¢'=¢@"‘, ¢*’=¢*@-"‘ (13153) corresponds ininﬁnitesimal form toagauge transformation oftheﬁrsttype, Eq.(l3.l50), with c=i, c*=—i. *The familiar gauge transformation oftheelectromagnetic ﬁeld, which addsa4-gradient A4,toAM. ispartofagauge transformation ofthesecond kindandisnotconsidered here. Chapter 13Formulations forContinuous Systems andFields Itisobvious thattheLagrangian density ofEq.(13.94) isinvariant under thetrans- formation (13.153). Hence. there isanassociated current density fortheKlein- Gordon ﬁeldthatcanbegiven as IF ..d.,41¢1,,=zq(tbclitqﬁ -¢dV), (13154) which isinagreement withtheconventional quantum-mechanical current density. Note thattheentire derivation oftheconserved charge current density depends upon thefactthattheﬁeldiscomplex. Thus, asmentioned above, arealﬁelddoes notleadtoacharge orcurrent density associated withtheﬁeld. Todescribe ﬁelds associated withcharged particles, wemust useapairofcomplex ﬁelds suchas¢ and¢*forthe(spin-less) Klein—Gordon particle. Note thatwhile Noether’s theorem proves thatacontinuous symmetry prop- ertyoftheLagrangian density leads toaconservation condition, theconverse isnottrue. There appear tobeconservation conditions thatcannot correspond toanysymmetry property. Themost prominent examples atthemoment arethe ﬁelds thathave soliton solutions, forexample, aredescribed bythesine-Gordon equation ortheKorteweg—deVries equation. Consider, forexample, theLagrangian density forthesine-Gordon equation, Eq.(13.107). Asxandtdonotappear explicitly, theLagrangian density isinvari- antunder translations ofspace andtimeinthemanner fulﬁlling theconditions of Noether’s theorem. Inaddition, there isasymmetry under aLorentz transforma- tion(inx,tspace). Noother symmetry isapparent. Wewould therefore expect no more thanthree conserved quantities from theapplication ofNoether’s theorem. Yetithasbeen demonstrated, bymethods lying outside theLagrangian descrip- tionofﬁelds, thatthere exists aninﬁnite number ofconserved quantities. That is tosay,aninﬁnite number ofdistinct functions F,-andG;thatarepolynomials of ¢,andderivatives canbefound forwhich (IF; dG,' ——- Z =, 13.155dt+dx O ( ) sothatthevolume integrals oftheF,-areconstant intime. Itappears thatthe presence ofsuchaninﬁnite setofconserved quantities isanecessary condition in order fortheﬁeldtodescribe solitons. Finally, wecaneasily deduce theversion ofNoether’s theorem thatshould apply todiscrete systems. Here thefourcoordinates ofspacetime arenolonger parametric variables onequal footing-—-the space coordinates revert totheir sta- tusasmechanical variables (orfunctions thereof), andonly timeremains toﬁll theroleofaparameter. Theaction integral, instead ofbeing afour-dimensional volume integral, I=/1L§dx4, 13.7 Noether’s Theorem 597 isaone-dimensional integral intasinEq.(2.1) which isHamilton’s principle: I=fLdr. Instead ofthecontinuously indexed ﬁeldvariables 1),,(x"), wehave thediscrete generalized coordinates qk(t).Itisstraightforward enough torecapitulate with these translations thesteps thatledtoNoether’s theorem. Wecould repeat inthis manner thearguments contained inEqs.(13.126) through (l3.148) asapplied to discrete systems. Buttheeffect oftheconversion issufﬁciently obvious andclear, thatwecanreadily seethetranslation need bedone directly only ontheﬁnal result, Eq.(13.148). Therules forthetranslation canbesununarized as £—+ L, x“orx”-—-> t. Up*'>qk. q/m, —->4,, (l3.l56) Further, allsums over4-valued Greek indices reduce tooneterm, int.Asaresult, thetransformations, Eq.(13.l43). under which theLagrangian istoexhibit form andscale invariance become at=e,X,, 8q;,=6,4/,,,. (13.15?) Equation (13.148), thestatement oftheconservation theorems resulting from the invariance, nowbecomes d 8L, 8LdtMM,‘ qk L)X, aqk\I/,4 _0. (13.158) Equation (13.158) isthestatement oftheconclusions ofNoether’s theorem foradiscrete mechanical system. Theexpression intheparentheses inEq.(13.158) isouroldfriend theJacobi integral hofEq.(2.53), orequivalently interms of(q,p),theHamiltonian. In- deed, wecanrecover theconservation ofhbyconsidering atransformation that involves adisplacement oftimeonly: X,=5r1, \I1,k=O. (l3.159) IftheLagrangian isnotanexplicit function oftime, thenclearly theform ofthe Lagrangian andthevalue oftheaction integral areunaffected bythistransfor- mation. ButNoether’s theorem, Eq.(13.l48), thensaysthatasaresult there isa conservation theorem Chapter 13Formulations forContinuous Systems andFields d8L——'—L=0, dr(841."") which isidentical withthefamiliar conclusion ofSection 2.6. Letussuppose further thataparticular coordinate qliscyclic. Then theLa- grangian andtheaction areinvariant under atransformation forwhich X,=O, \I/,1,=8145,] (l3.l60) andEq.(13.l58) immediately implies thesingle conservation statement d '— % ZOr dt aql I51=9-Of sothecanonical momentum isconserved. Thus, thetheorems ontheconservation bothofJacobi’s integral andofthegeneralized momentum conjugate toacyclic coordinate aresubsumed under Noether’s theorem asstated inEq.(l3.l58). Theconnection between symmetry properties ofamechanical system andcon- served quantities hasrunasathread throughout formulations ofmechanics as presented here. Having come fullcircle, asitwere, andrederived bysophisticated techniques symmetry theorems found intheﬁrstchapters, itseems anappropriate point atwhich toendourdiscussions. EXERCISES 1.(a)Thetransverse vibrations ofastretched string canbeapproximated byadiscrete system consisting ofequally spaced masspoints located onaweightless string. Show thatifthespacing isallowed togotozero, theLagrangian approaches the limit 1 2 an2L=— '—T-— d 2Il’”' (Bx) x forthecontinuous string, where Tistheﬁxed tension. What istheequation of motion ifthedensity 11,isafunction ofposition? (b)Obtain theLagrangian forthecontinuous string byﬁnding thekinetic andpo- tential energies corresponding totransverse motion. Thepotential energy canbe obtained fromthework donebythetension force instretching thestring inthe course ofthetransverse vibration. 2.(a)Describe theﬁeldofsound vibrations inagasintheHamiltonian formalism and obtain thecorresponding Hamilton equations ofmotion. (b)Generalizing themomentum expansion toavector ﬁeld, express theHamiltonian fortheacoustic modes ofagasinthemomentum representation. Exercises 599 3.Obtain Hamilton’s equations ofmotion foracontinuous system from themodiﬁed 4. 5. 6.Hamilton’s principle, following theprocedure ofSection 8.5. Show thatif1/;and(I/'*aretaken astwoindependent ﬁeldvariables, theLagrangian density I12 h ..4=Tvr -vr/»*+v¢*¢+—.<¢*=// —¢»¢*>87:m 4m leads totheSchrodinger equation n22 ihat/1—i—V V=———,8;rr2m ‘Z’+ll’ Znat anditscomplex conjugate. What arethecanonical momenta? Obtain theHamiltonian density corresponding toL. Show that 6G;=-Inil?av6x‘ isaconstant ofthemotion iftheHamiltonian density isnotanexplicit function of position. Thequantity G;canbeidentiﬁed asthetotallinear momentum oftheﬁeld along thexidirection. Thesimilarity ofthistheorem with theusual conservation theorem forlinear momentum ofdiscrete systems should beobvious. (a)Ina4-space thatisnotEuclidean, theD’Alembertian isdeﬁned as 22=V2=/.Lvl_ U 83x1/-6x” ' Heregl”isthecontravariant metric tensor, which intheﬂatspace ofspecial relativity isindeed thesame asgm). Forthemetric tensor oftrace +2instead of -2usedinEq.(7.33), ﬁndtheexplicit fonn oftheD’Alembertian sodeﬁned. (b)Asuitable Lagrangian forthecharged scalar meson ﬁeldinthismetric is L=%(gl1-"ﬂfi _”%¢¢*)_ 8x#8x" Show thatoneofthecorresponding ﬁeldequations is (1:12-r»%,>¢=(V2-r»3>¢=0. Show alsothatinlight ofpart(a)thisequation isactually identical with Eq.(13.99). 7.TotheLagrangian density forthescalar charged meson, Eq.(13.94), addthefollowing term torepresent theinteraction withanelectromagnetic ﬁeld: J')”A>. where 1'1=i(¢¢*,x —¢,2t¢*)- Chapter 13Formulations forContinuous Systems andFields What aretheﬁeldequations for¢and45*?What happens totheconserved currents andassociated conservation theorems? Suppose theLagrangian density inHamilton’s principle isafunction ofhigher deriva- tivesoftheﬁeldquantities 17,,: 5=c(7lp§ 7lp,p.§ TIp,;1.u§ Xx)- Assurning thevanishing ofthevariation attheendpoints, what istheform oftheﬁeld equations corresponding tosuchaLagrangian density? Consider ascalar ﬁeldquantity 17that,forsimplicity, isafunction only ofxandt. Suppose nowthattheHamiltonian density isafunction ofhigher spatial derivatives of17andJT,thatis, H=H07, 7l,x» 71',7T,x5 7T,xx)- What arethecorresponding Hamilton equations ofmotion? Show thattheKorteweg—deVries equation corresponds totheﬁeldequation forascalar ﬁeld(IrwithLagrangian density l oz v L=51/wt+gr?—5143,. where thesubscripts indicate derivatives withrespect tothevariables indicated, pro- vided iﬁisapotential function forthequantity ¢ofEq.(13.1 17): W ¢_8x' Consider aHamiltonian density in(x,t)space: H=713‘l‘%7l2,x "l'773,1: +%7T2,xx- Show thattheHamilton equations ofmotion correspond toaform oftheKorteweg- deVries equation, Eq.(13.1 17),if 11=¢(x.I) oo 71'=/l ¢(x',t)dx'. —oo Evaluate explicitly T?/c andT;1-forthesymmetrized stress-energy tensor ofthefree electromagnetic ﬁeldasgiven by A,,F* F),F1Tiivsym =Tl!-V_H?” =_'ZTu +cg/-W What canbesaidabout thephysical meaning ofthese components? Ina4-space with metric gw,oftrace +2,evaluate explicitly theelements ofthe covariant (mathematically speaking) tensor F,”oftheelectromagnetic ﬁeld. Also givetheelements ofthematrix withoneindex lifted andwithtwoindices lifted: Fri=SMF/xv? FM’=8mF/4v8pv- APPENDIXEuler Angles inAlternate Conventions and Cayley—Klein Parameters TheEuler angles asdeﬁned inSection 4.4arespeciﬁed byaninitial rotation about theoriginal zaxisthrough anangle ¢,asecond rotation about theintermediate x axisthrough anangle 9,andathird roation about theﬁnalzaxisthrough anangle 10.Thissequence isheredenoted asthe“xconvention,” referring tothechoice of thesecond rotation. Forthexconvention theCayley—Klein parameters intenns oftheEuler angles are - 6 - 9O,=el(1I/+¢)/2 cos_, 5=1-el(1//-¢)/2 Sin_, 2 2 --1<r—¢>/2 -9 -in»-¢>/2 9 y=ze S1115, 8=e cosi, Other conventions arepossible, andtwoinparticular have found frequent appli- cations inparticular ﬁelds. Fonnulas willbegiven hereforproperties ofageneral rotation interms oftheEuler angles ofthese twoaltemate conventions. yCONVENTION Theyconvention differs from thexconvention onlyinthatthesecond rotation isabout theintermediate yaxis. Transcription from thextotheyconvention isparticularly simple because 6retains itsmeaning inboth conventions andthe changes fortheother angles areeasily obtained. Inthexconvention, ¢isthe angle between thelineofnodes andthexaxis; intheyconvention, itisthesame angle measure totheyaxis.Similarly inthexconvention, 11/istheangle between thelineofnodes andthex’axis; while intheyconvention, itisthesame angle relative tothey’axis.Temprarily using subscripts toindicate theconvention used, these relations imply theconnection (cf.Fig.4.7) [Q.§ll\)~Fl¢x=¢y'l' lpx=1/ry__ (A-ly) or sin¢,,=cos(by sin11/,=—cos1//y cos¢,,=—sin(by cos10,=sintlry. (A.2y) 601 Appendix AEuler Angles inAlternate Conventions andCayley—Klein Parameters With thisrecipe weobtain thefollowing formulas interms oftheEuler angles in theyconvention: Rotation matrix. —sin1#sin¢+cosl9cos¢cos1/r sini/rcos¢+cos6sin¢cos1/r —cosrbsin6 =—cosrlrsin¢—cos6cos¢sinr0 cosr0cos¢—cos6sin¢sinrb sinr,0sint9 sin9cos¢ sin9sin¢ cos9 (A.3y) Thesame result canbeobtained bynoting thattheexchange ofyforxcorre- sponds toarotation ofthereference frames about thezaxisthrough anangle of -11’/2 or31:/2. Wecantherefore translate theAmatrix from xconvention toy convention byasimilarity transformation bytheorthogonal matrix G: 0-10 G=1O0 (A.4y) O O1 again leading toEq.(A.3y). Cayley—Klein parameters. Forthisconvention theCayley—Klein parameters are /‘\~t$- £2‘$~ 4%a=ei cosg ﬁ=ei(_)sin% __.-‘(£5381 5 _*‘(hf) 5 y- n2 8_e cos2. (A.5y) Euler parameters. Itirmnediately follows from thedeﬁnitions ofe()—exinSec- tion4.5andEq.(A.4y) thatintheyconvention theEuler parameters aregiven by IQQIQQ6 _ e()=cos¢%_<£cosE e2=cos%s1n— e1=sing sing e3=sin% cos—. (A.6y) Components ofangular velocity. Either bydirect useofthetranslation equations, (A.2y), orbyfollowing through thephysical meanings ofthecomponent parts of w,wecanobtain thefollowing components ofcoalong thebody axes inthey convention: cox’=—<psin9cos¢ +9sin¢ my=¢;isin9sini0 +§cos(0 wzr=cost)+ (A.7y) xyzConvention 603 Similarly, thecomponents oforalong thespace axesare cu,=—9sin¢ +tbsin9cos¢ wy=9cos¢ +sin9sin¢ (A.8y) cu,=i]rcos9 +<b. Finally, notethat ¢> 9cos =e0=cos% cos5 (A.9y) which isthesame asEq.(4.63) forthexconvention. xyzCONVENTION Inthisconvention eachrotation isabout adifferently labeled axis.Obviously, var- ioussequences ofrotations arestillpossible. Itappears thatmost U.S.andBritish aerodynamicists andpilots prefer thesequence inwhich theﬁrstrotation isthe yawangle ¢about azaxis, thesecond isthepitch angle 9about anintermediary yaxis, andthethird isabank orrollangle ¢about theﬁnalxaxis(orﬁgure axis ofthevehicle). Ofthethree elementary rotation matrices Dremains thesame as Eq.(4.43), Cappears as cos90—sin9 C= 0 1 0 , (A.lOxyz) sin9Ocos9 andBisthesame asEq.(4.44) (with urinplace of9,ofcourse). Theproduct BCD gives thefollowing formulas: cos9cos¢~ cos9sin¢ —sin9 A= sini//sin9cos¢—cos¢sin¢ sini!/sin9sin¢+cos1//cos¢ cos9sin¢Rotation matrix. (cos 1/1sin9cos¢+sin11/sin9 cos1/1sin9sin¢ —sinupcos¢cos9cosup (A.1lxyz) Cayley—Klein parameters. These parameters havetheform 9 9-or=8*=(cos gcos5—isingsin5)e“7’/2 6 . ,8=—y* =cosKsinQ+isinZcos—e_“i’/2. (A.12xyz)2 2 2 2 04 Appendix AEuler Angles inAlternate Conventions andCayley—Klein Parameters Euler parameters. From Section 4.5andEqs.(A.12xyz), itfollows thattheEuler parameters are l->*$U) MCI: we r\>"‘$ MO: we<13COS5-=€()=COS—CO —COS—+SlIl—S1l'l--S111- to-$'°$ IQ<blQ‘9~lQ'9~'oVJ l\->€|\>~$:5. wwbmco.,5. l\>‘$-l\J'$-e1=sin—cos—cos——c —s1 —s — e2=cos—sm—cos—+s1n—cos—sm— (A.l3xyz) e3=—sinfsingcosg+cosZcosgsing2 2 2 2 2 2 Note thatthecosine ofthetotal angle ofrotation nowhasadifferent form from either thexortheyconvention. Components ofangular velocity. Clearly 01¢,liesalong thebodyxaxis,w¢along thespace zaxis, andwealong theintermediate axis, andtherefore intheﬁnalyz plane. Theresulting components along body axesare 0),,’= —sin0 my=9cos1// +¢3cos0sin1// wz»=-9sin1p+cos6cos11/. (A.l4xyz) Similarly, thecomponents ofwalong thespace axesare 0),,=cos9cos¢ —9sin¢ my=cos9 sin¢ +9cos¢ cuz=—sin6. (A.l5xyz) Theprevious editions ofthiswork dealt withtheCayley-Klein parameters in more depth. APPENDIX Groups andAlgebras Aswehave seeninalmost every chapter ofthistext,invariances intheformu- lation ofclassical mechanics display themselves assymmetries intheequations ofmotion. Thisproperty isformally discussed inSection 13.7asNoether’s theo- rem.Newtonian mechanics wasfonnulated withtheexplicit assumption thatthe laws areinvariant under anyGalilean transformation toanother inertial frame. Inthespecial theory ofrelativity, thelaws areformulated tobeinvariant under Lorentz transformations between inertial frames. Thegeneral theory ofrelativity isformulated toremove therestriction ofusing inertial frames. These andother in- variances andtransformation properties thatwehavediscussed canbeunderstood interms ofgroups oftransformations. Inmany cases, physicists dealextensively withrepresentations ofgroups, rather thanthegroups themselves. sowewillput some stress onrepresentations. Forexample, thesetof3><3rotation matrices withdeterminant +1,which appear soextensively inthetext,isarepresentation ofthespecial orthogonal group inthree dimensions (denoted bySO(3)). Since the reader’s knowledge ofgroups maynotbeextensive, wewillbegin withbasics by deﬁning agroup andgivesome examples ofﬁnite groups. Weshall thendiscuss inﬁnite groups* andrepresentations. PROPERTIES OFGROUPS Agroup isasetofobjects called elements withaproduct operation andthefol- lowing deﬁning properties: 1.Closure—the product oftwoelements equals athird element inthegroup. Ifaandbareelements inthegroup, theproduct ab=cwhere cisalsoa member ofthegroup. 2.Multiplication isassociative—if a,b,andcaregroup members, a(bc) = (ab)c. 3.Thegroup contains aunitelement, I,called theidentity withtheproperty thatforallelements ofthegroup, a=aI=Ia. 4.Each element aofthegroup hasaninverse element, a_lwiththeproperty aa‘1 =a—1a =I. *Mathematicians atthispoint willuseadifferent terminology forinﬁnite groups. Weshallfollow the physicist’s convention ofreferring tobothﬁniteandinﬁnite collections ofelements asgroups. 605 Appendix BGroups andAlgebras TABLE B.l Multiplication Table fortheFour-Element Cyclic Abelian Group, C4 l —l i —i N.\~.i—lr—\ ha.~.I—lI-1 u-.v~|.>—l>—l i—ll—lhm.‘- >—4P—*v~|.Wu Agroup isabelian ifthemultiplication operation commutes; thatis,forall elements aandbofthegroup, ab=ba.Ifanyofthegroup elements failto commute, thenthegroup isnonabelian. Anexample ofaﬁnite abelian group is thesetofelements {l,—l,i,—i}where 1istheidentity, andi=x/:1 .This group hasfourelements, soitissaidtobeoforder h=4.Weshall usehforthe group order. Thisgroup multiplication table isshown inTable B.1. Each group element appears once andonlyonce ineach rowandineach col- umnofthemultiplication table. Thisgroup canbegenerated from oneelement, i, called thegenerator, withtheproperty i2=—l, i3=—1, i4=l, (13.1) soitiscalled C4,thecyclic group offourelements. Anycyclic group, C,,,of order h=nelements hasagenerator element Awiththeproperty thatthemth element ofthegroup, Am,isoftheform Am=Am, (B.2) where A"=I. (B3) Adihedral group, D",isagroup withh=2nmembers andtwogenerators A andFwiththeproperties A"=1and F2=1. (13.4) Asubgroup isacollection ofsome oftheelements ofalarger group thatby themselves form asmaller group. Forexample, inC4aswecanseefrom the multiplication table, theelements 1and—lform asubgroup. Twoelements band careconjugate withrespect toeachother ifforsome element ofthegroup, a, aba_1 =c. (B.5) Thecollection ofallelements “c”conjugate tobasarunsthrough alltheelements ofthegroup iscalled aclass. Allclasses aredisjoint subsets ofthegroup with eachelement belonging tooneandonlyoneclass. Forabelian groups, suchasthe oneshown inTable B.1,allelements aretheirownclass. Theidentity element, I, always belongs toaclass byitself. Theclass structure isimportant fornonabelian groups. Properties ofGroups 607 There aretwogroups withsixelements, thecyclic group C5andthedihedral group D3.Theelements ofD3areusually denoted byI,A,B,C,D,andF.The generator Ahastheproperty A3=I.Itgenerates theelement B AA=A2=B, (13.6) andA-1=BandB-1= A,since AB=BA=I. (3.7) Theelement F,hastheproperty F2=Iandgenerates theremaining twoele- ments CandDthrough multiplications ofAandB.Theelements C,DandF aretheirownreciprocals since F2=C2=D2=I;thatis, c-1=c, D_l=D, and F-1=F. (13.8) This isanonabelian group since, forexample, theelements AandCdonot commute AC=FCA=1.). (13.9) Thegroup multiplication table isshown inTable B.2. Thesubgroups are subgroup 1—>I,C subgroup 2—>I,D subgroup 3—>I,F subgroup4 —>I,A,B. Thesixelements divide intothreeclasses, classl I class 2A,B class 3C,D,F. TABLE B.2 TheMultiplication Table fortheDihedral Group, D3 1 A B 'c D F I I A B C D F A A B I F C D B B I A C C D F I A B D D F C B I A F F C D A B Illl ZZiii1_i-iii-‘_-lll|Ullll|l|'11llllIIl|Q 60 Appendix BGroups andAlgebras Note thatinTable B.2class 3appears only intheupper-right andthelower- leftquadrant ofthemultiplication table, while classes 1and2appear onlyinthe upper-left andlower-right. Thisshows therepresentations thatarepossible forD3. REPRESENTATIONS OFGROUPS Arepresentation ofagroup isasetofmatrices thatsatisﬁes themultiplication table ofthegroup.* Byarepresentation wemean what ismore precisely called aninequivalent irreducible representation, Pi,orasetofmXmmatrices that cannot besimultaneously decomposed intolower-order matrices. Atheorem in group theory states thatthenumber ofirreducible representations, k,isequal to thenumber ofclasses andthesumofthesquares ofthedimensions, l,-,ofthe irreducible representations, I‘;equals thegroup order, h.That is, k Z1?=h, (13.10)l=1 where histhenumber ofelements inthegroup, kisthenumber ofirreducible representations, andl,-isthedimension oftheithrepresentation. Forthegroup D3,k=3andh=6,soEq.(B.10) becomes zf+1§+z§=6, (B.ll) whose only solution isl,-=lg=1,I3=2.There is,asforallgroups, aone- dimensional identity representation, F1inwhich wemapeach element onto+1. Another one-dimensional representation ofD3isthesetI‘;={l,—l},where the mapping is{I.A,B}—>land {C.D,F}—>-1ascanbeseenfrom Table B.2. Thetwo-dimensional matrix representation, F3,canbegiven interms oftheunit matrix andthePauli matrices: 10 01 0—i 1 01_[,,1...-[,0]. 0]. _,],(B-12) with 1=1, A=—%(I—iO'2\/-3;), B=-§(1+ie2~/3), c=§(~/§e1+a3), D=—%(\/§0'1—O'3), F=a3. (13.13) Notice howthegroup elements inclass3involve only01anda3.Thus, they areindependent ofthematrices Iand02,asisexpected from thestructure ofthe *Mathematicians always mean matrices when theyrefer torepresentations. Some ﬁeldtheorists take amore general meaning. Representations ofGroups 609 multiplication table. However, since each representation hasanidentity element, thereisnosimple association between classes andrepresentations. Therepresentation ofagroup canbefaithﬁtl orunfaithful. Forafaithful ma- trixrepresentation, each element inthegroup isrepresented byaunique matrix. Inanunfaithful matrix representation, more thanoneelement inthegroup isrep- resented bythesame matrix. Therepresentations F1andF2ofD3areunfaithful, while F3isafaithful representation. Afaithful representation isanisomorphism oraone-to-one mapping ofthegroup elements ontothematrices oftherepresen- tation. Anunfaithful representation isahomomorphism oramany-to-one map- ping. Wehave discussed thedihedral group D3asanabstract entity, thatis,asa setofelements thatsatisfy agroup multiplication table, andwhich hasatwo- dimensional representation thatisasetofmatrices alsosatisfying thesame multi- plication table. Groups alsohavemathematical andphysical realizations innature. Forexample, thepennutation group ofthree numbers (123) isaD3group. Ithas theidentity (123), threetwofold cycles (213), (132), and(321), which correspond withtheelements C,D,andF,andtwothreefold cycles, (231) and(312), which correspond totheelements AandB.Aphysical realization ofthisgroup isthe symmetry operations ofanequilateral triangle. Theelements AandBare120° and240° rotations about acentered axisperpendicular totheplane ofthetriangle, andthereﬂection planes m1,mg,andm3,correspond totheelements C,D,andF ofthegroup. Thisissketched inFig.B.1.Wesaythattheabstract group D3,the threefold permutation group andtheinvariance group ofoperations ontheequi- lateral triangle areisomorphisms because there isaone-to-one mapping between theirelements. Asafurther example, letusconsider thequatemion group, Q,which isone oftheﬁvegroups oforder 8(8elements). Themultiplication table isnormally written asshown inTable B.3.Thisgroup has5classes 2 m3 ml 1 3 "'2 FIGURE B.l Equilateral triangle showing thethreemirror planes m,-. 0 Appendix BGroups andAlgebras TABLE B.3 TheMultiplication Table fortheQuatemion Group I —I e1 —e| e2 -e2 e3 -e3 '~< Nthr->-'~<'-4QI-4WW ~<*~<-—-—-N ¥<r—~—I —e1 e2 —e2 e3 —e3 -—I — I — —e2 82 —e3 e3 61 -21 — 63 -63 -62 82 —e1 — —I —-e3 e3 e2 —e2 e2 e2 —e2 —e3 e3 —I I e| —e1 —e2 —e2 e2 e3 —e3 I —I —e1 e1 e3 e3 —e3 e2 —e2 —e1 e1 —I I —e3 —e3 e3 -e2 e2 e1 —e1 I —I Class 1—>I Class 2->-1 Class 3—+:l:e1 Class 4—>:l:e2 Class 5->:l:e3 From Eq.(B.l0), wehave ﬁ+@+@+ﬁ+@=& which hasthesolution l1=l2 =l3=l4=1, and Z5=2. (B.14) Fortheone-dimensional representations, allelements canbemapped into+1, orthey canbemapped intotheone-dimensional representation 1"={l.-1} by{I,—I,e1,—e1} ->+1and{e2,—e2, e3,—e3} —>-1.Thetwo-dimensional faithful matrix representation haselements (cf.Eq.(B.l2)) I=I, —I=—I, :l:€1 =IFi0‘1, :l:€2 =I|Ii0‘2, and :l:€3 =I|IiO‘3. (B.15) Thus farwehaveconﬁned ourattention toﬁnite groups. However, therotations inthree-space andtheLorentz transformations areinﬁnite dimensional groups since therotation angles andtheboost velocities cantakeonvalues from the continuum. Thesetofallproper (determinant =+1)3><3rotation matrices areafaithful representation ofthespecial orthogonal group inthree dimensions, SO(3). Ifweaddtheinversion operation, weinclude theimproper rotations with determinant =-1andobtain thelarger orthogonal group 0(3). Thegroup SO(3) isasubgroup ofthegroup 0(3). ThesetofLorentz transformation matrices in onedirection constitutes agroup withthe0(3) asubgroup. Ifweallow boosts in twodirections, wehave amuch larger group ofinhomogeneous Lorentz transfor- mations. LieGroups andAlgebras 611 TABLE B.4 TheCharacter Table forD3 D3 cl 2c2 303 F1 1 1 1 F2 1 l -1 F3 2 —l 0 Thesumofthediagonal elements ofamatrix iscalled thetrace ofthematrix. Thetrace ofthematrix inanirreducible representation, I‘,-,iscalled thecharacter, Xi,ofthatmatrix. Thecharacter ofamatrix inarepresentation isdetermined by theclass; thatis,allthematrices ofarepresentation thatcorrespond tothesame class have thesame character. Forthedihedral group D3,therelation between the classes C,-ofthetwo-dimensional representation, F3,isgiven asfollows: Class C, Elements Character X,- Class l I +2 Class 2 A,B —l Class 3 C,D,F 0 Fortheone-dimensional representations F1andI‘;ofD3,thecharacters arethe same astheone-dimensional matrices. Thisinformation canbemost conveniently expressed inacharacter table. ForD3,thisisshown inTable B.4. InTable B.4,theheadings nC,,, onthecolumns givethenumber ofelements nintheclass Cmofthatrow.Thecharacters intheﬁrstrowforclass C1also givethedimensionality oftherepresentation. Therows ofthecharacter table are orthogonal toeachother, provided wetakeintoaccount thenumber ofelements in eachcolumn. Forexample, considering I";andF3,wehave 1><2+2 ><(1><—1)+ 3x(-1x0)=0.Asanapplication, inquantum mechanics the1“;’scanrepresent energy levels splitfrom aparent atomic state byanelectric ﬁeldenvironment of D3symmetry. LIEGROUPS AND ALGEBRAS Theterms Liegroup andtheassociated ideaofLiealgebra areused inseveral chapters. ALiegroup isamanifold, which isalsoagroup. Amanifold isacontin- uous geometric object; forexample, Euclidean space, thespacetime ofthespecial theory, andacircle ofradius 1inthecomplex plane areallmanifolds. Most of themanifolds considered inphysics arecontinuous manifolds.* Foramanifold tobeaLiegroup, there must exist agroup operation (termed multiplication) for *Acontinuous manifold isamanifold withtheconcept ofneamess. Thatis,forevery point, P,inthe manifold, there exist other points inthemanifold thatareasclose toPasdesired. Asthemathemati- cianswould say,forevery point, P,inthemanifold andgiven anye>0,thereexists another pointin themanifold thatiscloser toPthan2,nomatter howsmall 2. Appendix BGroups andAlgebras allpairs ofpoints inthemanifold, which isconsistent withthecontinuous nature ofthemanifold. Consider fourpoints inthemanifold a,b,c,anddanddenote thegroup operation ofaandcbyac.Consistent means, ifaandbareclose to each other andcanddarealsoclose toeach other, thenac,ad,bc,andbdare allclose toeachother. Ifwerestrict ourattention totheLiegroups thatphysicists arelikely toencounter, there areonlyafew.OnesetofLiegroup elements cor- responds torotations inodddimensions, forexample, thethree-dimensional ro- tation group O(3). Asecond setistherotations ineven dimensions, forexample, theLorentz group in4dimensions. Another setinvolves theunitary groups, for example, SU(2), which isthesetof2x2unitary matrices withdetenninant +1. Theﬁnalsetcontains thesymplectic groups (SeeSection 9.4).There arealsoﬁve special ﬁnite groups. Corresponding totheLiegroups areLiealgebras, which areﬁatvector spaces withaLiebracket orcommutator deﬁned forasetofvector ﬁelds, {r,-},which can serve asthebasis vectors ofthespace. These vectors satisfy [r,-,rj]=r,-r1-—rjr,-=c,-jkrk (summation convention) (B.16) where thec,-1-"(which clearly satisfy c,-1-"=—Cijk) arecalled thestructure con- stants ofthealgebra. AllLiealgebras must, bysymmetry, satisfy theJacobi iden- tity J(r,-, rj,rk)=[1,-,[1:j, rkl]+[r],[r1,,1:,-]]+[1:k, [1,-,1]-]]= O. (B.l7) Forexample, thePauli matrices satisfy Eqs.(B.l6) and(B.17)withthestructure constants c,-j"=2ie,-Jk,where €[jkistheLevi—Civita density symbol. They form aLiealgebra. There isadistinction between theelements oftheLiegroup andtheelements oftheLiealgebra. Themanifold oftheLiegroup isnotconceptually identical withtheﬁatvector space oftheLiealgebra. Therelation between theLiegroup andtheassociated Liealgebra isexponential. TheLiealgebra isthelogarithm of theLiegroup, andconversely theLiegroup istheexponential oftheLiealgebra inthefollowing sense. Letambeamember oftheLiegroup, then am=A"Z1<”m*"<), (13.18) where rkisabasis vector oftheLiealgebra. Theequal signisinterpreted asa one-to-one uniqueness. Forinﬁnite dimensional Liegroups andalgebras, thesum inEq.(B.18) isreplaced byanintegral andmisreplaced byacontinuous index. Eachquantity 6","isthekthcomponent (along thebasisvector rk)ofavector 6", ofthealgebra associated withthemthelement oftheLiegroup. Thevector 9is saidtoparameterize theLiegroup andtheLiealgebra. Anexample ofthegroup-algebra relationship isprovided bytheSU(2) rep- resentation oftherotation group. Thealgebra basis vectors aretheunitary Pauli matrices Eq.(B.12)which satisfy Eq.(B.16)(cf.page 412)withthestructure con- stants given above. Forarotation through theangle 6about thedirection ofthe LieGroups andAlgebras 613 unitvector n,wehavetherotation matrix Q(6,n)where nisaunitvector 6. .6Q=Icos5+zn-o's1n-5. (B.19) Thiscanbewritten intheform ofEq.(B.l8) Q=elf<9/2>"'°1. (13.20) Thisfollows fromtheexpansion oftheexponential inapower series. Anexpan- sionofthescalar product n-0'=nxcrx +nyay +nzcrz (B.2l) enables ustoidentify %nk6withtheparameter 6,,/‘ofEq.(B.18)andtoidentify rk ascrk.Thematrices Qareafaithful representation ofSU(2). TheuseofSU(2) was introduced intoclassical dynamics longbefore quantum mechanics wasdevised. Itwasusedbecause SU(2) notation allows aﬁnite rotation tobedescribed in terms ofasingle angle andasingle direction vector (cf.Eq.(B.21)). Foramore extended discussion, seeSection 4.5ofthe2ndedition ofthistext. Another example oftheLiegroup-Lie algebra relationship istheHeisenberg algebra which, inonedimension, hasthethreeelements x,pandI,andthethree commutators [X,p]=I=ih/272‘, [X,I]=0 (B22) [p,I]=0. Anassociated Liesubgroup comprises theinﬁnite setofelements ei°’Pwhich transform awavefunction Ix>inthequantum-mechanical coordinate represen- tation asfollows: e""P|x>=|a+x>, (12.23) where aisarealconstant. Another Liesubgroup comprises theeiﬂxoperators which transform awavefunction |p>inthequantum-mechanical momentum representation inthefollowing manner: eiﬁxlp >=I5+P>. (B-24) where ,6isreal.Theoverall Heisenberg Liegroup isformed bygroup multiplica- tionofthecorresponding subgroup elements el°‘Pwithell”. Formostphysical theories, there exists anaction thatremains unchanged in value forcertain continuous chances inthedynamical variables. This isused in Chapters 1,7,8,10,and13toderive dynamical equations oftheLagrange and Hamiltonian approaches. Wecannowseethatthesetoftransformations ofthe dynamical variables thatleavetheaction integral unchanged formarepresentation oftheinvariance group (often aLiegroup) ofthatphysical theory. Appendix BGroups andAlgebras CLIFFORD ALGEBRAS ThethreePaulimatrices ck,theirthreecounterparts icrk,the2><2unitmatrix Iand thematrix iItogether formanother typeofalgebra called aClifford algebra. The lowest order Clifford algebra contains thetwoelements iandl.Ahigher order Clifford algebra isformed from the4x4Dirac matrices y,-andtheir products. They,-canbeexpressed asdirect products ofPauli matrices andtheunitmatrix I asfollows: _ 0 0'1 _ 0 G2 _ 0 0'3 OlB-laolB-laolI0 InaPauli matrix Clifford algebra formalism thescalar (A-B)andcross (AxB) products combine intoasingle operation ABcalled ageometric product: AB= A-B+AxB.Thecoordinate vector iswritten inthefonnr=x0'1+yo;+zo'3, sothePauli matrices actasbasis vectors. Aquantity (S,V|Vp, S,,)deﬁned inthis algebra, called amultivector, hasonescalar component S,three vector compo- nents Vx,Vy,VZ,three pseudovector components from VP,andonepseudoscalar component SP.Several examples ofmultivectors andmultivector transfomrations are: energy-momentum 4-vector (0,Olp,E/c) (B.26a) electromagnetic ﬁeldtensor (0,E|cB,0) (B.26b) space rotation (cos6/2,Olnsin6/2,0) (B.26c) specialLorentz transformation (Hr—1}/211/2, -—B[{y +1}/2]‘/210, 0) (B.26d) identity transformation (1,010,0) (B.26e) Theﬁrstfourexpressions constitute various ways ofcombining thenonzero parts ofthefourterms S,V,VP,andSpinpairs. Forexample, theelectromagnetic ﬁelds BandEcombine together inamultivector inwhich Eisthevector part, cBisthepseudovector part,andthescalar andpseudoscalar components arezero. NotethatEq.(B.26c) reduces to(B.26c) inthelimit6Q0.Inthisfonnalism the product oftwosuccessive individual rotations about different axesautomatically provides theaxisdirection nandangle 6oftheequivalent single rotation, infor- mation which cannot bereadily obtained from theusual rotation matrix product operation. Thisconvenient successive rotation technique involving theuseofhalf angles wasdescribed inSection 4.5ofthesecond edition ofthepresent text,and isomitted inthepresent thirdedition tomake room fornewmaterial. TheClifford algebra approach wasdeveloped byHestenes inhisNewFoundations forClassi- calMechanics where hecalled itgeometric algebra (seeselected bibliography). Group Theory Classification ofElementary Particles 615 GROUP THEORY CLASSIFICATION OFELEMENTARY PARTICLES Thepower ofgroup theory isdemonstrated bythesimple unitary group SU(n) classiﬁcation schemes ofelementary particles. Webrieﬂy discuss thisforbaryons. Asmall submultiplet containing Nbaryons isclassiﬁed intenns ofanSU(2) representation byitsisospin number Iwhere N=21+1. (B.27) Forexample I=1/2fortheneutron, proton pairnandp,andI=1forthe sigma triplet E‘,2°,and2+.Eachparticle islabeled byitsm1value, where for agiven Ithem1values have integer spacings intherange —I5m15I.When thenexthigher unitary group SU(3) isinvoked anewquantum number called strangeness, s,isadded, andvarious SU(2) submultiplets withdifferent svalues group together inthelarger irreducible representations l‘,-ofSU(3). Each baryon hasthree quarks called up(u),down (d)andstrange (s)foratotalof33=27 combinations (e.g., aproton hastheuudgrouping), andtheSU(3) group theory classiﬁcation divides these 27intothree irreducible representations F1,Pgand 1"10,with F3appearing twice, andtherespective dimensionalities ofl“,-addas follows IT|+ITg+ITg+lT1()=l+8+8-I-lO=27 (B28) Figure B.2presents aplotofsversus m1fortheparticles oftheground stateSU(3) octetPgwhich combines fourSU(2) submultiplets: (n,p,I=1/2),(A0,I=O), (>3-,2°,2+,1=1),and(B-,a°,1=1/2).Ahigherorderclassiﬁcation of thebaryons interms ofthespecial unitary group SU(4) takes intoaccount afourth quark ccalled charm, andgroups together SU(3) multiplets interms oftheirtotal charm values. Nowthere arefourtypes ofquarks, u,d,s,andc,corresponding S ml | it ls’ | -1 +1 §- A°2° 5+ -9- -2 -:0:4 :4I-1 0-1 FIGURE B.2 Plotofstrangeness (s)ontheordinate versus isotopic spin(m1)onthe abscissa. Thestrangeness ranges from -2to0while theisotopic spinranges from -1to +1.Horizontal lines ofconstant strangeness contain SU(2) submultiplets. Appendix BGroups andAlgebras "ZZZ ‘-"I" r—I+"I' 3+ dcc ucc -=++ "W 0*c uuc ++ + c A’2 uuc -93}A tr_, Itr -1 St;20 dd 20 2 tiff A-dad A Q.A WA++ 3“A wi‘ti '1IIIo ___ 5° tr (11) (b) FIGURE B.3 Twoofthe20-fold supermultiplets oftheSU(4) classiﬁcation ofbaryons. Charm (c)isplotted vertically andstrangeness (s)andisotopic spin(m1)areplotted onthe horizontal plane. (a)hastheuncharmed ground-state octet, F8ofFig.B.2atthebottom. (b) istheplotofanother supermultiplet ofSU(4). (SeePhys. Rev., D54, Part1,1996, p.100.) to43=64baryon quark combinations. Figure B.3ashows aplotofthe20-fold SU(4) supermultiplet formed byhorizontal groupings ofSU(3) multiplets, with each particle labeled byitsquark composition. Inthelowest level weﬁndthe ground stateuncharmed baryons ofFig.B.2,thatisbaryons which contain only combinations ofthequarks u,d,ands.Themiddle level contains singly charmed particles, thatisbaryons withonecandtwoordinary quarks, andtheupper layer contains doubly charmed particles suchasQ2;withthequark content scc.Figure B.3b shows another oftheSU(4) supermultiplets. These classiﬁcation schemes areofmore thanacademic interest because they provide selection mlesforpredicting elementary particle interactions, suchasthe conservation ofstrangeness forstrong andelectromagnetic interactions, butnot forweak interactions. Mesons, eachofwhich contains aquark plusanantiquark, alsoconform toclassiﬁcation schemes bythesimple unitary groups SU(n). Selected Bibliography TEXTBOOKS ONCLASSICAL MECHANICS V.I.Arnold, Mathematical Methods ofClassical Mechanics, (Berlin: Springer- Verlag, 1989). Averyadvanced treatment ofthesubject. A.Arya, Introduction toClassical Mechanics, (Upper Saddle River, NJ:Prentice Hall, 1998). Undergraduate text. V.Barger andM.Olsson, Classical Mechanics." AModern Perspective, (New York: McGraw Hill, 1995). Undergraduate textthatcontains some discussion ofchaotic dynamics andaunique section onNewtonian Cosmology. H.C.Corben andP.Stehle, Classical Mechanics, (New York: Dover, 1994). Reprinted asaclassic. A.L.Fetter andJ.D.Walecka, Theoretical Mechanics ofParticles andContinua, (New York: McGraw Hill, 1980). Hasextensive discussion ofcontinuous sys- tems. G.Fowles andG.Cassiday, Analytical Dynamics, (Ft.Worth, TX:Saunders, 1999). Undergraduate textwithmany computer problems. L.Hand andJ.Finch, Analytical Mechanics, (Cambridge, England: Cambridge University Press, 1998). Covers standard topics, including Chaos, atalevel similar tothatofthepresent text. D.Hestenes, NewFoundations forClassical Mechanics, (Dordrecht, TheNether- lands: Kluwer, 1999). Anunconventional approach toclassical mechanics written inthemathematical language ofgeometric algebra. 1thasmany keen insights onthesubject. J.V.Jose andE.J.Saletan, Classical Dynamics, AContemporary Approach, (Cambridge, England: Cambridge University Press, 1998). Agood overall cov- erage ofthesubject. Much ofthetheory isexpressed interms ofconﬁguration manifolds andtangent bundles. L.D.Landau andE.M.Lifshitz, Mechanics, Volume IofCoarse inTheoretical Physics, (Oxford, England: Pergamon, 1976). Averyeconomical andpeda- gogic approach toMechanics. Contains many popular partially worked out examples. J.B.Marion andS.T.Thornton, Classical Dynamics ofParticles andSystems, (Ft. Worth, TX:Saunders, 1995). Astandard undergraduate textrecently updated withsome chaotic dynamics inthecontext ofnonlinear oscillations. 617 Selected Bibliography F.Scheck, Mechanics: From Newton ’sLaws toDeterministic Chaos, (Berlin: Springer-Verlag, 1990). Thisrecent textbook onclassical mechanics includes achapter onthegeometric aspects ofmechanics which develops thetheory in thelanguage ofmanifolds. There isalsoachapter onchaos. LAGRANGIAN FORMULATION Chapters 1to3 D.Hestenes, NewFoundations forClassical Mechanics, opcit.Ithasmany good examples from astronomy. Thethree-body problem, together with theEuler andLagrange solutions, areexplained verywell. L.D.Landau andE.M.Lifshitz, Mechanics, Volume IofCourse inTheoretical Physics, opcit.Averypedagogic approach tomechanics. K.R.Symon, Mechanics, (Reading, MA: Addison Wesley, 1971). Discusses the restricted three-body problem. RIGID BODIES Chapters 4and5 T.L.Chow, Classical Mechanics, (New York: Wiley, 1995). Provides anexcellent treatment ofthespinning top. D.Hestenes, NewFoundations forClassical Mechanics, opcit.Provides newin- sights intorigid body motion, including thesymmetric top. L.D.Landau andE.M.Lifshitz, Mechanics, opcit.Agood, pedagogic discussion ofrigid bodies. SMALL OSCILLATIONS Chapter 6 L.D.Landau andE.M.Lifshitz, Mechanics, opcit.Treatment includes damped, unharmonic, andnonlinear oscillations. C.P.Poole, Jr.,H.A.Farach andR.J.Creswick, Superconducitivity (Boston, Academic Press, 1995). Chapter 13discusses Josephson junctions andtheir mechanical analogues. Selected Bibliography 619 RELATIVITY Chapter 7 C.W.Misner, K.S.Thome, andJ.A.Wheeler, Gravitation, (SanFrancisco; Free- man, 1973). Acomplete introduction tothegeometric notation asapplied to boththespecial theory andthegeneral theory ofrelativity. B.F.Schutz, AFirst Course inGeneral Relativity, (Cambridge, England: Cam- bridge University Press, 1985). Theﬁrstfourchapters introduce thereader to theformalism ofthespecial theory ofrelativity inafonn thatcancany over tothegeneral theory. E.F.Taylor andJ.A.Wheeler, Spacetime Physics, (San Francisco: Freeman, 1992). Anexcellent discussion emphasizing thephysical thoughts behind and thephysical processes ofrelativity. HAMILTONIAN FORMULATION Chapters 8to10 I.Percival andD.Richards, Introduction toDynamics, (Cambridge, England: Cambridge University Press, NY,1982). Good treatment ofcanonical trans- fonnations, Hamilton-Jacobi equation, andaction-angle variables. CHAOS Chapter 11 H.Bai-Lin, Chaos, (Singapore: World Scientiﬁc, 1984). Theﬁrstpartofthebook consists oftengood introductory chapters thatexplain chaos. Chapter 2de- velops thetheory ofchaos from Hamilton’s equations andChapter 3discusses thelogistic equation. Thesecond andmain partofthebook isacollection of 41reprinted papers, many ofthem landmark articles inthedevelopment ofthe subject. R.H.Enns andG.McGuire, Nonlinear Physics withMaple ﬁarScientists and Engineers, (Boston: Birkhauser, 2000). Agood selection ofproblems witha diskofMaple programs. M.Hénon, Numerical Exploration ofHamiltonian Systems, Course 2inChaotic Behavior ofDeterministic Systems, LesHouches Ecole D’Eté dePhysique Théoretique, 1981, ed.G.Iooss, R.H.G.Hellennan andR.Stora, (New York: North Holland, 1983). This 114-page lecture isveryreadable. Itprovides one ofthebestexplanations oftheHénon—Heiles Hamiltonian, andcovers several other topics included inthepresent chapter. E.A.Jackson, Perspectives ofNonlinear Dynamics, (Cambridge, England: Cam- bridge University Press, 1990). Thistwo-volume setprovides areadable pre- Selected Bibliography sentation ofavariety ofcomplementary approaches totopics innonlinear dynamics. There areuseful discussions ofseveral topics covered inthischapter suchastheHénon—Heiles Hamiltonian, thelogistic equation, Liapunov expo- nents andPoincaré maps. Anexplanation isgiven oftheKAM theorem. S.A.Kauffman, TheOrigins ofOrder, (New York: Oxford University Press, 1993). Anintroduction totheconcepts ofcomplexity. W.Kinzel and G.Reents, Physics byComputer, (New York: Springer, 1998). Numerical solutions oflinear andnonlinear problems using Mathe- matica andC. H.O.Peitgen, H.Jiirgens andD.Saupe, Chaos andFractals, New Frontiers of Science, (Berlin: Springer-Verlag, 1992). This volume constitutes oneofthe bestavailable sources forinformation onfractals, anditdoes agood jobof explaining thechaotic behavior ofthelogistic equation. Thetextisverylong andwordy, butitcontains many beautiful ﬁgures offractals andtrajectories of attractors. L.E.Reichl, TheTransition toChaos, (Berlin: Springer-Verlag, 1992). Thethe- oryisdeveloped from theviewpoint ofclassical mechanics, using, forexample, action-angle variables. There aregood discussions ofPoincaré sections, Lia- punov exponents, theHénon—Heiles Hamiltonian, andKolmogorov’s approach forproving theKAM theorem. PERTURBATION THEORY Chapter 12 I.Percival andD.Richards, Introduction toDynamics, opcit.Good treatment of perturbation theory. CONTINUOUS SYSTEMS AND FIELDS Chapter 13 B.S.Dewitt, Dynamical theory ofgroups andﬁelds, inB.S.Dewitt andC.Dewitt (eds.) Relativity, Groups andTopology, (New York: Gordon andBreach, 1964). Anexcellent fonnalistic approach totheuseofgroups inmodern ﬁeldtheory. A.L.Fetter andJ.D.Walecka, Theoretical Mechanics ofParticles andContinua, opcit.Hasanextensive discussion ofcontinuous systems. C.W.Misner, K.S.Thome, andJ.A.Wheeler, Gravitation, opcit.This con- tains anexcellent discussion oftheconcepts ofﬁeldtheory inaRiemannian spacetime. Thematerial ispresented inmultiple level tracks. Selected Bibliography 621 APPENDIX B H.Goldstein, Classical Mechanics (Reading, MA: Addison Wesley, 1sted., 1950, 2ndEd.,1980). Theﬁrstandsecond editions ofthepresent work havethorough explanations ofthecorrespondence between the2X2complex unitary matrices ofSU(2) andthe3x3realorthogonal matrices of0(3), including theCayley- Klein parameters. These discussions involve applications ofClifford algebras toclassical mechanics. M.Hamennesh, Group Theory, (Reading, MA: Addison Wesley, 1962). Agood coverage ofgroup theory anditsapplications tophysical problems. Liegroups andalgebras arediscussed inChapter 8. D.Hestenes, NewFoundations forClassical Mechanics, op.-"it,Aclassical me- chanics textbook withextensive sections written intheformalism ofClifford algebra, illustrating theinsights tobegained bythisapproach. C.P.Poole, Jr.andH.A.Farach, Pauli-Dirac Matrix Generators ofClifford Al- gebras, Found. Phys, 12,719-738 (1982). Thisarticle provides background on therolethatClifford algebras canplayinclassical mechanics. M.Tinkham, Group Theory andQuantum Mechanics, (New York, McGraw Hill, 1964). Awellwritten introduction togroups. Chapter 5covers therotation group andangular momentum. Abel, N.H.,608.SeealsoAbelian insubject index Arnold, V.1.,484,487,489,617 Arya, A.,617 Atwood, G.,27,28 Bai-Lin, H.,489,619 Baker, G.L.,507,508,519 Barger, V.,617 Barone, A.,269 Bemoulli, J.,43 Bertrand, J.,89 Bessel, F.W.,127 Binet, A,203 Bohlin, 549 Bohr, N.H.D.,95,466 Boltzmann, L.E.,85,128 Boyle, R.,128 Bryan, 154 Cambel, A.B.,519 Cantor, G.F.L.P.,516,517,519, 522 Cassiday, G.,617 Carathéodory, C.,394 Cayley, A.,154,182,603,621 Chandler, S.C.,208,228 Chasles, M.,161,184,228 Chow, T.L.,618 Clausius, R.J.E.,84,128 Clifford, W.K.,614,621 Corben, H.C.,617 Coriolis, G.G.,174,326 Coulomb, C.A.de,111 Cramer, G.,149,263 Creswick, R.J.,265, 517, 618Author Index D’Alembert, J.,16,18,296,313, 548. Seealso D’Alembertian insubject index Delaunay, C.E.,477 Descartes, R.,26.Seealso Cartesian insubject index deVries, C.,587,596 Dewitt, B.S.,620 Dirac, P.A.M.,588,621 Dufﬁng, G.,523,524 Einstein, A.,139,276,326,327, 332,538 Enns, R.H.,619 Euclid, 278, 517. Seealso Euclidean insubject index Euler, L.,45,122,150,155,165, 196,197,200,209,234, 319,564,617 Farach, H.A.,265,517,618,621 Faraday, M.,297, 298 Feigenbaum, M.J.,506-515 Fermat, P.de,360 Fetter, A.L.,617,620 Finch, J.,617 Foucault, J.B.L.,179 Fourier, Baron J.B.J.,14,126, 259,274,460,545,551, 575 Fowles, G.,617 Galilei, G.,2.SeealsoGalilean in subject index Gibbs, J.W.,337Goldschimdt, 64 Goldstein, H.,621 Gollub, J.P.,507,508,519 Gordon, W.,584,585,596 Gram, 249 Hamermesh, M.,621 Hamilton, SirW.R.,34,44,45, 313,324,334,430,479, 488,562.Seealso Hamiltonian insubject index Hand, L.,617 Hausdorff, F.,517 Helmholtz, H.L.F.von,337 Hénon, M.,484,492,496,497, 619,621 Heiles, C.,484,492,496,497, 619,621 Helleman, R.H.G.,497, 619 Hertz, H.R.,361 Hestenes, D.,121.617, 618,621 Hooke, R.,52,91,317 Huygens, C.,132 Iooss, G.,497, 619 Jackson, E.A.,489,619 Jacobi, K.G.J.,61,334,338,361 390,398,426,430,479, 488,566,597 Jose, J.V.,617 Josephson, B.D.,265,271, 618 Jiirgens, H.,620 623 624 Kauffman, S.A.,620 Kelvin, Baron W.T.,85,338 Kinzel, I.W.,620 K1ein,F., 154,182,218,228,584, 585,596, 603,621 Kepler, J.,73,92,101,347,370, 414,445,466,470,484, 495,537 Kirchhoff, G.R.,66 Kolmogorov, A.N.,484, 487,489, 621 Korteweg, D.J.,587,596,600 Kronecker, L.,138,190 Lagrange, J.L.,14,123,198,200, 319,392,533, 561,564, 618.SeealsoLagrangian insubject index Landau, L.D.,617,618 Laplace, P.S.,marquis de102, 264,413 Larmor, SirJ.,231,318 Legendre, A.M.,224,334,539 Lenz, H.F.E.,102-104, 413 Levi—Civita, T.,169,410 Liapunov, M.A.,484,492,512, 519,621 Lie,M.S.,171,385,392,411, 613,621 Lifshitz, E.M.,617,61.8 Liouville, J.,418,483 Lissajous J.A.,83,258,439,358, 462,464 Lorentz, H.A.,22,131.,280,319, 328,415,580,612 Lorenz, E.N.,523,576 MacCu1lagh, 225 Mach, E.,324 Marion, J.B.,617 Maxwell, J.C.,54Author Index McGuire, G.,619 Minkowski, H.,278,287,290, 319,576,580,601 Misner, C.W.,619,620 Moser, J.,484,487,489 Napier, J.N.,476 Newton, SirI.,1,5,101,132,199, 299,526.Seealso Newtonian insubject index Nielsen, A.C.,30 Noether, A.E,344,566,589,597 Olsson, M.,617 Patemo, G.,269 Pauli, W.,610,614,621 Peitgen, H.O.,512, 513, 515, 620 Percival, I.,‘619,620 Planck, M.K.E.L.,380 Poincaré, J.H.,282,394,494, 524,545,621 Poinsot, L.,201,202,206,234 Poisson, S.D.,225,388,398,411, 532.Seealsosubject index Poole, C.P.,Jr.,265, 517,618,621 Ptolemy, C.,129 Raman, SirC.V.,258 Ray,J.,47 Rayleigh, Baron R.J.S.,23 Reents, G.,620 Reichl, L.E.,489,620 Ricci-Curbastro, G,327 Richards, D.,619,620 Riemann, G.F.B.,326,327,566, 620 Rossler, O.E.,523 Routh, E.J.,56,347 Runge, C.,102-104, 413 Rutherford, Baron E.,110,113Sa1etan,E. J.,617 Sander, L.N.,524 Saupe, D.,620 Scheck, F.,618 Schrodinger, E.,54,571,584,599 Schutz, B.F.,619 Schmidt, 249 Schwarzschild, K.,538 Shaw, R.,512 Sierpinski, W.,517,519,522 Sommerfeld, A.J.W.,218,228 Staeckel, 447 Stehle, P,617 Stokes, SirG.G.,20,24,52 Stora, R.,497 Symon, K.R.,618 Tait,P.G.,154 Taylor, B.,239 Taylor, E.F.,619 Thomas, L.H.,282,330 Thomson, SirW.,seeBaron (Lord) Kelvin Thorne, K.S.,619,620 Thornton, S.T.,617 Tinkham, M.,621 vanderPol,B.,490,491 Vinti, J.,463 Walecka, J.D.,617,620 Weber, W.E.,367 Wheatstone, SirC.,66 Wheeler, J.A.,324,619,620 Witten, I.H.,524 Young, T.,559 Zeeman, P,232 1-form, 289 charge, current, 295 covariant vector, 290 deﬁnition, 290 energy, momentum, 295 ﬁgure, 290 table, 290 4-vector energy, momentum, 295,300, 301 photon momentum, 304 table, 287 velocity, 286-288 4-velocity, 286-288 Abbreviated action, 354,434 Abelian group, 606 Acceleration, centripetal, 29,80 Acoustics, 53,237.239 Action, 356 abbreviated, 359,434 andreaction, 7 strong law,7,10 weak law,5 atadistance, 323,583 integral, 359,596 variable, 452 integral overorbit, 458 Action-angle variable, 430, 452-478, 619 celestial mechanics, 456 chaos, 485 completely separable, 457-466 degeneracy, 73,464,468 harmonic oscillator, 456 Kepler problem, 466-478 onedegree offreedom, 452-457 periodic motion, 452Subject Index perturbation, 541 proper variables, 481 Adiabatic invariance, 549-555 Algebra, 611 Clifford, 614,621 geometric, 617 Heisenberg, 613 Lie,611,612.SeealsoLie algebra Analogy, structural, 54 Analytical mechanics, 1 Angle variable, 455 Fourier expansion, 460 libration, 460 multiply periodic, 460 quasi-periodic, 461 rotation, 461 timedependence, 454,458, 460 Angular momentum 4-vector, 310 areal velocity, 73 canonical, 405 central force problem, 72 conservation, 3,72,73,571 total,7 deﬁnition, 2 density, total, 571 eigenvalue, 411 electromagnetic, 8 ellipsoid, 203 mechanical. 8,405 Poisson bracket, 408,411 relativistic, 309 rigidbody,rss spherical symmetry, 72 spin, 10 total,8Angular velocity inEuler angles, 602,615 Anharmonic oscillator, 545 Anomalistic year,131 Anomaly eccentric, 100 mean, 102 true,540 Antiproton, 304 Antiquark, 616 Aphelion, 484 Approximation, semiclassical, 115 Apsidal distance, 78,95,96 vector, 86 Areal velocity, 73 Ascending node, 472 Astronomy, medieval, 100 Attitude angle, 154 Attractor, 489,516,620 regular, 493 strange, 489,492,500 strange, Hénon-Heiles 500 Atwood’s machine, 27,28 Axis rigid body, 135 screw, 161 semimajor, 95,475 semiminor, 101 symmetry, 161 Azimuth, 209 Backward glory, 114 Bank angle, 154,603 Barrier, centrifugal, 112 Baryon. 615,616 Basis vector, 286 Bertrand’s theorem, 89,92 625 626 Bessel function, 126 Biform, 296 Bifurcation, 454,484.505.513, 514 diagram, 506,508,513,515 Bilinear, 219,388 form, 194 Binet ellipsoid, 203,204 Biot-Savart law,7 Bivector, 296 Black box, 121 Bohr quantum mechanics, 466 theory, 95 Boltzmann constant, 185 factor, 128 Boost. 280.SeealsoLorentz transformation Bounded motion. 80,484 Boyle law,128 andvirial, 84 Brachistochrone, 42,63 Calculus operational, 275 ofvariations, 36,43 fundamental lemma, 38 Canonical, 338 equations ofHamiltonian, 338 extended transformation, 371 invariant, 388 momentum, 55,314 relativistic, 322,323 perturbation theory, see Perturbation theory restricted transformation, 371 variables, 335,377 Canonical transformation, 348, 368-421, 619 active andpassive, 400,405 cyclic Hamiltonian, 369,377, 399,430,441 degeneracy, 464,470 equations, 368-375 examples, 375-377 explicit timedependence, 385, 397,402 generated byHamiltonian, 420Subject Index generating function, 373 group, 387 harmonic oscillator, 377-381 inﬁnitesimal, 385,402 invariant, phase space volume, 393,420 Poisson bracket, 389 Jacobi matrix, 382,394 parametric, 385,405,408 restricted, 371,381,382,387 symplectic, 381-388 tableof,373 Cantor set,516.519,522 Capacitance, 271 Carathéodory theorem, 394 Carousel, 183 Cartesian coordinates. 25,141 Catenary, 41.42,64 Cayley—Klein parameters, 154, 182.601,602 Celestial mechanics, 533 Center of energy, 312 force, 106 gravity, 185 mass, 5,6,185,312 momentum. 301,312 system, 301 Central force problem, 70-126. SeealsoKepler problem Centrifugal barrier, 112 effect, 126 Centripetal acceleration. 29,175 Chain rule, 18 Chandler wobble. 208,228 Chaos, 483-522, 617,619 attractor, 489-491 bifurcation, 505-509 damped harmonic oscillator, 505-509 dimensionality, 616-522 fractals, 516-522 Hénon-Heiles, 496-503, 506 Islands. 503-505 KAM theorem, 487-489 logistic equation, 509-516 motion, 491 onset, 492,501,503parametric oscillator, 508 resonance, 509 perturbation theory, 487-489 properties of,491 trajectory, 491,494,521,522 Character table, 611 Characteristic equation, 157 value, 156 Charge density, 588 Charged particle in electromagnetic ﬁeld, 23, 317,553 Charm, 615,616 Chasles’ theorem, 161,184 Class ofgroup, 607 Classical mechanics, I-600 Clifford algebra, 614,621 Closed orbit, 89.452 Colliding beam, 304 Collision elastic, 118,120,306 inelastic, 118 C-O-M, center ofrnomerilurn, 301 Commensurability, 463 condition, 464 Commensurate, 105,463 completely, 464 condition, 464 frequency, 462 m-fold. 464 Commutator, 171,411 quantum mechanics, 392,398 relations, 170 Conﬁguration space, 34,357 pointtransformation, 370 variation, 36 Congruence transformation, 245, 246,252 Conic section, 94.99 Conjugate momentum, 55,335, 351 Conservation differential theorem, 594 energy function, 62 momentum, 403 Conservation theorems, 7,55,72, 343,597 angular momentum, 3,344 total, 7 canonical momentum, 315,340 energy, 4,11,345,450 linear momentum, 2,6,344 system ofparticles, 6 Noether’s theorem, 589 Poisson bracket. 396,402 relation tosymmetry properties, 54-59 Conservative system, 4 Conserved current, 594,595 Constant ofmotion, 105,397,402, 403,415 algebraic, 418 central force, 105 Jacobi identity, 397,411 Poisson bracket, 398 Constraint, 12-16. 24 differential, 16 equation, 15 holonomic, 12 nonholonomic, 12 nonintegral, 16 rheonomous, 13 rigidbody, 12 rolling, 182 schleronomous. 13,25 semiholonomic, 46,49 virtual work, 16,17,48 weak, 321 Continuity conditions, 572 equation, 595 Continuous system. 265,558,568 Hamiltonian formulation, 572-577 Lagrangian density, 561-566 stress energy tensor, 566-572 transition fromdiscrete to continuous, 558-561 Contour integration, 469 Contraction, 290,295 oftensor. 191 Contravariant, 289 Control parameter, 503,506 logistic equation, 510 Coordinate basis, 286Subject Index Cartesian, 184 contraction, 295 cyclic, 55,343,369,445 generalized, 13,19,239 intemal, 272 mass weighted, 241,258 normal, 251,257,259 polar, 72 pseudo-Cartesian, 294 rotating, 175 Coriolis. 174-179 acceleration, 176 circulation ofﬂuiddynamics, 177 deﬂection, 176-178, 182 effect, 126,174-179, 326 onmeteorological phenomena, 177 force, 175 Foucault pendulum, 179 hemisphere, 178 pressure gradient, 176 Correspondence principle, 325, 390,392,398 Poisson bracket, 388,391.392, 398 Cosmological constant, 328 Cosmology, 617 Coulomb ﬁeld, 109,lll law,274 scattering, 110 Coupled electrical circuits, 53 Covariant deﬁnition, 277 equation, 297 Hamiltonian, 349,352 Lagrangian, 318,321,322,350, 352 principle, 325 relativistic, 577 vector. 289 Cramer’s rule, 149,263 Cross section highenergy limit, 127 Rutherford, 110 total. 110 Crossing theT,8627 Current conserved, 594.595 density, 588 elastic rod,567 ﬁeldﬂow, 568,571,594 ﬂow, RLcircuit, 51 Curvature scalar, 327 Cyclic coordinate, 55,343,369 Kepler problem, 445 group. 606 Cyclotron frequency, 318,553 resonance. 318 5-function, 588 8-variation, 38 5,-JKronecker delta, 138,181.190 A-variation, 357-359 D’Alembert characteristic. 548 principle, 16-20, 46,313 D’Alembertian, 296 Damping, 519 exponential, 262 vanderPolequation. 490 Deﬂection angle, scattering, 113 Degeneracy, 244,465 conditions, 465 exact, 547 Kepler problem, 470,484 proper, 547 vibrational modes and frequencies, 257 Degrees offreedom, 13.245,255, 342,427,541,549,563 Hamiltonian. 342 many, 457 molecular vibrations, 256 nparticles, 13 oscillator, 264 rigid body, 135 vibration, 257 Delaunay variables, 477 Delta 8-function, 588 5-variation, 38 Kronecker (6,7), 138,181,190 Dense quasi-periodic orbits. 491 628 Derivative, functional, 574 Deterministic, 483 Differential equation, inhomogeneous, 259 Diffusion, 524 Dihedral group, 606,607 Dilation oftime, 279 Dimension Cantor set,516 fractal, 516,517 Hausdorff, S17 Dipole moment gravitational, 226 magnetic, 185.230 Dirac 5-function, 588 Direction cosines, 136 orthogonality, 138 transformation, 139 Dissipation exponential damping, 263 forces, 259 function, 22-24, 53,63,261 Rayleigh, 23 Disturbing function, 533 Divergence, 295 4-divergence, 296 relativistic, 565 theorem, fourdimensional, 581, 593 Divergenceless, 581 Doppler effect, 329 Drag force, 24,52 Dualspace, 292 Dufﬁng oscillator, 523 inverted, 524 Dumbbell molecule, 347 Dynamic steady state, 267 e,1-kLevi—Civita density, permutation symbol, 169, 410 Earth equatorial bulge, 223 ﬁgure axis, 226 Lagrangian forprecession, 227 potential, 226.227 precession, 226 spinning ring, 229 torques, 237Subject Index Earth-Moon system, 124 Eccentric anomaly, 99 Eccentricity. 94,95,532 SunandMoon, 227 Ecliptic, 208,228 Eigenvalue, 156-158 angular momentum, 411 equation, 157 oscillations, 241 Euler’s theorem, 157 inertia tensor, 195,196 linear triatomic molecule, 254 problem, 157 transformation matrix, 160 Eigenvector, 247 indeterminacy, 258 inertia tensor, 196 linear combination, 248 orthononnal, 249 oscillations, 24-4 Eigenwerte (German for eigenvalue), 156 Einstein ﬁeldequations. 327,538 summation convention, 139 tensor, 327 velocity addition law,283,328 Elastic collision, 118,120,306 scattering, 120 solid, 563 wave, 560 Electric circuit equation, 264 Lagrangian, 53 Electromagnetic ﬁeld, 31,51,55,275, 571,587 Lagrangian, 350 Lagrangian, covariant, 352 potential, 342 radiation, 54 theory, 276 Elementary particle, 51.54.300, 615 Ellipse, 81 ﬁgure, 96 harmonic oscillator, 377 orbit equation, 484 phase space plot,98properties, 97 semimajor axis,95,475 shape, scale, orientation, 105 473 table, 97 Ellipsoid Binet, 203,204 inertia, 196,201 kinetic energy, 204,258 rigidbody, 185-188 moment ofinertia, 197 Ellipsoidal coordinates for Hamilton-Jacobi equation, 479 Elliptic function, 89 integral, 234 region forchaos, 504 Elsewhen, 279 Elsewhere, 279 Energy center of,312 conservation, 60 central force, 74,77 free,Gibbs, 337 free,Helmholtz, 337 function, 60-63, 314 conserved, 61-63 hypersurface, 494 potential, 4 Ensemble, 419 microcanonical, 421 Enthalpy, 336 Equant, 129 Equation ofmotion, 74 ofstate, gas,85 Equilateral triangle group, 609 Equilibrium generalized forces, 238 indifferent, 240 neutral, 240 stable, 238 statistical, 421 unstable, 239 Equinox, 539 precession, 223-230 Equipotential curve gravitation, 125 Hénon-Heiles, 498 Equivalence principle, 324,346 Ergotic hypothesis. 418 Escape velocity, 31 Ether, 566 Euclidean dimension, 518,519 space, 517 Euler equations, 198,199,234 derived from Lagrange’s equations, 200 heavy symmetrical top,210 symmetric body, 205 inhomogeneous function, 86 parameters, 155,182,602,603 solution ofthreebodyproblem. 122 theorem, 155-161 homogeneous functions, 320 Euler angles, 150-154, 196,601 angular velocity, 602,615 conventions, 154 ﬁgure, 152 inﬁnitesimal, 165 lefthanded, 152 SU(2) rotation, 412 timechanges, 210 x-convention, 154,601 xyz-convention, 154,603 y-convention, 154,601 Euler-Lagrange, 564 complex scalar ﬁeld, 583 electromagnetic ﬁeld, 587 equation, 45,64,65,319,354 relativistic equation, 564,588 Event, 279,311 Extremum path,40 problem, 39 surface area, 40 Faraday tensor, 297,298 FeigenbaumSubject Index Fermat’s principle, 360 Field canonical equations, 574 classical theory, 571 complex, 596 scalar, 583 deﬁnition, 566 elastic, 51 electromagnetic, 31,51,55, 275,571,587 elementary particle, 51 equation, Lagrange-Euler, 583 gravitational, 176,185,210,275 relativistic, 571 scalar 287,583 meson, 571,599 theory, 558-589 Hamiltonian formulation, 571-577 Noether’s theorem, 589-598 relativistic, 583-598 Schrodinger quantum theory, 576 spacetime, 566 vector, 286 velocity. 588 wave function, 571 Figure axis, 539 Fission, 120 Fluid dynamics, 419 perfect, 579 Fluxdensity, 107 Force central, 7,70 centrifugal, 176 cutoff, 111 driving, 259 effective, 80,94,175 electromagnetic, 259 external, 5 generalized, 19,21,58,238 gradient ofpotential, 10 gravitational, 93 diagram, logistic equation, 510, inertial, 5 513-515 number, 514 plot,506 point, 511intemal, 5,ll inverse square, 77,92 linear restoring, 83 longrange, 110629 Lorentz, 22,131,237, 317.350 Minkowski, 299,322 relativistic, 297 reversed effective, 18,80 strong, 299 weak, 299 Foucault pendulum, 179,183,184 Four-vector, see4-vector Four-velocity, see4-velocity Fourier series, 14,126,574 convergence, 545 multiple, 460 transform, 274 Fractal, 516,620 area, 521 dimension, 490 Sierpinski carpet, 519 geometry, 491 self-similarity, 505,514 Freeenergy Gibbs, 337 Helmholtz, 337 Frequency characteristic, 266 commensurate, 106 critical, 266 cyclotron, 318,553 driving, 490 imaginary, 244 Larmor, 231 resonant, 490 Friction, 24 atmosphere, 32 drag,24 electrical, 52 oscillating system, 262 rolling, 17 Functional, 287,293 derivative, 574,575 Future, 279 Galactic center, 496 Galaxy model, Hénon-Heiles, 496, 497,516 Galilean system, 2 transformation, 276-280 Gas,equation ofstate, 85 630 Gauge transformation, 595 general, 619 General relativity, 324 Generalized force 19,21,57, 58 mechanics, 65 Generating function, 371,372 canonical transformations, table of,373 chaos, 488 inﬁnitesimal canonical transformation (I.C.T.), 403 rotation, 404 Poisson bracket, 404,406 symplectic, 394 table, 373 Geodesic 40.324-326, 362 deviation, 325 Geoid, 176 Geostrophic wind, 178 Gibbs freeenergy, 337 Glory scattering, 114 Goldschmidt solution, 64,65 Gradient, 295 Gram—Schrnidt method, 249 Gravitational charge, 226 ﬁeld, 176.185,210,275 quadrupole moment, 226 Greek subscript convention. 286 Group abelian, 606 canonical transformation, 387 class, 607 conjugation, 606 cyclic, 606 deﬁnition, 605 dihedral, 606,607 generator, 606 Lorentz, 282,610 multiplication table, 606 properties, 387,605-611 quatemion, 610 representation, 608 rotation, 171 symmetry, 412 forsystem, 413Subject Index symplectic, 387,612 theory, 605 Gyration, radius of,198 Gyrocompass, 223 Gyromagnetic ratio, 230 Gyroscope inertia, 222 torque freemounting, 213 Hamilton’s principle, 34-36, 44-50, 313,324,355,562, 564 Lagrange’s equations derivation, 44,45 modiﬁed, 354.355,599 nonholonomic systems, 45-50 Hamiltonian, 334-353 astotalenergy, 339 covariant, 349,352 degrees offreedom, 342 density, 573.586 fonnulation continuous systems, 572 relativistic mechanics, 349 generates canonical transformation, 420 generator ofsystem motion, 399 Hénon-Heiles, 492,497,522 perturbation, 526 quantum mechanics, 613 symplectic, 576 Hamiltonian formulation, 334-363 advantages, 51-54 characteristic function, 434, 440-444 comparison ofcharacteristic and principal functions, 442-443 conservation theorems, 347-349 cyclic coordinates, 343-349 Hamilton equations ofmotion, 334-363. 368,397,402 derived from variational principle, 353 leastaction principle, 356-363 Legendre transformation derivation, 334-342principal function, 430-434, 433,528 compared withcharacteristic function, 442 relativistic formulation, 349-353 Routh procedure, 347-349 symplectic approach, 339-343 variational principle derivation, 353-356 Hamilton-Jacobi theory and equation, 334,430-451, 488,528, 549,619 central force, 448 chaos, 485 completely separable. 444 cyclic coordinates, 445-451 ellipsoidal coordinates, 479 harmonic oscillator, 434-439 method, 434-439 newconstant coordinates, 432 Kepler problem, 445-451 spherical coordinates, 451 separation ofvariables, 444-445 twomethods ofsolution, 442 Handedness convention, 169 Harmonic oscillator, 434-440, 485 action-angle variables, 455,456 485 adiabatic invariant, 550 canonical transformation, 377 constants ofmotion, 417 coordinate space plot,440 damped, 269 driven. 505,507 ellipse, 377 Feigenbaum plot,508 Hamilton-Jacobi, 434-440 isotropic, 82 threedimensional, 275 perturbation, 529,542 phase diagram, 380 Poisson brackets, 417 relativistic, 316 twodimensional, 415,416 anisotropic, 437 Heading angle, 154 Heisenberg algebra, 613 picture, 408 Helmholtz freeenergy, 337 Hénon-Heiles chaos, 484 equipotentials, 498 galaxy model, 516 Hamilton equations, 497 Hamiltonian, 492,496,497,522 islands inchaos, 502 Poincaré map.499-501 potential, 497 Hermitean matrix. 412 Herpolhode, 202,203 Hertz principle ofleastctuvature, 361 Hierarchy ofislands, 504,505 Highenergy physics, 300 Hodograph, 131 Holonomic, 12 constraint, 12 system, 199 Homogeneous function, 320 problem, 320,359 Homomorphisrn. 418,609 Hooke’s law,52,92,317,559 Hoop rolling, 50 vertical, 66 Huygens’ waves, 132 Hydrodynamic derivative, 419 Hyperbola, 81,316 Hyperbolic motion, 315 point, 504 region, 504 Hypersurface, 580 energy, 494 spacelike, 580 Hypocycloid, 64 Hysteresis, 270.271,523 I.C.T. (inﬁnitesimal canonical transformation), 385,386, 402,403,408,410,413 Identity transformation, 146,156 Ignorable, seecyclicSubject Index Imbedding inchaos, 514-516 Impact parameter. 107 Inclination, 532 Incomrnensurate, 548 frequency, pe1iod,462, 489,548 oscillator, 521 Inelastic collision, 118 Inertia ellipsoid, 197,201 tensor, 191 components, 195 diagonal, 196 eigenvalue, 195,196 eigenvector, 196 integral, 194 principal axes, 196 principal moments, 197 properties, 195 similarity transformation, 196 Inertial force, 5 system, deﬁnition, 2 Inﬁnitesimal canonical transformation, 385. 386,396,398,399,401 rotation, 163,166 Infrared spectroscopy, 258 Instability, 205 Integrability breakdown. S02 Integral invariants ofPoincaré, 394 Jacobi, 61 line,35 variation, 44 Integrating factor, 15 Invariable plane, 202 Invariance adiabatic, 549 condition, 594 group, 613 logistic equation, 484 Lorentz, 302 Poisson bracket, 388 rotation, 60 scale, 591 translation, 60 Inversion, 150,181 Islands inchaos, 502,503631 hierarchy, 504,505 various orders, 504 Isomorphism, 609 J-matrix, 342.382-389, 393 Poisson bracket. 388 Jabberwocky, 202 Jacobi determinant, 394 formofleastaction principle, 361 identity, 393,398.424,428 integral 61,566,597 Lagrange brackets, 424 Poisson bracket, 390 matrix ofcanonical transformation, 426 Josephson junction, 265,271,618 KAM (Kolmogorov-Arnold- Moser) theorem, 484, 487-492 Kamiltonian, 370 Kepler equation, 102,126,131 second law,73 third law,101,470 Kepler problem, inverse square lawpotential, 70-126, 347, 415 action variables, 471 action-angle variables, 466 closed orbits, conditions, 89-92 cyclic coordinate, 445 equations ofmotion, 72-76 equivalent onebody problem, 70-71 equivalent onedimensional problem, 76-83 inverse square law,92-96 Liealgebra, 414 motion intime,96 orbitequation, 86-89, 96-103 perturbation, 536 Poincaré map, 495,496 scattering, 106-121 spherical polarcoordinates, 467 symmetry group, 414 virial theorem, 472 632 Kinematics rigid body, 134,184 tools, 184 Kinetic energy ellipsoid, 203 rigid body, 184 rotational, 191 total,9 Kinetic theory, 85,112 Kirchhoff junction conditions, 66 Klein-Gordon equation, 585 ﬁeld, 585 particle, 596 Kolmogorov-Arnold-Moser (KAM) theorem, 484, 487-492 Korteweg—deVries equation, 596, 600 Kronecker delta (8,-J),138,181, 190 Laboratory frame, 302 system, transformation, 306 time, 279 Lagrange bracket, 392-394 fundamental, 393 calculus ofvariations, 36 equations, 16,21-23 derivation from I-lami1ton’s principle, 44,45 Euler equation derivation, 200 Nielsen form, 30 permrbation, 533 multipliers, 16,67 point, 124 solution ofthreebodyproblem, 123 undetermined multiplier, 198 Lagrangian applications, 24-29 central force, 71 conserved quantities, 566 covariant, 318,321,322, 352 deﬁnition, 21Subject Index density, 564,567,583 continuous system, 561-566 discrete system, 558-560 electromagnetic ﬁeld, 350 formulation versus Newtonian, 199 fromHamilton’s principle, 44 heavy symmetrical top,208 precession ofEarth, 227 relativistic, 312 rigid body, 185,199 separable, 185 Laplace transform, 264 Laplace-Runge—Lenz vector, 102-106, 131,429 Larmor frequency, 231 precession, 318 theorem. 232 LCcircuit, 51 Least action principle, 356,362 A—variation, 359 Jacobi form, 361 restrictions, 358 Legendre polynomial, 539 polynomial generating function, 224 twofold cycle, 510 transformation, 334,335,375, Longitude ofascending node, 474 549 Lorentz, 282 Levi—Civita density, (e,-J-1,) 169, boost, 284 410 condition, 297 force, 22,131,237,350 frame, 580 group, 282,610 invariance, 302,577 tenconstraints, 282 transformation, 280-265 boost, 282 equations forct’andr’, 281group, 411,412, 611-613 subgroup, 613 Light cone, 279,280 Lightlike, 278,304 Limit cycle, 489 ﬁgure, 491 vanderPolequation, 491 Lineofnodes, 150,473 Linear momentum, 1 particle, 1 system ofparticles, 6 total,6 Liouville theorem, 418-421, 428, 483 Lissajous ﬁgure, 83,258,439, 458,462 noncommensurate, 464 sketch, 440,463 Ljapunov, seeLiapunov Logistic equation, 509,620 control parameter, 510 Feigenbaum diagram, 510, 513-515 fourfold cycle, 510 iterations, 510 Liapunov exponent, 512,514 self-similarity, 514 Liapunov exponent, 491,519 damped pendulum, 519 diagram, 520 dimension, 521 logistic equation, 514,519 negative, 492 Sierpinski carpet, 519 solar system, 494 Libration, 452,455,460 Lie algebra, 171,412-415, 611-613 deﬁnition, 412 Kepler problem, 414 Poisson bracket, 392 structure constant, 413,612 scattering, 306 bracket, 171 Lorenz equations, 523 relations, 415 Lyapunov, seeLiapunovgeneral matrix, 281 homogeneous, 282 inhomogeneous. 282,610 invariance, 302 pure, 284 M—matrix, 382-389, 394 MacCullagh formula, 225 Mach’s principle, 324 Magnetic ﬁeld charge particle motion, 23, 317 uniform, 409 moment, 230 rigidity, 318 Manifold, 576,611,618 Mapping, 287 quadratic, 503 Mass center of.312 reduced, 71 weighted coordinates, 241 Matrix addition, 145 antisymmetric, 148,165 cofactor, 340 determinant, 159 hermitean. 412 inﬁnitesimal element, 164 inverse, 147 J-,342,383-389 M~,382-389, 394 multiplication, 144 orthogonal, 147 reciprocal, 147 rectangular, 147 skewsymmetric. 148 transpose, 147 unitary, 412 Maxwe11’s equations, 54.276, 297,350 covariant form, 298 Mean anomaly. 102 Mechanics, seeClassical mechanics Merry-go-round, 183 Meson, 331,616 scalar, 571,599 Metric Minkowski space, 287,580 matrix, 287 tensor, 327 MeV, deﬁnition, 32 Microcanonical ensemble, 421Subject Index Million electron volt,deﬁnition. 32 Minimum gravitational coupling principle, 325 surface ofrevolution, 40 Minkowski coordinate, 288 force, 299,322 space, 278,580 twodimensional, 287 Mixing, 516 property ofchaos, 491 Mode, normal, 252 Moderator, 120 Molecule internal coordinates, 272 linear triatomic, 272 pentatomic, 272 polyatomic, 258,259 rotation andvibration, 180 triatomic, 275 vibrating, 253,258 linear polyatomic, 558 Moment offorce, deﬁnition, 2 ofinertia, 191 about axisofrotation, 192 choice oforigin, 193 coefﬁcients, 187 ellipsoid, 197 integral, 194 operator, 188 parallel axes, 193,194 Momentum angular, 187,344 canonical, 55,314 center of,312 conjugate, 55,335,351 conservation, 403 density, 569,573,579 electromagnetic, 55 generalized, 55 linear, 1,6.24,344 representation, 576,598 Monochromatic light, 259 Monogenic, 34 Monopole. magnetic, 131,427633 Motion bounded, 80,484 chaotic, 491-493 equation, 74 hyperbolic, 315 periodic, 484 Multiplet. 615,616 Multiply periodic, 458,461 Multivector, 614 Napier’s rules, 476 Network, electrical, 264 Neutron scattering, 120 Newtonian equations ofmotion, 199 formulation versus Lagrangian, 199 mechanical corpuscles, 132 second law.1,299 third law,5 Nielsen form ofLagrange’s equations, 30 No-interaction theorem, 324, 353 Node ascending. 472 lineof,150,473 Noether’s theorem, 344,566,589, 594 conditions, 590 conserved current, 594 conserved quantities, 418 discrete, 596,597 statement of,594,595, 597 symmetry properties, 598 Non-Euclidean, 278 Nonabelian group, 606 Noncommensurate, 464 Nonholonomic system, 45 Noninertial system. 175 Normal behavior inchaos, 515 coordinates, 250,251 modes, 252,256 Number theory theorem, 463 Nutation, 215 heavy symmetrical top,209, 214 634 Subject Index 0(3)group, 610 Josephson junction, 271 Perturbation, 487 Oblateness Earth, 229 Moon, 229 Occupation number, 253 Onedimensional problem, equivalent, 76 One-form, seel-form Operational calculus, 275 Optics geometric, 112 meteorological, 114 Orbit bounded, 80 chaotic. 522 circular, 80,81,94 closed, 452 conditions for,89 commensurate, 106 degenerate, 106 elliptic, 94.95,484 equation, 99 ofstate, 86 integration, 93 hyperbolic, 94,110 inclination, 474 open, 452 osculating, 531 parabolic, 94 phase space, 452 quasi-periodic, 490 reﬂection symmetry. 87 regular, 522 satellite, 229 shape, scale, orientation, 105, 473 stable, 90 unbounded, 79 unstable, 90 Orbiting, 113 Orthogonal matrix, 147 transformation, 139 Orthogonality condition, 140 Oscillation, 238-265 eigenvalue equation, 241-249 forced, 259-265 freevibration frequencies, 249-253normal coordinates, 249-253 pendulum, damped anddriven, 265-27 1 potential expansion, 238-241 principal axistransformation, 241-249 triatomic molecule, 253-259 Oscillator anharmonic, 545 double, 486 parametric, 508 Parabola, 81,94.128 Parametric resonance, 505,508, 509 Parity, 590 Past,279 Pauli matrices, 412,612,614 Pendulum damped driven, 265 double, 14 equation, 267 hysteresis, 270 periodicity, 453 perturbation, 533action-angle variables, 541 adiabatic invariance, 549-555 degeneracy, 547,548 fastvariable, 547 ﬁrstorder, 530,534,537 Hamilton-Jacobi equation, 543 Hamiltonian, 526 harmonic oscillator, 529 Kepler problem, 536 n-thorder, 530 pendulum, 533 precession equinoxes, 539 Mercury. 538,539 satellite orbits, 539 second order. 534,544 secular. 532,535 slowvariable, 547 solar system, 532 theory, 229,338,483, 526-555 quantum, 527 timedependent, 527-533 examples, 533-541 timeindependent, 541-549 phase angle, 533 Phase space, 335,370,453,573 plane, 234 spherical, 83,428 Pentatomic molecule, 272 Periapsis, 99,108,540,541 Periastra, 474 Pericynthion, 99 Perigee, 474 Perihelion. 99,100,474,477. 484 Mercury, 332,538,539 Period doubling, 516 Periodic frequency, 455 motion, 452,484 libration, 452 rotation, 452 multiply, 458 orbits ofpendulum. 454 quasi, 461 Permutation group, 609 symbol (6,-jk), 169,173,181ellipse, 98 harmonic oscillator, 380 damped driven, plotof,507 uncoupled, 486,487 Kepler problem, 98 orbits, 454 point transformation, 370 regular orbits, Hénon-Heiles, 502 trajectory, 354 Photomeson production, 304 Photon, 253 Pitch angle, 154,603 Planck’s constant, 380 Poincaré integral invariants, 394 map, (orsection), 494,495 Hénon-Heiles, 499-501 Kepler problem, 495 transformation, 282 Poinsot’s construction, 201,202, 206,234 Point inﬂection, 42 Lagrange, 124 saddle, 124 transformation, 31,370,422 conﬁguration space, 370 phase space, 370 turning, 78 Poisson equation, 225 theorem, 398 Poisson bracket, 388-411 angular momentum, 408-411 applications, 396 canonically invariant, 390 conservation theorem, 402-404 correspondence principle, 390, 398 double, 390 equation ofmotion, 396-398. 407 fundamental, 389,411 generating function, 402-406 inﬁnitesimal canonical transformation (I.C.T.), 398-405 integral invariants ofPoincaré, 394 invariance, 388 Jacobi identity, 390 Jacobian determinant, 394 Lagrange bracket, 392 Liealgebra, 392 linear andangular momentum, 411 nested, 408 perturbation theory, 532 symmetry groups, 411-418 symplectic, 388,389 theorem, 411 Polar coordinate, 72 central force Lagrangian, 73 plane, 25 spherical, 32 Polhode, 202 Polyatornic molecule, 258,259 linear, 558 rotation andvibration, 180Subject Index Potential, 4 energy, 4 equilibrium, 239 total, 11 equivalent onedimensional, central force, 78 generalized, 22 gradient, 10 Hénon-Heiles, 497,498 hole, 82 integrable, 86 linear restoring force, 83 power law,86,87 scalar, 20 velocity dependent, 22-24 Power series, 43 Precession, 206 astronomical, 208,228 average frequency, 217 Earth, 207,226 equinoxes, 209,223-229 fastandslow, 219 force freemotion, 207 freebody, 205 heavy symmetrical top,209 Larmor, 231 magnetic ﬁeld, 230 Mercury, 332,538,539 orbital plane, 540 pseudoregular, 218 regular, 218 satellite, 228 system ofcharges, 230 Thomas, 282,330 Principal axistransformation, 241 Proper time, 279,310,321 Proton-neutron reaction, 304 Pseudoscalar, 614 Pseudotensor, 189 Pseudovector, 168,614 Ptolemaic system, 129 Qvalue, 304 Quadratic forms, diagonalization, 252 iterator, 509 mapping, 503 Quadrature, 75,211635 Quadrupole moment gravitational, 226 Sun,541 Quantization, 54 Quantum commutator, 392 corrections, 115 electrodynamics, 54 ﬁeldtheory, 576 Hamiltonian, 613 Heisenberg picture, 408 mechanics, 111 Bohr, 466 perturbation theory, 526 scattering, 120 theory, 290 transition from classical mechanics, 76 Quark, 615 Quasi- periodic, 461,490 static motion, 268 Quatemion group, 610 Radius gyration, 198 vector, 73 Rainbow scattering, 114 Raman spectroscopy, 258 Randomness, 483 Rayleigh’s dissipation function 23 Reactance, 53 Regularity, 488 breakdown, 488 Relativity, 276-328. 619 4-vector, 287 angular momentum, 309-312 collisions, 300-309 electromagnetism, 297-300 force, 297-300 general, 324-328, 538 Lagrangian, 312-324 metric tensor, 287,288,291 reduced mass, 71 spacetime, 278-280 special, 265,276-324 postulates, 277 636 Subject Index Representation faithful, 609,613 group, 608 irreducible, 608 momentum, 576 Repulsive centrifugal banier, 78 Residue, 469 Resonance, 260,548 deep, 549 parametric, 509 shallow, 549 transients, 260 vibrating system, 260 Resonant frequency oflinear triatomic molecule, 255 Reversed effective force, 80 Reversible process, 336 Rheonomous, 13 Ricci tensor, 327 Riemann surface, 469 tensor, 326,327 Rigid body, 11 angular momentum, 185-188 deﬁnition, 134-138 degrees offreedom, 134 equations ofmotion, 184, 198-200 Euler equations, 198-200 theorem, 155,156 heavy symmetrical topmotion. 208-223 kinematics, 134,184 Lagrangian, 199 motion, 134,155-174 nutating, 209,214 orientation, 169 rotation, 155-174 ﬁnite, 161-163 inﬁnite, 163-171 solving problems, 198 torque freemotion, 200-223 Rigidity, 318 Rollangle, 154,603 Rolling constraint, 14 disk, 15 hoop, 50Rossler equations, 523 Rotation, 141,452,455 active sense. 143 clockwise, 162 counterclockwise, 170 ﬁnite, 161 formula, 162,I70 generator, 171 group, 171 inﬁnitesimal, 162,163 instantaneous axis, 172 kinetic energy, 191 matrix, 142 passive, 169 sense, 143 proper, 158 trace, 160 vector, 59 Routh Kepler problem, 348 procedure, 56,347 Routhian, 348 Rutherford cross section, 110 scattering, 131 Satellite artiﬁcial, 229 close, 229 orbiting Earth, 474 orbits, 223,229 Scalar, 189.293 curvature, 327 ﬁeld, 287 meson, 571 ﬁeld, 599 potential, 20 product. Minkowski space, 288, 290,291 scale invariance, 591 transformation, 370 Scattering, 106,306 angle, 112,308,309 center ofmass, 116 cross section 107 deﬂection angle, 114 differential cross section, 107, 119 elastic, 118,120,306glory, 114 inelastic, 118 laboratory coordinates, 115-121 neutron, 120 rainbow, 112 Rutherford, 111,131 Schrodinger equation, 54,584,599 Schwarzschild solution ofEinstein ﬁeldequations. 538 Scleronomous, 13,25 Screening, nucleus, 111 Screw motion, 161 symmetry axis,161 Secular change, 531 equation, 157,244 linear triatomic molecule, 254 perturbation, 532,535 Self-similarity, 505,514 fractal, 516-519 logistic equation, 513-515 Semiclassical approximation, 115 Semiholonomic, 46,48,49 Semimajor axis,95,475 Semiminor axis, 101 Sensitivity toinitial conditions, 491 Separation constant, 445 Siderial day,175 year, 538 Sierpinski carpet, 517-519, 522 fractal dimension, 518 sponge, 522 Sigma elementary particle, 615 Similarity transformation, 149. 158,189 trace, 160 Simultaneity, 580 Sine-Gordon equation. 585 ﬁeld, 585 SO(3) group, 413,418, 610 SO(4) group, 414 SO(n) group, 418 SOHO, 126 Solar day,175 Soliton, 587.596 Sound vibrations ingas,598 Space conﬁguration, 34.357 dual, 292 ﬁlling, 521 Minkowski, 278,290 Spacelike, 278,580 Spacetime, 278 interval, 278 Special relativity. 276 postulates, 277 Spherical triangle, 181,476 Spinangular momentum, 10 Spiraling, 113 Stability, 205 marginal, 493 Staeckel conditions. 446,447 Stationary path, 37 value, 35 Steady state, dynamic, 267 Stochastic. 483 Stokes’ law,24,52 Strange attractor, 489,492,500 dimension, 521 fractal dimension, 520 Hénon-Heiles, 500,501 Strangeness, 615 Stress energy tensor, 566,570,589 conservation, 595 properties, 578 symmetrize, 572,600 tensor, 570 Strong lawofaction andreaction, 7 nuclear force, 299 Structure analogy. 54 constant, 412,413,612 SU(2) group, 413, 418, 612, 615, 616,621 SU(3)group, 418,615 SU(4) group, 616 SU(n) group. 418,615,616 Subgroup, 606 Submultiplet, 615Subject Index Summation convention, 138,169, 186 Superconductivity, 618 Supermultiplet, 615,616 Susceptance. 53 Symmetry groups, 411-418 mechanical systems, 411-418 properties, 60 spherical. 60,72 Symplectic, 343,381 approach, 339.343 canonical transfonnation, 381, 382 condition, 384,387,422 generating function, 394 group, 387,612 Hamilton’s equations, 343 matrix, 384 Poisson bracket, 388,397 System continuous, 568 discrete, 558 vector, 409,410,413 Tachyon, 278 Tait-Bryan angles, 154 Tardyon, 278 Taylor series, 239 potential expansion, 482 Temperature, deﬁnition, 85 Tensor, 188-191 alternating, 169 Cartesian, 189 deﬁnition, 293 ﬁrstrank, 189 inertia, 191-198 isotropic ofrank3,169 metric, 286 moment ofinertia, 191-198 product, 294 properties, 188 rank.293 second rank. 188 slots,293 unit, 190 wedge product, 295 zerorank, I89 Thermodynamics, 336637 Thomas frequency, 285 precession, 282,330 Three body problem, 121-126, 617 Euler solution, 122 Lagrange solution, 123 restricted, 124,133 Threshold energy, 302-305 Time dilation, 279 Timelike, 278 Top Euler equations, 210 fast,215,221 heavy symmetrical, 200,208, 482 withonepoint ﬁxed, 208 motion, 208,212 sleeping, 221 symmetric, 618 tippie, 221 uniform, 221 Topological dimension, 518 Torque. 2 critical. 266 damping, 266 gravitational, 223 pendulum, 266 Torus, 487,492 Tourdeforce, 407 Trace ofsimilarity transformation, 160 Transfonnation active sense, 143 canonical, 368-421 inﬁnitesimal, 396 restricted, 371,382 congruence, 245,246, 252 equation, 13 extended canonical, 371 formal properties, 144 Galilean, 281 gauge, 595 generating function, 371 identity, 146,156,395 improper, 151,168 inﬁnitesimal, 165 canonical (I.T.C.), 396 638 Subject Index Transformation (cont) fast,547 ﬁeld, 588 Legendre, 375,549 examples, 375 linear, 187 Lorentz, 280 matrix, 144 elements, 140 operator, 142 orthogonal, 139-150, 184 passive sense, 143 point, 31,370,422 principal axis, 241 proper, 151 restricted canonical, 371,382 rigid body rotation, 139-155 scale, 370 similarity, 149, 158, 180, 189, 244 Transient, 260 Translational mode, 272 Triatomic molecule, 275 Triple cross product, 186 Tuming angles, 213 Twin paradox, 285 Ultrarelativistic, 303 region, 308 Undetemiined multipliers of Lagrange, 46,363 Unitary matrix, 412slow,547 Variation, 354 8-type, 38,44 A-type, 357,359 integral, 44 lineintegral, 35 Variational Hamiltonian, 353 principle, 5,34-43, 51 Vector 4-vector energy, momentum, 295,300, 301 photon momentum, 304 table, 287 velocity, 286-288 addition, 163 axial, 168 conserved, 104 covariant, 289 ﬁeld,table, 287 ﬁrstranktensor, 189 fluxdensity, 569 Minkowski space, 286 polar, 167 radius. 73 rateofchange, 171-174 system, 409,410,413 tangent, 286,326 Unstable moment ofinertia axis, Velocity 205 vanderPol equation, 490 limit cycle, 491 Variable canonical, 335addition law,282 angular, 172,187 critical, 221 rigid body, 172 areal, 73 critical angular, 221 escape, 31four-, 286 generalized, 25,319 Vibration anharmonic, 255 forced, 259,264 free,250,253 modes, 261 linear triatomic molecule, 253 longitudinal mode, 257 number ofnormal modes, 255 transverse mode, 257 Virial Clausius, 84,128 theorem, 83-86, 94,472 Vrtual displacement, 16,20 work principle, 17 Viscosity, 51,265 Wavefunction, 613 Weak nuclear force, 299 Weber’s electrodynarnics, 367 Wedge product, 295,296 Wheatstone bridge. 66 Witten andSander diffusion model, 524 Wobble, Chandler, 208,228 Work, 9 Yawangle, 154,603 Year, anomalistic, 131 Young’s modulus, 559,560 Zeeman effect, 232 fr ~<~ rte°a ex r ° ».O Q) 9 \ W tic