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Graduate-level textbook by V.I. Arnold, translated by K. Vogtmann and A. Weinstein, second edition with 1989 copyright. The preface describes coverage of differential equations and phase flows, oscillations, rigid body motion, variational principles, and the Hamiltonian formalism with symplectic geometry. Appendices treat Riemannian geometry, ideal fluids, perturbation theory, short-wave asymptotics, and caustics. It is a published book kept among downloaded math methods references, not Phil's own work.

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Graduate Texts inMathematics Editorial Board J.H.Ewing F.W.Gehring P.R.Halmos Graduate Texts inMathematics l 2 3 4 5 \OOO\lC7‘\ l0 ll l2 13 l4 l5 I6 17 18 Q19 20 21 22 23 24 25 26 27 28 29 30 31 32TAKEUTI/ZARING. Introduction toAxiomatic SetTheory. 2nded. Oxroav. Measure andCategory. 2nded. SCI-IAEFFER. Topological Vector Spaces. HILTON/STAMMBACH. ACourse in Homological Algebra. MAC LANE. Categories fortheWorking Mathematician. Huoi-iEs/PrPER. Projective Planes. SERRE. ACourse inArithmetic. TAKEUTI/ZARING. Axiometic SetTheory. HUMPHREYS. Introduction toLieAlgebras andRepresentation Theory. COHEN. ACourse inSimple Homotopy Theory. CONWAY. Functions ofOneComplex Variable. 2nded. BEALS. Advanced Mathematical Analysis. ANDERSON/FULLER. Rings andCategories of Modules. 2nded. GOLUBITSKY/GUILEMIN. Stable Mappings and Their Singularities. BERBERIAN. Lectures inFunctional Analysis andOperator Theory. WINTER. TheStructure ofFields. R()SENBLA'l'I‘. Random Processes. 2nded. HALMOS. Measure Theory. HALMOS. AHilbert Space Problem Book. 2nded. HUSEMOLLER. Fibre Bundles. 3rded. HUMPHREYS. Linear Algebraic Groups. BARNES/MACK. AnAlgebraic introduction to Mathematical Logic. GREUB. Linear Algebra. 4thed. HOLMES. Geometric Functional Analysis and ltsApplications. HEWITT/STROMBERG. Real andAbstract Analysis. MANES. Algebraic Theories. KELLEY. General Topology. ZARISKI/SAMUEL. Commutative Algebra. Vol l. ZARISKI/SAMUEL. Commutative Algebra. Vol ll. JACOBSON. bectures inAbstract Algebra l. Basic Concepts. Mcoason. Lectures inAbstract Algebra II. Linear Algebra. JACOBSON. Lectures inAbstract Algebra III. Theory ofFields andGalois Theory.HIRSCH. Differential Topology. SPITZER. Principles ofRandom Walk. 2nded. WERMER. Banach Algebras andSeveral Complex Variables. 2nded. KELLEY/NAMIOKA etal.Linear Topological Spaces. M()NK. Mathematical Logic. GRAUERT/FR1T2sCHE. Several Complex Variables. ARvEs()N. Aninvitation toC*-Algebras. KEMENY/SNELL/KNAPP, Denumerable Markov Chains. 2nded. APOSTUL. Modular Functions andDirichlet Series inNumber Theory. 2nded. SERRE. Linear Representations ofFinite Groups. GlLLMAN/.lERlS()N. Rings ofContinuous Functions. KENDIG. Elementary Algebraic Geometry. LoEvE. Probability Theory l.4thed. LQEVE. Probability Theory ll.4thed. MOISE. Geometric Topology inDimensions 2 and3. SACHS/WU. General Relativity for Mathematicians. GRUENBERG/WEIR. Linear Geometry. 2nded. EDWARDS. Fennat’s LastTheorem. KLINGENBERG. ACourse inDifferential Geometry. HARTSHORNE. Algebraic Geometry. MANIN. ACourse inMathematical Logic. GRAVER/WATKINS. Combinatorics with Emphasis ontheTheory ofGraphs. BROWN/PEARCY. introduction toOperator Theory l:Elements ofFunctional Analysis. MASSEY. Algebraic Topology: An lntroduction. CROWELIJFOX. lntroduction toKnot Theory. KOBLITZ. p-adic Numbers, p-adic Analysis, andZeta-Functions. 2nded. LANG. Cyclotomic Fields. ARNOLD. Mathematical Methods inClassical Mechanics. 2nded. WHITEHEAD. Elements ofl-lomotopy Theory. KARGAP()L()V/MERLZIAK()V. Fundamentals of theTheory ofGroups. BOLLOBAS. Graph Theory. EDWARDS. Fourier Series. Vol.l.2nded. continued aﬁer index V.I.Arnold Mathematical Methods of Classical Mechanics Second Edition Translated byK.Vogtmann andA.Weinstein With 269Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest V.I.Arnold Department of Mathematics Steklov Mathematical Institute Russian Academy of Sciences Moscow 117966 GSP-1 Russia Editorial Board J.H.Ewing Department of Mathematics Indiana University Bloomington, IN47405 U.S.A.K.Vogtmann Department of Mathematics Comell University Ithaca, NY14853 U.S.A. F.W.Gehring Department of Mathematics University ofMichigan AnnArbor, MI48109 U.S.A.A.Weinstein Department of Mathematics University ofCalifornia atBerkeley Berkeley, CA94720 U.S.A. P.R.Halmos Department of Mathematics Santa Clara University Santa Clara, CA95053 U.S.A. Mathematics Subject Classiﬁcations (I991): 70HXX, 70D05, 58-XX Library ofCongress Cataloging-in-Publication Data Amol 'd,V.I.(Vladimir Igorevich), I937- [Matematicheskie metody klassicheskoi mekhaniki. English] Mathematical methods ofclassical mechanics IV.I.Amol 'd; translated byK.Vogtmann andA.Weinstein.—2nd ed. p.cm.—(Graduate texts inmathematics ;60) Translation of:Mathematicheskie metody klassicheskoi mekhaniki. Bibliography: p. Includes index. ISBN 0-387-96890-3 l.Mechanics, Analytic. I.Title. II.Series. QA805.A68l3 I989 53l’.0l'5l5—-dcl9 88-39823 Title oftheRussian Original Edition: Matematicheskie metody klassicheskoi" mekhaniki. Nauka, Moscow, 1974. Printed onacid-free paper ©1978, 1989 bySpringer-Verlag New York Inc. Allrights reserved. This work maynotbetranslated orcopied inwhole orinpartWithout the written permission ofthepublisher (Springer-Verlag, 175Fifth Avenue, New York, NY10010, U.S.A.), except forbriefexcerpts inconnection with reviews orscholarly analysis. Useinconnec- tionwithanyform ofinformation storage andretrieval, electronic adaptation, computer software, orbysimilar ordissimilar methodology nowknown orhereafter developed isforbidden. Printed andbound byR.R.Donnelley andSons, I-Iarrisonburg, Virginia. Printed intheUnited States ofAmerica. 9876543(Third corrected printing.) ISBN 0-387-96890-3 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-96890-3 Springer-Verlag Berlin Heidelberg New York Preface Many different mathematical methods andconcepts areused inclassical mechanics: differential equations andphase ﬂows, smooth mappings and manifolds, Liegroups andLiealgebras, symplectic geometry andergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them sohard tostudy. inthis book weconstruct themathematical apparatus ofclassical mechanics from thevery beginning; thus, thereader isnotassumed tohave anyprevious knowledge beyond standard courses inanalysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) andlinear algebra (linear operators, quadratic forms). With thehelp ofthisapparatus, weexamine allthebasic problems in dynamics, including thetheory ofoscillations, thetheory ofrigid body motion, andthehamiltonian formalism. Theauthor hastried toshow the geometric, qualitative aspect ofphenomena. Inthis respect thebook is closer tocourses intheoretical mechanics fortheoretical physicists than to traditional courses intheoretical mechanics astaught bymathematicians. Aconsiderable part ofthebook isdevoted tovariational principles and analytical dynamics. Characterizing analytical dynamics inhis“Lectures on thedevelopment ofmathematics inthenineteenth century,” F.Klein wrote that ..aphysicist, forhisproblems, canextract from these theories only verylittle, andanengineer nothing.” Thedevelopment ofthesciences inthe following years decisively disproved thisremark. Hamiltonian formalism layatthebasis ofquantum mechanics andhasbecome oneofthemost often used tools inthemathematical arsenal ofphysics. After thesigniﬁcance of symplectic structures andHuygens’ principle forallsorts ofoptimization problems wasrealized, Hamilton’s equations began tobeused constantly in V Preface engineering calculations. Ontheother hand, thecontemporary development ofcelestial mechanics, connected with therequirements ofspace exploration, created newinterest inthemethods andproblems ofanalytical dynamics. Theconnections between classical mechanics andother areas ofmathe- matics andphysics aremany andvaried. Theappendices tothisbook are devoted toafewofthese connections. Theapparatus ofclassical mechanics isapplied to:thefoundations ofriemannian geometry, thedynamics of anideal ﬂuid, Kolmogorov’s theory ofperturbations ofconditionally periodic motion, short-wave asymptotics forequations ofmathematical physics, andtheclassiﬁcation ofcaustics ingeometrical optics. These appendices areintended fortheinterested reader andarenotpart oftherequired general course. Some ofthem could constitute thebasis of special courses (forexample, onasymptotic methods inthetheory ofnon- linear oscillations oronquasi-classical asymptotics). The appendices also contain some information ofareference nature (forexample, alistofnormal forms ofquadratic hamiltonians). While inthebasic chapters ofthebook the author hastried todevelop alltheproofs asexplicitly aspossible, avoiding references toother sources, theappendices consist onthewhole ofsummaries ofresults, theproofs ofwhich aretobefound inthecited literature. The basis forthebook was ayear-and-a-half-long required course inclassical mechanics, taught bytheauthor tothird- and fourth-year mathematics students atthemathematics-mechanics faculty ofMoscow State University in1966-1968. Theauthor isgrateful toI.G.Petrovsky, who insisted thatthese lectures bedelivered, written up,and published. Inpreparing these lectures for publication, theauthor found very helpful thelecture notes ofL.A.Buni- movich, L.D.Vaingortin, V.L.Novikov, andespecially, themimeographed edition (Moscow State University, 1968) organized byN.N.Kolesnikov. The author thanks them, andalsoallthestudents andcolleagues whocommuni- cated their remarks onthemimeographed text; many ofthese remarks were used inthepreparation ofthepresent edition. The author isgrateful to M.A.Leontovich, forsuggesting thetreatment ofconnections bymeans ofa limit process, andalsotoI.I.Vorovich andV.I.Yudovich fortheir detailed review ofthemanuscript. V.ARNOLD Thetranslators would liketothank Dr.R.Barrar forhishelp inreading theproofs. Wewould alsoliketothank many readers, especially TedCourant, forspotting errors intheﬁrsttwoprintings. Berkeley, I981 K.VOGTMANN A.Wsmsrsm vi Preface tothesecond edition Themain part ofthisbook waswritten twenty years ago. The ideas and methods ofsymplectic geometry, developed inthisbook, have now found many applications inmathematical physics andinother domains ofapplied mathematics, aswellasinpure mathematics itself. Especially, theshort-wave asymptotical expansions theory hasreached avery sophisticated‘ level, with many important applications tooptics, wave theory, acoustics, spectroscopy, andeven chemistry; thisdevelopment wasparallel tothedevelopment ofthe theories ofLagrange andLegendre singularities, that is,ofsingularities of caustics and ofwave fronts, oftheir topology and their perestroikas (in Russian metamorphoses were always called “perestroikas,” asin“Morse perestroika” fortheEnglish “Morse surgery”; now thattheword perestroika hasbecome international, wemay preserve theRussian term intranslation andarenotobliged tosubstitute “metamorphoses” for“perestroikas” when speaking ofwave fronts, caustics, andsoon). Integrable hamiltonian systems have been discovered unexpectedly inmany classical problems ofmathematical physics, andtheir study hasledtonew results inboth physics andmathematics, forinstance, inalgebraic geometry. Symplectic topology hasbecome oneofthemost promising andactive branches of“global analysis.” Animportant generalization ofthePoincare “geometric theorem” (see Appendix 9)was proved byC.Conley and E.Zehnder in1983. Asequence ofworks (byM.Chaperon, A.Weinstein, J.-C. Sikorav, M.Gromov, Ja.M.Eliashberg, Ju.Tchekanov, A.Floer, C.Viterbo, H.Hofer, andothers) marks important progress inthisvery living domain. One may hope that thisprogress willlead totheproof ofmany known conjectures insymplectic andcontact topology, andtothediscovery ofnew results inthisnewdomain ofmathematics, emerging from theproblems of mechanics andoptics. vii Preface tothesecond edition The present edition includes three new appendices. They represent the modern development ofthetheory ofraysystems (thetheory ofsingularity andofperestroikas ofcaustics andofwave fronts, related tothetheory of Coxeter reﬂection groups), thetheory ofintegrable systems (thegeometric theory ofelliptic coordinates, adapted totheinﬁnite-dimensional Hilbert space generalization), andthetheory ofPoisson structures (which isageneral- ization ofthetheory ofsymplectic structures, including degenerate Poisson brackets). Amore detailed account ofthepresent state ofperturbation theory may be found inthebook, Mathematical Aspects ofClassical andCelestial Mechanics byV.I.Arnold, V.V.Kozlov, andA.I.Neistadt, Encyclopaedia ofMath. Sci., Vol. 3(Springer, 1986); Volume 4ofthisseries (1988) contains asurvey “Symplectic geometry” byV.I.Arnold andA.B.Givental’, anarticle by A.A.Kirillov ongeometric quantization, andasurvey ofthemodern theory ofintegrable systems byS.P.Novikov, I.M.Krichever, andB.A.Dubrovin. Formore details onthegeometry ofraysystems, seethebook Singularities ofDiﬂerentiable Mappings byV.I.Arnold, S.M.Gusein-Zade, andA.N. Varchenko (Vol. 1,Birkhéiuser 1985; vol.2,Birkhiiuser, 1988). Catastrophe Theory byV.I.Arnold (Springer, 1986) (second edition) contains along annotated bibliography. Surveys onsymplectic andcontact geometry andontheir applications may befound intheBourbaki seminar (D.Bennequin, “Caustiques mystiques”, February, I986) andinaseries ofarticles (V.I.Arnold, First steps ofsymplectic topology, Russian Math. Surveys, 41(1986); Singularities ofraysystems, Russian Math. Surveys, 38(1983); Singularities invariational calculus, Modern Problems ofMath., VINITI, 22(1983) (translated inJ.Soviet Math.); and O.P.Shcherbak, Wave fronts and reﬂection groups, Russian Math. Surveys, 43(1988)). Volumes 22(1983) and 33(1988) oftheVINITI series, “Sovremennye problemy mathematiki. Noveishie dostijenia,” contain adozen articles onthe applications ofsymplectic andcontact geometry andsingularity theory to mathematics andphysics. Bifurcation theory (both forhamiltonian andformore general systems) isdiscussed inthetextbook Geometrical Methods oftheTheory ofOrdinary Diﬂerential Equations (Springer, 1988) (this newedition ismore complete than thepreceding one). The survey “Bifurcation theory anditsapplications in mathematics andmechanics” (XVIIth International Congress ofTheoretical andApplied Mechanics inGrenoble, August, 1988) alsocontains newinfor- mation, asdoes Volume SoftheEncyclopaedia ofMath. Sci.(Springer, 1989), containing thesurvey “Bifurcation theory” byV.I.Arnold, V.S.Afraimovich, Ju.S.Iljashenko, and L.P.Shilnikov. Volume 2ofthisseries, edited by D.V.Anosov andJa.G.Sinai, isdevoted totheergodic theory ofdynamical systems including those ofmechanics. Thenewdiscoveries inallthese theories have potentially extremely wide applications, butsince these results were discovered rather recently, they are viii Preface tothesecond edition discussed only inthespecialized editions, andapplications areimpeded by thediﬂiculty ofthemathematical exposition fornonmathematicians. Ihope that thepresent book willhelp tomaster these new theories notonly to mathematicians, butalsotoallthose readers whousethetheory ofdynamical systems, symplectic geometry, and thecalculus ofvariations-—in physics, mechanics, control theory, andsoon.Theauthor would liketothank Dr. T.Tokieda forhishelp incorrecting errors inprevious printings andfor reading theproofs. December 1988 V.I.Arnold ix Translator’s preface tothesecond edition This edition contains three newappendices, originally written forinclusion in aGerman edition. They describe work bytheauthor andhisco-workers on Poisson structures, elliptic coordinates with applications tointegrable sys- tems, andsingularities ofraysystems. Inaddition, numerous corrections to errors found bytheauthor, thetranslators, andreaders have been incorpo- rated intothetext. Contents Preface v Preface tothesecond edition vii Part I NEWTONIAN MECHANICS l Chapter 1 Experimental facts 3 1.Theprinciples ofrelativity anddeterminacy 3 2.Thegalilean group andNewton’s equations 4 3.Examples ofmechanical systems 1l Chapter 2 Investigation oftheequations ofmotion 15 4.Systems with onedegree offreedom 15 5.Systems with twodegrees offreedom 22 6.Conservative force ﬁelds 28 7.Angular momentum 30 8.Investigation ofmotion inacentral ﬁeld 33 9.Themotion ofapoint inthree-space 42 IO.Motions ofasystem ofnpoints 44 ll.Themethod ofsimilarity 50 PartII LAGRANGIAN MECHANICS 53 Chapter 3 Variational principles 55 12.Calculus ofvariations 55 13.Lagrange’s equations 59 Contents 14.Legendre transformations 15.Hamilton’s equations 16.Liouville’s theorem Chapter 4 Lagrangian mechanics onmanifolds 17.Holonomic constraints 18.Differentiable manifolds 19.Lagrangian dynamical systems 20.E.Noether’s theorem 21.D’Alembert’s principle Chapter 5 Oscillations 22.Linearization 23.Small oscillations 24.Behavior ofcharacteristic frequencies 25.Parametric resonance Chapter 6 Rigid Bodies 26.Motion inamoving coordinate system 27.Inertial forces andtheCoriolis force 28.Rigid bodies 29.Euler’s equations. Poinsot‘s description ofthemotion 30.Lagrange’s top 31.Sleeping topsandfasttops Part III HAMILTONIAN MECHANICS Chapter 7 Differential forms 32.Exterior forms 33.Exterior multiplication 34.Differential forms 35.Integration ofdifferential forms 36.Exterior differentiation Chapter 8 Symplectic manifolds 37.Symplectic structures onmanifolds 38.Hamiltonian phase ﬂows andtheir integral invariants 39.TheLiealgebra ofvector ﬁelds 40.TheLiealgebra ofhamiltonian functions 41.Symplectic geometry 42.Parametric resonance insystems with many degrees offreedom 43.Asymplectic atlas Chapter 9 Canonical formalism 44.Theintegral invariant ofPoincaré—Cartan 45.Applications oftheintegral invariant ofPoincaré—Cartan 46.Huygens’ principle 47.TheHamilton—Jacobi method forintegrating Hamilton’s canonical equations 48.Generating functions Chapter 10 Introduction toperturbation theory 49.Integrable systems 50.Action-angle variables 51.Averaging 52.Averaging ofperturbations Appendix l Riemannian curvature Appendix 2 Geodesics ofleft-invariant metrics onLiegroups and thehydrodynamics ofideal ﬂuids Appendix 3 Symplectic structures onalgebraic manifolds Appendix 4 Contact structures Appendix 5 Dynamical systems with symmetries Appendix 6 Normal forms ofquadratic hamiltonians Appendix 7 Normal forms ofhamiltonian systems near stationary points andclosed trajectories Appendix 8 Theory ofperturbations ofconditionally periodic motion, andKolmogorov’s theoremContents 219 225 229 233 233 240 248 258 266 271 271 279 285 291 301 318 343 349 371 381 385 399 Contents Appendix 9 Poincaré’s geometric theorem, itsgeneralizations and applications 416 Appendix 10 Multiplicities ofcharacteristic frequencies, andellipsoids depending onparameters 425 Appendix ll Short wave asymptotics 438 Appendix 12 Lagrangian singularities 446 Appendix 13 TheKorteweg—de Vries equation 453 Appendix 14 Poisson structures 456 Appendix 15 Onelliptic coordinates 469 Appendix 16 Singularities ofraysystems 480 Index 511 PART I NEWTONIAN MECHANICS Newtonian mechanics studies themotion ofasystem ofpoint masses inthree-dimensional euclidean space. The basic ideas and theorems of newtonian mechanics (even when formulated interms ofthree-dimensional cartesian coordinates) areinvariant with respect tothesix-dimensional‘ group ofeuclidean motions ofthisspace. Anewtonian potential mechanical system isspeciﬁed bythemasses ofthepoints andbythepotential energy. Themotions ofspace which leave thepotential energy invariant correspond tolaws ofconservation. Newton’s equations allow onetosolve completely aseries ofimportant problems inmechanics, including theproblem ofmotion inacentral force ﬁeld. ~ 1Andalsowith respect tothelarger group ofgalilean transformations ofspace-time. Experimental facts Inthischapter wewrite down thebasic experimental facts which lieatthe foundation ofmechanics: Galileo’s principle ofrelativity and Newton’s differential equation. Weexamine constraints ontheequation ofmotion imposed bytherelativity principle, andwemention some simple examples. lTheprinciples ofrelativity anddeterminacy Inthisparagraph weintroduce anddiscuss thenotion ofaninertial coordinate system. The mathematical statements ofthisparagraph areformulated exactly inthenextparagraph. Aseries ofexperimental facts isatthebasis ofclassical mechanics? We listsome ofthem. ASpace andtime Ourspace isthree-dimensional andeuclidean, andtime isone-dimensional. BGalileo’s principle ofrelativity There exist coordinate systems (called inertial) possessing thefollowing twoproperties: 1.Allthelaws ofnature atallmoments oftime arethesame inallinertial coordinate systems. 2.Allcoordinate systems inuniform rectilinear motion with respect toan inertial onearethemselves inertial. 2Allthese “experimental facts“ areonly approximately trueandcanberefuted bymore exact experiments. Inorder toavoid cumbersome expressions, wewillnotspecify thisfrom now on andwewillspeak ofourmathematical models asiftheyexactly described physical phenomena. 3 11Experimental facts Inother words, ifacoordinate system attached totheearth isinertial, then anexperimenter onatrain which ismoving uniformly inastraight line with respect totheearth cannot detect themotion ofthetrain byexperiments conducted entirely inside hiscar. Inreality, thecoordinate system associated with theearth isonly approxi- mately inertial. Coordinate systems associated with thesun, thestars, etc. aremore nearly inertial. CNewt0n’s principle ofdeterminacy The initial state ofamechanical system (the totality ofpositions and velocities ofitspoints atsome moment oftime) uniquely determines allof itsmotion. Itishard todoubt thisfact,since welearn itvery early. Onecanimagine aworld inwhich todetermine thefuture ofasystem onemust alsoknow the acceleration attheinitial moment, butexperience shows usthatourworld isnotlikethis. 2Thegalilean group andNewton’s equations Inthisparagraph wedeﬁne andinvestigate thegalilean group ofspace-time transformations. Then weconsider Newton‘s equation andthesimplest constraints imposed onitsright-hand side bytheproperty ofinvariance with respect togalilean transformations.’ ANotation Wedenote thesetofallrealnumbers byR.Wedenote byR"ann-dimen- sional realvector space. a a+b i} Figure 1Parallel displacement Aﬂine n-dimensional space A"isdistinguished from R"inthat there is “noﬁxed origin.” Thegroup R"actsonA"asthegroup ofparallel displace- ments (Figure 1): a—>a+b, aeA",beR",a+beA". [Thus thesumoftwopoints ofA"isnotdeﬁned, buttheir difference isdeﬁned andisavector inlR".] 3Thereader who hasnoneed forthemathematical formulation oftheassertions ofSection I canomit thissection. 4 2;Thegaliliean group andNewton’s equations Aeuclidean structure onthevector space R"isapositive deﬁnite symmetric bilinear form called ascalar product. The scalar product enables oneto deﬁne thedistance ﬂ(x,y)=llx—yll=\/(X—y,X—y) between points ofthecorresponding aﬂine space A".Anaffine space with this distance function iscalled aeuclidean space andisdenoted byE". BGalilean structure Thegalilean space-time structure consists ofthefollowing three elements: 1.The universe—a four-dimensional afﬁne“ space A4.The points ofA‘ arecalled world points orevents. Theparallel displacements oftheuniverse A“constitute avector space IR‘. 2.Time—a linear mapping t:IR“—>Rfrom thevector space ofparallel displacements oftheuniverse tothereal“time axis.” Thetime interval from event aeA4toevent beA4isthenumber t(b—a)(Figure 2).If t(b—a)=0,then theevents aandbarecalled simultaneous. 3A v a A4 telm-ii Figure 2Interval oftime t The setofevents simultaneous with agiven event forms athree- dimensional afﬁne subspace inA“.Itiscalled aspace ofsimultaneous events A3. Thekernel ofthemapping tconsists ofthose parallel displacements of A4which takesome (and therefore every) event intoanevent simultaneous with it.This kernel isathree-dimensional linear subspace R3ofthevector space R4. Thegalilean structure includes onefurther element. 3.Thedistance between simultaneous events />(a,b)= |l4—bl|=\/(a—b,a—b) a,bE/13 isgiven byascalar product onthespace R3.This distance makes every space ofsimultaneous events intoathree-dimensional euclidean space E3. ‘Formerly, theuniverse wasprovided notwith anaffine, butwith alinear structure (thegeo- centric system ofthe universe). 5 1:Experimental facts Aspace A“,equipped with agalilean space-time structure, iscalled a galilean space. Onecanspeak oftwoevents occurring simultaneously indifferent places, buttheexpression “two non-simultaneous events a,beA“ occurring at oneandthesame place inthree-dimensional space ”hasnomeaning aslong aswehave notchosen acoordinate system. Thegalilean group isthegroup ofalltransformations ofagalilean space which preserve itsstructure. Theelements ofthisgroup arecalled galilean transformations. Thus, galilean transformations areaffine transformations ofA4which preserve intervals oftime andthedistance between simultaneous events. EXAMPLE. Consider thedirect product3 R><R3ofthetaxiswith athree- dimensional vector space R3;suppose R3hasaﬁxed euclidean structure. Such aspace hasanatural galilean structure. Wewillcallthisspace galilean coordinate space. Wemention three examples ofgalilean transformations ofthisspace. First, uniform motion with velocity v: g1(t,x)=(t,x+vt) VteR,xeR3. Next, translation oftheorigin: g2(t,x)=(t+s,x+s) VteR,xeR3. Finally, rotation ofthecoordinate axes: g3(t,x)=(t,Gx), VteR,xeR3, where G:R3—>R3isanorthogonal transformation. PROBLEM. Show that every galilean transformation ofthespace RxR3 canbewritten inaunique wayasthecomposition ofarotation, atranslation, andauniform motion (g=g1Og20g3)(thus thedimension ofthegalilean group isequal to3+4+3=10). PROBLEM. Show that allgalilean spaces areisomorphic toeach other“ and, inparticular, isomorphic tothecoordinate space RxR3. LetMbeaset.Aone-to-one correspondence <p1:M —>R><R3iscalled agalilean coordinate system onthesetM.Acoordinate system (p2moves uniformly with respect tocplif<p1><p2_1: R><R3->R><R3isagalilean transformation. Thegalilean coordinate systems cplandgo;giveMthesame galilean structure. 3Recall that thedirect product oftwosetsAandBisthesetofordered pairs (a,b),where aeAandbEB.Thedirect product oftwo spaces (vector, affine, euclidean) hasthestructure ofa space ofthe same type. 6That is,there isaone-to-one mapping ofonetotheother preserving thegalilean structure. 6 2:The galilean group andNewtOn‘s equations CMotion, velocity, acceleration Amotion inR”isadifferentiable mapping x:I—>R”,where Iisaninterval ontherealaxis. Thederivative in=§ =limll_"(‘°3’hi7"(’°)ER” 0) l=tQ h**0 h iscalled thevelocity vector atthepoint toeI. Thesecond derivative .. d3x"("3=Ff=1‘Q iscalled theacceleration vector atthepoint to. Wewillassume thatthefunctions weencounter arecontinuously differ- entiable asmany times asnecessary. Inthefuture, unless otherwise stated, mappings, functions, etc.areunderstood tobedifferentiable mappings, functions, etc.Theimage ofamapping X:I—>R”iscalled atrajectory or curve inR”. PROBLEM. Isitpossible forthetrajectory ofadifferentiable motion onthe plane tohave theshape drawn inFigure 3?Isitpossible fortheacceleration vector tohave thevalue shown? ANSWER. Yes.No. X Figure 3Trajectory ofmotion ofapoint Wenowdeﬁne amechanical system ofnpoints moving inthree-dimensional euclidean space. Letx:R—>R3beamotion inR3.Thegraph7 ofthismapping isacurve inRxR3. Acurve ingalilean space which appears insome (and therefore every) galilean coordinate system asthegraph ofamotion, iscalled aworld line (Figure 4). IThegraph ofamappingf: A—>Bisthesubset ofthedirect product AxBconsisting ofall pairs (a,f(a)) with aEA. 7 1:Experimental facts \ . >R Figure 4World lines Amotion ofasystem ofnpoints gives, ingalilean space, nworld lines. Inagalilean coordinate system they aredescribed bynmappings x,:R—>R3, i=1,...,n. The direct product ofncopies ofR3iscalled theconﬁguration space ofthesystem ofnpoints. Our nmappings x,-:R—>R3deﬁne onemapping x:R—>R” N=3n ofthetime axisintotheconﬁguration space. Such amapping isalsocalled amotion ofasystem ofnpoints inthegalilean coordinate system onRxR3. DNewton’s equations According toNewton’s principle ofdeterminacy (Section 1C)allmotions ofasystem areuniquely determined bytheir initial positions (x(t0) 6R”) andinitial velocities (x(t0) eR”). Inparticular, theinitial positions andvelocities determine theacceleration. Inother words, there isafunction F:R”xR”><R—>R”such that (1) it=F(x,>2,t). Newton used Equation (1)asthebasis ofmechanics. Itiscalled Newton’s equation. Bythetheorem ofexistence and uniqueness ofsolutions toordinary differential equations, thefunction Fandtheinitial conditions x(t0) and x(tO) uniquely determine amotion.3 Foreach speciﬁc mechanical system theform ofthefunction Fisdeter- mined experimentally. From themathematical point ofview theform ofF foreach system constitutes thedeﬁnition ofthatsystem. EConstraints imposed bytheprinciple ofrelativity Galileo’s principle ofrelativity states that inphysical space-time there isa selected galilean structure (“the class ofinertial coordinate systems”) having thefollowing property. 3Under certain smoothness conditions, which weassume tobefulﬁlled. Ingeneral, amotion isdetermined byEquation (1)only onsome interval ofthetime axis. Forsimplicity wewill assume thatthisinterval isthewhole time axis, asisthecasein most problems inmechanics. 8 2:The galilean group andNewton's equations 1’ .\’ +- I I Figure 5Galileo’s principle ofrelativity Ifwesubject theworld lines ofallthepoints ofanymechanical system3 tooneandthesame galilean transformation, weobtain world lines ofthe same system (with newinitial conditions) (Figure 5). This imposes aseries ofconditions ontheform oftheright-hand sideof Newton’s equation written inaninertial coordinate system: Equation (1) must beinvariant with respect tothegroup ofgalilean transformations. EXAMPLE 1.Among thegalilean transformations arethetime translations. Invariance with respect totime translations means that“the laws ofnature remain constant,” i.e.,ifx=<p(t)isasolution toEquation (1),then forany seR,x=<p(t+s)isalsoasolution. From thisitfollows thattheright-hand sideofEquation (1)inaninertial coordinate system does notdepend onthetime: ii=<D(x, )2). Remark. Differential equations inwhich theright-hand sidedoes depend ontime arise inthefollowing situation. Suppose that wearestudying part Iofthemechanical system I+II. Then theinfluence ofpart IIonpart Icansometimes bereplaced byatime variation ofparameters inthesystem ofequations describing themotion of partI.Forexample, theinﬂuence ofthemoon ontheearth canbeignored in investigating themajority ofphenomena ontheearth. However, inthestudy of thetides thisinﬂuence must betaken intoaccount; onecanachieve thisby introducing, instead oftheattraction ofthemoon, periodic changes inthe strength ofgravity onearth. 9Informulating theprinciple ofrelativity wemust keep inmind that itisrelevant only to closed physical (inparticular, mechanical) systems, i.e.,thatwemust include inthesystem all bodies whose interactions playaroleinthestudy ofthe given phenomena. Strictly speaking, we should include inthesystem allbodies intheuniverse. Butweknow from experience thatone candisregard theeffect ofmany ofthem: forexample, instudying themotion ofplanets around thesunwecandisregard theattractions among thestars, etc. Ontheother hand, inthestudy ofabody inthevicinity ofearth, thesystem isnotclosed iftheearth isnotincluded; inthestudy ofthe motion ofanairplane thesystem isnotclosed if itdoes notinclude theairsurrounding theairplane, etc.Inthefuture, theterm “mechanical system" willmean aclosed system inmost cases, andwhen there isanon-closed system in question thiswillbeexplicitly stated (cf.,forexample, Section 3). 9 l:Experimental facts Equations with variable coefficients canappear alsoastheresult offormal operations inthesolution ofproblems. EXAMPLE 2.Translations inthree-dimensional space aregalilean trans- formations. Invariance with respect tosuch translations means thatspace ishomogeneous, or“has thesame properties atallofitspoints." That is, ifx,-=<p,(t)(i =1,...,n)isamotion ofasystem ofnpoints satisfying (I), then foranyreR3themotion(p,(t) +r(i=1,...,n)alsosatisﬁes Equation (1). From thisitfollows thattheright-hand sideofEquation (1)intheinertial coordinate system candepend only onthe“relative coordinates” xj—x,,. From invariance under passage toauniformly moving coordinate system (which does notchange xiorxj—x,,,butadds toeach x1-aﬁxed vector v)it follows that theright-hand side ofEquation (1)inaninertial system of coordinates candepend only ontherelative velocities if: —Xki k Xlt})s iﬁjﬂ k=1, -"7 n- EXAMPLE 3.Among thegalilean transformations aretherotations inthree- dimensional space. Invariance with respect tothese rotations means that space isisotropic; there arenopreferred directions. Thus, if(pi:R—>R3(i =1,...,n)isamotion ofasystem ofpoints satis- fying (1),andG:R3->R3isanorthogonal transformation, then themotion G(p,: R—>R3(i, ...,n)alsosatisﬁes (1).Inother words. F(GX, G =GF(x, it), where Gxdenotes (Gxl, ...,Gxn), xieR3. PROBLEM. Show thatifamechanical system consists ofonly onepoint, then itsacceleration inaninertial coordinate system isequal tozero (“Newton’s ﬁrstlaw”). Hint. ByExamples 1and2theacceleration vector does notdepend on x,x,ort,andbyExample 3thevector Fisinvariant with respect torotation. PROBLEM. Amechanical system consists oftwo points. Attheinitial moment their velocities (insome inertial coordinate system) areequal tozero. Show that thepoints willstay onthelinewhich connected them attheinitial moment. PROBLEM. Amechanical system consists ofthree points. Attheinitial moment their velocities (insome inertial coordinate system) areequal tozero. Show thatthepoints always remain intheplane which contained them atthe initial moment. PROBLEM. Amechanical system consists oftwopoints. Show that forany initial conditions there exists aninertial coordinate system inwhich the twopoints remain inaﬁxed plane. 10 3:‘Examples ofmechanical systems PROBLEM. Show that mechanics “through thelooking glass” isidentical toours. Hint. Inthegalilean group there isareﬂection transformation, changing theorientation ofR3. PROBLEM. Istheclass ofinertial systems unique? ANSWER. No.Other classes canbeobtained ifonechanges theunits oflength andtime orthedirection oftime. 3Examples ofmechanical systems Wehave already remarked thattheform ofthefunction FinNewton's equation (1)isdetermined experimentally foreach mechanical system. Here areseveral examples. Inexamining concrete systems itisreasonable nottoinclude alltheobjects oftheuniverse inasystem. Forexample, instudying themajority ofphenomena taking place ontheearth we canignore theinﬂuence ofthemoon. Furthermore, itisusually possible todisregard theeﬂect oftheprocesses wearestudying onthemotion oftheearth itself; wemayeven consider acoordi- natesystem attached totheearth as“ﬁxed.” Itisclear thattheprinciple ofrelativity nolonger imposes theconstraints found mSection 2forequations ofmotion written insuch acoordinate system. Forexample, near theearth there isadistinguished direction, thevertical. AExample I."Astone falling totheearth Experiments show that (2) .t=—g, where gz9.8m/s3 (Galileo)* where xistheheight ofastone above thesurface oftheearth. Ifweintroduce the“potential energy” U=gx,then Equation (2)can bewritten intheform ..__dQx_ dx. IfU:E”—>Risadifferentiable function oneuclidean space, then wewill denote by8U/dx thegradient ofthefunction U.IfE”=E'"x xE"" isadirect product ofeuclidean spaces, then wewilldenote apoint xeE” by(xl,...,x,,),andthevector 6U/dx by(dU/dxl, ...,6U/dxk). Inparticular, ifx1,...,xNarecartesian coordinates inE”,then thecomponents ofthe vector 0U/dx arethepartial derivatives 6U/dxl, ...,6U/dx~. Experiments show that theradius vector ofthestone with respect to some point 0ontheearth satisﬁes theequation 6(3) if=—a%, where U=—(g, x) "Inthisandother sections, themass ofaparticle istaken tobe1. ll 1:Experimental facts Thevector intheright-hand sideisdirected towards theearth. Itiscalled thegravitational acceleration vector g.(Figure 6.) " g // , Figure 6Astone falling totheearth BExample 2:Falling from great height Like allexperimental facts, thelawofmotion (2)hasarestricted domain of application. According toamore precise lawoffalling bodies, discovered byNewton, acceleration isinversely proportional tothesquare ofthedistance from thecenter oftheearth: .. réx=—g-r2’ where r=ro+x(Figure 7). r r0 x Figure 7Theearth’s gravitational ﬁeld This equation canalso bewritten intheform (3),ifweintroduce the potential energy kU=—— k=gré,r inversely proportional tothedistance tothecenter oftheearth. PROBLEM. Determine with what velocity astone must bethrown inorder that itﬂyinﬁnitely farfrom thesurface ofthe earth.‘° ANSWER. 211.2km/sec. ‘°This istheso-called second cosmic velocity v2.Ourequation does nottakeintoaccount the attraction ofthesun.Theattraction ofthe sunwillnotletthestone escape from thesolar system ifthevelocity ofthestone withrespect totheearth islessthan16.6km/sec. I2 3:Examples ofmechanical systems CExample 3:Motion ofaweight along aline under theaction ofaspring Experiments show that under small extensions ofthespring theequation ofmotion oftheweight willbe(Figure 8) .56=—ot2x. til \ Figure 8Weight onaspring This equation canalso bewritten intheform (3)ifweintroduce the potential energy ozzxz Ifwereplace ouroneweight bytwoweights, then itturns outthat, under thesame extension ofthespring, theacceleration ishalfaslarge. Itisexperimentally established that foranytwobodies theratio ofthe accelerations $61/362 under thesame extension ofaspring isﬁxed (does not depend ontheextent ofextension ofthespring oronitscharacteristics, but only onthebodies themselves). Thevalue inverse tothisratio isbydeﬁnition theratio ofmasses: 561 m2 . 562 ml Foraunitofmass wetake themass ofsome ﬁxed body, e.g.,oneliterof water. Weknow byexperience thatthemasses ofallbodies arepositive. The product ofmass times acceleration mic’does notdepend onthebody, and isacharacteristic oftheextension ofthespring. This value iscalled the force ofthespring acting onthebody. Asaunitofforce, wetake the“newton.” Ifoneliterofwater issuspended onaspring atthesurface oftheearth, thespring acts with aforce of9.8 newtons (=1kg). DExample 4.'Conservative systems LetE3"=E3><---><E3betheconﬁguration space ofasystem ofnpoints intheeuclidean space E3.LetU:E3"—>IRbeadifferentiable function and letml,...,m,,bepositive numbers. 13 l:Experimental facts Deﬁnition. Themotion ofnpoints, ofmasses ml,...,m,,,inthepotential ﬁeldwith potential energy Uisgiven bythesystem ofdifferential equations . 5U . m;Xl= —'-'“‘ l=1,...,n.dxl Theequations ofmotion inExamples 1to3have thisform. Theequations ofmotion ofmany other mechanical systems canbewritten inthesame form. Forexample, thethree-body problem ofcelestial mechanics isproblem (4) inwhich mlm; m2m3 m3m1 U=— A —E — . "X1_X2" "X2_X3" "X3_X1" Many different equations ofentirely different origin canbereduced to form (4),forexample theequations ofelectrical oscillations. Inthefollowing chapter wewillstudy mainly systems ofdifferential equations intheform (4). l4 Investigation oftheequations ofmotion Inmost cases (forexample, inthethree-body problem) wecanneither solve thesystem ofdifferential equations norcompletely describe thebehavior ofthesolutions. Inthischapter weconsider afewsimple butimportant problems forwhich Newton’s equations canbesolved. 4Systems with onedegree offreedom lnthisparagraph westudy thephase ﬂowofthe differential equation (1).Alook atthegraph of thepotential energy isenough foraqualitative analysis ofsuch anequation. Inaddition, Equation (1)isintegrated byquadratures. ADeﬁnitions Asystem withonedegree offreedom isasystem described byonedifferential equation (1) 5c'=f(x) xen. Thekinetic energy isthequadratic form* T=5x1. Thepotential energy isthefunction um=—lxft/:>dc. Thesign inthisformula istaken sothat thepotential energy ofastone is larger ifthestone ishigher offtheground. Notice that thepotential energy determines f.Therefore, tospecify a system oftheform (1)itisenough togive thepotential energy. Adding a constant tothepotential energy does notchange theequation ofmotion (1). *seefootnote onp.ll. 15 2:Investigation ofthe equations ofmotion Thetotal energy isthesum E=T+U. Ingeneral, thetotal energy isafunction, E(x,>2),ofxand>2. Theorem (The lawofconservation ofenergy). The total energy ofpoints moving according totheequation (1)isconserved: E(x(t), x(t)) isindependent oft. PROOF. %(T+u)=>2s%+%x=>z(§e-f(x))=0. CI BPhase_ﬂow Equation (1)isequivalent tothesystem oftwoequations: (2) X=yy'=f(x)- Weconsider theplane with coordinates xandy,which wecallthephase plane ofEquation (1).Thepoints ofthephase plane arecalled phase points. The right-hand sideof(2)determines avector ﬁeldonthephase plane, called the phase velocity vector ﬁeld. Asolution of(2)isamotion (p:R—>R2ofaphase point inthephase plane, such thatthevelocity ofthemoving point ateach moment oftime is equal tothephase velocity vector atthelocation ofthephase point atthat moment.“ Theimage oftpiscalled thephase curve. Thus thephase curve isgiven by theparametric equations x=<t>(t) y=¢(t)- PROBLEM. Show that through every phase point there isoneandonly one phase curve. Hint. Refer toatextbook onordinary differential equations. Wenotice that aphase curve could consist ofonly onepoint. Such a point iscalled anequilibrium position. The vector ofphase velocity atan equilibrium position iszero. The lawofconservation ofenergy allows onetoﬁndthephase curves easily. Oneach phase curve thevalue ofthetotal energy isconstant. Therefore, each phase curve liesentirely inoneenergy level setE(x,y)=h. CExamples EXAMPLE 1.Thebasic equation ofthetheory ofoscillations is 56=—-x. ‘1Here weassume forsimplicity thatthesolution (pisdeﬁned onthewhole time axisR. l6 4:Systems with onedegree offreedom J. A I O O I ' \ OeeO 0 0 >x I O 0 0 O Figure 9Phase plane oftheequation 56=—x Inthiscase(Figure 9)wehave: 122 Vx2 X2 x2 T—? L“? E-?+€. Theenergy level setsaretheconcentric circles andtheorigin. The phase velocity vector atthephase point (x,y)hascomponents (y,—x). Itis perpendicular totheradius vector andequal toitinmagnitude. Therefore, themotion ofthephase point inthephase plane isauniform motion around 0:x=rl,c0s(<p0 —t),y=rosin(rp0 —t).Each energy level setisaphase curve. EXAMPLE 2.Suppose thatapotential energy isgiven bythegraph inFigure 10.Wewilldraw theenergy level sets%y2+U(x) =E.Forthis,thefollowing facts arehelpful. 1.Anyequilibrium position of(2)must lieonthexaxisofthephase plane. Thepoint x-5,y=0isanequilibrium position if6isacritical point ofthepotential energy, i.e.,if(GU/ax)|,,=; =0. 2.Each level setisasmooth curve inaneighborhood ofeach ofitspoints which isnotanequilibrium position (this follows from theimplicit function theorem). Inparticular, ifthenumber Eisnotacritical value of thepotential energy (i.e.,isnotthevalue ofthepotential energy atoneof itscritical points), then thelevel setonwhich theenergy isequal toE isasmooth curve. Itfollows that inorder tostudy theenergy level curve, weshould turn ourattention tothecritical andnear-critical values ofE.Itisconvenient heretoimagine alittle ballrolling inthepotential well U. For example, consider thefollowing argument: “Kinetic energy is nonnegative. This means that potential energy islessthan orequal tothe total energy. The smaller thepotential energy, thegreater thevelocity.” This translates to:“The ballcannot jump outofthepotential well, rising I7 2:Investigation oftheequations ofmotion U A E,- E2._ E3- 5,,_ E5- ‘ >X X A E1 E2 E3 E4 E58 O >.\’ Figure IOPotential energy andphase curves higher than thelevel determined byitsinitial energy. Asitfallsintothewell, theballgains velocity.” Wealsonotice thatthelocal maximum points ofthe potential energy areunstable, buttheminimum points arestable equilibrium positions. PRQBLEM. Prove this. PROBLEM. How many phase curves make uptheseparatrix (ﬁgure eight) curve, corresponding tothelevel E2? ANSWER. Three. PROBLEM. Determine theduration ofmotion along theseparatrix. ANSWER. Itfollows from theuniqueness theorem thatthetime isinﬁnite. PROBLEM. Show thatthetime ittakes togofrom xltox2(inonedirection) isequal to ‘Z dx 18 4:Systems with onedegree offreedom U U >X _.._._ ___ ___ >X (a) (b) Figure llPotential energy PROBLEM. Draw thephase curves, given thepotential energy graphs in Figure ll. ANSWER. Figure 12. X X /':F"'_"'_,_X X \i— X /_— ta) (b) Figure 12Phase curves PROBLEM. Draw thephase curves forthe“equation ofanideal planar pendulum”: >'c'=—sin x. PROBLEM. Draw thephase curves forthe“equation ofapendulum ona rotating axis”: it=—sin x+M. Remark. Inthese twoproblems xdenotes theangle ofdisplacement ofthe pendulum. Thephase points whose coordinates differ by21:correspond to thesame position ofthependulum. Therefore, inaddition tothephase plane, itisnatural tolookatthephase cylinder {x(mod 21:),y}. PROBLEM. Find thetangent lines tothebranches ofthecritical level corre- sponding tomaximal potential energy E=U(§)(Figure 13). ANSWER. y=i./—U"(§)(x —5). 19 2:Investigation oftheequations ofmotion U A E X yA E X Figure 13Critical energy levellines PROBLEM. LetS(E) bethearea enclosed bytheclosed phase curve cor- responding totheenergy level E.Show that theperiod ofmotion along thiscurve isequal to T_dS 'dE' PROBLEM. LetEl,bethevalue ofthepotential function ataminimum point 5.Find theperiod T},=liml;_,llo T(E) ofsmall oscillations inaneighbor- hood ofthepoint 5. ANSWER. 21:/./U”(§). PROBLEM. Consider aperiodic motion along theclosed phase curve corre- sponding totheenergy level E.Isitstable inthesense ofLiapunov?“ ANswER. No.13 DPhase ﬂow LetMbeapoint inthephase plane. Welook atthesolution tosystem (2) whose initial conditions att=0arerepresented bythepoint M.Weassume thatanysolution ofthesystem canbeextended tothewhole time axis. The value ofoursolution atanyvalue oftdepends onM.Wedenote theresulting phase point (Figure 14)by M(t) =g'M. Inthisway wehave deﬁned amapping ofthephase plane toitself, g‘:R2—>R2.Bytheorems inthetheory ofordinary differential equations, ‘ZForadeﬁnition, see,e.g.,p.I55ofOrdinary Diﬂerential Equations byV.I.Arnold, MIT Press, 1973. '3Theonly exception isthecasewhen theperiod does notdepend ontheenergy. 20 4:Systems with onedegree offreedom S 2' g +s) M Figure 14Phase flow themapping g‘isadiffeomorphism (aone-to-one differentiable mapping with adifferentiable inverse). Thediffeomorphisms g’,teR,form agroup: g‘*‘=g‘cg‘.The mapping goistheidentity (g°M =M),andg“'isthe inverse ofg‘.The mapping g:R><R2—>R2,deﬁned byg(t,M)=g'M is differentiable. Allthese properties together areexpressed bysaying thatthe transformations g‘form aone-parameter group ofdiﬂeomorphisms ofthephase plane. This group isalso called thephase ﬂow, given bysystem (2)(or Equation (1)). EXAMPLE. The phase ﬂow given bytheequation x"=—xisthegroup g‘ ofrotations ofthephase plane through angle taround theorigin. PROBLEM. Show that thesystem with potential energy U=—x“ does not deﬁne aphase ﬂow. PROBLEM. Show thatifthepotential energy ispositive, then there isaphase ﬂow. Hint. Usethelawofconservation ofenergy toshow thatasolution can beextended without bound. PROBLEM. Draw theimage ofthecircle x2+(y~1)’<§under theaction ofatransformation ofthephase ﬂow fortheequations (a)ofthe“inverse pendulum,” x=xand(b)ofthe“nonlinear pendulum,” 56=—sin x. ANSWER. Figure 15. y y I -11‘ 7|‘ X >X fa) (bl Figure 15Action ofthephase ﬂowonacircle 21 2:Investigation oftheequations ofmotion 5Systems with twodegrees offreedom Analyzing ageneral potential system with twodegrees offreedom isbeyond thecapability ofmodern science. Inthisparagraph welook atthesimplest examples. ADeﬁnitions Byasystem with twodegrees offreedom wewillmean asystem deﬁned by thedifferential equations (1) it=f(X), XEE2, where fisavector ﬁeld ontheplane. Asystem issaid tobeconservative ifthere exists afunction U:E2—>R such that f=—8U/6X. Theequation ofmotion ofaconservative system then hastheform“ it=—6U/6X. BThelawofconservation ofenergy Theorem. Thetotal energy ofaconservative system isconserved, i.e., E2;?=O,where E=%X2+U(X), x2=(X,>2). PROOF. dE/dt =(X,ii)+(t3U/6X, X)=(ii+(5U/6X), X)=Obytheequation ofmotion. El Corollary. Ifattheinitial moment thetotal energy isequal toE,then all trajectories lieintheregion where U(X) 3E,i.e.,apoint remains inside thepotential wellU(xl, x2)3Eforalltime. Remark. Inasystem with onedegree offreedom itisalways possible to introduce thepotential energy U(X)=—_lxf(€)d€. Forasystem with twodegrees offreedom thisisnotso. PROBLEM. Find anexample ofasystem oftheform X=f(X),XeE2,which is notconservative. CPhase space Theequation ofmotion (1)canbewritten asthesystem: 3f1=)’1 X2=}’2 (2) , av , av }’1=-K Y2=—E 1‘Incartesian coordinates ontheplane E2,iil=—0U/dxl and562=—5U/0x2. Z2 5:Systems with twodegrees offreedom The phase space ofasystem with twodegrees offreedom isthefour- dimensional space withcoordinates xl,x2,yl,andyz. Thesystem (2)deﬁnes thephase velocity vector ﬁeld infour space aswell as‘5thephase ﬂowofthesystem (aone-parameter group ofdiffeomorphisms offour-dimensional phase space). Thephase curves of(2)aresubsets offour- dimensional phase space. Allofphase space ispartitioned intophase curves. Projecting thephase curves from four space tothexl,x2plane gives the trajectories ofourmoving point inthexl,x;plane. These trajectories are alsocalled orbits. Orbits canhave points ofintersection even when thephase curves donotintersect oneanother. Theequation ofthelawofconservation ofenergy -2 2 2E=X3+U(x)=5-? +U(xl,x2) deﬁnes athree-dimensional hypersurface infour space: E(xl, x2,yl,yz)= E0;thissurface, rtEl,,remains invariant under thephase flow: g‘rtEo =rcfo. Onecould saythatthephase ﬂowﬂows along theenergy level hypersurfaces. Thephase velocity vector ﬁeld istangent atevery point to1:50.Therefore, 1250isentirely composed ofphase curves (Figure 16). F2 nEo .Vt xl xz Figure 16Energy level surface andphase curves EXAMPLE 1(“small oscillations ofaspherical pendulum”). LetU= +xi). Thelevel setsofthepotential energy inthexl,x2plane willbeconcentric circles (Figure 17). The equations ofmotion, >2,=—xl, it",=-x2, areequivalent tothe system -221=y1 X2=}’2 Y1: —X1 Y2=—X2- This system decomposes into two independent ones; inother words, each ofthecoordinates xlandx2changes with time inthesame wayasin asystem with onedegree offreedom. '5With theusual limitations. 23 2:Investigation oftheequations ofmotion X2 X1 Figure 17Potential energy levelcurves foraspherical pendulum Asolution hastheform xl=clcost+c;sinr x2=c;,cost+c4sint yl=-clsint+c2 cost y2= —c,sint+c4cost. Itfollows from thelawofconservation ofenergy that E=tot+Yi)+%(xi+xi)=const. i.e.,thelevel surface ttlloisasphere infourspace. PROBLEM. Show that thephase curves aregreat circles ofthissphere. (A great circle istheintersection ofasphere with atwo-dimensional plane passing through itscenter.) PROBLEM. Show thatthesetofphase curves onthesurface 1:50forms atwo- dimensional sphere. Theformula w=(xl+iyl)/(x2 +iyz)gives the“Hopf map” from thethree sphere 1:50tothetwosphere (the complex w-plane completed bythepoint atinﬁnity). Our phase curves arethepre-images ofpoints under theHopf map. PROBLEM. Find theprojection ofthephase curves onthexl,x2plane (i.e., draw theorbits ofthemotion ofapoint). EXAMPLE 2(“Lissajous ﬁgures”). Welook atonemore example ofaplanar motion (“small oscillations withtwodegrees offreedom”): xl=-—xl 552=—w2x2. Thepotential energy is U=§-x2li+§w2x§. From thelawofconservation ofenergy itfollows that, ifattheinitial moment oftime thetotal energy is %(>?i+Xi)+Ufxi,X1)=E, then allmotions willtake place inside theellipse U(xl, x2)5E. 24 5:Systems with twodegrees offreedom Oursystem consists oftwoindependent one-dimensional systems. There- fore, thelawofconservation ofenergy issatisﬁed foreach ofthem separately, i.e.,thefollowing quantities arepreserved E1=iii+ix? E2=iii+%w2><§ (E=E1+E2)- Consequently, thevariable xlisbounded bytheregion |xl|3Al,Al= ./2El(0), andx2oscillates within theregion Ixzl3A2.The intersection ofthese tworegions deﬁnes arectangle which contains theorbits (Figure 18). X2 M‘ /12 /l1 bx] V Figure I8Theregions U3E,Ul3EandU23E PROBLEM. Show that this rectangle isinscribed intheellipse U3E. The general solution ofour equations isxl=Alsin(t +(pl), x2= A2sin(cut +(pl); amoving point independently performs anoscillation withfrequency 1andamplitude Alalong thehorizontal andanoscillation with frequency coandamplitude A2along thevertical. Consider thefollowing method ofdescribing anorbit inthexl,x2plane. Welook atacylinder with base 2Alandaband ofwidth 2A2. Wedraw on theband asinewave with period 21tAl/cu andamplitude A2andwind the band onto thecylinder (Figure 19).Theorthogonal projection ofthesinusoid X22A, ACAP ~ A2 XI Figure 19Construction ofaLissajous ﬁgure7EEﬁ./ 25 2:Investigation oftheequations ofmotion wound around thecylinder onto thexl,xlplane gives thedesired orbit, called aLissajous ﬁgure. Lissajous ﬁgures canconveniently beseen onanoscilloscope which dis- plays independent harmonic oscillations onthehorizontal andvertical axes. Theform ofaLissajous ﬁgure very strongly depends onthefrequency cu. Ifco=1(the spherical pendulum ofExample 1),then thecurve onthe cylinder isanellipse. The projection ofthisellipse onto thexl,xlplane depends onthedifference (pl—(plbetween thephases. For(pl=(plweget asegment ofthediagonal oftherectangle; forsmall (pl—(plwegetan ellipse close tothediagonal andinscribed intherectangle. For(pl—(pl=1:/2 wegetanellipse with major axes xl,xl;as(pl—(plincreases from rt/2 torttheellipse collapses onto thesecond diagonal; as(pl—-(plincreases further thewhole process isrepeated from thebeginning (Figure 20). X2 >_X'1 Figure 20Series ofLissajous ﬁgures withtn=1 Now letthefrequencies beonly approximately equal: wz1.Thesegment ofthecurve corresponding to03t32nisveryclose toanellipse. Thenext loop also reminds oneofanellipse, buthere thephase shift (pl—(plis greater than intheoriginal by21r(co —1).Therefore, theLissajous curve with cuz1isadistorted ellipse, slowly progressing through allphases from collapsed onto onediagonal tocollapsed onto theother (Figure 21). Ifoneofthefrequencies istwice theother (cu=2),then forsome particular phase shift theLissajous ﬁgure becomes adoubly traversed arc(Figure 22). X2 >X1 Figure 21Lissajous ﬁgure withtoz1 26 5:Systems with twodegrees offreedom PROBLEM. Show thatthiscurve isaparabola. Byincreasing thephase shift (pl—(plwegetinturn thecurves inFig.23. Ingeneral, ifoneofthefrequencies isntimes bigger than theother (co=n), then among thegraphs ofthecorresponding Lissajous ﬁgures there isthe graph ofapolynomial ofdegree n(Figure 24);thispolynomial iscalled a Chebyshev polynomial. ‘<2 >XI Figure 22Lissajous ﬁgure withtn=2 X22 X2 A X1 X1 X2 X2 A 1 XI Figure 23Series ofLissajous ﬁgures with co=2 X2 X2 X2 < ‘ X1 xi Figure 24Chebyshev polynomials 27 2:Investigation oftheequations ofmotion PROBLEM. Show thatifto=m/n,then theLissajous ﬁgure isaclosed algebraic curve; butiftoisirrational, then theLissajous ﬁgure ﬁllstherectangle every- where densely. What does thecorresponding phase trajectory ﬁllout? 6Conservative force ﬁelds Inthissection westudy theconnection between work andpotential energy. AWork ofaforce ﬁeld along apath Recall thedeﬁnition ofthework byaforce Fonapath S.Thework ofthe constant force F(forexample, theforce with which weliftupaload) onthe 1M2 Flps M1 Figure 25Work oftheconstant force Falong thestraight path S A path S=MlMlis,bydeﬁnition, thescalar product (Figure 25) A=(F,S)=|F||S| -cos (p. Suppose wearegiven avector ﬁeld Fandacurve Iofﬁnite length. We approximate thecurve Ibyapolygonal linewith components AS,anddenote byFlthevalue oftheforce atsome particular point ofASl;then thework of theﬁeld Fonthepath Iisbydeﬁnition (Figure 26) A=limZ(F,.,As,). IAS-‘l-*0 Inanalysis courses itisproved that iftheﬁeld iscontinuous andthepath rectiﬁable, then thelimit exists. Itisdenoted byll(F,dS). Fl I A51" Figure 26Work oftheforce ﬁeldFalong thepath I 28 6:Conservative force ﬁelds BConditions foraﬁeld tobeconservative Theorem. Avector ﬁeld Fisconservative ifandonly ifitswork along any path MlMldepends onlyontheendpoints ofthepath, andnotontheshape ofthepath. PROOF. Suppose thatthework ofaﬁeld Fdoes notdepend onthepath. Then MU(M)=-l(F.ds> M0 iswelldeﬁned asafunction ofthepoint M.Itiseasy toverify that 6UF=__dx’ i.e.,theﬁeld isconservative and Uisitspotential energy. Ofcourse, the potential energy isdeﬁned only uptotheadditive constant U(M0), which canbechosen arbitrarily. Conversely, suppose that theﬁeld Fisconservative and that Uisits potential energy. Then itiseasily veriﬁed that f:(F, dS)=~U(M) +U(Ml,), i.e.,thework does notdepend ontheshape ofthepath. Cl PROBLEM. Show thatthevector ﬁeldFl=xl,Fl=—xlisnotconservative (Figure 27). C/‘Q->~£-it/2’<-—o Figure 27Anon-potential ﬁeld PROBLEM. Istheﬁeldintheplane minus theorigin given byFl=xl/(xf +xl), Fl=—xl/(xf +x§)conservative? Show that aﬁeld isconservative ifand only ifitswork along anyclosed contour isequal tozero. CCentral ﬁelds Deﬁnition. Avector ﬁeld intheplane E2iscalled central with center at0, ifitisinvariant with respect tothegroup ofmotions“ oftheplane which ﬁx0. "‘Including reﬂections. 29 2:Investigation oftheequations ofmotion PROBLEM. Show thatallvectors ofacentral ﬁeldlieonraysthrough 0,and thatthemagnitude ofthevector ﬁeldatapoint depends onlyonthedistance from thepoint tothecenter oftheﬁeld. Itisalsouseful tolook atcentral ﬁelds which arenotdeﬁned atthepoint 0. EXAMPLE. The newtonian ﬁeld F=—k(r/|r|3) iscentral, buttheﬁeld in theproblem inSection 6Bisnot. Theorem. Every central ﬁeld isconservative, anditspotential energy depends only onthedistance tothecenter oftheﬁeld, U=U(r). PROOF. According totheprevious problem, wemay setF(r)=<D(r)e,, where ristheradius vector with respect to0,risitslength andtheunit vector e,=r/|r| itsdirection. Then Ml r(Ml) [Ml (F,dS)=J2 <D(r)dr, l’fMtl andthisintegral isobviously independent ofthepath. El PROBLEM. Compute thepotential energy ofthenewtonian ﬁeld. Remark. Thedeﬁnitions andtheorems ofthisparagraph canbedirectly carried over toaeuclidean space E"ofanydimension. 7Angular momentum Wewillseelaterthattheinvariance ofanequation ofamechanical problem withrespect tosome group oftransformations always implies aconservation law.Acentral ﬁeld isinvariant with respect tothegroup ofrotations. Thecorresponding ﬁrstintegral iscalled theangular momen- tum. Deﬁnition. Themotion ofamaterial point (with unitmass) inacentral ﬁeld onaplane isdeﬁned bytheequation P=<l>(r)¢,, where ristheradius vector beginning atthecenter oftheﬁeld 0,ris itslength, ande,itsdirection. Wewillthink ofourplane aslying inthree- dimensional oriented euclidean space. Deﬁnition. Theangular momentum ofamaterial point ofunitmass relative tothepoint 0isthevector product M=[r,1"]. Thevector Misperpendicular toourplane andisgiven byonenumber: M=Mn,where n=[el,el]isthenormal vector, elandelbeing an oriented frame intheplane (Figure 28). 30 7:Angular momentum M II ('2 Figure 28Angular momentum Remark. Ingeneral, themoment ofavector a“applied atthepoint r” relative tothepoint 0is[r,a];forexample, inaschool statics course one studies themoment offorce. [The literal translation oftheRussian term for angular momentum is“kinetic moment.” (Trans. note)] AThelawofconservation ofangular momentum Lemma. Letaandbbetwovectors changing withtimeintheoriented euclidean space R2.Then d . - 2; has :La: +has PROOF. This follows from thedeﬁnition ofderivative. El Theorem (The lawofconservation ofangular momentum). Under motions inacentral ﬁeld, theangular momentum Mrelative tothecenter ofthe ﬁeld Odoes notchange withtime. PROOF. Bydeﬁnition M=[r,i].Bythelemma, M=[i",1"]+[r,l‘].Since theﬁeldiscentral itisapparent from theequations ofmotion thatthevectors i‘andrarecollinear. Therefore M=0. [II BKepler’s law The lawofconservation ofangular momentum was ﬁrst discovered by Kepler through observation ofthemotion ofMars. Kepler formulated this lawinaslightly different way. Weintroduce polar coordinates r,toonourplane with pole atthecenter oftheﬁeld 0.Weconsider, atthepoint rwith coordinates (|r|=r,(p), twounitvectors: e,,directed along theradius vector sothat r=re,, ande,,,perpendicular toitinthedirection ofincreasing (p.Weexpress the velocity vector i"interms ofthebasis e,,ea,(Figure 29). Lemma. Wehavetherelation I"=fe,+r(pe,,. 31 2:Investigation oftheequations ofmotion ell, r re er I‘ 0 \P Figure 29Decomposition ofthevector i"interms ofthebasis e,,ea, PROOF. Clearly, thevectors e,andell,rotate with angular velocity (p.i.e., e,=(peg eq,=-(pe,, Differentiating theequality r=re,gives us r=re,+re,=re,+r(pe,l,. Cl Consequently, theangular momentum is M=tr.*1=tr.ie]+tr.nu.)=r[r,en=r2¢[@,.ei- Thus, thequantity M=r2(pispreserved. This quantity hasasimple geometric meaning. r(t+At) rtr) |l"" 4"‘.,1 I.illlll Figure 30Sectorial velocity Kepler called therateofchange ofthearea S(t)swept outbytheradius vector thesectorial velocity C(Figure 30): dSC-Z. The lawdiscovered byKepler through observation ofthemotion ofthe planets says: inequal times theradius vector sweeps outequal areas, so thatthesectorial velocity isconstant, dS/dt =const. This isoneformulation ofthelawofconservation ofangular momentum. Since AS=S(t+Ar)—S(t)=lr2¢At +o(At), 32 8:Investigation ofmotion inacentral ﬁeld thismeans thatthesectorial velocity dSC=i =12'=1Mdt 2'(PI ishalftheangular momentum ofourpoint ofmass 1,andtherefore constant. ExAMPLE. Some satellites have veryelongated orbits. ByKepler’s lawsuch asatellite spends most ofitstime inthedistant part ofitsorbit, where the magnitude of(pissmall. 8Investigation ofmotion inacentral ﬁeld Thelawofconservation ofangular momentum letsusreduce problems about motion ina central ﬁeldtoproblems withonedegree offreedom. Thanks tothis,motion inacentral ﬁeldcan becompletely determined. AReduction toaone-dimensional problem Welookatthemotion ofapoint (ofmass 1)inacentral ﬁeldontheplane: r=-'23-‘; U=U(r). Itisnatural tousepolar coordinates r,(p. Bythelawofconservation ofangular momentum thequantity M= q'2(t)r2(t) isconstant (independent oft). Theorem. Forthemotion ofamaterial point ofunitmass inacentral ﬁeld thedistance from thecenter oftheﬁeld varies inthesame wayasrvaries intheone-dimensional problem withpotential energy V(r) =U(r) + PROOF. Differentiating therelation shown inSection 7(|"=re,+r(,be,,,), weﬁnd i==(F-rqi12)e, +(2r(p+r('l>)e,,,. Since theﬁeldiscentral, ‘2l_Ql, ﬁrFﬁr'2 Therefore theequation ofmotion inpolar coordinates takes theform .. _ 8U _ ,_r—r(p2=———- 2r(p+r(p=0.ﬁr 33 2:Investigation oftheequations ofmotion But, bythelawofconservation ofangular momentum, ,_M (p_rzv where Misaconstant independent oft,determined bytheinitial conditions. Therefore, ,_ 8U M2 ,, 5V M2 f'=_7r'l-I‘? Of l'=*“?r, WhereV=U-l-5;; Thequantity V(r)iscalled theeﬂective potential energy. El Remark. Thetotal energy inthederived one-dimensional problem r2 E1 Z ‘E + isthesame asthetotal energy intheoriginal problem E—I22+U(r) _2 l since l'.2 —r'.2 r2¢2 _'12 M2 2_2+ 2_2+2r1' BIntegration oftheequation ofmotion The total energy inthederived one-dimensional problem isconserved. Consequently, thedependence ofrontisdeﬁned bythequadrature __ _ : drr-,/2(E V(r)) Jdt J.‘f(E _V(r)). Since (b=M/r2, d(p/dr —(M/r2)/l /2(E —V(r)), andtheequation ofthe orbit inpolar coordinates isfound byquadrature, (P:I‘ M/r2 dr l./2(E-V(r))- CInvestigation oftheorbit Weﬁxthevalue oftheangular momentum atM.Thevariation ofrwith time iseasy tovisualize, ifonedraws thegraph oftheeffective potential energy V(r)(Figure 31). LetEbethevalue ofthetotal energy. Allorbits corresponding tothegiven EandMlieintheregion V(r) 3E.Ontheboundary ofthisregion, V=E, 34 8:Investigation ofmotion inacentral ﬁeld V E >r rmin rm“ Figure 31Graph oftheeffective potential energy i.e.,r=0.Therefore, thevelocity ofthemoving point, ingeneral, isnotequal tozero since (p=160forMaé0. Theinequality V(r) 3Egives oneorseveral annular regions intheplane: 0srminsrSrmaxs(X2' <rmll,<r,,,_,,,,<oo,then themotion isbounded andtakes place inside theringbetween thecircles ofradius rmlnandr,,,,,,l. Pericenter re Apocenter Figure 32Orbit ofapoint inacentral ﬁeld Theshape ofanorbit isshown inFigure 32.Theangle (pvaries mono- tonically while roscillates periodically between rmlnand r,,,,,,,. The points where r=r,,,l,,arecalled pericentral, andwhere r=rm“, apocentral (ifthe center istheearth—perigee andapogee; ifitisthesun—perihelion and aphelion; ifitisthemoon—perilune andapolune). Each oftherays leading from thecenter totheapocenter ortotheperi- center isanaxisofsymmetry oftheorbit. Ingeneral, theorbit isnotclosed: theangle between thesuccessive pericenters andapocenters isgiven bytheintegral J'"""‘ M/r2dr rfnjn./2(E —V(r)) Theangle between twosuccessive pericenters istwice asbig.(I): 35 2:Investigation oftheequations ofmotion 6%‘:W’up‘all AMIiwil\tlI/4" Theorbit isclosed iftheangle (Discommensurable with 21:,i.e.,if(D= 21c(m/n), where mandnareintegers. Itcanbeshown thatiftheangle (Disnotcommensurable with 21:,then the orbit iseverywhere dense intheannulus (Figure 33). Ifr,,,,,,=rm”, i.e.,Eisthevalue ofVataminimum point, then theannulus degenerates toacircle, which isalsotheorbit. PROBLEM. Forwhich values ofozismotion along acircular orbit intheﬁeld with potential energy U=r“,-250t<oo,Liapunov stable? ANSWER. Only forat=2. For values ofEalittle larger than theminimum ofVtheannulus r,,,,,,5r5rm,“willbevery narrow, andtheorbit willbeclose toacircle. Inthecorresponding one-dimensional problem, rwillperform small oscilla- tions close totheminimum point ofV. PROBLEM. Find theangle (Dforanorbit close tothecircle ofradius r. Hint. Cf.Section Dbelow. Wenow look atthecase rm,=oo.Iflim,a,,o U(r) =lim,_,,.,o V(r) = U0,,<oo,then itispossible fororbits togooﬁtoinﬁnity. Iftheinitial energy Eislarger than U,then thepoint goes toinﬁnity with ﬁnite velocity foo= ,/2(E —Um). Wenotice that ifU(r)approaches itslimit slower than r'2, then theellective potential Vwillbeattracting atinﬁnity (here weassume that thepotential Uisattracting atinﬁnity). If,asr—>0,|U(r)| does notgrow faster than M2/2r2, then rm,">0and theorbit never approaches thecenter. If,however, U(r) +(M2/Zrz) —>—oo asr—>0,then itispossible to“fall intothecenter oftheﬁeld.” Falling into thecenter oftheﬁeld ispossible even inﬁnite time (forexample, intheﬁeld U(r) =-1/r3). PROBLEM. Examine theshape ofanorbit inthecase when thetotal energy isequal tothevalue oftheeffective energy Vatalocal maximum point. 36 8:Investigation ofmotion inacentral ﬁeld DCentral ﬁelds inwhich allbounded orbits are closed Itfollows from thefollowing sequence ofproblems that there areonly two cases inwhich allthebounded orbits inacentral ﬁeld areclosed, namely, U=arz, aZO and U=—§, kz0. 7' PROBLEM 1.Show that theangle (Dbetween thepericenter andapocenter isequal tothesemiperiod ofanoscillation intheone-dimensional system with potential energy W(x) =U(M/x)+(x2/2). Hint. Thesubstitution x=M/rgives (D=J‘-xmax dx x..,,,,,/2(E-W)‘ PROBLEM 2.Find theangle (Dforanorbit close tothecircle ofradius r. ANSWER. (DzCDC“=1c(M/r2. /V"(r)) =rt./U’/(3U' +rU"). PROBLEM 3.Forwhich values ofUisthemagnitude of(Dd,independent ofthe radius r? ANSWER. U(r)=ar“(oz2-2,aat0)andU(r)=blogr. Itfollows that (Dd,=1:/,/oz+2(the logarithmic case corresponds to at=0).Forexample, forat=2wehave (Dd,=rt/2,andforor=-1wehave (Dcir =ni- PROBLEM 4.Letinthesituation ofproblem 3U(r)—> ooasr—>oo.Find limE_,w (I>(E, M). ANSWER. 1:/2. Hint. Thesubstitution x=yxmax reduces (Dtotheform =‘ dy i=12 _1_ (DJim.../2(w*(1) -w*(y))’ W(Y)2+xiiiUyxmlx' AsE—>oowehave xma,—>ooandymin—>0,andthesecond term inW*can bediscarded. 37 2:Investigation oftheequations ofmotion PROBLEM 5.LetU(r)=-1<r"'*,0 <5<2.Find<1>,,=lirnE_,_0 <r>. ANSWER. (D0=(,1,dx/,/x”—x2=1:/(2—B).Note thatCD0does notdepend onM. PROBLEM 6.Find allcentral ﬁelds inwhich bounded orbits exist andareall closed. ANSWER. U=arzorU=—k/r. Solution. Ifallbounded orbits areclosed, then, inparticular, (Dd,= 2r:(m/n) =const. According toProblem 3,U=av-‘(oz 2—2),orU=blnr (oz=0).Inboth cases (Dd,=1:/./oz+2.If0:>0,then according toProblem 4,lim£_.w (I>(E, M)=1:/2. Therefore, (D6,,=1:/2, o:=2.If0:<O,then according toProblem 5,limE_._,, (E, M)=1:/(2+oz).Therefore, 1:/(2+(1)=r:/\/2 +oz,oz=—l.Inthecase on=0weﬁnd(Dd,=1:/\fL which isnotcommensurable with 21:.Therefore, allbounded orbits canbe closed only inﬁelds where U=arzorU=—k/r. Intheﬁeld U=arz, a>0,alltheorbits areclosed (these areellipses with center at0,cf.Example 1,Section 5).Intheﬁeld U=—k/r allbounded orbits arealsoclosed and alsoelliptical, aswewillnow show. EKepler’s problem This problem concerns motion inacentral ﬁeld with potential U=—k/r andtherefore V(r) =—(k/r) +(M2/2r2) (Figure 34). Bythegeneral formula =I__”£L'.’i_(P./2(E-V(r)). V >r Figure 34Effective potential oftheKepler problem 38 8:Investigation ofmotion inacentral ﬁeld Integrating, weget M k r M (P=ﬂl'C COS ii. 2E+k2M2 Tothisexpression weshould have added anarbitrary constant. We willassume itequal tozero; thisisequivalent tothechoice ofanorigin of reference fortheangle rpatthepericenter. Weintroduce thefollowing notation: M2 2EM2 Now weget(p=arccos((p/r) —1)/e, i.e., Pr=ii.1+ecos(p This istheso—called focal equation ofaconic section. Themotion isbounded (Figure 35)forE<0.Then e<I,i.e.,theconic section isanellipse. The number piscalled theparameter oftheellipse, andetheeccentricity. Kepler’s ﬁrstlaw, which hediscovered byobserving themotion ofMars, consists ofthefactthattheplanets describe ellipses, with thesunatonefocus. C I > ”b L V 1—e P I+e Figure 35Keplerian ellipse Ifweassume that theplanets move inacentral ﬁeld ofgravity, then Kepler’s ﬁrstlawimplies Newton’s lawofgravity: U=—(k/’r) (cf.Section 2Dabove). The parameter and eccentricity arerelated with thesemi-axes bythe formulas 11 P 211 2a—l—e+l+e_l~e2’ i.e., _ P a_le2’ e=c/a=,/a2 —b2/a, where c=aeisthedistance from thecenter to thefocus (cf.Figure 35). 39 2:Investigation oftheequations ofmotion Remark. Anellipse with small eccentricity isvery close toacircle." Ifthedistance from thefocus tothecenter issmall ofﬁrstorder, then the difference between thesemi-axes isofsecond order: b=a,/1—e2z a(1—%e2). Forexample, intheellipse with major semi-axes of10cmand eccentricity 0.1,thedifference ofthesemi-axes is0.5mm, andthedistance between thefocus andthecenter is1cm. The eccentricities ofplanets’ orbits arevery small. Therefore, Kepler originally formulated hisﬁrstlawasfollows: theplanets move around the sunincircles, butthesunisnotatthecenter. Kepler’s second law,thatthesectorial velocity isconstant, istrueinany central ﬁeld. Kepler’s third lawsays thattheperiod ofrevolution around anelliptical orbit depends only onthesizeofthemajor semi-axes. Thesquares oftherevolution periods oftwoplanets ondifferent elliptical orbits have thesame ratio asthecubes oftheir major semi-axes.“ PROOF. Wedenote byTtheperiod ofrevolution andbySthearea swept outbytheradius vector intime T.2S=MT, since M/2 isthesectorial velocity. Butthearea oftheellipse, S,isequal torcab,soT=2r:ab/M. Since M2/k k azi-—i:——~i M2 2E MI? || 2(ff0ITl¢1=P/(1-6)),-'111d b_M2 1_M Tk' M_I/—’./211~:|I 2'El then T=2r:(k/(, /2lE|)3); but2lE| =k/a,soT=21:03/2k_1/2. El Wenote that thetotal energy Edepends only onthemajor semi-axis a oftheorbit andisthesame forthewhole setofelliptical orbits, from acircle ofradius atoalinesegment oflength 2a. PROBLEM. Attheentry ofasatellite intoacircular orbit atadistance 300km from theearth thedirection ofitsvelocity deviates from theintended direction by1°towards theearth. How istheperigee changed? ANswER. Theheight oftheperigee islessbyapproximately 110km. '7Letadrop ofteafallintoaglass oftea close tothecenter. Thewaves collect atthesymmetric point. The reason isthat, bythefocal deﬁnition ofanellipse, waves radiating from onefocus of theellipse collect attheother. '8Byplanets wemean herepoints inacentral ﬁeld. 40 8:Investigation ofmotion inacentral ﬁeld Figure 36Anorbit which isclose tocircular Hint. Theorbit differs from acircle only tosecond order, andwecandis- regard thisdifference. The radius hastheintended value since theinitial energy hastheintended value. Therefore, wegetthetrue orbit (Figure 36) bytwisting theintended orbit through 1°. PROBLEM. How does theheight oftheperigee change iftheactual velocity islm/sec lessthan intended? PROBLEM. Theﬁrst cosmic velocity isthevelocity ofmotion onacircular orbit ofradius close totheradius oftheearth. Find themagnitude ofthe ﬁrstcosmic velocity vlandshow thatv2=\/20, (cf.Section 3B). ANSWER. 8.1km/sec. PROBLEM.” During hiswalk inouter space, thecosmonaut A.Leonov threw thelenscapofhismovie camera towards theearth. Describe themotion of thelenscapwith respect tothespaceship, taking thevelocity ofthethrow as10m/sec. ANSWER. Thelenscapwillmove relative tothecosmonaut approximately inanellipse with major axisabout 32kmandminor axisabout 16km.The center oftheellipse willbesituated 16kminfront ofthecosmonaut inhis orbit, andtheperiod ofcirculation around theellipse willbeequal tothe period ofmotion around theorbit. Hint. Wetake asourunitoflength theradius ofthespace ship’s circular orbit, andwechoose aunitoftime sothattheperiod ofrevolution around this orbit is21:.Wemust study solutions toNewton’s equation .. Ir=—- r3’ close tothecircular solution with ro=1,(p0=:.Weseek those solutions intheform r=r0+rl (p=(p0+(/)1 r1<l,(,01<l. '9Thisproblem istaken from V.V.Beletskii‘s delightful book.“ Notes ontheMotion ofCelestial Bodies," Nauka, 1972. 41 2:Investigation oftheequations ofmotion Bythetheorem onthedifferentiability ofasolution with respect toits initial conditions, thefunctions r,(t) and <p,(t) satisfy asystem oflinear differential equations (equations ofvariation) uptosmall amounts which areofhigher than ﬁrstorder intheinitial deviation. Bysubstituting theexpressions forrand(pinNewton’s equation, weget, after simple computation, thevariational equations intheform 5:1=371+ 2¢1 (fit=_2f1- After solving these equations forthegiven initial conditions (r,(O) = <p,(0) =¢,(0) =0,f,(0) =—(l/800)), wegettheanswer given above. Disregarding thesmall quantities ofsecond order gives aneffect ofunder 1/800 oftheoneobtained (i.e., ontheorder of10meters ononeloop). Thus thelenscapdescribes a30kmellipse inanhour-and-a-half, returns tothespace shiponthesideopposite theearth, andgoes pastatthedistance ofafewtensofmeters. Ofcourse, inthiscalculation wehave disregarded thedeviation oftheorbit from acircle, theeffect offorces other than gravity, etc. 9Themotion ofapoint inthree-space Inthisparagraph wedeﬁne theangular momentum relative toanaxisandweshow that, for motion inanaxially symmetric ﬁeld, itisconserved. Alltheresults obtained formotion inaplane canbeeasily carried overtomotions inspace. AConservative ﬁelds Weconsider amotion intheconservative ﬁeld .. 5Ur=——,at where U=U(r), reE3. Thelawofconservation ofenergy holds: E:7=0,where E=gr’+U(r). BCentral ﬁelds Formotion inacentral ﬁeldthevector M=[r,t]doesnotchange: dM/dt = 0. Every central ﬁeldisconservative (this isproved asinthetwo-dimensional case), and M %—= [l",l"] -1-[l',i‘] =0, since i'=—(6U/6r), andthevector an/at iscollinear withrsince theﬁeldis central. 42 9;Themotion ofapoint inthree-space Corollary. Formotion inacentral ﬁeld, every orbit isplanar. PROOF. (M,r)=([r,1"],r)=0;therefore r(t)LM,and since M=const., allorbits lieintheplane perpendicular toM.2° 1:1 Thus thestudy oforbits inacentral ﬁeld inspace reduces totheplanar problem examined intheprevious paragraph. PROBLEM. Investigate motion inacentral ﬁeld inn-dimensional euclidean space. CAxially symmetric ﬁelds Deﬁnition. Avector ﬁeld inE3hasaxial symmetry ifitisinvariant with respect tothegroup ofrotations ofspace which ﬁxevery point ofsome axis. PROBLEM. Show thatifaﬁeld isaxially symmetric andconservative, then its potential energy hastheform U=U(r,z),where r,(p,andzarecylindrical coordinates. Inparticular, itfollows from thisthatthevectors oftheﬁeld lieinplanes through thezaxis. Asanexample ofsuch aﬁeld wecantake thegravitational ﬁeld created byasolid ofrevolution. ez F Or / r. 0| Figure 37Moment ofthevector Fwith respect toanaxis Letzbetheaxis, oriented bythevector e,inthree-dimensional euclidean space E3;Favector intheeuclidean linear space R3;0apoint onthezaxis; r=x—0e[R3theradius vector ofthepoint xeE3relative to0(Figure 37). Deﬁnition. The moment M,relative tothezaxis ofthevector Fapplied atthepoint ristheprojection onto thezaxisofthemoment ofthevector Frelative tosome point onthisaxis: M,=(e...[r,F1)- 2°ThecaseM=Oislefttothereader. 43 2:Investigation oftheequations ofmotion Thenumber M,does notdepend onthechoice ofthepoint 0onthe zaxis. Infact, ifwelook atapoint 0’ontheaxis, then byproperties ofthe triple product, M’,=(e,,[r’,F])=([e,, r'],F)=([e,, r],F)=M,. Remark. M,depends onthechoice ofthedirection ofthezaxis:ifwechange e,to—e,, then M,changes sign. Theorem. Foramotion inaconservative ﬁeld withaxial symmetry around the zaxis, themoment ofvelocity relative tothezaxisisconserved. PROOF. M,=(e,,[r,r]).Since i"=F,itfollows that randilieinaplane passing through thezaxis, andtherefore [r,r]isperpendicular toe,. Therefore, M,=(8,.[11fl)+(ez,[13F1)=0- U Remark. This proof works foranyforce ﬁeld inwhich theforce vector F liesintheplane spanned byrande,. 10Motions ofasystem ofnpoints Inthisparagraph weprove thelawsofconservation ofenergy. momentum, andangular momen- tumforsystems ofmaterial points inE3. AInternal andexternal forces Newton’s equations forthemotion ofasystem ofnmaterial points, with masses m,andradius vectors r,eE3aretheequations ii1,2,-..,n. Thevector F,iscalled theforce acting onthei-thpoint. Theforces F,aredetermined experimentally. Weoften observe ina system that fortwo points these forces areequal inmagnitude andact inopposite directions along thestraight linejoining thepoints (Figure 38). Ft; F/i iO—-P 4-——‘ Figure 38Forces ofinteraction Such forces arecalled forces ofinteraction (example: theforce ofuniversal gravitation). Ifallforces acting onapoint ofthesystem areforces ofinteraction, then thesystem issaid tobeclosed. Bydeﬁnition, theforce acting onthei-th point ofaclosed system is Ff 1' Z i=1 jii 44 10:Motions ofasystem ofnpoints Thevector F,-jistheforce with which thej-thpoint actsonthei-th. Since theforces F,-jandFJ,areopposite (FU=—-F,-i),wecanwrite them intheform F,1-=f,,-e,1-,where f»,-=fj,isthemagnitude oftheforce ande,-j istheunitvector inthedirection from thei-thpoint tothej-thpoint. Ifthesystem isnotclosed, then itisoften possible torepresent theforces acting onitintheform F,=ZF,,-+F;, where F,-,-areforces ofinteraction andF1-(r,-) istheso-called external force. Figure 39Internal andexternal forces EXAMPLE. (Figure 39)Weseparate aclosed system intotwoparts, IandII. Theforce F,applied tothei-thpoint ofsystem Iisdetermined byforces of interaction inside system Iandforces acting onthei-thpoint from points ofsystem II,i.e., 1eIj¢i Fistheexternal force withrespect tosystem I. BThelawofconservation ofmomentum Deﬁnition. Themomentum ofasystem isthevector P=_§1m;I",-. Theorem. Therateofchange ofmomentum ofasystem isequal tothesum ofallexternal forces acting onpoints ofthesystem. PROOF-dP/dl =Zi'=1mift =Zi'=i Ft=Zi.jFij 'f'ZrFi=ZrFl;Zt.jFtj = 0,since forforces ofinteraction F,j-—=-F,-,-. El Corollary I.Themomentum ofaclosed system isconserved. Corollary 2.Ifthesumoftheexterior forces acting onasystem isperpendicular tothexaxis, then theprojection P,ofthemomentum onto thexaxis is conserved :P,=const. 45 2:Investigation oftheequations ofmotion Deﬁnition. Thecenter ofmass ofasystem isthepoint m-r-FL; 2'"-" PROBLEM. Show thatthecenter ofmass iswelldeﬁned, i.e.,does notdepend onthechoice oftheorigin ofreference forradius vectors. Themomentum ofasystem isequal tothemomentum ofaparticle lying at thecenter ofmass ofthesystem andhaving mass Zm,. Infact,(Zm,)r=Z(m,r,-), from which itfollows that(Zm,-)i'=Zm,i',-. Wecannowformulate thetheorem about momentum asatheorem about themotion ofthecenter ofmass. Theorem. Thecenter ofmass ofasystem moves asifallmasses were concen- trated atitandallforces were applied toit. PROOF. (Zm,)1"=P.Therefore, (Zm,-)f=dP/dt =Z,F,-. El Corollary. Ifasystem isclosed, then itscenter ofmass moves uniformly andlinearly. CThelawofconservation ofangular momentum Deﬁnition. Theangular momentum ofamaterial point ofmass mrelative tothe point0, isthemoment ofthemomentum vector relative to0: M=[r,mi]. Theangular momentum ofasystem relative to0isthesumoftheangular momenta ofallthepoints inthesystem: M= [r,,m,-l",]. Theorem. Therateofchange oftheangular momentum ofasystem isequal tothesumofthemoments oftheexternal forces” acting onthepoints of thesystem. PROOF. dM/dt =XL, [i-,-,m,-I",-] +XL, [r,-,m,-i‘,-]. The ﬁrst terti isequal tozero, andthesecond isequal to F.1= Fr;+1%)]=tn.F21. u-. byNewton’s equations. Z‘Themoment offorce isalsocalled thetorque [Trans. note]. 46 10:Motions ofasystem ofnpoints Thesumofthemoments oftwoforces ofinteraction isequal tozero since F__F‘so[rmF +[rji Fji] _[(71_rj):Fij] =0- ij_ J"’ U ' — Therefore, thesum ofthemoments ofallforces ofinteraction isequal tozero: II Therefore, dM/dt =21;, [r,,F,]. [3 Corollary 1(The lawofconservation ofangular momentum). Ifthesystem isclosed, thenM=const. Wedenote thesum ofthemoments oftheexternal forces byN= Zi=1 [Tn Then, bythetheorem above, dM/dt =N,from which wehave Corollary Z.Ifthemoment oftheexternal forces relative tothezaxis is equal tozero, thenM,isconstant. DThelawofconservation ofenergy Deﬁnition. Thekinetic energy ofapoint ofmass mis m|"2 T Z T Deﬁnition. Thekinetic energy ofasystem ofmass points isthesum ofthe kinetic energies ofthepoints: "m~|"-2T=Z_'_', r=I2 where them,arethemasses ofthepoints andI",aretheir velocities. Theorem. Theincrease inthekinetic energy ofasystem isequal tothesumof thework ofallforces acting onthepoints ofthesystem. PROOF. I‘ II§=im.»<i.-J.) =Zt-t.m.r.~> =Z<t..F.~>.It1 1 1 -. u-. Therefore, T-1<r0>=f"i,—§dr= if'<r,.F.->di= in B ft) i=1 IQ 1 471 l 1l : 2:Investigation oftheequations ofmotion Theconﬁguration space ofasystem ofnmass points inE3isthedirect product ofneuclidean spaces:E3" =E3x xE3.Ithasitselfthe structure ofaeuclidean space. Letr=(r,,...,r,,)betheradius vector ofapoint intheconﬁguration space, andF=(F1,...,F,,)theforce vector. Wecanwrite thetheorem above intheform T(t,) —-T(t0) =fr(m(F, dr)=fl1(l", F)dt. "(fol 10 Inother words: The increase inkinetic energy isequal tothework ofthe“force” F onthe“path” r(t)inconﬁguration space. Deﬁnition. Asystem iscalled conservative iftheforces depend onlyonthe location ofapoint inthesystem (F=F(r)), andifthework ofFalong anypathdepends onlyontheinitial andﬁnalpoints ofthepath: M2J.(F,dr)=(D(M,,M2). M1 Theorem. Forasystem tobeconservative itisnecessary andsufficient that there exist apotential energy, i.e.,afunction U(r) such that GUF-F5 PROOF. Cf.Section 6B. El Theorem. Thetotal energy ofaconservative system (E=T+U)ispreserved under themotion .'E(t,) =E(t0). PROOF. Bywhat wasshown earlier, l'(!i)re.)—1(:.>=f(F.an=v(r<i.,>>-v<r<i.>>. B P110) Letalltheforces acting onthepoints ofasystem bedivided intoforces of interaction andexternal forces: 11 i#=j where F,,=—F,, f,,e,,. Proposition. Iftheforces ofinteraction depend only ondistance, f-,= f,-,(|r, —r,-I),then theyareconservative. 48 10:Motions ofasystem ofnpoints PROOF. Ifasystem consists entirely oftwopoints iandj,then, asiseasily seen, thepotential energy oftheinteraction isgiven bytheformula I‘v.~,~m=f/,,-(map. Wethenhave 6Uij(iri—rji) _ airi—rj| _"iii. -"~""—@i-.— O Therefore, thepotential energy oftheinteraction ofallthepoints willbe U(r) : Ui](iri “'T,"l)- D Iftheexternal forces arealso conservative, i.e.,F}=—(6U§-/6|’,-), then thesystem isconservative, anditstotal potential energy is U(r)=ZU,,+Z0;. i>j i Forsuch asystem thetotal mechanical energy 4.2 i i>j t isconserved. Ifthesystem isnotconservative, then thetotal mechanical energy isnot generally conserved. Deﬁnition. Adecrease inthemechanical energy E(t0) —E(t1) iscalled an increase inthenon-mechanical energy E’: E'(51) —E150) IE00) -‘E01)- Theorem (The lawofconservation ofenergy). Thetotal energy H=E+E’ isconserved. This theorem isanobvious corollary ofthedeﬁnition above. Itsvalue lies inthefactthatinconcrete physical systems, expressions forthesizeofthe non-mechanical energy canbefound interms ofother physical quantities (temperature, etc.). EExample: Thetwo-body problem Suppose thattwopoints with masses m,andm,interact with potential U, sothattheequations ofmotion have theform .. 5U .. 5U "1171 =-5‘: m2|'2 =—5:2“, U=U(l|'1'“ r2|)- 49l 2:Investigation oftheequations ofmotion Theorem. Thetime variation ofr=r,—r2inthetwo-body problem isthe same asthatforthemotion ofapoint ofmass m=mlmz/(ml +ml)ina ﬁeld withpotential U(|r|). Wedenote byrotheradius vector ofthecenter ofmass: ro= (m,r, +m2r2)/(m, +m2).Bythetheorem ontheconservation ofmomentum, thepoint r0moves uniformly andlinearly. Wenow look atthevector r=r,—r2.Multiplying theﬁrst ofthe equations ofmotion bym2,thesecond byml,andcomputing, weﬁndthat m,m2i‘ =—(m, +m2)(6U/dr), where U=U(|r, —Tgl)=U(|r|). Inparticular, inthecase ofaNewtonian attraction, thepoints describe conic sections with fociattheir common center ofmass (Figure 40). Figure 40Thetwobody problem PROBLEM. Determine themajor semi-axis oftheellipse which thecenter of theearth describes around thecommon center ofmass oftheearth andthe moon. Where isthiscenter ofmass, inside theearth oroutside? (The mass ofthemoon is1/81times themass oftheearth.) 11Themethod ofsimilarity Insome cases itispossible toobtain important information from theform oftheequations of motion without solving them. byusing themethods ofsimilarity anddimension. Themain idea inthese methods istochoose achange ofscale (oftime, length, mass. etc.) under which the equations ofmotion preserve their form. AExample Letr(t)satisfy theequation m(d3r/dt3) =-(av/at). Wesett,=atand m,=azm. Then r(t,) satisﬁes theequation m,-(dzr/dtf) =—(8U/dr). In other words: Ifthemass ofapoint isdecreased byafactor of4,then thepoint cantravel thesame orbit inthesame force ﬁeld twice asfast.” 23Here weareassuming that Udoes notdepend onm.Intheﬁeld ofgravity. thepotential energy Uisproportional tom,and therefore theacceleration does notdepend onthemass m ofthemoving point. 50 11:Themethod ofsimilarity BAproblem Suppose thatthepotential energy ofacentral ﬁeldisahomogeneous function ofdegree v: U(o:r) =o:“U(r) foranya>0. Show that ifacurve yistheorbit ofamotion, then thehomothetic curve ayisalsoanorbit (under theappropriate initial conditions). Determine theratio ofthecirculation times along these orbits. Deduce from thisthe isochronicity oftheoscillation ofapendulum (v=2)andKepler’s third law (v=-1). PROBLEM. Iftheradius ofaplanet isoztimes theradius oftheearth andits mass ,8times thatoftheearth, ﬁndtheratio oftheacceleration oftheforce ofgravity andtheﬁrstandsecond cosmic velocities tothecorresponding quantities fortheearth. ANSWER. y=Bot'2, 5=,/B/oz. Forthemoon, forexample, oi=1/3.7 and,3=I/81. Therefore, theaccel- eration ofgravity isabout 1/6that oftheearth (yz1/6), andthecosmic velocities areabout I/5those fortheearth (5zI/4.7). PROBLEM.23 Adesert animal hastocover great distances between sources of water. How does themaximal time theanimal canrundepend onthesize Lofthe animal? ANSWER. Itisdirectly proportional toL. Solution. Thestore ofwater isproportional tothevolume ofthebody, i.e.,L3;theevaporation isproportional tothesurface area, i.e.,L2.Therefore, themaximal timeofarunfrom onesource toanother isdirectly proportional toL. Wenotice that themaximal distance ananimal canrunalso grows proportionally toL(cf.thefollowing problem). PROBLEM.“ I-low does therunning velocity ofananimal onlevel ground anduphill depend onthesizeLoftheanimal? ANswER. Onlevelground ~L°,uphill ~L‘1. 23J.M.Smith. Mathematical Ideas inBiology. Cambridge University Press. 1968. 3‘lbid. 51 2:Investigation oftheequations ofmotion Solution. The power developed bytheanimal isproportional toL2 (thepercentage used bymuscle isconstant atabout 25%,theother 75‘X,of thechemical energy isconverted toheat; theheat output isproportional tothebody surface, i.e.,L2,which means that theeffective power ispro- portional toL2). The force ofairresistance isdirectly proportional tothesquare ofthe velocity andthearea ofacross-section; thepower spent onovercoming itistherefore proportional tov2L2v. Therefore, v3L2 ~L2,sov~L°.In fact, therunning velocity onlevel ground, nosmaller forarabbit than for ahorse, inpractice does notspeciﬁcally depend onthesize. Thepower necessary torunuphill ismgv~L3v; since thegenerated power is~L2,weﬁndthatv~L‘1.Infact,adogeasily runs upahill,while a horse slows itspace. PROBLEM. 24“How does theheight ofananimal’s jump depend onitssize? ANSWER. ~L°. Solution. Forajump ofheight honeneeds energy proportional toL311, andthework accomplished bymuscular strength Fisproportional toFL. Theforce Fisproportional toL2(since thestrength ofbones isproportional totheir section). Therefore, L3h~L2L, i.e.,theheight ofajump does not depend onthesizeoftheanimal. Infact, ajerboa andakangaroo canjump toapproximately thesame height. 14"Ibid. 52 PART II LAGRANGIAN MECHANICS Lagrangian mechanics describes motion inamechanical system bymeans of theconﬁguration space. Theconﬁguration space ofamechanical system has thestructure ofadifferentiable manifold, onwhich itsgroup ofdiffeo- morphisms acts. Thebasic ideas andtheorems oflagrangian mechanics are invariant under thisgroup,23 even ifformulated interms oflocal coordinates. Alagrangian mechanical system isgiven byamanifold (“conﬁguration space”) andafunction onitstangent bundle (“the lagrangian function”). Every one-parameter group ofdiffeomorphisms ofconﬁguration space which ﬁxes thelagrangian function deﬁnes aconservation law(i.e., aﬁrst integral oftheequations ofmotion). Anewtonian potential system isaparticular case ofalagrangian system (theconﬁguration space inthiscaseiseuclidean, andthelagrangian function isthedifference between thekinetic andpotential energies). Thelagrangian point ofview allows ustosolve completely aseries of important mechanical problems, including problems inthetheory ofsmall oscillations andinthedynamics ofarigid body. 23Andeven under larger groups oftransformations, which alsoaffect time. Variational principles Inthischapter weshow thatthemotions ofanewtonian potential system areextremals ofavariational principle, “Hamilton’s principle ofleast action.” This facthasmany important consequences, including aquick method forwriting equations ofmotion incurvilinear coordinate systems, and a series ofqualitative deductions-—for example, atheorem onreturning toa neighborhood oftheinitial point. Inthischapter wewilluseann-dimensional coordinate space. Avector insuch aspace isasetofnumbers x=(x,,...,x,,).Similarly, {if/ox means (df/6x,, ...,Elf/6x,,), and(a,b)=a,b1 + +a,,b,,. 12Calculus ofvariations Forwhat follows, wewillneed some facts from thecalculus ofvariations. Amore detailed exposition canbefound in“ACourse intheCalculus ofVariations” byM.A.Lavrentiev and L.A.Lusternik, M.L.,1938, orG.E.Shilov, “Elementary Functional Analysis," MIT Press, 1974. Thecalculus ofvariations isconcerned with theextremals offunctions whose domain isaninﬁnite-dimensional space: thespace ofcurves. Such functions arecalled functionals. Anexample ofafunctional isthelength ofacurve intheeuclidean plane: ify={(t,X)Ix(t)=x,tost51,},then<1>(y)=j,’;./1 +x3at Ingeneral, afunctional isanymapping from thespace ofcurves tothe realnumbers. Weconsider an“approximation” y’toy,y’={(t,x):x=x(t)+h(t)}. Wewillcallity’=y+h.Consider theincrement of<1),<l>(y+h)—(D(y) (Figure 41). 55l l 1 ...__.._........ l 1. i4 .1 -l 3:Variational principles X 7| X] 1'0 7 10 I r, ’f Figure 41Variation ofacurve AVariations Deﬁnition. Afunctional (Discalled diﬂerentiable2° if(D(y+h)—<D(y) = F+R,where Fdepends linearly onh(i.e.,foraﬁxed y,F(h,+hz)= F(h,)+F(I12)andF(ch)=cF(h)), andR(h, y)=O(h2) inthesense that, forIhl<sand Idh/dtl <s,wehave |R|<C52. The linear part ofthe increment, F(h), iscalled thediﬂerential. Itcanbeshown that if(Disdifferentiable, itsdifferential isuniquely deﬁned. Thedifferential ofafunctional isalsocalled itsvariation, andhis called avariation ofthecurve. EXAMPLE. Lety={(t,x):x=x(t),to3t3t,}beacurve inthe(t,x)-plane; x=dx/dt; L=L(a,b,c)adifferentiable function ofthree variables. We deﬁne afunctional (Dby It@111=ftote.x(t).on lo IncaseL=./1+b2,wegetthelength ofy. Theorem. Thefunctional (y) =I};L(x, x,t)dt isdijferentiable, and its derivative isgiven bytheformula '1atdat at'1F"”=l in-aal""‘+(@l ) to PROOF. 11 _on+ll)-(y)=I[L(x+h,X+h,t)-L(x,x,t)]dt I0 '=aL at. ,_fm[5h aXh]dt+O(h)-F(h)+R, 2°Weshould specify theclass ofcurves onwhich 11>isdeﬁned andthelinear space which con- tains h.One could assume. forexample, thatboth spaces consist oftheinﬁnitely differentiable functions. 56 12:Calculus ofvariations where 1| 8 . F(h) =I h+ h)dt and R=O(h2). Integrating byparts, weﬁndthat "0L. “d6L 6L“max t-—LhE(ai)dt+(haX)'o. El Deﬁnition. Anextremal ofadifferentiable functional (y)isacurve ysuchthat F(h) =0forallh. (Inexactly thesame waythatyisastationary point ofafunction ifthe differential isequal tozero atthatpoint.)BExtremals Theorem. The curve y:x=x(t) isanextremal ofthefunctional (I>(y) = If;L(x,x,t)dtonthespace ofcurves passing through thepoints x(t,,) =xo andx(t,) =x,,ifand only if 8L 6L% —E=0along thecurve x(t). Lemma. Ifacontinuous function f(t),to5tgt,satisﬁes ff;f(t)h(t)dt =0 foranycontinuous” function h(t)withh(t,,) =h(t1)=0,thenf(t)E0. h t*—-d 1* t*+d Yo A It 3 Figure 42Construction ofthefunction h PROOF OFTHELEMMA. Letf(t*) >0forsome t*,to<t*<t,.Since fis continuous, f(t)>cinsome neighborhood Aofthepoint t*:to<t*— d<t<t*+d<t1.Leth(t)besuch thath(t)=0outside A,h(t)>0inA, andh(t)=1inA/2(i.e., forts.t.t*—§d<t<t*+éd).Then, clearly, jig,f(t)h(t) 2dc>0(Figure 42).This contradiction shows thatf(t*)=0 forall1*,to<1*<t,. El PROOF OFTHETHEOREM. Bythepreceding theorem, =-d6L aL aL '1Fh=— — ——hd —~h() [at ﬁx]'+(axi 22Oreven foranyinﬁnitely differentiable function h. 571 1 l Y. 1l A___T_..5.,,____t 3:Variational principles Theterm after theintegral isequal tozerosince h(t,,) =h(t,) =0.Ifyisan extremal, then F(h)=0forallhwith h(t,,) =h(t,) =0.Therefore, fhf(r)h(r)dt =0. d0L 8L forallsuch h.Bythelemma, f(t)E0.Conversely, iff(t)E0,then clearly F(h)20. Elwhere EXAMPLE. Weverify thattheextremals oflength arestraight lines. Wehave: 6L 6L >2 d x2=“““*’ F“ afvrm di(¢r.wl=° >2 ./1+x2 CTheEuler-Lagrange equation d6L 6L_0 dtas dx— iscalled theEuler—Lagrange equation forthefunctional=c x=c, x=c,t+c2. Deﬁnition. Theequation It (D=IL(x,x,t)dt. lo Now letxbeavector inthen-dimensional coordinate space IR",y= {(t,x):x=x(t),to3t3t,} acurve inthe (n+1)-dimensional space R><R",andL:R"xR”><JR—+[Rafunction of2n+Ivariables. Asbefore, weshow: Theorem. Thecurve yisanextremal ofthefunctional <l>(y) =_l§,‘,L(x, 12,t)dt onthespace ofcurves joining (to,xo)and(t,,x1),ifand only ifthe Euler— Lagrange equation issatisﬁed along y. This isasystem ofnsecond-order equations, andthesolution depends on 2narbitrary constants. The 2nconditions x(t,,) =X0,x(t,) =X,areused forﬁnding them. PROBLEM. Citeexamples where there aremany extremals connecting two given points, andothers where there arenone atall. 58 13:Lagrange‘s equations DAnimportant remark Thecondition foracurve ytobeanextremal ofafunctional does notdepend onthechoice ofcoordinate system. Forexample, thesame functional—length ofacurve—is given incartesian andpolar coordinates bythedifferent formulas I1 11 (D,,,, :J‘./xi+xidt (D,,,,,=f./r2+r2:/‘>2dt. I0 lg Theextremals arethesame—straight lines intheplane. Theequations of lines incartesian andpolar coordinates aregiven bydifferent functions: x,=x,(t), x2=x,(t), andr=f(t), (p=(p(t). However, both these vector functions satisfy theEuler—Lagrange equation d5L 8L_0 ataxdxT only, intheﬁrst case, when x,,,,, =x,,x2andL,,,, =,/x2+>25,andin thesecond casewhen x,,,,,=r,(pandL,,,,,=./r2+r2¢12. Inthiswaywecaneasily describe inanycoordinates adifferential equa- tionforthefamily ofallstraight lines. PROBLEM. Find thedifferential equation forthefamily ofallstraight lines intheplane inpolar coordinates. 13Lagrange’s equations Here weindicate thevariational principle whose extremals aresolutions ofNewton’s equations ofmotion inapotential system. Wecompare Newton’s equations ofdynamics d 3U<1) Etm.-r.-)+°7=0 withtheEuler—Lagrange equation d6L 0L—0ItE_Eli_' AHamilton’s principle ofleast action Theorem. Motions ofthemechanical system (1)coincide with extremals of thefunctional I1 (I)(y) =ILdt, whereL= T-— U isthedifference between thekinetic andpotential energy. 59l l1 =15! It|. ifll it1 I1 l .11 l “‘"“*M--*"" t l l l l 2 ll 3:Variational principles PROOF. SinceU=U(r)andT=Zm,1",2/2,wehaveat/at, =ar/at, =m,t, andat/at, =—0U/8r,. El Corollary. Let(q,,...,q3,,)beanycoordinates intheconﬁguration space of asystem ofnmass points. Then theevolution ofqwithtimeissubject tothe Euler-Lagrange equations dEL 5L Wh€l'€L= T— PROOF. Bythetheorem above, amotion isanextremal ofthefunctional ILdt.Therefore, inanysystem ofcoordinates theEuIer—Lagrange equation written inthatcoordinate system issatisﬁed. El Deﬁnition. Inmechanics weusethefollowing terminology: L(q,1'],t)=T—U istheLagrange function orlagrangian, q,arethegeneralized coordinates, q,are generalized velocities, at/aq, =p,are generalized momenta, 6L/oq, aregeneralized forces, fl;L(q,t'], t)dtistheaction, (d(6L/651,)/dt) —(dL/5410 =0areLagrange’s equations. Thelasttheorem iscalled “Hamilton’s form oftheprinciple ofleast motion” because inmany cases theaction q(t)isnotonly anextremal but isalsoaminimum value oftheaction functional ff;Ldt. BThesimplest examples EXAMPLE 1.Forafreemass point inE3, mt"2 L: ii‘T 2, incartesian coordinates q,=r,weﬁnd m. . .L=5(qi+615+qt)- Here thegeneralized velocities arethecomponents ofthevelocity vector, thegeneralized momenta p,=mq,arethecomponents ofthemomentum vector, and Lagrange’s equations coincide with Newton’s equations dp/dt =0.The extremals arestraight lines. Itfollows from Hamilton’s principle thatstraight lines arenotonly shortest (i.e.,extremals ofthelength _l§,‘,./qt +41%+4;dt)butalsoextremals oftheaction jig,(qt+qt+q§)dt. PROBLEM. Show thatthisextremum isaminimum. EXAMPLE 2.Weconsider planar motion inacentral ﬁeldinpolar coordinates q,=r,q,=rp.From therelation 1"=re,+<t'>re,,, weﬁndthekinetic energy 60 14:Legendre transformations T=§mi'2 =%m(r2 +r22) and thelagrangian L(q, ('1)=T(q, q)—U(q), where U=U(q,). Thegeneralized momenta willbep=at/aq, i.e., P1=mi F2=mr2¢- TheﬁrstLagrange equation p,=0L/oql takes theform ...,av mr=mr(p —-5;. Wealready obtained thisequation inSection 8. Since qz=(pdoes notenter intoL,wehave 6L/oqz =0.Therefore, the second Lagrange equation willbep2=0,p,=const. This isthelawof conservation ofangular momentum. Ingeneral, when theﬁeld isnotcentral (U=U(r,<p)),weﬁnd pg= —6U/dip. This equation canberewritten intheform d(M, e,)/dt =N,where N=([r,F],e,)andF=-av/at. (The rateofchange inangular momentum relative tothezaxisisequal tothemoment oftheforce Frelative tothe zaxis.) Infact,wehave dU=(<'iU/0r)dr +(<3U/6go)d(p =—~(F, dr)=—(F, e,)dr — r(F,e,,,)d(p; therefore, —0U/dtp =r(F,e,,,)=r([e,, F],6,)=([r,F],e,). This example suggests thefollowing generalization ofthelawofcon- servation ofangular momentum. Deﬁnition. Acoordinate q,iscalled cyclic ifitdoes notenter into the lagrangian: 0L/dq, =0. Theorem. Thegeneralized momentum corresponding toacyclic coordinate is conserved: p,=const. PROOF. ByLagrange’s equation dp,/dt =5L/oq, =0. U I4Legendre transformations TheLegendre transformation isavery useful mathematical tool: ittransforms functions ona vector space tofunctions onthedual space. Legendre transformations arerelated toprojective duality andtangential coordinates inalgebraic geometry andtheconstruction ofdual Banach spaces inanalysis. They areoften encountered inphysics (forexample, inthedeﬁnition of thermodynamic quantities). ADeﬁnition Lety=f(x)beaconvex function, f”(x)>0. TheLegendre transformation ofthefunction fisanew function gofa newvariable p,which isconstructed inthefollowing way (Figure 43).We draw thegraph offinthex,yplane. Letpbeagiven number. Consider the 61 3:Variational principles y f(x) 11 g(p) Xx(p) Figure 43Legendre transformation straight liney=px.Wetakethepoint x=x(p)atwhich thecurve isfarthest from thestraight lineinthevertical direction: foreach pthefunction px— f(x)=F(p,x)hasamaximum with respect toxatthepoint x(p). Now we deﬁne g(p) =F(p,x(p)). The point x(p) isdeﬁned bytheextremal condition 8F/ﬁx =0,i.e., f'(x)=p.Since fisconvex, thepoint x(p)isunique.“ PROBLEM. Show thatthedomain ofgcanbeapoint, aclosed interval, orarayiffisdeﬁned onthewhole xaxis.Prove thatiffisdeﬁned onaclosed interval, thengisdeﬁned onthewhole p axis. BExamples EXAMPLE 1.Letf(x) =x2.Then F(p,x)=px—x2,x(p) ==,1;p,g(p)=§p2. EXAMPLE 2.Letf(x)=mx2/2. Then g(p) =p2/2m. EXAMPLE 3.Letf(x)=x“/at. Then g(p)=pl’/B, where (1/oz) +(1/B) =1 (at>1,,8>1). 1' 3 8' P1 2 3 I P0 PxP0 2Pr I Figure 44Legendre transformation taking anangle toalinesegment EXAMPLE 4.Letf(x)beaconvex polygon. Then g(p)isalsoaconvex polygon, inwhich thevertices off(x)correspond totheedges ofg(p),andtheedges of f(x)tothevertices ofg(p). Forexample, thecorner depicted inFigure 44is transformed toasegment under theLegendre transformation. 22Ifitexists. 62 14:Legendre transformations CInvolutivity Letusconsider afunction fwhich isdifferentiable asmany times asnecessary, withf”(x)>0.Itiseasy toverify thataLegendre transformation takes convex functions toconvex functions. Therefore, wecanapply ittwice. Theorem. The Legendre transformation isinvolutive, i.e.,itssquare isthe identity: ifunder theLegendre transformation fistaken tog,then the Legendre transform ofgwillagain bef. PROOF. Inorder toapply theLegendre transform tog,with variable p,we must bydeﬁnition lookatanewindependent variable (which wewillcallx), construct thefunction G(x,iv)=Xv—g(p). andﬁnd thepoint p(x) atwhich Gattains itsmaximum: 8G/op =0,i.e., g'(p) =x.Then theLegendre transform ofg(p)willbethefunction ofx equal toG(x, p(x)). Wewillshow thatG(x, p(x)) =f(x).Tothisendwenotice thatG(x,p)= xp—g(p)hasasimple geometric interpretation: itistheordinate ofthe point with abscissa xonthelinetangent tothegraph off(x)with slope p J’ f(x)P X0 X Y Figure 45Involutivity oftheLegendre transformation (Figure 45).Forﬁxed p,thefunction G(x,p)isalinear function ofx,with 6G/ox =p,andforx=x(p) wehave G(x,p)=xp—g(p) =f(x)bythe deﬁnition ofg(p). Letusnow ﬁxx=xoandvary p.Then thevalues ofG(x,p)willbethe ordinates ofthepoints ofintersection ofthelinex=xowith thelinetangent tothegraph off(x)with various slopes p.Bytheconvexity ofthegraph it follows thatallthese tangents liebelow thecurve, andtherefore themaximum ofG(x,p)foraﬁxed x(p,,) isequal tof(x)(and isachieved forp=p(X0) = f'(><o))- El 63l l l i _..»-.,.-r-.--lupi- I 1 l 1 1 1 1 l 3:Variational principles J. g(p) f(x) .\. Figure 46Legendre transformation ofaquadratic form Corollary.29 Consider agiven family ofstraight lines y=px—g(p). Then itsenvelope hastheequation y=f(x), where fistheLegendre transform ofg. DYoung’s inequality Deﬁnition. Two functions, fandg,which aretheLegendre transforms of oneanother arecalled dual inthesense ofYoung. Bydeﬁnition oftheLegendre transform, F(x,p)=px—f(x)isless than orequal tog(p)foranyxandp.From thiswehave Young’s inequality: PX3f(x)+g(p)- EXAMPLE 1.Iff(x)=%x2, then g(p) =%p2andweobtain thewell-known inequality px3%x2+%p2forallxandp. EXAMPLE 2.Iff(x) =x“/oi, then g(p) =pf/,8, where (1/0:) +(1/B) =1,and weobtain Young’s inequality px3(x3/oz) +(pﬁ/,3) forallx>0,p>0, <1>1,/t >1,and(1/oz) +(1//1)=1. EThecase ofmanyvariables Now letf(X)beaconvex function ofthevector variable x=(xl,...,x,,) (i.e., thequadratic form ((Zi2f/r'ix2)dx,dx) ispositive deﬁnite). Then the Legendre transform isthe function g(p)ofthevector variable p=(p,,...,p,,), deﬁned asabove bytheequalities g(P) =F(p,x(p)) =max, F(p,x),where F(P,X)=(P.X)—f(X)andP=5./‘/@X- Alloftheabove arguments, including Young’s inequality, canbecarried over without change tothiscase. PROBLEM. Letf:IR"->[Rbeaconvex function. LetR"*denote thedual vector space. Show that theformulas above completely deﬁne themapping g:lR"*—>IR(under thecondition thatthelinear form dfI,ranges over allof R"when xranges over R"). 29Onecaneasily seethatthisisthetheory of“Clairaufs equation “ 64 15:Hamilton’s equations PROBLEM. Letfbethequadratic form f(x)=Zf,-xix,-. Show that its Legendre transform isagain aquadratic form g(p)=ig;jpip1,andthatthe values ofboth forms atcorresponding points coincide (Figure 46): f(X(P)) =Q(P) and y(P(X)) =f(X)- 15Hamilton’s equations Bymeans ofaLegendre transformation, alagrangian system ofsecond-order differential equations isconverted intoaremarkably symmetrical system of2nﬁrst-order equations called ahamiltonian system ofequations (orcanonical equations). AEquivalence ofLagrange’s andHamilton’s equations Weconsider thesystem ofLagrange’s equations p=6L/éiq, where p= 0L/81], withagiven lagrangian function L:IR"xR"xIR—>IR,which wewill assume tobeconvex” withrespect tothesecond argument 1']. Theorem. Thesystem ofLagrange’s equations isequivalent tothesystem of 2nﬁrst-order equations (HamiIt0n’s equations) __ 6H __8H q_ 7 where H(p, q,t)=pi]—L(q,1'],t)istheLegendre transform ofthelagrang- ianfunction viewed asafunction of1']. PROOF. Bydeﬁnition, theLegendre transform ofL(q,q,t)with respect to1'] isthefunction H(p) =pi]—L(t'|), inwhich 1'1isexpressed interms ofp bytheformula p=0L/06], andwhich depends ontheparameters qandt. Thisfunction Hiscalled thehamiltonian. Thetotal differential ofthehamiltonian 6H OH 0H=— —d —-d dH apdp+aq q+at t isequal tothetotal differential ofpq—Lforp=BL/61']: _ 0L 0L Both expressions fordHmust bethesame. Therefore, __aH aH__aL @__ai q_6p 041- ﬁq 0t_ 6t' 3°Inpractice thisconvex function willoften beapositive deﬁnite quadratic form. 65‘i I E1lr ms.-.._——_;: 2It i g.ll’ ii1. ii il i i 1 ~iH|lli 3:Variational principles Applying Lagrange’s equations 1')=6L/dq, weobtain Hamilton’s equa- tions. Wehave seen that, ifq(t)satisﬁes Lagrange’s equations, then (p(t), q(t)) satisﬁes Hamilton’s equations. The converse isproved inananalogous manner. Therefore, thesystems ofLagrange andHamilton areequivalent. El Remark. Thetheorem justproved applies toallvariational problems, not justtothelagrangian equations ofmechanics. BHamilton’s function andenergy EXAMPLE. Suppose now that theequations aremechanical, sothat the lagrangian hastheusual form L=T—U,where thekinetic energy Tisa quadratic form with respect toq; T= an-£1,-c'1J-, where ai,=a,-,(q, t)andU=U(q). Theorem. Under thegiven assumptions, thehamiltonian Histhetotal energy H=T+U. Theproof isbased onthefollowing lemma ontheLegendre transform of aquadratic form. Lemma. Thevalues ofaquadratic form f(X)andofitsLegendre transform g(p)coincide atcorresponding points: f(X)=g(p). EXAMPLE. Fortheform f(x)=x2thisisawell-known property ofatangent toaparabola. For theform f(x)=%mx2 wehave p=mxand g(p)= p2/2)’)? =mxz/2 =f(x). PROOF OFTHELEMMA ByEuler’s theorem onhomogeneous functions (af/am =2f.Therefore, g(p(x)) =px—f(X)=(of/am —f=2f(X)— f(X)=f(X) El PRooE OFTHETHEOREM. Reasoning asinthelemma, weﬁndthatH=pq— L=2T—(T—U)=T+U. El EXAMPLE. Forone-dimensional motion ..__<9£ Inthiscase T=%q2, U=U(q), p=q,H=%p2+U(q) and Hamilton’s equations take theform 4=P -__<ll p— aq‘ 66 15:Hamilton's equations This example makes iteasy toremember which ofHamilton’s equations hasaminus sign. Several important corollaries follow from thetheorem ontheequivalence oftheequations ofmotion toahamiltonian system. Forexample, thelawof conservation ofenergy takes thesimple form: Corollary l.dH/dt =5H/dt. Inparticular, forasystem whose hamiltonian function doesnotdepend explicitly ontime(6H/0t =0),thelawofconserva- tionofthehamiltonian function holds: H(p(t), q(t)) =const. PROOF. Weconsider thevariation inHalong thetrajectory H(p(t), q(t),t). Then, byHamilton’s equations, dH__6H _6H +6H5H+6H_6H D dttop oq 6qap at_6t' CCyclic coordinates When considering central ﬁelds, wenoticed thataproblem could bereduced toaone-dimensional problem bytheintroduction ofpolar coordinates. It turns outthat, given anysymmetry ofaproblem allowing ustochoose a system ofcoordinates qinsuch away that thehamiltonian function is independent ofsome ofthecoordinates, wecanﬁndsome ﬁrstintegrals and thereby reduce toaproblem inasmaller number ofcoordinates. Deﬁnition. Ifacoordinate qldoes notenter into thehamiltonian function H(p1,p2,...,p,;q1,..., q,,;t),i.e.,6H/dql =0,then itiscalled cyclic (theterm comes from theparticular caseoftheangular coordinate ina central ﬁeld). Clearly, thecoordinate qliscyclic ifandonly ifitdoes notenter intothe lagrangian function (6L/dql =0).Itfollows from thehamiltonian form of theequations ofmotion that: Corollary 2.Letq,beacyclic coordinate. Then plisaﬁrst integral. Inthis casethevariation oftheremaining coordinates withtimeisthesame asina system withthen—1independent coordinates qz,...,q,,andwithhamilton- ianfunction H(pZ!"‘1pn>q2>"'!q7|It1c)s depending ontheparameter c=pl. PROOF. Wesetp’=(p2,...,p,,)and q’=(q2,..., q,,).Then Hamilton’s equations take theform ‘1'_@ .4.J31dtq'5])’ at"1'ap, 1/__‘ll’ 1‘._0dtp_aq' dt"“' 67 3:Variational principles Thelastequation shows thatpl=const. Therefore, inthesystem ofequations forp’andq’,thevalue ofplenters only asaparameter inthehamiltonian function. After thissystem ofZn—2equations ISsolved, theequation forql takes theform 5gt.=rm.whererm=-5Hm.pm.qt».I) andiseasily integrated. El Almost allthesolved problems inmechanics have been solved bymeans ofCorollary 2. Corollary 3.Every closed system withtwodegrees offreedom (n=2)which has acyclic coordinate isintegrable. PROOF. Inthiscase thesystem forp’andq’isone-dimensional andisim- mediately integrated bymeans oftheintegral H(p’, q’)=c. El 16Liouville’s theorem Thephase ﬂowofHamilton’s equations preserves phase volume. Itfollows, forexample, thata hamiltonian system cannot beasymptotically stable. Forsimplicity welook atthecasein which thehamiltonian function does notdepend explicitly onthetime: H=H(p,q). AThephase ﬂow Deﬁnition. The2n-dimensional space with coordinates pl,...,p,,;ql,...,q,, iscalled phase space. EXAMPLE. Inthecasen=lthisisthephase plane ofthesystem it=——5U/ox, which weconsidered inSection 4. Just asinthis simplest example, theright-hand sides ofHamilton’s equations give avector ﬁeld: ateach point (p,q)ofphase space there isa 2n-dimensional vector (—6H/iiq, 6H/dp). Weassume thatevery solution of Hamilton’s equations canbeextended tothewhole time axis.“ Deﬁnition, The phase ﬂow istheone-parameter group oftransformations ofphase space 9'1(M0).q(0))*—>(P(I).q(t)). where p(t)and q(t) aresolutions ofHamilton’s system ofequations (Figure 47). PROBLEM. Show that{g‘}isagroup. 3‘Forthisitissulﬁcient, forexample, thatthelevel setsofHbecompact. 68 16:Liouville’s theorem f gt (v(t).q(t)) (P(0). q(0)) Figure 47Phase ﬂow 17 I l BLiouville stheorem Theorem I.Thephase ﬂow preserves volume: foranyregion Dwehave (Figure 48) 1 volume ofg‘D=volume ofD. Wewillprove thefollowing slightly more general proposition also duetoLiouville. %3@ Figure 48Conservation ofvolume Suppose wearegiven asystem ofordinary differential equations it=f(x),x=(xl,...,x,,),whose solution maybeextended tothewhole timeaxis. Let{g'}bethecorresponding group oftransformations: (1) g‘(x) =X+f(x)t +O(t2), (t—>0). LetD(0)bearegion inx-space andv(0)itsvolume; v(t)=volume ofD(t) D(t) =g‘D(0). Theorem 2.IfdivfEO,theng’preserves volume: v(t)=v(0). CProof Lemmal. (dv/dt)I,=0 =lbw, divfdx (dx=dxl---dx,,). PROOF. Foranyt,theformula forchanging variables inamultiple integral gives 6Iv(t)=Idetixdx.0(0) ﬁx Calculating 0g'x/dx byformula (1),weﬁnd dg'x at 2F-E+&t+O(t) ast—>0. 69 3:Variational principles Wewillnowuseawell-known algebraic fact: Lemma 2.Foranymatrix A=(al1-), det(E +At)=1+ttrA+O(t’), t->0, where trA=2§'=l allisthetrace ofA(thesumofthediagonal elements). (The proof ofLemma 2isobtained byadirect expansion ofthedeter- minant: wegetlandnterms int;theremaining terms involve t2,t3,etc.) Using this, wehave 6g'X (if 2detg-1+ mg +0(1). Buttr(if/6x =Z§'=l (if,/6x, =divf.Therefore, v(t)=J[1+tdivf +O(t2)]dx, D(0) which proves Lemma 1. Cl PROOF OFTHEOREM 2.Since r=toisnoworse than t=0,Lemma 1canbe written intheform dvim =I divfdx dtI=Io Dtto) ’ andifdivf E0,dv/dt E0. Cl Inparticular, forHamilton’s equations wehave . 6 5H) 6<6”)df=——— +‘— E0.Wdph@q@q"ﬁn This proves Liouville’s theorem (Theorem 1). l:l PROBLEM. Prove Liouville’s formula W=VVOel"""' fortheWronskian determinant ofthelinear system X=A(t)X. Liouville’s theorem hasmany applications. PROBLEM. Show that inahamiltonian system itisimpossible tohave asymptotically stable equilibrium positions andasymptotically stable limit cycles inthephase space. e Liouville’s theorem hasparticularly important applications instatistical mechanics. 70 16:Liouville’s theorem Liouville’s theorem allows onetoapply methods ofergodic theory” to thestudy ofmechanics. Weconsider only thesimplest example: DPoincaré’s recurrence theorem Letgbeavolume-preserving continuous one-to-one mapping which maps abounded region Dofeuclidean space onto itself: gD=D. Then inanyneighborhood Uofanypoint ofDthere isapoint xeU which returns toU,i.e.,g"x6Uforsome n>O. U X] Figure 49Thewayaballwillmove inanasymmetrical cupisunknown; however Poincaréfs theorem predicts thatitwillreturn toaneighborhood oftheoriginal position. This theorem applies, forexample, tothephase ﬂow g’ofatwo-dimen- sional system whose potential U(xl,x2)goes toinﬁnity as(xl,x2)——>oo;in thiscasetheinvariant bounded region inphase space isgiven bythecondition (Figure 49) D= {p,q:T+ U$E}. Poincaré’s theorem can bestrengthened, showing that almost every moving point returns repeatedly tothevicinity ofitsinitial position. This is oneofthefewgeneral conclusions which canbedrawn about thecharacter ofmotion. Thedetails ofmotion arenotknown atall,even inthecase GUX=—E, where x=(xl,x2). Thefollowing prediction isaparadoxical conclusion from thetheorems ofPoincare and Liouville: ifyouopen apartition separating achamber containing gasandachamber with avacuum, then after awhile thegas molecules willagain collect intheﬁrstchamber (Figure 50). Theresolution oftheparadox liesinthefactthat“awhile” maybelonger than theduration ofthesolar system’s existence. 32Cf,forexample, thebook: Halmos, Lectures onErgodic Theory, 1956 (Mathematical Society ofJapan. Publications. No.3). 71 3:Variational principles Figure 50Molecules return totheﬁrstchamber. D U@ gU til’2211 Figure 51Theorem onreturning PROOF OFPoiNcARE’s THEOREM. Weconsider theimages oftheneighborhood U(Figure 51): U,gU,g2U,...,g"U,... Allofthese have thesame volume. Ifthey never intersected, Dwould have inﬁnite volume. Therefore, forsome k20andI20,withk>I, g"Ung'U=,éQ. Therefore, g"“U nUaéQ.Ifyisinthisintersection, then y=g"x, with xeU(n=k—I).ThenxeUandg"xeU(n=k—l). [1 EApplications ofPoincaré’s theorem EXAMPLE 1.LetDbeacircle andgrotation through anangle a.Ifat= 21t(m/n), theng"istheidentity, andthetheorem isobvious. Ifatisnotcommen- surable with 21:,then Poincaré’s theorem gives V5>0,Eln:|g"x ~xl<6 (Figure 52). fl X‘sX 8,, e2X 23X Figure 52Dense setonthecircle 72 16:Liouville‘s theorem Iteasily follows that Theorem. Ifozaé2rr(m/n), then thesetofpoints g"xisdense“ onthecircle (k=l,2,...). PROBLEM. Show thatevery orbit ofmotion inacentral ﬁeld with U=r4is either closed ordensely ﬁllstheringbetween twocircles. EXAMPLE 2.LetDbethetwo-dimensional torus and(pland(p2angular coordinates onit(longitude andlatitude) (Figure 53). ‘P2 WI Figure 53Torus Consider thesystem ofordinary differential equations onthetorus (Pi=11 (P2=<12- Clearly, divf=0andthecorresponding motion Qt?($1, (P2)“*(Q91'l'alt,(P2‘l'aztl preserves thevolume d<pldrpz. From Poincare’s theorem itiseasy todeduce Theorem. Ifoil/(7.2isirrational, thenthe“winding Iine” onthetorus, g‘(<,ol, (pl), isdense inthetorus. PROBLEM. Show thatifwisirrational, then theLissajous ﬁgure (x=cost, y=coswt)isdense inthesquare Ixl31,lyl31. EXAMPLE 3.LetDbethen-dimensional torus T",i.e.,thedirect product“ ofncircles: D=S‘ xS1><---><S‘=T". i--mf--ma Apoint onthen-dimensional torus isgiven bynangular coordinates rp=(<pl,...,<p,,).Letoi=(al,...,<1,,),andletg’bethevolume-preserving transformation g'jT"—+T" q)—~>(|)-l-Eli. 33AsetAisdense inBifthere isapoint ofAinevery neighborhood ofevery point ofB. 3‘Thedirect product ofthe setsA,B,...isthesetofpoints (a,b,...),with aeA,bEB,.... 73 3:Variational principles PROBLEM. Under which conditions on:1arethefollowing setsdense :(a)the trajectory {g'tp}; (b)thetrajectory {g"tp} (tbelongs tothegroup ofreal numbers ER,ktothegroup ofintegers Z). The transformations inExamples lto3areclosely connected to mechanics. Butsince Poincaré’s theorem isabstract, italsohasapplications unconnected with mechanics. EXAMPLE 4.Consider theﬁrstdigits ofthenumbers 2“:1,2,4,8,1,3,6,1,2, 5,1,2,4,.... PROBLEM. Does thedigit 7appear inthissequence? Which digit appears more often, 7or8?How many times more often‘? 74 Lagrangian mechanics onmanifolds Inthischapter weintroduce theconcepts ofadifferentiable manifold and itstangent bundle. Alagrangian function, given onthetangent bundle, deﬁnes alagrangian “holonomic system” onamanifold. Systems ofpoint masses with holonomic constraints (e.g., apendulum orarigid body) are special cases. 17Holonomic constraints Inthisparagraph wedeﬁne thenotion ofasystem ofpoint masses with holonomic constraints. AExample Letybeasmooth curve intheplane. Ifthere isavery strong force ﬁeld ina neighborhood ofy,directed towards thecurve, then amoving point will always beclose to1,‘.Inthelimit caseofaninﬁnite force ﬁeld, thepoint must remain onthecurve y.Inthiscase wesaythat aconstraint isputonthe system (Figure 54). Toformulate thisprecisely, weintroduce curvilinear coordinates qland qlonaneighborhood ofy;qlisinthedirection ofyandqzisdistance from thecurve. Weconsider thesystem with potential energy U~=Nqi+U0(q1. qt). depending ontheparameter N(which wewilllettend toinﬁnity) (Figure 55). Weconsider theinitial conditions ony: 411(0): q(t) ¢l1(O) = q2(0) =0 (l2(0) =O- 75I 4 4:Lagrangian mechanics onmanifolds ‘12 ~ ﬁll \\\\\\h Figure 54Constraint asaninﬁnitely strong ﬁeld U U0 7 Figure 55Potential energy UN Denote byql=<p(t,N)theevolution ofthecoordinate qlunder amotion with these initial conditions intheﬁeld UN. Theorem. Thefollowing limit exists, asN—>oo: lim(i.N)=i//(i)- N->00 Thelimit ql=i//(t)satisﬁes Lagrange’s equation d6L* _dL,,, dl691 _5411’ where L,(q,,q,) =Tlq2:q'2=0 —U0|q,=o (Tisthekinetic energy of motion along y). Thus, asN~>oo,Lagrange’s equations forqlandqzinduce Lagrange’s equation forql=t//(t). Weobtain exactly thesame result ifwereplace theplane bythe3n- dimensional conﬁguration space ofnpoints, consisting ofamechanical system with metric dsz=Zjlzl mldrlz(themlaremasses), replace thecurve y byasubmanifold ofthe3n-dimensional space, replace qlbysome coordinates qlony,andreplace qzbysome coordinates qlinthedirections perpendicular toy.Ifthepotential energy hastheform U=U0(q1.q1) +Nqi, then asN—>oo,amotion onyisdeﬁned byLagrange’s equations with the lagrangian function L=|<=Tlq2=¢'l2=0 —U9lq2=0' 76 18:Dillerentiable manifolds BDeﬁnition ofasystem with constraints Wewillnotprove thetheorem above,“ butneither willweuseit.Weneed itonly tojustify thefollowing. Deﬁnition. Letybeanm-dimensional surface inthe3n-dimensional con- ﬁguration space ofthepoints rl,...,r,,with masses ml,...,m,,. Let q=(ql,...,q,,,)besome coordinates onyzrl=r,-(q). The system described bytheequations =3%L=time +um) iscalled asystem ofnpoints with 3n—mideal holonomic constraints. Thesurface yiscalled theconﬁguration space ofthesystem withconstraints. Ifthesurface yisgiven byk=3n—mfunctionally independent equations fl(r) =0,...,f,,(r) =0,then wesaythat thesystem iscon- strained bytherelations fl=O,...,ﬂl=0. Holonomic constraints alsocould have been deﬁned asthelimiting case ofasystem with alarge potential energy. Themeaning ofthese constraints in mechanics liesintheexperimentally determined factthatmany mechanical systems belong tothisclass more orlessexactly. From now on,forconvenience. Wewillcallideal holonomic constraints simply constraints. Other constraints willnotbeconsidered inthisbook. 18Differentiable manifolds Theconﬁguration space ofasystem with constraints isadifferentiable manifold. Inthispara- graph wegivetheelementary facts about differentiable manifolds. ADeﬁnition ofadiﬂerentiable manifold AsetMisgiven thestructure ofadifferentiable manifold ifMisprovided withaﬁnite orcountable collection ofcharts, sothatevery point isrepresented inatleast onechart. Achart isanopen setUintheeuclidean coordinate space q=(ql,...,q,,), together with aone-to-one mapping (pofUonto some subset ofM, (p:U-><pU<:M. Weassume thatifpoints pandp’intwocharts UandU’have thesame image inM,then pandp’have neighborhoods VcUandV’CU’with the same image inM(Figure 56).Inthiswaywegetamapping 0"1(pIV—>V’. This isamapping oftheregion Voftheeuclidean space qonto theregion V’oftheeuclidean space q’,anditisgiven bynfunctions ofnvariables, 35Theproof isbased onthefactthat, duetotheconservation ofenergy, amoving point cannot move further from ythan cN'‘/2,which approaches zero asN—>00. 77w 4:Lagrangian mechanics onmanifolds ii‘'6\\\\‘-(¢.'.§IV, l.\\“' q 0'w Figure 56Compatible charts q’=q’(q), (q=q(q’)). The charts Uand U'arecalled compatible ifthese functions aredifferentiable.“ Anatlas isaunion ofcompatible charts. Two atlases areequivalent if their union isalsoanatlas. Adifferentiable manifold isaclass ofequivalent atlases. Wewillconsider only connected manifolds.” Then thenumber nwillbethesame forall charts; itiscalled thedimension ofthemanifold. Aneighborhood ofapoint onamanifold istheimage under amapping cp:U~>Mofaneighborhood oftherepresentation ofthispoint inachart U. Wewill assume that every two different points have non-intersecting neighborhoods. BExamples EXAMPLE l.Euclidean space IR"isamanifold, with anatlas consisting ofonechart. EXAMPLE 2.Thesphere S2={(x,y,z):xz+yz+:2=l}hasthestructure ofamanifold. with atlas, forexample, consisting oftwocharts (Ul,(pl-,i=1,2)instereographic projection (Figure 57).An analogous construction applies tothen-sphere Sn = {(x1, -.-,xn+|):Zxi2 I Eb” \"1 U, ' Figure 57Atlas ofasphere EXAMPLE 3.Consider aplanar pendulum. Itsconﬁguration space—the circle S‘~isamanifold. Theusual atlas isfurnished bytheangular coordinates (pfIR‘—>S‘.Ul=(-n, rt),U2=(0,211) (Figure 58). EXAMPLE 4.Theconﬁguration space ofthe“spherical” mathematical dimensional sphere S2(Figure 58). 36Bpendulum isth ydifferentiabl (letwo- ehere wemean rtimes continuously differentiable: th 3r3ac)isimmaterial (wemaytaker=ac,forex 7Amanifold isconnected ifiteexact val ample). cannot bedi' ' 78ueofr vided intot ''' wodisjoint open subsets. 18:Diﬂerentiable manifoldsoi... Figure 58Planar, spherical anddouble planar pendulums EXAMPLE 5.Theconﬁguration space ofa“planar double pendulum” isthedirect product oftwo circles, i.e.,thetwo-torus T2=S‘xS‘(Figure 58). EXAMPLE 6.Theconﬁguration space ofaspherical double pendulum isthedirect product of twospheres, S2xS2. EXAMPLE 7.Arigid linesegment inthe(ql.ql)-plane hasforitsconﬁguration space themani- foldR2xS‘,with coordinates ql,ql,ql(Figure 59).Itiscovered bytwocharts. ‘T2 ‘ls ‘T1 Figure 59Conﬁguration space ofasegment intheplane EXAMPLE 8.Arigid right triangle OAB moves around thevertex O.Theposition ofthe triangle isgiven bythree numbers: thedirection OA6S2isgiven bytwonumbers, andifOAisgiven, onecanrotate OBeS‘around theaxisOA(Figure 60). Connected with theposition ofthetriangle OAB isanorthogonal right-handed frame, el=OA/IOAI, e2=OB/IOBI, e3=[el,el].Thecorrespondence isone-to-one: therefore the position ofthetriangle isgiven byanorthogonal three-by-three matrix with determinant 1. A B 0 Figure 60Conﬁguration space ofatriangle Thesetofallthree-by-three matrices isthenine-dimensional space R9.Sixorthogonality conditions select outtwothree-dimensional connected manifolds ofmatrices with determinant +land—l.Therotations ofthree-space (determinant +1)form agroup, which wecall80(3). Therefore, theconﬁguration space ofthetriangle OAB isS0(3). PROBLEM. Show thatS0(3) ishomeomorphic tothree-dimensional realprojective space. 79z > I l l ‘a t i l it t 1 ,1tft ?l 4:Lagrangian mechanics onmanifolds Deﬁnition. Thedimension oftheconﬁguration space iscalled thenumber of degrees offreedom. EXAMPLE 9.Consider asystem ofkrods inaclosed chain with hinged joints. PROBLEM. How many degrees offreedom does thissystem have? EXAMPLE ll).Embedded manifolds. WesaythatMisanembedded k-dimensional sub-manifold of euclidean space IR"(Figure 61)ifinaneighborhood Uofevery point XeMthere aren—kfunc- tionsfl; U—>IR,fl: U—»IR,...,ﬁ,_,,: U—>IRsuch thattheintersection ofUwith Misgiven by theequations fl=O,...,f,,_,, =0,andthevectors gradfl, gradf§,_,, atXarelinearly independent. X11 U ll'l X1 Figure 61Embedded submanifold ItiseasytogiveMthestructure ofamanifold, i.e.,coordinates inaneighborhood ofx(how?). Itcanbeshown thatevery manifold canbeembedded insome euclidean space. InExample 8, 50(3) isasubset ofIR9. PROBLEM. Show that50(3) isembedded inIR9,andatthesame time, that50(3) isamanifold. CTangent space IfMisak-dimensional manifold embedded inE",then atevery point x wehave ak-dimensional tangent space TMX.Namely, TM,,istheorthogonal complement to{grad fl,...,grad f,,_k} (Figure 62). The vectors ofthe tangent space TM, based atxarecalled tangent vectors toMatX.Wecan alsodeﬁne these vectors directly asvelocity vectors ofcurves inM: >2=Iim$9-li"@ where(p(O)=X,¢(t)EM. t—>0 t TMX M Ell Figure 62Tangent space 80 18:Dilfcrentiable manifolds The deﬁnition oftangent vectors canalso begiven inintrinsic terms, independent oftheembedding ofMintoE". Wewillcalltwocurves X=q>(t)andX=\l!(t)equivalent if(p(0) =\l!(O) =X andlim,_ll (tp(t) —\lt(t))/t =0insome chart. Then thistangent relationship istrueinanychart (prove thisI). Deﬁnition. Atangent vector toamanifold Matthepoint Xisanequivalence class ofcurves (p(t), with q)(O) =X. Itiseasy todeﬁne theoperations ofmultiplication ofatangent vector byanumber andaddition oftangent vectors. Thesetoftangent vectors toMatXforms avector space TM,,. This space isalsocalled thetangent space toMatX. Forembedded manifolds thedeﬁnition above agrees with theprevious deﬁnition. Itsadvantage liesinthefact that italso holds forabstract manifolds, notembedded anywhere. Deﬁnition. LetUbeachart ofanatlas forMwith coordinates ql,...,q,,. Then thecomponents ofthetangent vector tothecurve q=(p(t)arethe numbers ﬁl,...,in,where §,-=(dip,-/dt)|,=0. DThetangent bundle " Theunion ofthetangent spaces toMatthevarious points, UXEM TMX,has anatural differentiable manifold structure, thedimension ofwhich istwice thedimension ofM. This manifold iscalled thetangent bundle ofMandisdenoted byTM. A point ofTMisavector §,tangent toMatsome point X.Local coordinates onTM areconstructed asfollows. Letql,...,q,,belocal coordinates on M,and5l,...,6,,components ofatangent vector inthiscoordinate system. Then the2nnumbers (ql,...,q,,,Cl,...,éf,,)givealocal coordinate system onTM.Onesometimes writes dq,for5,. Themapping p:TM —>Mwhich takes atangent vector §tothepoint XeMatwhich thevector istangent toM(§eTM,,),iscalled thenatural projection. Theinverse image ofapoint XeMunder thenatural projection, p‘‘(X),isthetangent space TM,,.This space iscalled theﬁber ofthetangent bundle over thepoint X. ERiemannian manifolds IfMisamanifold embedded ineuclidean space, then themetric oneuclidean space allows ustomeasure thelengths ofcurves, angles between vectors, volumes, etc.Allofthese quantities areexpressed bymeans ofthelengths of tangent vectors, that is,bythepositive deﬁnite quadratic form given on every tangent space TMX(Figure 63): TM.—>IR5-»<§.§>- 81I I l I I l l I '1 _,_....;_-.._q_'. l I l -i 4:Lagrangian mechanics onmanifolds dx Mx € Figure 63Riemannian metric Forexample, thelength ofacurve onamanifold isexpressed using thisform asl(";)= ll;./(dx, dx). or,ifthecurve isgiven parametrically. ;':[10,tl]~>M,t—>x(t)EM.then Io)=Ii:./<_>1.T6dr. Deﬁnition. Adifferentiable manifold with aﬁxed positive deﬁnite quadratic form (Q,Q)onevery tangent space TM,liscalled aRiemannian manifold. Thequadratic form iscalled theRiemannian metric. Remark. LetUbeachart ofanatlas forMwith coordinates ql,...,q,,. Then aRiemannian metric isgiven bytheformula dsz=_ai1(q)dqidqj an=air, -s. pun where dq,-arethecoordinates ofatangent vector. The functions al,-(q) areassumed tobedifferentiable asmany times as necessary. FThederivative map Letf:M—>Nbeamapping ofamanifold Mtoamanifold N.fiscalled diﬂerentiable ifinlocal coordinates onMandNitisgiven bydifferentiable functions. Deﬁnition. Thederivative ofadifferentiable mapping f:M—>Natapoint XEMisthelinear map ofthetangent spaces f,l,,lITM, —>TNfl,l,, which isgiven inthefollowing way(Figure 64): Letv6TMX. Consider acurve q):IR—>Mwith q)(0) =X,andvelocity vector (dip/dt)l,=o =v.Then fnv isthevelocity vector ofthecurve fQq):IR—>N, r..v=%':0r<¢<r>>. 82 19:Lagrangian dynamical systems Figure 64Derivative ofamapping PROBLEM. Show that thevectorf,,,,lv does notdepend onthecurve (p,butonly onthevector v. PROBLEM. Show thatthemapfl,,,: TM, —>TN,l,, islinear. PROBLEM. LetX=(xl,...,x,,,)becoordinates inaneighborhood ofxeM,andy=(yl,...,_v,,) becoordinates inaneighborhood ofyEN.Let§bethesetofcomponents ofthevector v,and 1|thesetofcomponents ofthe vector fﬂv. Show that 5y . 5y.-ll-é;§- I-6» 'lt—;(7j§,~- Taking theunion ofthemappings fl,forallX,wegetamapping ofthewhole tangent bundle f*:TM->TN f*v=f*,lv forveTM,l. PROBLEM. Show thatf* isadifferentiable map. PROBLEM. Letf: M—>N,g: N—>K,andh=g3]:M—>K.Show thathl,=g*fl. 19Lagrangian dynamical systems Inthisparagraph wedeﬁne lagrangian dynamical systems onmanifolds. Systems withholonomic constraints areaparticular case. ADeﬁnition ofalagrangian system LetMbeadifferentiable manifold, TMitstangent bundle, andL:TM—>IR adifferentiable function. Amap 7:IR->Miscalled amotion inthelagrangian system with conﬁguration manifold Mandlagrangian function Lif7isan extremal ofthefunctional om=fitiidr. where 7isthevelocity vector 'Y(t)eTM,,l,). EXAMPLE. LetMbearegion inacoordinate space with coordinates q=(ql,...,q,,). The lagrangian function L:TM ->Rmay bewritten intheform ofafunction L(q,q) oftheZn coordinates. Asweshowed inSection I2,theevolution ofcoordinates ofapoint moving with timesatisﬁes Lagrange’s equations. 83I r'4'----~-~-—-.:.=—_.-—-$_.._..n n I I 5.r i L I 3I ._,..._._.._...__:.=..A....:l_my“ mmr--r--- 4:Lagrangian mechanics onmanifolds Theorem. Theevolution ofthelocal coordinates q=(ql,...,q,,)ofapoint v(t) under motion inalagrangian system onamanifold satisﬁes theLagrange equations d5L_8L ataq‘aq’ where L(q,q)istheexpression forthefunction L:TM—>IRinthecoordinates qand(1onTM. Weoften encounter thefollowing special case. BNatural systems LetMbeaRiemannian manifold. Thequadratic form oneach tangent space, T=§(v,v) v6TM,, iscalled thekinetic energy. Adifferentiable function U1M—>Riscalled a potential energy. Deﬁnition. Alagrangian system onaRiemannian manifold iscalled natural ifthelagrangian function isequal tothedifference between kinetic and potential energies: L=T—U. EXAMPLE. Consider two mass points mland "12joined byalinc segment oflength Iinthe (x,y)-plane. Then aconﬁguration space ofthree dimensions M=R2><S‘cR2xlR2 isdeﬁned inthefour-dimensional Conﬁguration space R2xR2oftwofreepoints (x,,y,)and (X2,yz)bythecondition t/(x1 —x2)2 +(y,—_v,)2 =l(Figure 65). y "NW Figure 65Segment intheplaneX There isaquadratic form onthetangent space tothefour-dimensional space (xl,xl,y,,y2): ‘l’ + + Ourthree-dimensional manifold, asitisembedded inthefour-dimensional one,isprovided with aRiemannian metric. Theholonomic system thusobtained iscalled inmechanics alinesegment ofﬁxed length inthe(xiY)-plane. Thekinetic energy isgiven bytheformula 58+? .t1+;»2 Tznjli-ilz 1+m2 _ 84 19:Lagrangian dynamical systems CSystems with holonomic constraints InSection 17wedeﬁned thenotion ofasystem ofpoint masses with holo- nomic constraints. Wewillnow show thatsuch asystem isnatural. Consider theconﬁguration manifold Mofasystem with constraints as embedded inthe3n-dimensional conﬁguration space ofasystem offree points. The metric onthe3n-dimensional space isgiven bythequadratic form X§'=1mit‘;". The embedded Riemannian manifold Mwith potential energy Ucoincides with thesystem deﬁned inSection 17orwith thelimiting case ofthesystem with potential U+Nq§, N—>oo,which grows rapidly outside ofM. DProcedure forsolving problems with constraints 1.Determine the conﬁguration manifold and introduce coordinates q1,...,qk(inaneighborhood ofeach ofitspoints). 2.Express thekinetic energy T=Z%mir",2 asaquadratic form inthe generalized velocities T=ixaij(q)qiqj- 3.Construct thelagrangian function L=T—U(q) andsolve Lagrange’s equations. EXAMPLE. Weconsider themotion ofapoint mass ofmass 1onasurface ofrevolution inthree- dimensional space. Itcanbeshown thattheorbits aregeodesics onthesurface. Incylindrical coordinates r,tp,zthesurface isgiven (locally) intheform r=r(z)orz=z(r).Thekinetic energy hastheform (Figure 66) T=il-*2+J72+$2)=%[(1+til)?’ +r2(Z)¢2] incoordinates (pandz,and T=%(>?2+Y2+52)=%[(1+Z12)? +r22] incoordinates randtp.(Wehave used theidentity .\"2+_1"2=fl+rzrbz.) Thelagrangian function Lisequal toTInboth coordinate systems goisacyclic coordinate. Thecorresponding momentum isprescrvcd; pa,=r2¢isnothing other than thez-component of Z 3%}7/ Figure 66Surface ofrevolution 85F. tl t + L3:-L—-,1..'1 Mlt t I tI 1 t E I i ,..'.-,Js.7:4;.;s=.t-1it t ll 4:Lagrangian mechanics onmanifolds angular momentum. Since thesystem hastwodegrees offreedom. knowing thecyclic coordinate tpissufﬁcient forintegrating theproblem completely (cf.Corollary 3,Section 15). Wecanobtain more easily aclear picture oftheorbits byreasoning slightly diﬁerently. Denote byattheangle ofthe orbit withameridian. Wehave rd)=lvlsin1,where |v|isthemag- nitude ofthevelocity vector (Figure 66). Bythelawofconservation ofenergy, H=L=Tispreserved. Therefore, irl=const. so theconservation lawforpwtakes theform rsinat=const (“C1airaut's theorem"). Thisrelationship showsthat themotion takes placein theregion|sin 1|5l.i.e..r 2r0sin:0. Furthermore. theinclination oftheorbit from themeridian increases astheradius rdecreases. When theradius reaches thesmallest possible value. r=rosin:10,theorbit isreﬂected and returns totheregion with larger r(Figure 67). r=rsin01 p‘ 0 0 r=rosin010 Figure 67Geodesics onasurface ofrevolution PROBLEM. Show that thegeodesics onaconvex surface ofrevolution aredivided intothree classes: meridians, closed curves, andgeodesics dense inaringr2c. PROBLEM. Study thebehavior ofgeodesics onthesurface ofatorus ((r—R)’+:2=p2). ENon-autonomous systems Alagrangian non-autonomous system differs from theautonomous systems, which wehave been studying until now, bytheadditional dependence ofthe lagrangian function ontime: L:TM><lR—>lRB L=L(q,q,t). Inparticular, both thekinetic andpotential energies candepend ontime ina non-autonomous natural system: T:TM><R—+R U:M><lR—+lFR T=T(q,q,t) U=U(q,t). Asystem ofnmass points, constrained byholonomic constraints depen- dent ontime, isdeﬁned with thehelp ofatime-dependent submanifold ofthe conﬁguration space ofafreesystem. Such amanifold isgiven byamapping i:M><lR—>E3" i(q,t)=x, which, foranyﬁxed teR,deﬁnes anembedding M—>E3".Theformula of section Dremains truefornon-autonomous systems. 86 l9:Lagrangian dynamical systems Z VIno X Figure 68Bead onarotating circle EXAMPLE. Consider themotion ofabead along avertical circle ofradius r(Figure 68)which rotates with angular velocity waround thevertical axispassing through thecenter 0ofthe circle. Themanifold Misthecircle. Letqbetheangular coordinate onthecircle, measured from thehighest point. Letx.y.andzbecartesian coordinates inE3with origin 0andvertical axisz.Let(,0bethe angle oftheplane ofthecircle with theplane x0z. Byhypothesis, (,0=wt.The mapping i:M><ER—>E’isgiven bytheformula i(q,t)=(rsinqcoswt,rsinqsinwt,rcosq). From thisformula (or,more simply, from an“inﬁnitesimal right triangle”) weﬁndthat m ' vT=5(w2r2 Slﬂzq+rzqz) U=mgrcosq. Inthiscasethelagrangian function L=T-Uturns outtobeindependent oft.although the constraint does depend ontime. Furthermore, thelagrangian function turns outtobethesame asintheone-dimensional system with kinetic energy M To =Y dz M Zmrz, andwith potential energy V=Acosq—Bsin2 q, A=mgr,B=€o>2r2. Theform ofthephase portrait depends ontheratio between AandB.For2B<A(i.e.,fora rotation ofthecircle slow enough thatwzr<g),thelowest position ofthebead (q=1:)is V A —1r 1r 31r . I0 ’‘I 51 1 f /WA*->. L _>— TH’\ti l/\ Figure 69Effective potential energy andphase plane ofthebead 87l l | I ll |l !l l + 4 1 4:Lagrangian mechanics onmanifolds stable andthecharacteristics ofthe motion aregenerally thesame asinthecaseofamathematical pendulum (0)=0). For2B>A,i.e.,forsufﬁciently fastrotation ofthecircle, thelowest position ofthebead becomes unstable; ontheother hand, twostable positions ofthebead appear onthecircle. where cosq=—A/2B =—g/wzr. Thebehavior ofthe bead under allpossible initial conditions isclear from theshape ofthephase curves inthe(q,:1)-plane (Figure 69). 20E.Noether’s theorem Various laws ofconservation (ofmomentum, angular momentum, etc.)areparticular cases of onegeneral theorem: toevery one-parameter group ofdiﬁeomorphisms oftheconﬁguration manifold ofalagrangian system which preserves thelagrangian function, there corresponds a ﬁrstintegral ofthe equations ofmotion. AFormulation ofthetheorem LetMbeasmooth manifold, L:TM—>Rasmooth function onitstangent bundle TM. Leth:M—>Mbeasmooth map. Deﬁnition. Alagrangian system (M,L)admits themapping hifforanytangent vector vETM, L(h*v)=L(v). EXAMPLE. LetM={(x,, x2,x,)}, L=(m/2)(Xf + +25)—U(x2, x3).Thesystem admits thetranslation h1(x,, xl,x3)—>(x,+s,X2,x3)along thexlaxisanddoes notadmit, generally speaking, translations along thex2axis. Noether’s theorem. Ifthesystem (M,L)admits theone-parameter group of diﬂeomorphisms h’:M—>M,seR,then thelagrangian system ofequations corresponding toLhasaﬁrst integral I:TM—>R. Inlocal coordinates qonMtheintegral Iiswritten intheform 6Ldh‘(q) I(1l,1l)=%“a;** _ BProof First, letM=IR"becoordinate space. Let(p:R—>M,q=<p(t)beasolution toLagrange’s equations. Since hf,preserves L,thetranslation ofasolution, h‘0(p:IR—>Malsosatisﬁes Lagrange’s equations foranys.38 Weconsider themapping (D:[Rx[Rt—>IR",given byq=(D(s,t)=h‘((p(t)) (Figure 70). Wewilldenote derivatives with respect totbydots andwith respect tos byprimes. Byhypothesis at, 8L 01.-(1) 0=_<gt.l=_.<t>/+__<t>»,5s dq éiq 38Theauthors ofseveral textbooks mistakenly assert thattheconverse isalsotrue, i.e.,thatif h‘takes solutions tosolutions, then hj,preserves L. 88 20:E.Noether’s theorem q(t) 4 qtstI)=h’(q(I)) q .hi(q) h‘((1) Figure 70Noether’s theorem where thepartial derivatives ofLaretaken atthepoint q=<D(s, t),q= (l>(s, t). Aswestated above, themapping (l>|s:c,,,,,,: [R—>R"forany ﬁxed s satisﬁes Lagrange’s equation 68L - 6L -5;[aq(‘F(s, I),¢(s,0)]e56(<l>(S, I),¢(S,I))- Weintroduce thenotation F(s,t)=(dL/5q)(<l>(s, t),(s, t))andsubstitute ar/at for6L/dq in(1). Writing q’asdq’/dt, weget FIL 5L0:1 1+ i»_i‘”~tLﬂ Udtaqq aqat“—dt0qq "at" Remark. The ﬁrst integral I=(6L/0q)q' isdeﬁned above using local coordinates q.Itturns outthatthevalue ofI(v)does notdepend onthechoice ofcoordinate system q. Infact,Iistherateofchange ofL(v)when thevector veTM, varies inside TM,,with velocity (d/ds)ls:Oh‘x. Therefore, I(v)iswelldeﬁned asafunction ofthetangent vector v6TM,,.Noether’s theorem isproved inthesame way when Misamanifold. CExamples EXAMPLE l.Consider asystem ofpoint masses with masses m,-: X2 L=Zmt?‘ —U(X) xi=xilel +xtzez ‘I’xi3e3$ constrained bytheconditions f,-(x) =O.Weassume thatthesystem admits translations along theelaxis: h‘:x,-—>x,-+sel foralli. Inother words, theconstraints admit motions ofthesystem asawhole along theelaxis, andthepotential energy does notchange under these. 89l t l l l l I i I I I I I t II Il éj it if l 4:Lagrangian mechanics onmanifolds ByNoether’s theorem weconclude: Ifasystem admits translations along thee1axis, then theprojection ofitscenter ofmass onthee1axismoves linearly anduniformly. Infact, (d/ds)|,=0hsx,- =e1.According totheremark attheendofB,the quantity 5L .I=20* el=Zm,-xi, ispreserved, i.e.,theﬁrst component P1ofthemomentum vector ispre- served. Weshowed thisearlier forasystem without constraints. EXAMPLE 2.Ifasystem admits rotations around thee,axis, then theangular momentum with respect tothisaxis, Ml = (Lxis ml,ki]: el) I isconserved. Itiseasy toverify thatifh‘isrotation around thee,axisbytheangle s, then (d/ds)|,=0h‘x,- =[el,x,-],from which itfollows that I=ZLelaXi]=Z(mm.tet,x.1>=Z(ts.mist].91)- PROBLEM 1.Suppose thataparticle moves intheﬁeld oftheuniform helical linex=cos(p. y=sin(p,z=ctp.Find thelawofconservation corresponding tothishelical symmetry. ANSWER. Inanysystem which admits helical motions leaving ourhelical lineﬁxed, thequantity I=cP3+M3isconserved. PROBLEM 2.Suppose thatarigid body ismoving under itsown inertia. Show thatitscenter of mass moves linearly anduniformly. Ifthecenter ofmass isatrest,then theangular momentum with respect toitisconserved. PROBLEM 3.What quantity isconserved under themotion ofaheavy rigid body ifitisﬁxed at some point O?What if,inaddition, thebody issymmetric with respect toanaxispassing through 0‘? PROBLEM 4.Extend Noether’s theorem tonon-autonomous lagrangian systems. Hint. LetM1=MxRbetheextended conﬁguration space (thedirect product ofthe conﬁguration manifold Mwith thetime axisIR). Deﬁne afunction L,:TM, —>[Rby dz L—:dr i.e.,inlocal coordinates q,tonM1wedeﬁne itbytheformula dqdt dq/dr dt L 9t1W ~*7 = L s ‘W s —- -‘(Qatdr) lqat/attlat Weapply Noether‘s theorem tothelagrangian system (M,, L,). 90 21:D‘Alembert's principle IfL,admits thetransformations h‘;M,~+M,,weobtain aﬁrstintegral I,:TM, —>IR. Since Ltlt=_\'L,dt.thisreduces toaﬁrstintegral I:TM><IR—>IRoftheoriginal system. lf.inlocalcoordinates (q,t)on M,,we haveI,=l,(q,t,dq/dr,dt/dr),then I(q,t'].t) =1,(q,t,q, 1). Inparticular, ifLdoes notdepend ontime, L,admits translations along time, h’(q, t)= (q,t+s).Thecorresponding ﬁrstintegral Iistheenergy integral. 21D’Alembert’s principle Wegivehereanewdeﬁnition ofasystem ofpoint masses withholonomic constraints andprove itsequivalence tothedeﬁnition given inSection 17. AExample Consider theholonomic system (M, L),where Misasurface inthree- dimensional space {x}: L=§mit2 —U(X). Inmechanical terms, “the mass point xofmass mmust remain onthesmooth surface M Consider amotion ofthepoint, x(t).IfNewton’s equations mil+(5U/dx) =0were satisﬁed, then intheabsence ofexternal forces (U=0)thetra- jectory would beastraight lineandcould notlieonthesurface M. From thepoint ofview ofNewton, thisindicates thepresence ofanew force “forcing thepoint tostayonthesurface.” Deﬁnition. Thequantity ..6UR=mx+—dx iscalled theconstraint force (Figure 71). R x(t) M l S Figure 71Constraint force Ifwetake theconstraint force R(t)intoaccount, Newton’s equations are obviously satisﬁed: 6U ..=____ Rmx ax+. Thephysical meaning oftheconstraint force becomes clear ifweconsider oursystem with constraints asthelimit ofsystems with potential energy U+NU, asN—>1:.where U,(x) = p3(X, M).Forlarge Ntheconstraint potential NU, produces arapidly changing force 91 4:Lagrangian mechanics onmanifolds F=—NdU,/ﬁx; when wepass tothelimit (N—»x)theaverage value oftheforce Funder oscillations ofxnear MisR.Theforce Fisperpendicular toM.Therefore. theconstraint force Risperpendicular toM:(R,§)=0forevery tangent vector F, BFormulation oftheD’Alembert—Lagrange principle Inmechanics, tangent vectors totheconﬁguration manifold arecalled virtual variations. TheD’Alembert~Lagrange principle states: __+au§ _0 "IX ax, — foranyvirtual variation Q,orstated differently, thework oftheconstraint force onanyvirtual variation iszero. Forasystem ofpoints X,with masses m,-theconstraint forces R,aredeﬁned byR,=m,-ii, +(5U/dx,~), andD’Alembert’s principle hastheform Z(R,-,§,) =0,orZ((m,ii,- +(dU/dx,-), §,)=0,i.e.,thesum oftheworks ofthecon- straint forces onanyvirtual variation {§,}eTM, iszero. Constraints with theproperty described above arecalled ideal. Ifwedeﬁne asystem with holonomic constraints asalimit asN—>I,thentheD‘Alembert_ Lagrange principle becomes atheorem: itsproof issketched above forthesimplest case. Itispossible, however, todeﬁne anideal holonomic constraint using theD'Alembert- Lagrange principle. Inthiswaywehave three deﬁnitions ofholonomic systems withconstraints: 1.Thelimit ofsystems with potential energies U+NU, asN—+ac. 2.Aholonomic system (M.L),where Misasmooth submanifold oftheconﬁguration space ofasystem without constraints andListhelagrangian. 3.Asystem which complies with theD’Alembert-Lagrange principle. Allthree deﬁnitions aremathematically equivalent. Theproof ofthe implications (I)=-(2)and(I)=~(3)issketched above andwillnotbegiven infurther detail. Wewillnowshow that(2)-=>(3). CTheequivalence oftheD’Alembert—Lagrange principle andthevariational principle LetMbeasubmanifold ofeuclidean space, McIR”,andx:IR-+Macurve, with x(t,,) =xo,x(t,) =x,. Deﬁnition. Thecurve xiscalled aconditional extremal oftheaction functional I1 X2 <1)=JIO ~—U(X)}dt, ifthedifferential 511)isequal tozero under thecondition thatthevariation consists ofnearby curves” joining X0tox,inM. 3°Strictly speaking, inorder todeﬁne avariation 6(1),onemust deﬁne onthesetofcurves nearx onMthestructure ofaregion inavector space. This canbedone using coordinates onM; however, theproperty ofbeing aconditional extremal does notdepend onthechoice ofaco- ordinate system. 92 21:D‘Alembert‘s principle Wewillwrite (1) 5,,,<l> =0. Clearly, Equation (1)isequivalent totheLagrange equations 5L 5L X2 d5,,-5 L~;—v<><> X-xii). insome local coordinate system qonM. Theorem. Acurve x:[R—>Mc[RNisaconditional extremal oftheaction (i.e.,satisﬁes Equation (1))ifandonly ifitsatisﬁes D’Alembert’s equation (2) +‘iii.Q)=0,vigeTM,. Lemma. Letf:{trto3t3t,}—>[RNbeacontinuous vectorﬁeld. Iﬂfor every continuous tangent vector ﬁeld Q,tangent toMalong x(i.e.,h(t)ETM,0), with§(t)=Ofor t=to,t,),wehave it If(t)§(t)dt =0, Io thentheﬁeld f(t)isperpendicular toMatevery point x(t)(i.e.,(f(t), h)=0 forevery vector heTM,,,))(Figure 72). ' M Figure 72Lemma about thenormal ﬁeldF‘as Theproof ofthelemma repeats theargument which weused toderive the Euler—Lagrange equations inSection 12. PROOF orTI-IETHEOREM. Wecompare thevalue of(Donthetwocurves x(t) andx(t)+¢';(t),where §(t,,) =§(t,) =0.Integrating byparts, weobtain “_- 5U “,_8U5(1)-‘[0 <x§—5;E,)dt- —L (x+5;)§dt. 93 4:Lagrangian mechanics onmanifolds Itisobvious from thisformula“ thatEquation (1),5M(D =0,isequivalent tothecollection ofequations (3) £1(ii+f%)§dt=0. foralltangent vector ﬁelds §(t)e TM,,,, with §(t,,) =§(t,) =0.Bythe lemma (where wemust setf=it+(dU/dx)) thecollection ofequations (3) isequivalent totheD’Alembert-Lagrange equation (2). Cl DRemarks Remark I.Wederive theD’Alembert—Lagrange principle forasystem ofn points xie[R3,i=1,...,n,with masses m,-,with holonomic constraints, from theabove theorem. Inthecoordinates 2={i,=\/Ere}, thekinetic energy takes theform T= =5&2. Bythetheorem, theextremals oftheprinciple ofleast action satisfy the condition <iTi+%g,§)=0 (theD’Alembert—Lagrange principle forpoints inIR3": the3n-dimensional reaction force isorthogonal tothemanifold Minthemetric T).Returning tothecoordinates xi,weget .. 5U .. 8U0= + mt.)=;+E. i.e.,theD’Alembert—Lagrange principle intheform indicated earlier: the sumofthework ofthereaction forces onvirtual variations iszero. Remark 2.TheD’Alembert—Lagrange principle canbegiven inaslightly different form ifweturn tostatics. Anequilibrium position isapoint X0which istheorbit ofamotion: x(t)=x,,. Suppose that apoint mass moves along asmooth surface Munder the inﬂuence oftheforce f=—dU/ax. Theorem. Thepoint X0inMisanequilibrium position ifandonly iftheforce isorthogonal tothesurface atX02(f(x0), Q)=OforallE,6TM,O. This follows from theD’Alembert—Lagrange equations inview ofthe factthatii=0. Deﬁnition. —mii iscalled theforce ofinertia. ‘°Thedistance ofthepoints x(t)+h(t)from Missmall ofsecond-order compared with §(t). 94 21:D’Alembert’s principle Now theD’Alembert—Lagrange principle takes theform: Theorem. Iftheforces ofinertia areadded totheacting forces, Xbecomes an equilibrium position. PROOF. D’Alembert’s equation expresses thefact, asinthepreceding theorem, that xisanequilibrium position ofasystem with forces —mii +f. Cl Entirely analogous statements aretrueforsystems ofpoints: Ifx={x,} areequilibrium positions, then thesumofthework oftheforces acting onthe virtual variations isequal tozero. Iftheforces ofinertia —m,-i~i,(t) areadded totheacting forces, then theposition x(t)becomes anequilibrium position. Now aproblem about motions can bereduced toaproblem about equilibrium under actions ofother forces. Remark 3.Uptonow wehave notconsidered cases when theconstraints depend ontime. Allthat wassaid above carries over tosuch constraints without anychanges. EXAMPLE. Consider abead sliding along arodwhich istilted atanangle oz tothevertical axisandisrotating uniformly with angular velocity toaround Z Figure 73Bead onarotating rod thisaxis(itsweight isnegligible). Forourcoordinate qwetake thedistance from thepoint 0(Figure 73).Thekinetic energy andlagrangian are: L=T=émvz =§mq2 +émcozrl, r=qsinoz. Lagrange’s equation: mtj=mwzq sinzOt. Theconstraint force ateach moment isorthogonal tovirtual variations (i.e.,tothedirection oftherod), butisnotatallorthogonal totheactual trajectory. Remark 4.Itiseasy toderive conservation laws from theD’Alembert— Lagrange equations. Forexample, iftranslation along thex,axis§,=e,is 95 4:Lagrangian mechanics onmanifolds among thevirtual variations, thenthesumofthework oftheconstraint forces onthisvariation isequal tozero: Z(Ri» 91)= Rn91)=0- Ifwenowconsider constraint forces asexternal forces, then wenotice thatthe sumoftheﬁrstcomponents oftheexternal forces isequal tozero. This means thattheﬁrstcomponent, P,,ofthemomentum vector ispreserved. Weobtained thissame result earlier from Noether’s theorem. Remark 5.Weemphasize once again thattheholonomic character ofsome particular physical constraint oranother (toagiven degree ofexactness) isa question ofexperiment. From themathematical point ofview, theholonomic character ofaconstraint isapostulate ofphysical origin; itcanbeintroduced invarious equivalent forms, forexample, intheform oftheprinciple ofleast action (I)ortheD’AlemberteLagrange principle (2),but, when deﬁning theconstraints, theterm always refers toexperimental facts which gobeyond Newton’s equations. Remark 6.Ourterminology differs somewhat from thatused inmechanics textbooks, where theD’Alembert—Lagrange principle isextended toawider class ofsystems (“non-holonomic systems with ideal constraints”). Inthis book wewillnotconsider non~holonomic systems. Weremark only thatone example ofanon-holonomic system isasphere rolling onaplane without slipping. Inthetangent space ateach point ofthe conﬁguration manifold ofa non-holonomic system there isaﬁxed subspace towhich thevelocity vector must belong. Remark 7.Ifasystem consists ofmass points connected byrods, hinges, etc.,then theneed mayarise totalkabout theconstraint force ofsome partic- ularconstraint. Wedeﬁned thetotal “constraint force ofallconstraints” R,forevery mass point m,-.Theconcept ofaconstraint force foranindividual constraint is impossible todeﬁne, asmaybealready seenfrom thesimple example ofabeam resting onthree columns. Ifwetrytodeﬁne constraint forces ofthecolumns, R,,R2,R3bypassing toalimit (considering thecolumns asvery rigid springs), then wemay become convinced that theresult depends onthe distribution ofrigidity. R? P Figure 74Constraint force onarod 96 21:D’Alembert’s principle Problems forstudents areselected sothatthisdiﬁiculty does notarise. PROBLEM. Arodofweight P,tilted atanangle of60°totheplane ofatable, begins tofall withinitial velocity zero(Figure 74).Find theconstraint force ofthe table attheinitial moment, considering thetable as(a)absolutely smooth and(b)absolutely rough. (Intheﬁrstcase, the holonomic constraint holds theendofthe rodontheplane ofthe table, andinthesecond case, atagiven point.) 97i l i ._:in-..<_-4;,-..-L. it 4 Oscillations Because linear equations areeasy tosolve andstudy, thetheory oflinear oscillations isthemost highly developed area ofmechanics. Inmany non- linear problems, linearization produces asatisfactory approximate solution. Even when thisisnotthecase, thestudy ofthelinear part ofaproblem is often aﬁrststep, tobefollowed bythestudy oftherelation between motions inanonlinear system andinitslinear model. 22Linearization Wegiveherethedeﬁnition ofsmall oscillations. AEquilibrium positions Deﬁnition. Apoint X0iscalled anequilibrium position ofthesystem (1) 6;—:=f(x), XEIR" ifx(t)EX0isasolution ofthissystem. Inother words, f(x0) =0,i.e., thevector ﬁeld f(x)iszero atX0. EXAMPLE. Consider thenatural dynamical system with lagrangian function L(q,<1)=T—U.where T=iZai.(q)qté,- 20andU=U(q)= ddL 0L Z ~—=——, = ,..., ,,. Lagrange’s equations canbewritten intheform ofasystem of2nﬁrst- order equations ofform (1).Wewilltrytoﬁndanequilibrium position: 98 22:Linearization Theorem. Thepoint q=q0,1']=('10willbeanequilibrium position ifandonly ifq0=Oandq0isacritical point ofthepotential energy, i.e., 8U(3) — =0.5*!... PROOF. Wewrite down Lagrange’s equations d6T 6T 0U dtaq dq dq From (2)itisclear that, forq=0,wewillhave 6T/dq =0andat/aq =0. Therefore, q=q0isasolution incase(3)holds andonlyinthatcase. El BStability ofequilibrium positions Wewillnow investigate motions with initial conditions close toanequi- librium position. Theorem. Ifthepoint q0isastrict local minimum ofthepotential energy U, thentheequilibrium q=q0isstable inthesense ofLiapunov. PROOF. LetU(q0) =h.For sufficiently small t-:>0,theconnected com- ponent oftheset{q:U(q) sh+a}containing q0will beanarbitrarily small neighborhood ofq0(Figure 75).Furthermore, theconnected com- ponent ofthecorresponding region inphase space p,q,{p,q:E(p, q)5 h+s},(where p=ar/aq isthemomentum and E=T+ Uisthetotal energy) willbeanarbitrarily small neighborhood ofthepoint p=O,q=q0. Buttheregion {p,q:E3h+a}isinvariant with respect tothephase ﬂow bythelawofconservation ofenergy. Therefore, forinitial conditions p(0), q(0)close enough to(0,q0),every phase trajectory (p(t), q(t)) isclose to (0.tIt>)- El U h‘i'€ Y h __ pq P E<h+e Q Figure 75Stable equilibrium position 991 1 I 1 1 l ll5 l f NHrt»‘Adi—.-Z:-wru-:1-acct‘.3t | l ‘~"""“‘%‘..“K"s:-':r2z:€ t i Ir >1Oscillations PROBLEM. Cananequilibrium position q=q0,p=0beasymptotically stable? PROBLEM. Show thatinananalytic system with onedegree offreedom anequilibrium position q0which isnotastrict local minimum ofthepotential energy isnotstable inthesense of Liapunov. Produce anexample ofaninﬁnitely differentiable system where thisisnottrue. Remark. Itseems likely that inananalytic system with ndegrees of freedom, anequilibrium position which isnotaminimum point isunstable; butthishasnever been proved forn>2. CLinearization ofadiﬁerential equation Wenow turn tothegeneral system (1).Instudying solutions of(1)which are close toanequilibrium position X0,weoften usealinearization. Assume that X0=0(thegeneral case isreduced tothisonebyatranslation oftheco- ordinate system). Then theﬁrstterm oftheTaylor series forfislinear: df f(x) =AX+R2(X), A=a—x andR2=O(X2), 0 where thelinear operator Aisgiven incoordinates x,,...,x,,bythematrix a,-J-: 5.A(X), =Za,-J-xj; an= I Deﬁnition. Thepassage from system (1)tothesystem dy(4) E=Ay (XER".yETR3) iscalled thelinearization of(1). PROBLEM. Show that linearization isawell-deﬁned operation: theoperator Adoes notdepend onthecoordinate system. Theadvantage ofthelinearized system isthat itislinear andtherefore easily solved: Alt’ y(t)=e’“y(0), where e"=E+At+7 + . Knowing thesolution ofthelinearized system (4),wecansaysomething about solutions oftheoriginal system (1).Forsmall enough X,thedifference between thelinearized andoriginal systems, R2(X), issmall incomparison with X.Therefore, foralong time, thesolutions y(t), x(t)ofboth systems with initial conditions y(0)=X(0) =X0remain close. More explicitly, we caneasily prove thefollowing: Theorem. Forany T>Oandforanys>0there isa5>0such that if |x(O)l <5,then |x(t) —y(t)] <soforalltintheinterval 0<t<T. 100 22:Linearization DLinearization ofalagrangian system Wereturn again tothelagrangian system (2)andtrytolinearize itina neighborhood oftheequilibrium position q=qo.Inorder tosimplify the formulas, wechoose acoordinate system sothatqo=0. Theorem. Inorder tolinearize thelagrangian system (2)inaneighborhood of theequilibrium position q=0,itissujﬁcient toreplace thekinetic energy TZ value at q= 0, T2=%Zaijqiqj: at;=aij(0)» andreplace thepotential energy U(q)byitsquadratic part 62U U2=%Zbijqiqja bi]= F0- PROOF. Wereduce thelagrangian system totheform (1)byusing thecanonical variables pandq: _an _an _n——E 11-55, H(|>,q)—T+U- Since p=q=0isanequilibrium position, theexpansions oftheright-hand sides inTaylor series atzero begin with terms that arelinear inpandq. Since theright-hand sides arepartial derivatives, these linear terms are determined bythequadratic terms H2oftheexpansion forH(p,q). But H2isprecisely thehamiltonian function ofthesystem with lagrangian L2=T2—U2,since, clearly, H2=T2(p) +U2(q). Therefore, thelinearized equations ofmotion aretheequations ofmotion forthesystem described inthetheorem with L2=T2—U2. El EXAMPLE. Weconsider thesystem with onedegree offreedom: T=%a(q)¢i2, U=U(q)- Letq=qobeastable equilibrium position: (5U/dq) |q=q0 =0,(52U/5q2)|q=q0 >0(Figure 76). U2 U (I P (I Figure 76Linearization lO1 5:Oscillations Asweknow from thephase portrait, forinitial conditions close toq=qo, p=0,thesolution isperiodic with period Tdepending, generally speaking, ontheinitial conditions. Theabove twotheorems imply Corollary. The period Iofoscillations close totheequilibrium position qo approaches thelimit to=21:/coo, (where (of,=b/a,b=(62U/c7q2)|q=,m, anda=a(q0)) astheamplitudes oftheoscillations decrease. PROOF. Forthelinearized system, T2=%aq2 andU2=%bq2 (taking qo=0). Thesolutions toLagrange’s equation 5;=—w§q have period to=21:/we: q=clcoswot+c2sinwot foranyinitial amplitude. E] ESmall oscillations Deﬁnition. Motions inalinearized system (L2=T2-U2)arecalled small oscillations“ near anequilibrium q=qo.Inaone-dimensional problem thenumbers toandwoarecalled theperiod andthefrequency ofsmall oscillations. PROBLEM. Find theperiod ofsmall oscillations ofabead ofmass lonawire y=U(x)ina gravitational ﬁeldwith g=1,nearanequilibrium position x=xo(Figure 77). U V m mg X0_ — >X Figure 77Bead onawire Solution. Wehave U=mgy=U(x) 6 2 T=§mv2 =§[1+ ].\'"2.fix Letx0beastable equilibrium position: (5U/6x)|,° =O;(c“'2U/6x2)|,m >0.Then thefrequency ofsmall oscillations, w,isdeﬁned bytheformula (ﬁxll“62U (U2 = 4*? since. forthelinearized system, T2=M2andU2=%wZq’ (q=x—xo). ‘1Iftheequilibrium position isunstable, wewilltalk about “unstable small oscillations“ even though these motions maynothave anoscillatory character. 102 23:Small oscillations PROBLEM. Show thatnotonly asmall oscillation, butanymotion ofthebead isequivalent toa motion insome one-dimensional system with lagrangian function L=%q2—V(q). Hint. Take length along thewireforq. 23Small oscillations Weshow herethatalagrangian system undergoing small oscillations decomposes intoadirect product ofsystems with onedegree offreedom. AAproblem about pairs offorms Wewillconsider inmore detail theproblem ofsmall oscillations. Inother words, weconsider asystem whose kinetic and potential energies are quadratic forms (1) T=it/1<i,¢i) U=%(Bq.q) qER",tiEW Thekinetic energy isapositive deﬁnite form. Inorder tointegrate Lagrange’s equations, wewillmake aspecial choice ofcoordinates. Asweknow from linear algebra, apairofquadratic forms (Aq,q),(Bq,q), theﬁrstofwhich ispositive deﬁnite, canbereduced toprincipal axes bya linear change ofcoordinates :42 Q=C11 Q=(Q1,~--,Q..)- Inaddition, thecoordinates Qcanbechosen sothat theform (Aq,q)de- composes intothesumofsquares (Q,Q).LetQbesuch coordinates; then, since Q=Cq,wehave 1n. 1n (2) T=-ZQ.-2 U=-Z/1.-Q52.-=1 2i=1 Thenumbers /l,arecalled theeigenvalues oftheform Bwithrespect toA. PROBLEM. Show thattheeigenvalues ofBwithrespect toAsatisfy thechar- acteristic equation (3) detlB—/lA|=0, alltheroots ofwhich are,therefore, real(thematrices AandBaresymmetric andA>O). BCharacteristic oscillations Inthecoordinates Qthelagrangian system decomposes intonindependent equations (4) Qt=_/liQi- ‘2Ifonewants to,onecanintroduce aeuclidean structure bytaking theﬁrstform asthescalar product, andthen reducing thesecond form totheprincipal axes byatransformation which is orthogonal with respect tothiseuclidean structure. 103l Y |F <t l ll Fl r .~_t|-nae-*~r‘——-taut“--at 1. 1 1 i i 5:Oscillations Therefore wehave proved: Theorem. Asystem performing small oscillations isthedirect product ofnone- dimensional systems performing small oscillations. Fortheone-dimensional systems, there arethree possible cases: Case 1:/l=wz>0;thesolution isQ=C1coswt+C2sinwt(oscillation) Case 2:A=0;thesolution isQ=C2+C2t(neutral equilibrium) Case 3:,1=—k2 <0;thesolution isQ=C,cosh kt+C2sinhkt (instability) Corollary. Suppose oneoftheeigenvalues of(3)ispositive." /1=wz>O.Then system (1)canperform asmall oscillation oftheform (5) q(t)=(C1coswt+C2sinwt)§, where §isaneigenvector corresponding to/1(Figure 78): B§=/lA§. q Q2 2 Q1 ‘I1 Figure 78Characteristic oscillation This oscillation istheproduct oftheone-dimensional motion Q,= C,cosw,-t+C2sinwitandthetrivial motion Qj=0(jati). Deﬁnition. The periodic motion (5)iscalled acharacteristic oscillation of system (1),andthenumber wiscalled thecharacteristic frequency. Remark. Characteristic oscillations arealso called principal oscillations ornormal modes. Anonpositive Aalsohaseigenvectors; wewillalsocallthe corresponding motions “characteristic oscillations,” although they arenot periodic; thecorresponding “characteristic frequencies” areimaginary. PROBLEM. Show that thenumber ofindependent realcharacteristic oscil- lations isequal tothedimension ofthelargest positive deﬁnite subspace for thepotential energy §(Bq, q). 104 23:Small oscillations Now theresult may beformulated asfollows: Theorem. Thesystem (1)hasncharacteristic oscillations, thedirections of which arepairwise orthogonal with respect tothescalar product given by thekinetic energy A. PROOF. Thecoordinate system Qisorthogonal with respect tothescalar product (Aq,q)by(2). I1 CDecomposition intocharacteristic oscillations Itfollows from theabove theorem that: Corollary. Every small oscillation isasumofcharacteristic oscillations. Asum ofcharacteristic oscillations isgenerally notperiodic (remember theLissajous ﬁguresl). Todecompose amotion into asum ofcharacteristic oscillations, itis sufﬁcient toproject theinitial conditions q,tionto thecharacteristic direc- tions Q,andsolve thecorresponding one-dimensional problems (4). Therefore, theLagrange equations forsystem (1)canbesolved inthe following way. Weﬁrst look forcharacteristic oscillations oftheform q=e“"'§. Substituting these intoLagrange’s equations 4_/1 :_dtq Bq. weﬁnd (B—w2A)§ =O. From thecharacteristic equation (3)weﬁndrteigenvalues 2.,=wf.Tothese there correspond npairwise orthogonal eigenvectors 5,2.Ageneral solution inthecaseA#50hastheform II q(t)=ReZC,e"“’*'§,,. k=1 Remark. This result isalso true when some ofthe,1aremultiple eigen- values. Thus, inalagrangian system, asopposed toageneral system oflinear differential equations, resonance terms oftheform tsinwt,etc.donotarise, even inthecaseofmultiple eigenvalues. DExamples EXAMPLE l.Consider thesystem oftwo identical mathematical pendulums oflength I,=l2=l andmass m,=m2=linagravitational ﬁeld with g=1.Suppose that thependulums are connected byaweightless spring whose length isequal tothedistance between thepoints of suspension (Figure 79).Denote byq,andq2theangles ofinclination ofthependulums. Then l0551 r ‘I t | t 1 5:Oscillations I l L qt 0000000 Figure 79Identical connected pendulums forsmall oscillations, T=tot +4;)andU=§(q§ +q§+a(q, —q2)2), where §ot(q, —q2)2 isthepotential energy oftheelasticity ofthespring. Set qt‘l’(l2 qr_‘I2 Q1 :7? and Q2 _Qt+Q2 _Q1"Q2q.---e andqt--1.\/5 ﬂ andboth forms arereduced toprincipal axes: T=%(Qi+Q3) U=%(wiQi +w%Q§)- where w,=landw2=./l+211(Figure 80).Sothetwocharacteristic oscillations areas follows (Figure 81):Then 1.Q2=0,i.e.,q,=q2;both pendulums move inphase with theoriginal frequency l,andthe spring hasnoeffect; 2.Q,=0,i.e.,ql=-q2: thependulums move inopposite phase with increased frequency (02>lduetotheaction ofthe spring. ‘T2 A Qt n-Q1 2U=I Q2 __i___ \/I+2ot Figure 80Conﬁguration space oftheconnected pendulums Figure 81Characteristic oscillations oftheconnected pendulums 106 23:Small oscillations Now letthespring beveryweak: at<1.Then aninteresting effect called exchange ofenergy occurs. EXAMPLE 2.Suppose thatthependulums areatrestattheinitial moment, andoneofthem is given velocity q,=v.Wewillshow thatafter some time Ttheﬁrstpendulum willbealmost stationary. andalltheenergy willhave gone tothesecond. Itfollows from theinitial conditions thatQ,(0) =Q2(0) =O.Therefore, Q,=c,sint,and Q2=C‘;Sll'lwtwithw=oi+21z1+1(1e1).ButQ,(0)=Q2(0)=tyﬁ.Therefore, c,=l’/\/5 andc2=v/w\/6, andoursolution hastheform v_ I, v_ l_q,=—(sin! +~sinwt) q2=—(sint —vsinwt)2 w 2 w or,disregarding theterm v(l—(1/w))sin wt,which issmall since atis. v_ , _,q,z§(s1nt+ sinwt)=vcos ctsinwt, v2 _ ,,q2z5(sint —sinwt)=—vcoswtsin st, w—l at w+l¢,~=i—:— w’=~1zl. 2 2 2 Thequantity szat/2issmall, since atis;therefore q,undergoes anoscillation offrequency w’xlwith slowly changing amplitude vcosat(Figure 82). After time T=rt/2e2rt/oz,essentially only thesecond pendulum willbeoscillating; after 2T,again only theﬁrst,etc.(“beats”) (Figure 83). ‘I2 qt Figure 82Beats: trajectories intheconﬁguration space Q1 Q; —>[ >[ Figure 83Beats 107it 1. I. ‘l:lyl _‘.".‘.'.ii\i0|n<1-—"""" l 5 J t }i lx s x l t i r ll I‘ l ..:;an=w- 5:Oscillations / ‘ii I -7KT I1 m2 "11 Figure 84Connected pendulums ‘T2 ‘I1 Figure 85Potential energy ofstrongly connected pendulums EXAMPLE 3.Weinvestigate thecharacteristic oscillations oftwodifferent pendulums (m.9*m2, ll9%I2,g=1),connected byaspring withenergy 2ot(q, —q2)2(Figure 84).How dothecharac- teristic frequencies behave asoz—>0orasor—>oo? Wehave T=i(militii +M11542) q’ 42@= U=mill 51+"1212; +i(ql —¢lz)2- Therefore (Figure 85), A=(m1lf 0 B: m,l1+ot —<x 0 rnztg ‘(X "1212 +Q andthecharacteristic equation hastheform I —AI2 —det(B—AA)= ""‘+°‘ ”"‘ “ 2=0 '"(X m2l2-l-Ol—lm2l2 or all—(bo+b,.'x)). +(co+clot) =0, where a=m,m2lfl§ bo=m1l,m2l2(l, +l2) bl=m|lf +m2l§ co=m,m2l1l2 c,=m,l, +m2l2. This istheequation ofahyperbola inthe(at,/1)-plane (Figure 86).Asat—>0(weak spring) the frequencies approach thefrequencies offreependulums (wf_2 =lj); asat—>ac.oneofthe 108 23:Small oscillations a 7\=wz 0012oio5 Figure 86Dependence ofcharacteristic frequencies onthestiffness ofthespring / "12 "11 Figure 87Limiting caseofpendulums connected byaninﬁnitely stiffspring frequencies tends to1:,while theother approaches thecharacteristic frequency wxofapendu- lumwith twomasses ononerod(Figure 87): 2 m,l1 +m2l2w=L. ‘U m1l%+ mzlg PROBLEM. Investigate thecharacteristic oscillations ofaplanar double pendulum (Figure 88). PROBLEM. Find theshape ofthe trajectories ofthe small oscillations ofapoint mass ontheplane, sitting inside anequilateral triangle andconnected byidentical springs tothevertices (Figure 89). / 11 "11 12 "12 Figure 88Double pendulum Figure 89System with aninﬁnite setofcharacteristic oscillations 109l 4'..=.‘.i§: . 1‘; l1. 2; iigt it __..a..... —1.=-.-.-:;:;.a.~.\.-nvt~.--=--—-r l ll1 5:Oscillations Solution. Under rotation by120°thesystem ismapped onto itself. Consequently, alldirec- tions arecharacteristic, andboth characteristic frequencies arethesame: U=%w2(x2 +yz). Therefore, thetrajectories areellipses (cf.Figure 20). 24Behavior ofcharacteristic frequencies Weprove here theRayleigh-Courant-Fisher theorem onthebehavior ofcharacteristic fre- quencies ofasystem under increases inrigidity andunder imposed constraints. ABehavior ofcharacteristicfrequencies under a change inrigidity Consider asystem performing small oscillations, with kinetic andpotential energies T=%(Aq,q)>0 and U=%(Bq,q)>0 forallq,q¢0. Deﬁnition. Asystem with thesame kinetic energy, andanewpotential energy U’,iscalled more rigid ifU’=%(B'q, q)2%(Bq, q)=Uforallq. Wewish tounderstand how thecharacteristic frequencies change under anincrease intherigidity ofasystem. PROBLEM. Discuss theone-dimensional case. Theorem 1.Under anincrease inrigidity, allthecharacteristic frequencies areincreased, i.e.,ifwl3w23---3w,,arethecharacteristicfrequencies ofthelessrigid system, andw’,3w’23 3wj,arethecharacteristic frequencies ofthe more rigid system, thenw,3w’1;w23w’2;...;w,,3w§,. This theorem hasasimple geometric meaning. Without lossofgenerality wemay assume thatA=E,i.e.,thatweareconsidering theeuclidean struc- turegiven bythekinetic energy T=gq,ti).Toeach system weassociate the ellipsoids E:(Bq,q)=1andE’:(B’q, q)=1. Itisclear that Lemma l.Ifthesystem U’ismore rigid than U,then thecorresponding ellipsoid E’liesinside E. Itisalsoclear that Lemma 2.Themajor semi-axes oftheellipsoid aretheinverses ofthechar- acteristic frequencies witw,=1/a,-. Therefore, Theorem 1isequivalent tothefollowing geometric proposition (Figure 90). ll0 24:Behavior ofcharacteristic frequencies aim Figure 90Thesemi-axes oftheinside ellipse aresmaller. Theorem 2.Iftheellipsoid Ewith semi-axes a12a22 2a,,contains the ellipsoid E’with semi-axes a'12a'22 2(11,,both ellipses having the same center, thenthesemi-axes oftheinside ellipsoid aresmaller: a2za’1,a2 2-_a’2,...,a,,2a§,. EXAMPLE. Under anincrease intherigidity atofthe spring connecting thependulums ofExample 3,Section 23.thepotential energy grows, andbyTheorem l,thecharacteristic frequencies grow: dw,-/dot >0. Now consider thecasewhen therigidity ofthespring approaches inﬁnity, at—>ac.Then in thelimit thependulums arerigidly connected andwegetasystem with onedegree offreedom; thelimiting characteristic frequency wmsatisﬁes w,<wm<w2. BBehavior ofcharacteristic frequencies under the imposition ofaconstraint Wereturn toageneral system with ndegrees offreedom, andletT=2-(q,q) andU=%(Bq, q)(qeR")bethekinetic andpotential energies ofasystem performing small oscillations. A01. Figure 91Linear constraint LetlR"'1 cR”bean(n—1)-dimensional subspace inR"(Figure 91). Consider thesystem with n—1degrees offreedom (qEIR“1)whose kinetic andpotential energies aretherestrictions ofTandUtoR“1.Wesaythat thissystem isobtained from theoriginal byimposition ofalinear constraint. Letw,3w23---3(0,,bethencharacteristic frequencies oftheoriginal system, and wl_€01';S S011.-1 the(n—l)characteristic frequencies ofthesystem with aconstraint. lll1 I l l l t l l I 1 l , l r l I E t 1 5:Oscillations (.01 (02 OJ" O$O$ GO I I I0)] (4)2 b)n—I Figure 92Separation offrequencies Theorem 3.The characteristic frequencies ofthesystem with aconstraint separate thecharacteristicfrequencies oftheoriginal system (Figure 92): mlswlSw2sw2s"‘Swn—1Swh—1Swn~ ByLemma 2thistheorem isequivalent tothefollowing geometric propo- sition. Theorem 4.Consider thecross-section ofthen-dimensional ellipsoid E= {qz(Bq,q)=1}withsemi-axes a,2a22 2a,,byahyperplane W“ through itscenter. Then thesemi-axes ofthis(n—1)-dimensional ellip- soid—the cross-section E’—separate thesemi-axes oftheellipsoid E’ (Figure 93): a,za',2a2za’22---2a,,_, zaj,_,za,,. that\ Figure 93Thesemi-axes oftheintersection separate thesemi-axes oftheellipsoid CExtremal properties ofeigenvalues Theorem 5.Thesmallest semi-axis ofanycross-section oftheellipsoid Ewith semi-axes al2a22---2a,,byasubspace IR“islessthan orequal toah: ak=max min l|xl| {Rk} x6lRknE (the upper bound isattained onthesubspace spanned bythesemi-axes a,2a22 zat). Prtoor.“ Consider thesubspace R""‘+ 1spanned bytheaxesa,,2a,,.,, 2~~- 2a,,.Itsdimension isn—k+1.Therefore, itintersects R“.Letxbeapoint oftheintersection lying ontheellipsoid. Then l|x||3ak,since xe[l§’"""". 43Itisuseful tothink ofthe casen=3,k=2. lI2 25:Parametric resonance Since l3||x||,where listhelength ofthesmallest semi-axis oftheellipsoid EnR“,lmust benolarger than ah. l:l PROOF orTHEOREM 2.The smallest semi-axis ofevery k-dimensional section oftheinner ellipsoid R“nE’islessthan orequal tothesmallest semi-axis ofR“nE.ByTheorem 5, a},=max min ||xl|3max min l|x||=ah. l:l {llllk} xellll"nE' {[Rl"} xellll"nI:' PRo0i= orTHEOREM 4.Theinequality aj,3akfollows from Theorem 5, since inthecalculation ofakthemaximum istaken over alarger set.Toprove theinequality al.2a,.+1, weintersect lR"“ with anyk+1-dimensional subspace R“1.Theintersection hasdimension greater than orequal tok. Thesmallest semi-axis oftheellipsoid E’nlR"*1 isgreater than orequal to thesmallest semi-axis ofErsR“1.ByTheorem 5, a],= max min l|xl|z max min llxll [lRkClR"_l}XElRkﬁE' {llR“*1ClR"}xelR"*lnE' 2 max min |lxl|=a,,,1. l:l {[Rl"*1¢lR"}xe[Rl"*lr\E Theorems land3follow directly from those justproven. PROBLEM. Show that ifweincrease thekinetic energy ofasystem without decreasing thepotential energy (forexample, weincrease themass onagiven spring), then every characteristic frequency decreases. PROBLEM. Show thatunder theorthogonal projection ofanellipsoid lying inonesubspace of euclidean space onto another subspace, allthesemi-axes aredecreased. PROBLEM. Suppose thataquadratic form /1(5)oneuclidean space R"isacontinuously differen- tiable function oftheparameter c.Show thatevery characteristic frequency depends differen- tiably onc,andﬁnd thederivatives. ANSWER. Let/11,...,/1,,betheeigenvalues ofA(0). Toevery eigenvalue A,ofmultiplicity v,there corresponds asubspace IR".Thederivatives oftheeigenvalues ofA(a)at0areequal tothe eigenvalues oftherestricted form B=(dA/dz-:)l,=0 onR“. lnparticular, ifalltheeigenvalues ofA(0)aresimple. then their derivatives areequal tothe diagonal elements ofthematrix Binthecharacteristic basis forA(0). ltfollows from thisproblem that when aform isincreased, itseigenvalues grow. Inthisway weobtain newproofs ofTheorems 1and2. PROBLEM. How does thepitch ofabellchange when acrack appears inthebell? 25Parametric resonance Iftheparameters ofasystem vary periodically with time. then anequilibrium position canbe unstable, even ifitisstable foreach ﬁxed value ofthe parameter. This instability iswhat makes it possible toswing onaswing. 113I ll It l I i 2 i 1 l 1 1 wn-.._...u.,...~»- ... S l K. ll it l li gt =-m|-='r.-;‘=§:a'-'-......*-*'-“"'- W5-r;§'n:'-ii:\»1'..17 5:Oscillations ADynamical systems whose parameters vary periodically with time EXAMPLE 1.Aswing: thelength oftheequivalent mathematical pendulum I(t)varies periodically with time: l(t+T)=l(t)(Figure 94). / I / Figure 94Swing EXAMPLE 2.Apendulum inaperiodically varying gravitational ﬁeld (for example, themoon) isdescribed byHill’sequation: (1) q"=—w’(r)q w(r+T)=w(r) EXAMPLE 3.Apendulum suspended from apoint which periodically oscillates vertically isalsodescribed byanequation oftheform (1). Forsystems with periodically varying parameters theright-hand sideof theequations ofmotion areperiodic functions oft.Theequations ofmotion canbewritten intheform ofasystem ofﬁrst-order ordinary differential equaﬁons (2) X=f(x,t) f(X,t+T)=f(x,t), XEIR" with periodic right-hand sides. Forexample, Equation (1)canbewritten as thesystem (3) ii2w2x1}w(t +T)=(v(t). BThemapping ataperiod Recall thegeneral properties ofthesystem (2).Wedenote byg‘:IR"—>lR"‘the mapping taking XeIR"tothevalue attime t,g‘x=(p(t), ofthesolution (pof system (2)with initial conditions q>(0) =X(Figure 95). Themappings g‘donotform agroup: ingeneral, gz+s ¢gigs ¢gsgtl PROBLEM. Show that{g‘}isagroup ifandonly iftheright-hand sides fdonot depend ont. PROBLEM. Show that, ifTistheperiod off,then gT”‘ =g‘-gT and, in particular, g"T=(gT)", sothatthemappings g"T(naninteger) form agroup. 114 25:Parametric resonance Rf! AX‘) 0 rO ’A X . >1U T Figure 95Mapping ataperiod Themapping gr:R"—>R"plays animportant roleinwhat istocome; we willcallitthemapping ataperiod andwilldenote itby A:IR"—>IR" AX(0) =x(T). EXAMPLE. Forthesystems {xi =X2 {XI=X1 XZ= —~X| X2: —.\'2, which canbeconsidered periodic with anyperiod T,themapping Aisarotation orahyper- bolic rotation (Figure 96). X2 X2 x H in51.~“‘§~\\ :sR O— ZPX] a-X1 Figure 96Rotation andhyperbolic rotation Theorem. 1.Thepoint X0isaﬁxed point ofthemapping A(AX0=X0)ifandonlyifthe solution withinitial conditions X(0) =X0isperiodic withperiod T. 2.Theperiodic solution x(t)isLiapunov stable (asymptotically stable) ifand onlyiftheﬁxed point X0ofthemapping AisLiapunov stable (asymptoti- cally stable)“ 3.Ifthesystem (2)islinear, i.e.,f(X,t)=f(t)xisalinear function ofX, thenAislinear. 4.Ifthesystem (2)ishamiltonian, then Apreserves volume: detA*=1. “Aﬁxed point x0ofthe mapping AisLiapunov stable (respectively, asymptotically stable) if Vs>0,36>0suchthatifIx—x0|<6,then|A"x —A"x0| <sforall0<n<1:(respec- tively, A"x—A"x0 —>0asn—>ac). ll5l 1 l ll i1 i v l ‘llwt .__*-:.:..’;_T.‘:_- 1!?l it4% JE iti,LL i 1 A '1‘ i l '1 '1 La-;"f'ﬁ“_“e->*=*.:‘_':2"_‘ t l 4 l i‘ *2I! 5:Oscillations PROOF. Assertions (1)and (2)follow from therelationship gT"‘ =g‘A. Assertion (3)follows from thefactthatasumofsolutions ofalinear system isagain asolution. Assertion (4)follows from Liouville’s theorem. I1 Weapply thetheorem above tothemapping Aofthephase plane {(x1, x2)} onto itself, corresponding totheequation (1)andthesystem (3).Since (3)is linear andhamiltonian (H=%w2xf +%x§), Wegctt Corollary. Themapping Aislinear, andpreserves area (detA=1).Thetrivial solution ofEquation (1)isstable ifandonlyifthemapping Aisstable. PROBLEM. Show thatarotation oftheplane isastable mapping, anda hyperbolic rotation isunstable. CLinear mappings oftheplane toitself which preserve area Theorem. LetAbethematrix ofalinear mapping oftheplane toitself which preserves area (detA=1).Then themapping Aisstable if|trAI<2,and unstable ifltr Al>2(tr A=an+an). PROOF. LetA,andA2betheeigenvalues ofA.They satisfy thecharacteristic equation /12—(trA)/1+1=0with real coefﬁcients /ll+/12=trAand A1-/12=detA=1.Theroots /11and/12ofthisrealquadratic equation are realforltrA|>2andcomplex conjugate for|trAI<2. Intheﬁrstcase oneoftheeigenvalues hasabsolute value greater than 1, and onehasabsolute value lessthan 1;themapping Aisahyperbolic rotation andisunstable (Figure 97). 7\1A2 4 II O- 0 I . 0‘, 0A2 Figure 97Eigenvalues ofthemapping A Inthesecond case theeigenvalues lieontheunitcircle (Figure 97): 1:'l1"l2 =’11'I1=l’1ll2~ The mapping Aisequivalent toarotation through angle at(where /110 = ei“’),i.e.,itmaybereduced toarotation bymeans ofanappropriate choice of coordinates ontheplane. Therefore, itisstable. El Inthisway, every question about thestability ofthetrivial solution ofan equation oftheform (1)isreduced tocomputation ofthetrace ofthematrix 116 25:Parametric resonance A.Unfortunately, thecalculation ofthistrace canbedone explicitly only in special cases. Itisalways possible toﬁndthetrace approximately bynumeri- cally integrating theequation ontheinterval 03t5T.Intheimportant casewhen (v(t)isclose toaconstant, some simple general arguments canhelp. DStrong stability Deﬁnition. The trivial solution ofahamiltonian linear system isstrongly stable ifitisstable, andifthetrivial solution ofevery suﬂiciently close linear hamiltonian system isalsostable.“ Thetwotheorems above imply: Corollary. Ifltr Al<2,thenthetrivial solution isstrongly stable. PROOF. If|trAl<2,then amapping A’corresponding toasufficiently close system willalsohave |trA’!<2. U Letusapply thistoasystem with almost constant (only slightly varying) coefﬁcients. Consider, forexample, theequation (4) iiI—o)2(l +sa(t))x, s<1 where a(t+27:)=a(t), e.g., a(t)=cost (Figure 98)(apendulum whose frequency oscillates near towith small amplitude andperiod 2rt).‘6 wz 2nT=— V >1 Figure 98Instantaneous frequency asafunction oftime Wewillrepresent each system oftheform (4)byapoint intheplane of parameters s,to>0.Clearly, thestable systems with |trAI<2form an open setinthe(co,s)-plane; sodotheunstable systems with |trAl>2 (Figure 99). Theboundary ofstability isgiven bytheequation |trAl=2. Theorem. Allpoints ontheco-axis except theintegers and half—integers at=k/2,k-=0,1,2,...correspond tostrongly stable systems (4). ‘*5Thedistance between twolinear systems with periodic coefﬁcients, .i=B1(t)x, X=B2(t)x, isdeﬁned asthemaximum over tofthedistance between theoperators B,(t) andB1(t). “°Inthecasea(t)=cost, Equation (4)iscalled Matliieuk" equation. ll7l I=l '1.‘. 4 1 i - li» -.-.-.~.4...-=w=<._-.~.<=r.v:-- i i I i i i v *1 l M S I‘:1 ,....,,_‘_..l._,.__zr_ 5:Oscillations 6 2 I .3 ew Figure 99Zones ofparametric resonance Thus, thesetofunstable systems canapproach thecu-axis only atthe points w=k/2.Inother words, swinging aswing bysmall periodic changes ofthelength ispossible only inthecase when oneperiod ofthechange in length isclose toawhole number ofhalf-periods ofcharacteristic oscillations —aresult wellknown experimentally. Theproof ofthetheorem above isbased onthefactthatfore=0,Equation (4)hasconstant coefficients andisclearly solvable. PRQBLEM. Calculate thematrix ofthetransformation Aafter period T=21: inthebasis x,Scforsystem (4)with s=0. Solution. Thegeneral solution is: x=c,coscot+c2sinwt. Thesolution with initial conditions x=1,x=Ois: x=coswt x=-tosincut. Thesolution with initial conditions x=0,x=1is: 1.x=—sincot x=coscut. 0.) ANSWER. 1.cos21cm —sin21ta> A= 5‘) . —cosinZttco cosZnai Therefore, ltrAI=|2cos2am] <2iftoaék/2,k=0,1,...,and the theorem follows from thepreceding corollary. Amore careful analysis“ shows that ingeneral (and fora(t)=cost) theregion ofinstability (shaded inFigure 99)infactapproaches theto-axis near thepoints to=k/2,k=1,2,... . 47Cf.,forexample, theproblem analyzed below. l18 25:Parametric resonance Thus, fortozk/2,k=1,2, thelowest equilibrium position ofthe idealized swing (4)isunstable and itswings under anarbitrarily small periodic change oflength. This phenomenon iscalled parametric resonance. Acharacteristic property ofparametric resonance isthatitisstrongest when thefrequency ofthevariation oftheparameter v(inEquation (4),v=l) istwice thecharacteristic frequency w. Remark. Theoretically, parametric resonance canbeobserved forthe inﬁnite collection ofcases C0/l’zk/2,k=1,2,....Inpractice, itisusually observed only when kissmall (k=1,2,andmore rarely, 3).Thereason is that: 1.Forlarge ktheregion ofinstability approaches theco-axis inaverynarrow “tongue” andtheresonance frequencies tomust satisfy very rigid bounds (~ell“,where 6e(O,1)depends onthewidth oftheanalyticity band forthe function a(t)in(4)). 2.Theinstability itself isweak forlarge k,since itrAl—2issmall andthe eigenvalues areclose tolforlarge k. 3.Ifthere isanarbitrarily small amount offriction, then there isaminimal value skoftheamplitude inorder forparametric resonance tobegin (fors lessthan thistheoscillation diesout). Askgrows, ckgrows quickly (Figure 100). 6 >1»EQ Figure 100 Inﬂuence offriction onparametric resonance Wealsonotice thatforEquation (4)thesizeofxgrows without bound in theunstable case. Inrealsystems, oscillations attain only ﬁnite amplitudes, since forlarge xthelinear equation (4)itself loses inﬂuence, andwemust consider thenonlinear effects. PROBLEM. Find theshape oftheregion ofstability inthei:.to-plane forthesystem described by theequations (1)+t: 0<t<TI=—./‘mix rm= <1<2is<i —II TI 7! {(1) /"(t+Zn)=/"(1). Solution. Itfollows from thesolution ofthe preceding problem thatA=A2/1,,where 1 A,,=6*53" —w,,s,, ck q=costtwk, s,‘=sin1tw,.w,_3 =(L)i4:. 1195i I i 3 i I F i ~.<»_...._.__.___.._._..,_.,,,~. i 1 it ...-.~=<-.'e..a...-' ?§ s > 4. — -Eii ‘_1 il- iii 1'1 I I 5:Oscillations Therefore, theboundary ofthezone ofstability hastheequation (1)1 to; (5) ill"/4| =25152 —( +‘)$i$z =2-(02 0), Since 8<1,wehave to,/cu; =(cu+s)/(w —e)z1,Weintroduce thenotation col wz f+—=2(1+A).(U2 co, Then, asiseasily computed, A=(282/(U2) +O(s“) <1.Using therelations 2c,c, = cos211s+cos21:0)andZslsz =cos21:2-cos21:w, werewrite Equation (5)intheform —Acos27158+(2+A)cos 211w =1-2 OI’ (6) 02 2+Acos21ts a cs 1zco=—-i2+A -2+Acos2rr.s (6b) cos21:00 =ii2+A Intheﬁrstcasecos21:a) 21.Therefore, weset co=k+a,Ia| <1 cos2rrw=cos21ia= 1—2ﬂ2a2+O(d4), Werewrite Equation (6a)intheform A cos211a) =1— m(l —cos21:2) or21:20: +O(a“) =Artzez +O(s"). Substituting inthevalue A=(262/(oz) +O(iz“), weﬁnd +22 (2)_ k+1:2 (2 = —— .. L0= -- OB. a _w2+0£,1e, _k2+ ) Equation (6b)issolved analogously; fortheresult weget at=+—_a +0a, k1+E ()2 1c(k+%) Therefore theanswer hastheform depicted inFigure 101. L 220I 2 2 _w Figure 101 Zones ofparametric resonance forf=atie.6 120 25:Parametric resonance EStability ofaninverted pendulum with vertically oscillating point ofsuspension PROBLEM. Can thetopmost, usually unstable, equilibrium position ofa pendulum become stable ifthepoint ofsuspension oscillates inthevertical direction (Figure 102)? me 2.1 a Parabola r 21 I Figure 102 Inverted pendulum with oscillating point ofsuspension Letthelength ofthependulum bel,theamplitude oftheoscillation ofthe point ofsuspension bea<l,theperiod ofoscillation ofthepoint ofsuspen- sion21,and, moreover, inthecourse ofevery half-period lettheacceleration ofthepoint ofsuspension beconstant andequal toic(then c=8a/:2). It turns outthatforfastenough oscillations ofthepoint ofsuspension (1<1) thetopmost equilibrium becomes stable. Solution. Theequation ofmotion canbewritten intheform ii=(col1d2)x(thesignchanges after time r),where co’=g/landdz=c/I.Iftheoscillation ofthesuspension isfastenough, thendz>wz(dz=8a/Ir’). Asintheprevious problem, A=A2A,,where A,= l< A,= Q1 1chkt —shkt cosQt ~sinQt kshkt chkt —QsinQt cosQt k2=dZ+a>2, Q2=d2—w2. Thestability condition |trAl<2therefore hastheform kQ (7) ‘2CIlkTCOSQT+(6—F)ShkTSlﬂQT <2 Wewillshow thatthiscondition isfulﬁlled forsufﬁciently fastoscillations ofthepoint of suspension, i.e.,when c>g.Weintroduce thedimensionless variables 12,it: €=s2<l §=tﬁ<1.C 121i i 1 i -1211 ...rz.":'.<;.s‘.‘;~}2%iL!::*<~a*.~':iiri'<=....**-"‘.._-_.-;‘<.;.w.~mi.-1.~.<.~.f;1‘_::?;E <£1 ii}, ‘U1;: f1;; i=h Iii: -‘€ﬁ§ I;ﬁK3*;t;:l*1“gt31it .f5 L0 5:Oscillations Then kt=2,/it./1+ #2 Qt=2\/2e./1- if kQ l+,u2 l—t:2§:t=\/,"_7;"\/r11F=2"’+°‘"‘l- Therefore, forsmall canditwehave thefollowing expansion with error o(e“+ii‘): chkr= 1+4t:2(l +,u2)+§c‘+'"" cosQt=1—4a1(1—;i2)+§s4+-'" k Q - 226-I shktsinQr=l6c;i +--- sothestability condition (7)takes theform 2(1—16:-:4 +#3‘-is“ +88z}.l2 +---)+1682/.12 <2, i.e.,disregarding thesmall higher-order terms, §l6t-:4 23211222 orits1:./2/3, org/c52a/3!. This condition canberewritten as N> 3I022 I _-—co—z .—,64a wa where N=1/22isthenumber ofoscillations ofthepoint inoneunitoftime. Forexample, ifthe length ofthependulum lis20cm,andtheamplitude oftheoscillation ofthepoint ofsuspension ais1cm,then l980N20.22 20z31(oscillations persecond). Forexample, thetopmost position isstable ifthefrequency ofoscillation ofthepoint of suspension isgreater than 40persecond. 122 Rigid bodies Inthischapter westudy indetail some very special mechanical problems. These problems aretraditionally included inacourse onclassical mechanics, ﬁrstbecause they were solved byEuler andLagrange, andalsobecause we liveinthree-dimensional euclidean space, sothat most ofthemechanical systems with aﬁnite number ofdegrees offreedom which wearelikely to encounter consist ofrigid bodies. 26Motion inamoving coordinate system Inthisparagraph wedeﬁne angular velocity. AMoving coordinate systems Welook atalagrangian system described incoordinates q,tbythelagrangian function L(q,ti,t).Itwilloften beuseful toshift toamoving coordinate system Q=Q(q, t). Towrite theequations ofmotion inamoving system, itissufficient to express thelagrangian function inthenewcoordinates. Theorem. Ifthetrajectory y:q=q>(t)ofLagrange’s equations d(5L/dq)/dt = (‘L/Pq iswritten asy:Q=(h(t) inthelocal coordinates Q,t(where Q= Q(q, t)),thenthefunction (h(t)satisﬁes Lagrange’s equations d(8L’/0Q)/dt = 0L’/0Q, where I/(Q, Q,t)=L(q,tj, t). PR00i=. Th_e trajectory yisanextremal: 51,L(q, 1'],t)dt=0.Therefore, 6I,L’(Q, Q,t)dt=0and(D(t)satisﬁes Lagrange’s equations. III 123>1..._.__.__c”......_.__k.___ l ( I i i ii.,1l ':;;1t-.1-i¢.<:¢2wr-r’.'.t::';'::‘:*:$-"~"1Q—.*_-'.".¢~.=;': it U Tl1; 1'i .~ I -Ii —'i;l‘..'_‘1‘-i§r}_h¢-.1-it J ii 6:Rigid bodies BMotions, rotations, andtranslational motions Weconsider, inparticular, theimportant casewhere qisthecartesian radius vector ofapoint relative toaninertial coordinate system k(which wewill callstationary), andQisthecartesian radius vector ofthesame point relative toamoving coordinate system K. Deﬁnition. LetkandKbeoriented euclidean spaces. Amotion ofKrelative tokisamapping smoothly depending ont: D,:K—>k, which preserves themetric andtheorientation (Figure 103). .\/a /‘C’ Figure 103 Themotion D,decomposed astheproduct ofarotation B,andtransla- tionC, Deﬁnition. Amotion D,iscalled arotation ifittakes theorigin ofKtothe origin ofk,i.e.,ifD,isalinear operator. Theorem. Every motion D,canbeuniquely written asthecomposition ofa rotation B,:K—>kandatranslation C,:k—>k: D,=C,B,, where C,q=q+r(t),(q,rek). PROOF. Wesetr(t)=D,0,B,=C,‘‘D,.Then B,0=0. El Deﬁnition. Amotion D,iscalled translational ifthemapping B,:K —>k correspondingto itdoes notdepend ont:B,=B0=B,D,Q =BQ+r(t). Wewillcallkastationary coordinate system, Kamoving one, and q(t)ektheradius-vector ofapoint moving relative tothestationary system; if (1) q(t)=DiQ(I) =BiQ(t) +r(t) (Figure 104), Q(t)iscalled theradius vector ofthe point relative tothemoving system. Warning. Thevector B,Q(t) ekshould notbeconfused with Q(t)e K— they lieindifferent spaces! 124 26:Motion inamoving coordinate system BiQ(!) k q(t) Q(t) r(t) K Figure 104 Radius vector ofapoint withrespect tostationary (q)andmoving (Q) coordinate systems CAddition ofvelocities Wewillnowexpress the“absolute velocity” 1']interms oftherelative motion Q(t) andthemotion ofthecoordinate system, D,.Bydifferentiating with respect totinformula (1)weﬁndaformula fortheaddition ofvelocities (2) q=BQ+BQ+i-. Inorder toclarify themeaning ofthethree terms in(2),weconsider the following special cases. Thecaseoftranslational motion (B=0) InthiscaseEquation (2)gives q=BQ+i'.Inother words, wehave shown Theorem. Ifthemoving system Khasatranslational motion relative tok,then theabsolute velocity isequal tothesumoftherelative velocity andthe velocity ofthemotion ofthesystem K: (3) V=vi+V0» where v=qekistheabsolute velocity, v’=BQ6kistherelative velocity (distinctfrom eKl) v0=i'ekisthevelocity ofmotion ofthemoving coordinate system. DAngular velocity Inthecase ofarotation ofKtherelationship between therelative andab- solute velocities isnotsosimple. Weﬁrstconsider thecasewhen ourpoint is atrestinK(i.e., =0)andthecoordinate system Krotates (i.e., r=0). Inthiscase themotion ofthepoint q(t)iscalled atransferred rotation. EXAMPLE. Rotation withﬁxed angular velocity wek.LetU(t): k—>kbethe rotation ofthespace karound theto-axis through theangle |o)|t. Then B(t)=U(t)B(0) iscalled auniform rotation ofKwithangular velocity 0). 125‘lr l '11.,1 ii iii ls§1 l I r i Ii I 4 1. \‘ r2IiiiLiii 1 a ﬁe-\~—r,~q.._pg i t,. t‘ v. iii 6:Rigid bodies (.0 <1 q O Figure 105 Angular velocity Clearly, thevelocity ofthetransferred motion ofthepoint qinthiscaseis given bytheformula (Figure 105) ll=Iw.q]- Wenow turn tothegeneral caseofarotation ofK(r=0, =0). Theorem. Atevery moment oftime t,there isavector m(t)eksuch thatthe transferred velocity isexpressed bytheformula (4) ii=Io.q]. ‘mek- The vector toiscalled theinstantaneous angular velocity; clearly, itis deﬁned uniquely byEquation (4). Corollary. Suppose thatarigid body Krotates around astationary point 0of thespace k.Then atevery moment oftimethere exists aninstantaneous axis ofrotation thestraight lineinthebody passing through Osuch thatthe velocity ofitspoints atthegiven moment oftime isequal tozero. The velocity oftheremaining points isperpendicular tothisstraight lineandis proportional tothedistance from it. Theinstantaneous axisofrotation inkisgiven byitsvector 0);inKthe corresponding vector isdenoted byQ=B"‘to6K;Qiscalled thevector of angular velocity inthebody. EXAMPLE. Theangular velocity oftheearth isdirected from thecenter totheNorth Pole; its length isequal toZ1:/3600 -24sec” z7.3-10" sec“'. . PROOF orTHETHEOREM. By(2)wehave it=EQ- Therefore, ifweexpress Qinterms ofq,wegetq=BB“q =Aq,where A=BB'1:k —>kisalinear operator onk. 126 26:Motion inamoving coordinate system Lemma 1.Theoperator Aisskew-symmetric: A’+A=0. PRooE. Since B:K->kisanorthogonal operator from oneeuclidean space toanother, itstranspose isitsinverse: B‘=B‘1:k—>K.Bydifferentiating therelationship BB‘=Ewith respect tot,weget BB‘+BB‘=01212"‘+(BB“)' =0. III Lemma 2.Every skew—symmetric operator Aonathree-dimensional oriented euclidean space istheoperator ofvector multiplication byaﬁxed vector: Aq=[(0,q]forallqe[R3. PROOF. Theskew-symmetric operators from [R3to[R3form alinear space. Itsdimension is3,since askew-symmetric 3><3matrix isdetermined byits three elements below thediagonal. Theoperator ofvector multiplication by0)islinear andskew-symmetric. Theoperators ofvector multiplication byallpossible vectors tointhree- space form alinear subspace ofthespace ofallskew-symmetric operators. Thedimension ofthissubspace isequal to3.Therefore, thesubspace of vector multiplications isthespace ofallskew-symmetric operators. El CONCLUSION orTHEPROOF OFTHETHEOREM. ByLemmas 1and2, <1=Aq=Iw.q]~ E1 Incartesian coordinates theoperator Aisgiven byanantisymmetric matrix; wedenote itselements byi(01‘2, 3: 0 —(1)3 002 A = (1)3 0 *(01 - -(U2 (1)1 0 Inthisnotation thevector to=colel +0),e2+023e3willbeaneigenvector with eigenvalue 0.Byapplying Atothevector q=qlel +qzez +q3e3, weobtain byadirect calculation Aq=Iw.<1]- ETransferred velocity Thecaseofpurely rotational motion Suppose ‘now that thesystem Krotates (r=0),andthat apoint inK ismoving (Q9*0).From (2)weﬁnd(Figure 106) i=BQ+BQ= [Mi+v'. Inother words, wehave shown I27*;_,__.Q-$1...._..__._.,N_=,_,_ l 4‘ 12 5l ir 5. 1. il1.11. ii ,_. 1. t t it 1. 1l 1 l i 2 1 -é 1, 1.i i 6:Rigid bodies (U U: U vn q O Figure 106 Addition ofvelocities Theorem. Ifamoving system Krotates relative toOek, then theabsolute velocity isequal tothesum oftherelative velocity andthetransferred velocity: v=v’+vn, where v=q6kistheabsolute velocity (5) v’=BQekistherelative velocity v,,=BQ=[(1),q]ekisthetransferred velocity ofrotation. Finally, thegeneral case canbereduced tothetwocases above, ifwe consider anauxiliary system K1which moves bytranslation with respect to kand with respect towhich Kmoves byrotating around OEK1. From formula (2)onecanseethat v=v’+v,,+v0, where v=qekistheabsolute velocity, v’=BQekistherelative velocity, v,,=BQ=[(0,q—r]ekisthetransferred velocity ofrotation, and v0=rekisthevelocity ofmotion ofthemoving coordinate system. PROBLEM. Show thattheangular velocity ofarigid body does notdepend on thechoice oforigin ofthemoving system Kinthebody. PROBLEM. Show thatthemost general movement ofarigid body isahelical movement, i.e.,thecomposition ofarotation through angle (paround some axisandatranslation byhalong it. PROELEM. Awatch liesonatable. Find theangular velocity ofthe hands ofthe watch: (a)relative totheearth, (b)relative toaninertial coordinate system. 128 27: lnertial forces and theCoriolis force Him. lfwe aregiven three coordinate systems k.K,,andK2.then theangular velocity ofK2 relative tokisequal tothesumoftheangular velocities ofK,relative tokandofK2relative toK,,since its+A,t+--»)(E+ /in+~-)=E+(/11+ A2)t+ 27Inertial forces andtheCoriolis force Theequations ofmotion inanon-inertial coordinate system differ from theequations ofmotion inaninertial system byadditional terms called inertial forces. This allows ustodetect experi- mentally thenon-inertial nature ofasystem (forexample. therotation oftheearth around its axis). ACoordinate systems moving bytranslation Theorem. Inacoordinate system Kwhich moves bytranslation relative toan inertial system k,themotion ofamechanical system takes place asifthe coordinate system were inertial, butonevery point ofmass manadditional “inertial force” acted: F=—mi,where Fistheacceleration ofthesystem K. PROOF. IfQ=q—r(t),thenmQ=mii—mi.Theeﬁect ofthetranslation of thecoordinate system isreduced inthisway totheappearance ofanaddi- tional homogeneous force ﬁeld~mW, where Wistheacceleration ofthe origin. El it m(g—F) //l\\ 7777777777777 Figure 107 Overload EXAMPLE I.Atthemoment oftakeoff, arocket hasacceleration i‘directed upward (Figure I07). Thus, thecoordinate system Kconnected totherocket isnotinertial, andanobserver inside can detect theexistence ofaforce ﬁeldmWandmeasure theinertial force. forexample. bymeans of weighted springs. Inthiscasetheinertial force iscalled overloizd.* EXAMPLE 2.When jumping from aloft,aperson hasacceleration g,directed downwards. Thus, thesumoftheinertial force andtheforce ofgravity isequal tozero; weighted springs show that theweight ofanyobject isequal tozero, sosuch astate iscalled weighilessness. Inexactly the same way_ weightlessness isobserved inthefreeballistic ﬂight ofasatellite since theforce of inertia isopposite tothegravitational force oftheearth. EXAMPLE 3.llthepoint ofsuspension ofapendulum moves with acceleration Wtt). then the pendulum moves asiftheforce ofgravity gwere variable andequal tog—W(t). *Translator’s note. Theword overload istheliteral translation oftheRussian term peregruzka. There does notseem tobeanEnglish term forthisparticular kind ofinertial force. I29l iI l li ,1».-.~n=-,1. .1- iiiif it i 6 l Ii 6:Rigid bodies BRotating coordinate systems LetB,:K—>kbearotation ofthecoordinate system Krelative tothesta- tionary coordinate system k.Wewilldenote byQ(t)6Ktheradius vector of amoving point inthemoving coordinate system, andbyq(t)=B,Q(t) 6k theradius vector inthestationary system. Thevector ofangular velocity in themoving coordinate system isdenoted, asinSection 26,byQ.Weassume that themotion ofthepoint qinkissubject toNewton’s equation mij= T(q,(1)- Theorem. Motion inarotating coordinate system takes place asifthree addi- tional inertial forces acted onevery moving point Qofmass m: 1.theinertial force ofrotation :m[§2, Q], 2.theCoriolis force.‘ 2m[Q, Q],and 3.thecentrifugal force: m[Q, [(2, Thus mo=F—m[§1,Q] —2min,Q1—mm,[9,Q11, where BF(Q,Q)=f(BQ,(Bot)- The ﬁrstoftheinertial forces isobserved only innonuniform rotation. Thesecond andthird arepresent even inuniform rotation. Q l9,Ql I’ -[9, [9,Qll Q O Figure 108 Centrifugal force ofinertia The centrifugal force (Figure 108) isalways directed outward from the instantaneous axis ofrotation Q;ithasmagnitude Iﬂlzr, where risthe distance tothisaxis.Thisforce doesnotdepend onthevelocity oftherelative motion, andactseven onabody atrestinthecoordinate system K. TheCoriolis force depends onthevelocity Inthenorthern hemisphere oftheearth itdeﬂects every body moving along theearth totheright, and every falling body eastward. I30 27:Inertial forces andtheCoriolis force PROOF orTHETHEOREM. Wenotice that foranyvector XEK wehave BX=B[Q, X].Infact, bySection 26,BX=[(0,X]=[B9, BX]. This is equal toB[Q, X]since theoperator Bpreserves themetric andorientation, andtherefore thevector product. Since q=BQweseethatq=BQ+BQ=B(Q +[0,Q]). Differenti- ating once more, weobtain ii=3(Q+[QtQ])+B(Q+[51,Q]+[9,Q1)=B(l_Q,(Q+i_n,Q1)1 +o+[Q01+tn,Q1)=B(Q+ztn,Q]+tn,in,Q1]+tn,Q])- iii (We again used therelationship BX=B[Q, X];this time X= + [9,Q]-) Wewillconsider inmore detail theeﬂ'ect oftheearth’s rotation onlaboratory experiments. Since theearth rotates practically uniformly, wecantake Q=0.Thecentrifugal force hasits largest value attheequator, where itattains Qzp/g z(7.3x10-5)’ -6.4x10°/9.8 z3/I000 theweight. Within thelimits ofalaboratory itchanges little, sotoobserve itonemust travel some distance. Thus, within thelimits ofalaboratory therotation oftheearth appears only in theform oftheCoriolis force: inthecoordinate system Qassociated totheearth, wehave, with good accuracy. d. .EMQ=ms+Zm[Q. 9] (thecentrifugal force istaken intoaccount ing). EXAMPLE I.Astone isthrown (without initial velocity) intoa250mdeep mine shaft atthe latitude ofLeningrad. How fardoes itdeviate from thevertical? Wesolve theequation Q=2+2[Q.0] bythefollowing approach, taking Q<l.Weset(Figure 109) Q=Qt+Q1. whereQ,(0)=Q2(0)=0andQ,=Q,(0)+git/2.ForQ,,wethenget _ 3 2 ZQ2=2[8l-91+ Om’) Q1~';ten1=§ih,ni h- Q,(0) Q >~N E 8 Q,(I) Figure 109 Displacement ofafalling stone byCoriolis force 1311 I l ll itI ii IE! i l i. E'. li i 6:Rigid bodies From thisitisapparent thatthestone lands about 2 27l|h||QlcosA:.-——-250-7-l0‘5-lm 24cm3 3 2 totheeast. PROBLEM. Byhow much would theCoriolis force displace amissile ﬁred vertically upwards at Leningrad from falling back onto itslaunching pad, ifthemissile rose lkilometer? EXAMPLE 2(The Foucault pendulum). Consider small oscillations ofanideal pendulum, taking intoaccount theCoriolis force. Letex,e,.,ande,betheaxesofacoordinate system associated totheearth, with e:directed upWat'd5, ande,ande,inthehorizontal plane (Figure I10). ln Q Figure 110 Coordinate system forstudying themotion ofaFoucault pendulum theapproximation ofsmall oscillations, z"=0(incomparison with itand_i-);therefore, the horizontal component oftheCoriolis force willbe2my§2,e, —2m.€Q,e,.. From thiswegetthe equations ofmotion {ii=—w2x +2y'Q,, (Q,=IﬂlsinA0,where 3.0isthelatitude) it=—w2y —ZXQU Ifwesetx+iy=w,then w=)2+iy,ﬁt=ii+ii‘,andthetwoequations reduce toone complex equation +i'2Q,w +rozw =0. Wesolve ii:M=e",A2+2i'Q,A +of=0,A=—iQ, 11"/Q5 +012.ButQf<(1)2.Therefore. \/Q3 +lI)2=cu+0(Qfl. from which itfollows, bydisregarding Qf,that Z2—iQ: iiw or,tothesame accuracy, W=e—i'ﬂ,i((,1eimi +(,2e—|mI). ForQ:=0wegettheusual harmonic oscillations ofaspherical pendulum. Weseethatthe effect ofthe Coriolis force reduces toarotation ofthe whole picture withangular velocity —Q,_ where |Q,| =IQIsinlo. Inparticular, ifthe initial conditions correspond toaplanar motion (y(0) =_t"(0)=0),then theplane ofoscillation willberotating with angular velocity —Q, with respect totheearth‘s coordinate system (Figure lII). Atapole, theplane ofoscillation makes oneturn inatwenty-four-hour day(and isﬁxed withrespect toacoordinate system notrotating withtheearth). Atthelatitude ofMoscow (56°) theplane ofoscillation turns 0.83ofarotation inatwenty-four-hour day.i.e.,12.5‘ inanhour. l32 28:Rigid bodies Figure 111 Trajectory ofaFoucault pendulum PROBLEM. Ariver ﬂows with velocity 3km/hr. Forwhat radius ofcurvature ofariver bend isthe Coriolis force from theearth’s rotation greater than thecentrifugal force determined bytheﬂow oftheriver? ANSWER. Theradius ofcurvature must beleast ontheorder of10kmforariver ofmedium width. Thesolution ofthisproblem explains why alarge river inthenorthern hemisphere (for example, theVolga inthemiddle ofitscourse), undermines thebase ofitsright bank, while a river liketheMoscow River, with itsabrupt bends ofsmall radius, undermines either theleftor right (whichever isoutward from thebend) bank. 28Rigid bodies Inthisparagraph Wedeﬁne arigid body anditsinertia tensor, inertia ellipsoid, moments of inertia, andaxes ofinertia. ATheconﬁguration manifold ofarigid body Deﬁnition. Arigid body isasystem ofpoint masses, constrained byholonomic relations expressed bythefactthatthedistance between points isconstant: Theorem. The conﬁguration manifold ofarigid body isasix-dimensional manifold, namely, R3><SO(3) (thedirect product ofathree-dimensional space [R3andthegroup S0(3) ofitsrotations), aslong asthere arethree points inthebody notinastraight line. PROOF. Letxl,x2,andX3bethree points ofthebody which donotlieina straight line. Consider theright-handed orthonormal frame whose ﬁrst vector isinthedirection ofx2—x,,andwhose second isontheX3sideinthe x1x2x3-plane (Figure 112). Itfollows from theconditions Ix,—xj-I=r,-j (i=1,2,3),that thepositions ofallthepoints ofthebody areuniquely determined bythepositions ofx,, x2,andx3,which aregiven bytheposition oftheframe. Finally, thespace offrames inR3is{R3><S0(3), since every frame isobtained from aﬁxed onebyarotation andatranslation.“ 1:1 ‘8Strictly speaking, theconﬁguration space ofarigid body isR3xO(3), and R3><SO(3) is onlyoneofthe twoconnected components ofthismanifold, corresponding totheorientation of thebody. 133l iii J l‘ i l M t lit 4l ' lg. n_<4‘l|?.fil:yy... ‘r-erg.4....;. M.‘ i- ."., ‘i l .'; . ' ii;. l‘ ' 1 -.-=_-.=eA;-.¢e::=_-\_E’f~_%".1. ii.;. -4-=-of;-:- l l 1 6:Rigid bodies 6'2 X? 91Xi X2 63 Figure 112 Conﬁguration manifold ofarigid body PROBLEM. Find theconﬁguration space ofarigid body, allofwhose points lieonaline. ANSWER. R3><S1. Deﬁnition. Arigid body with aﬁxed point Oisasystem ofpoint masses con- strained bythecondition xl=Oinaddition toconditions (1). Clearly, itsconﬁguration manifold isthethree-dimensional rotation group S0(3). BConservation laws Consider theproblem ofthemotion ofafreerigid body under itsowninertia, outside ofanyforce ﬁeld. Foran(approximate) example wecanusethe rolling ofaspaceship. The system admits alltranslational displacements: they donotchange thelagrangian function. ByNoether’s theorem there exist three ﬁrstintegrals: thethree components ofthevector ofmomentum. Therefore, wehave shown Theorem. Under thefree motion ofarigid body, itscenter ofmass moves uniformly andlinearly. Now wecanlook ataninertial coordinate system inwhich thecenter of inertia isstationary. Then wehave Corollary. Afreerigid body rotates about itscenter ofmass asifthecenter of mass were ﬁxed atastationary point O. Inthisway, theproblem isreduced totheproblem, with three degrees of freedom, ofthemotion ofarigid body around aﬁxed point 0.Wewillstudy thisproblem inmore detail (notnecessarily assuming that0isthecenter of mass ofthebody). The lagrangian function admits allrotations around O.ByNoether’s theorem there exist three corresponding ﬁrstintegrals: thethree components ofthevector ofangular momentum. Thetotal energy ofthesystem, E=T, 134 28:Rigid bodies isalsoconserved (here itisequal tothekinetic energy). Therefore, wehave shown Theorem- Intheproblem ofthemotion ofarigid body around astationary point O,intheabsence ofoutside forces, there arefour ﬁrst integrals: M,,,MY, Mz,andE. From thistheorem wecangetqualitative conclusions about themotion without anycalculation. Theposition andvelocity ofthebody aredetermined byapoint inthe six-dimensional manifold TSO(3)—the tangent bundle oftheconﬁguration manifold S0(3). Theﬁrstintegrals M,,,My’M2,andEarefourfunctions on TSO(3). Onecanverify thatinthegeneral case(ifthebody does nothave any particular symmetry) these four functions areindependent. Therefore, the fourequations Mx=C1 My=C2 MZZC3 deﬁne atwo-dimensional submanifold V,inthesix-dimensional manifold TSO(3). This manifold isinvariant: iftheinitial conditions ofmotion giveapoint onV,,then foralltime ofthemotion, thepoint inTSO(3) corresponding to theposition andvelocity ofthebody remains inV,. Therefore, V,admits atangent vector ﬁeld (namely, theﬁeld ofvelocities ofthemotion onTSO(3)); forC4>0thisﬁeld cannot have singular points. Furthermore, itiseasy toverify that V,iscompact (using E)andorientable (since TSO(3) isorientable)/w Intopology itisproved thattheonly connected orientable compact two- dimensional manifolds arethespheres with nhandles, n20(Figure 113). Ofthese, only thetorus (n=1)admits atangent vector ﬁeld without singular points. Therefore, theinvariant manifold V,isatwo-dimensional torus (or several tori). Wewillseelater thatonecanchoose angular coordinates (p1,(p2,(mod 21:) onthistorus such thatamotion represented byapoint ofV,isgiven bythe equations <12,=w1(c), (p2=co2(c). 49Thefollowing assertions areeasy toprove: 1.Letf,,...,ﬂ,:M—>[Rbefunctions onanoriented manifold M.Consider thesetVgiven by theequations f,=c1,...,ﬂ, =ck.Assume that thegradients offl,...,f,, arelinearly independent ateach point. Then Visorientable. 2.Thedirect product oforientable manifolds isorientable. 3.Thetangent bundle TSO(3) isthedirect product [R3><S0(3). Amanifold whose tangent bundle isadirect product iscalled parallelizable. Thegroup S0(3) (like every Liegroup) is parallelizable. 4.Aparallelizable manifold isorientable. Itfollows from assertions 1-4that50(3), TSO(3), andV,areorientable. 135‘hm 1 E1 ll. l I l i1 ,. l. 9 it.lf!'= 1:llt it‘; llll F ‘¢.l .il‘ill 7'I.".T?"C".'7< ‘I; v‘ '1 iill if.; -'i1.1 t i 1. ‘l l 6:Rigid bodies ooood Figure 113 Two-dimensional compact connected orientable manifolds Inother words, arotation ofarigid body isrepresented bythesuper- position oftwo periodic motions with (usually) different periods: ifthe frequencies co,and(02arenon-commensurable, then thebody never returns toitsoriginal state ofmotion. Themagnitudes ofthefrequencies (U1andcu; depend ontheinitial conditions C. CTheinertia operator“) Wenow goontothequantitative theory and introduce thefollowing notation. Letkbeastationary coordinate system andKacoordinate system rotating together with thebody around thepoint 0:inKthebody isatrest. OJ m V q 0 Figure 114 Radius vector and vectors ofvelocity, angular velocity and angular momentum ofapoint ofthebody inspace Every vector inKiscarried over tokbyanoperator B.Corresponding vectors inKandkwillbedenoted bythesame letter; capital forKandlower case fork.So,forexample (Figure 114), qekistheradius vector ofapoint inspace; QeKisitsradius vector inthebody, q=BQ; v=qekisthevelocity vector ofapoint inspace; VeKisthesame vector inthebody, v=BV; toEkistheangular velocity inspace; QeKistheangular velocity inthebody, 0)=BQ; mekistheangular momentum inspace; M6Kistheangular momentum inthebody, m=BM. Since theoperator B:K—>kpreserves themetric and orientation, it preserves thescalar andvector products. 5°Often called theinertia tensor (translators note). 136 28:Rigid bodies Bydeﬁnition ofangular velocity (Section 26), V=[(9,q]. Bydeﬁnition oftheangular momentum ofapoint ofmass mwith respect toO, m=[iimi]=mlq.lw,<11]- Therefore, M=m[Q,[9,Q1]- Hence, there isalinear operator transforming QtoM: A:K—>K AQ=M. This operator stilldepends onapoint ofthebody (Q)anditsmass (m). Lemma. Theoperator Aissymmetric. PROOF. Inview oftherelation ([a,b],c)=([c,a],b)wehave, foranyXand YinK, (AX,Y)=m([Q. [X,Q1].Y)=m([Y. Q],[X,Q]). andthelastexpression issymmetric inXandY. Cl Bysubstituting thevector ofangular velocity QforXandYandnoticing that[K2,Q]2 =V2=v2,weobtain Corollary. Thekinetic energy ofapoint ofabody isaquadratic form with respect tothevector ofangular velocity Q,namely: T=%(AQ,Q)=govt,Q). Thesymmetric operator Aiscalled theinertia operator (ortensor) ofthe point Q. Ifabody consists ofmany points Q,with masses m,,then bysumming we obtain Theorem. Theangular momentum Mofarigid body withrespect toastationary point 0depends linearly ontheangular velocity Q,i.e.,there exists alinear operator A:K —>K,AQ=M.Theoperator Aissymmetric. Thekinetic energy ofabody isaquadratic form withrespect totheangular velocity Q, T=%(AQ,Q)=givi,Q). PROOF. Bydeﬁnition, theangular momentum ofabody isequal tothesum oftheangular momenta ofitspoints: M=ZM,=ZA,-Q=AQ, whereA=ZA,-. 137\ .4—-—--—-—~—~ﬂ:'-viil i ll -wr»—nu'e1. __.-_--‘an.-£6.-1_e=-_ /—*.'.'€:-:ﬁ#‘""'"‘"* ,. l .i, i- I i 6:Rigid bodies Since bythelemma theinertia operator A,-ofevery point issymmetric, theoperator Aisalsosymmetric. Forkinetic energy weobtain, bydeﬁnition, T=z7l=Zi(Mt,Q)=§(M.Q)=i(A9.9)- El DPrincipal axes Like every symmetric operator, Ahasthree mutually orthogonal char- acteristic directions. Letel,e2,ande3eKbetheir unit vectors andI1,I2, andI3their eigenvalues. Inthebasis ei,theinertia operator andthekinetic energy have aparticularly simple form: Mt: [int T=2(I,Qf +1295+1,95). Theaxes e,arecalled theprincipal axes ofthebody atthepoint O. Finally, ifthenumbers I1,I2,andI3arenotalldifferent, then theaxes e,- arenotuniquely deﬁned. Wewillfurther clarify themeaning oftheeigen- values I1,I2,andI3. Theorem. Forarotation ofarigid body ﬁxed atapoint 0,withangular velocity Q=Qe(Q=IQI) around theeaxis, thekinetic energy isequal to T=21,92, where I,=Zmirf andr,-isthedistance ofthei-thpoint totheeaxis(Figure 115). Q=ﬁe Figure 115 Kinetic energy ofabody rotating around anaxis PRooF. Bydeﬁnition T= m,-viz; but|v,|=Qr,-,soT= m,-r,2)Q2. U Thenumber I,depends onthedirection eoftheaxisofrotation Qinthe body. Deﬁnition. I,iscalled themoment ofinertia ofthebody with respect tothe eaxis: _ 2It—2mir,-. 138 28:Rigid bodies Bycomparing thetwoexpressions forTweobtain: Corollary. Theeigenvalues Iioftheinertia operator Aarethemoments of inertia ofthebody with respect totheprincipal axes e,-. ETheinertia ellipsoid Inorder tostudy thedependence ofthemoment ofinertia I,upon thedirec- tionoftheaxiseinabody, weconsider thevectors e/\/Z, where theunit vector eruns over theunitsphere. Theorem. Thevectors e/\/T, form anellipsoid inK. PRooF. Ifn=e/,/Z, thenthequadratic formT=2(/in, Q)isequalto5 Therefore, {Q}isthelevel setofapositive deﬁnite quadratic form, i.e.,an ellipsoid. El Onecould saythatthisellipsoid consists ofthose angular velocity vectors Qwhose kinetic energy isequal to2. Definition. Theellipsoid {S}:(AQ, Q)=1}iscalled theinertia ellipsoid ofthe body atthepoint 0(Figure 116). Body Ellipsoid ofinertia Figure 116 Ellipsoid ofinertia Interms oftheprincipal axes ei,theequation oftheinertia ellipsoid has theform Therefore theprincipal axes oftheinertia ellipsoid aredirected along the principal axes oftheinertia tensor, andtheir lengths areinversely proportionalto,5. Remark. Ifabody isstretched outalong some axis, then themoment of inertia with respect tothisaxis issmall, andconsequently, theinertia el- lipsoid isalso stretched outalong thisaxis; thus, theinertia ellipsoid may resemble theshape ofthebody. Ifabody hasanaxisofsymmetry oforder kpassing through 0(sothatit coincides with itself after rotation byZrc/k around theaxis), then theinertia ellipsoid alsohasthesame symmetry with respect tothisaxis. Butatriaxial 139l 6:Rigid bodies ellipsoid does nothave axesofsymmetry oforder k>2.Therefore, every axis ofsymmetry ofabody oforder k>2isanaxis ofrotation oftheinertia ellipsoid and, therefore, aprincipal axis. EXAMPLE. Theinertia ellipsoid ofthree points ofmass matthevertices ofanequilateral triangle with center 0isanellipsoid ofrevolution around anaxisnormal totheplane ofthetriangle (Figure 117). Figure 117 Ellipsoid ofinertia ofanequilateral triangle Ifthere areseveral such axes, then theinertia ellipsoid isasphere, andany axisisprincipal. PROBLEM. Draw thelinethrough thecenter ofacube such thatthesumofthesquares ofits distances from thevertices ofthecube is:(a)largest, (b)smallest. Wenow remark that theinertia ellipsoid (ortheinertia operator orthe moments ofinertia I1,I2,and I3)completely determines therotational characteristics ofourbody: ifweconsider twobodies with identical inertia ellipsoids, then foridentical initial conditions theywillmove identically (since they have thesame lagrangian function L=T). Therefore, from thepoint ofview ofthedynamics ofrotation around 0, thespace ofallrigid bodies isthree-dimensional, however many points com- pose thebody. Wecaneven consider the“solid rigid body ofdensity p(Q),” having in mind thelimit asAQ—>Oofthesequence ofbodies with aﬁnite number of points Q,with masses p(Q,)AQi (Figure 118)or,what amounts tothesame thing, anybody with moments ofinertia 1.=mp<Q>#<Q>dQ, where risthedistance from Qtotheeaxis. LIIIl=.IIII|llnly‘!!'[>l9Q,- iir ‘,' Figure 118 Continuous solid rigid body 140 28:Rigid bodies EXAMPLE. Find theprincipal axes andmoments ofinertia oftheuniform planar plate |x|5a, lylsb,z=0with respect to0. Solution. Since theplate hasthree planes ofsymmetry, theinertia ellipsoid hasthesame planes ofsymmetry and,therefore, principal axesx,y,andz.Furthermore, a b 2 ma: I_,= xpdxdy=—. —n —b 3 Inthesame way I_mb2_ v— 3* Clearly, I,=I,+1,.. PROBLEM. Show thatthemoments ofinertia ofanybody satisfy thetriangle inequalities 13$I2+I1 I2SI1+l3 and I1sI2+I3, andthatequality holds only foraplanar body. PROBLEM. Find theaxes andmoments ofinertia ofahomogeneous ellipsoid ofmass mwith semiaxes a,b,andcrelative tothecenter 0. Hint. First look atthesphere. PROBLEM. Prove Steiner’s theorem: Themoments ofinertia ofanyrigid body relative totwoparallel axes, oneofwhich passes through thecenter ofmass, arerelated bytheequation I=IO+mrz, where misthemass ofthebody, risthedistance between theaxes, andI0 isthemoment ofinertia relative totheaxispassing through thecenter of mass. Thus themoment ofinertia relative toanaxispassing through thecenter ofmass islessthan themoment ofinertia relative toanyparallel axis. PROBLEM. Find theprincipal axes andmoments ofinertia ofauniform tetrahedron relative to itsvertices. PROBLEM. Draw theangular momentum vector Mforabody with agiven inertia ellipsoid rotating with agiven angular velocity Q. ANSWER. Misinthedirection normal totheinertia ellipsoid atapoint ontheQaxis(Figure l19). Q M Figure 119 Angular velocity, ellipsoid ofinertia andangular momentum 141l"ll‘. it ii l i E ii1 .~<_-=@a:a¢::;:,=....-.E2.,.»;.~_»r.=>=~:.—.=rJ. ll 1 2 ,- 1 i. t'i I 4i l I l1 El I v r 1 t ‘mg.,_.___,-Vt,._m......_-_-__:..¢......_.-_-_- .i_ 6:Rigid bodies / I /Ix / / I // \_,/ Figure 120 Behavior ofmoments ofinertia asthebody becomes smaller PROBLEM. Apiece iscutoffarigid body ﬁxed atthestationary point O.How aretheprincipal moments ofinertia changed? (Figure 120). ANSWER. Allthree principal moments aredecreased. Hint. Cf.Section 24. PROBLEM. Asmall mass sisadded toarigid body with moments ofinertia I,>I2>I3atthe point Q=xlel +x2e2+x,e3. Find thechange inI,ande,with error O(.s2). Solution. Thecenter ofmass isdisplaced byadistance oforder s.Therefore, themoments of inertia oftheoldbody with respect totheparallel axespassing through theoldandnewcenters ofmass differ inmagnitude ofOrder £2.Atthesame time, theaddition ofmass changes the moment ofinertia relative toanyﬁxed axisbyorder s.Therefore, wecandisregard thedisplace- ment ofthecenter ofmass forcalculations with error O(s2). Thus, after addition ofasmall mass thekinetic energy takes theform T=To+%6[9, Q1’+0(9), where '1},=%(l,(2} +129% +1,52%) isthekinetic energy oftheoriginal body. Welook forthe eigenvalue I,(s)andeigenvector e,(s) oftheinertia operator intheform ofaTaylor series ins. Byequating coefficients ofeintherelation A(a)e,(s) =I1(s)e,(s), weﬁndthat, within error O(s2): X I,(a)zI,+s(x§+x§)and e,(s) ze,+s(—ﬂ£2— e2+»i3~e3). I2"_Il I3_Il From theformula forI,(s) itisclear thatthechange intheprincipal moments ofinertia (tothe ﬁrstapproximation ins)isasifneither thecenter ofmass northeprincipal axeschanged. The formula fore,(s) demonstrates how thedirections oftheprincipal axes change: thelargest principal axisoftheinertia ellipsoid approaches theadded point, andthesmallest recedes from it.Furthermore, theaddition ofasmall mass ononeoftheprincipal planes oftheinertia ellipsoid rotates thetwoaxes lying inthisplane anddoes notchange thedirection ofthe third axis. Theappearance ofthedifferences ofmoments ofinertia inthedenominator isconnected with thefactthatthemajor axes ofanellipsoid ofrevolution arenotdeﬁned. Iftheinertia ellipsoid isnearly anellipsoid ofrevolution (i.e.,I,2I2)thentheaddition ofasmall mass could strongly turntheaxese,ande2intheplane spanned bythem. 29Eu1er’s equations. Poinsot’s description ofthemotion Here westudy themotion ofarigid body around astationary point intheabsence ofoutside forces andthesimilar motion ofafreerigid body. Themotion turns outtohave twofrequencies. AEuler’s equations Consider themotion ofarigid body around astationary point 0.LetMbe theangular momentum vector ofthebody relative to0inthebody, Qthe 142 29:Euler’s equations. Poinsot’s description ofthemotion angular velocity vector inthebody, andAtheinertia operator (AQ =M); thevectors QandMbelong tothemoving coordinate system K(Section 26). Theangular momentum vector ofthebody relative toOinspace, m=BM, ispreserved under themotion (Section 28B). Therefore,the vectorM inthebody (MeK)must move sothatm =B,M(t) doesnotchange when tchanges. Theorem dM1 —= _ () dt [1\/L9] PROOF. Weapply formula (5),Section 26forthevelocity ofthemotion of the“point” M(t) GKwith respect tothestationary space k.Weget m=BM+[@,m]=B(M+[0,M]). Butsince theangular momentum mwith respect tothespace ispreserved (ti1=0),M+[Q,M]=0. E] Relation (1)iscalled theEuler equations. Since M=AQ, (1)canbe viewed asadifferential equation forM(orforQ).If Q=Q181+Q2€2+Q3€3 M=M1e1 arethedecompositions ofQandMwith respect totheprincipal axes at0, then M,-=I,Q,-and(1)becomes thesystem ofthree equations dM dM dM<2)7‘=a.M2M. 7’=a2M3M. 7,,—’=a3M.M,. Wherea1=(I2 _I3)/1213,42 =(I3“I1)/I3Ilsanda3 =(I1_ I2)/I1[29Or9 intheform ofasystem ofthree equations forthethree components ofthe angular velocity, dQ I17; =(I2_I3)Q2Q3, dQIt72=<13—10039,, dQ I3Z3 =(I1_I2)Q1Q2-dt Remark. Suppose thatoutside forces actonthebody, thesum ofwhose moments with respect to0isequal toninthestationary coordinate system andNinthemoving system (n=BN). Then ~ andtheEuler equations take theform dM—=M,Q N.dt [ 1+ 143E i- lt A_....a._...~,..._ i l.‘ i It ii ll :1 l I\ T.ll1.1..Ii .1.1 it ~..+.._-s.,_'___:-.-‘ 1 ii 6:Rigid bodies BSolutions oftheEuler equations Lemma. TheEuler equations (2)have twoquadratic ﬁrst integrals M2 2 2 2E=—Il+%/I—2+¥3- and M2=M%+M§+M§. 1 2 3 PROOF. Eispreserved bythelawofconservation ofenergy, andM2bythe lawofconservation ofangular momentum m,since m2=M21M2. El Thus, Mliesintheintersection ofanellipsoid andasphere. Inorder to study thestructure ofthecurves ofintersection wewillﬁxtheellipsoid E>0andchange theradius Mofthesphere (Figure 121). 6'2 1,,idi“\\I l 81 “ll .7 V’‘\\‘»§=2lllv’ 93 Figure 121 Trajectories ofEu1er’s equation onanenergy level surface Weassume that I1?i2 >I3.The semiaxes oftheellipsoid willbe ,/2EI, >./2EI2 >\/2EI3. Iftheradius Mofthesphere islessthan the smallest semiaxes orlarger than thelargest (M<,/2EI3 orM>./2EI1), then theintersection isempty, andnoactual motion corresponds tosuch values ofEandM.Iftheradius ofthesphere isequal tothesmallest semi- axes, then theintersection consists oftwopoints. Increasing theradius, so that\/2E1, <M<./ZEIZ ,wegettwocurves around theends ofthe small- estsemiaxes. Inexactly thesame way, iftheradius ofthesphere isequal tothelargest semiaxes wegettheir ends, andifitisalittle smaller weget two closed curves close totheends ofthelargest semiaxes. Finally, if M=./2E1 2,theintersection consists oftwocircles. Each ofthesixends ofthesemiaxes oftheellipsoid isaseparate trajectory oftheEuler equations (2)—a stationary position ofthevector M.Itcorre- sponds toaﬁxed value ofthevector ofangular velocity directed along one oftheprincipal axes e,;during such amotion, Qremains collinear with M. Therefore, thevector ofangular velocity retains itsposition toinspace collinear with m:thebody simply rotates with ﬁxed angular velocity around theprincipal axisofinertia ei,which isstationary inspace. 144 29:Euler’s equations. Poinsot’s description ofthemotion Deﬁnition. Amotion ofabody, under which itsangular velocity remains constant (co=const, Q=const) iscalled astationary rotation. Wehave proved: Theorem. Arigid body ﬁxed atapoint Oadmits astationary rotation around anyofthe three principal axes el,e2,ande3. If,asweassumed, I1>I2>I3,then theright-hand side oftheEuler equations does notbecome 0anywhere else,i.e.,there arenoother stationary rotations. Wewillnow investigate thestability (inthesense ofLiapunov) ofsolu- tions totheEuler equations. Theorem. Thestationary solutions M=M,e1andM=M3e3oftheEuler equations corresponding tothelargest andsmallest principal axes are stable, while thesolution corresponding tothemiddle axis (M=M292) isunstable. PROOF. Forasmall deviation oftheinitial condition from Mlel orM3e3, thetrajectory willbeasmall closed curve, while forasmall deviation from Mzez itwillbealarge one. El PROBLEM. Arestationary rotations ofthebody around thelargest andsmallest principal axes Liapunov stable? ANSWER. No. CPoinsofs description ofthemotion Itiseasy tovisualize themotion oftheangular momentum andangular velocity vectors inabody (MandQ)—they areperiodic ifMaé./2EI,-. Inorder toseehowabody rotates inspace, welook atitsinertia ellipsoid. E={o;(/to, Q)=1}c K, where A:Q—>Misthesymmetric operator ofinertia ofthebody ﬁxed at0. Atevery moment oftime theellipsoid Eoccupies aposition B,Einthe stationary space k. Theorem (Poinsot). Theinertia ellipsoid rolls without slipping along astation- aryplane perpendicular totheangular momentum vector m(Figure 122). PROOF. Consider aplane rtperpendicular tothemomentum vector mand tangent totheinertia ellipsoid B,E. There aretwosuch planes, andatthe point oftangency thenormal totheellipsoid isparallel tom. 1451 6:Rigid bodies m J 0’ / _ Figure 122 Rolling oftheellipsoid ofinertia ontheinvariable plane~r TI Buttheinertia ellipsoid Ehasnormal grad AQ,Q)=2/{Q =2Matthe point Q.Therefore, atthepoints ii=in/\/ﬁ" ofthe0)axis, thenormal to B,Eiscollinear with m. Sotheplane rtistangent toB,Eatthepoints iiontheinstantaneous axisofrotation. Butthescalar product ofCwith thestationary vector mis equal toi(1/\/ﬁ")(m, om)=i\/F, andistherefore constant. Sothe distance oftheplane rtfrom 0does notchange, i.e.,rtisstationary. Since thepoint oftangency liesontheinstantaneous axisofrotation, its velocity isequal tozero. This implies that theellipsoid B,Erolls without slipping along rt. 1:1 Translator’s remark: Theplane 1tissometimes called theinvariable plane. Corollary. Under initial conditions close toastationary rotation around the large (orsmall) axis ofinertia, theangular velocity always remains close toitsinitial position, notonly inthebody (Q)butalsoinspace (co). Wenow consider thetrajectory ofthepoint oftangency inthestationary plane rt.When thepoint oftangency makes anentire revolution ontheellip- soid, theinitial conditions arerepeated except that thebody hasturned through some angle oiaround themaxis. The second revolution willbe exactly liketheﬁrst; ifat=2rt(p/q), themotion iscompletely periodic; if theangle isnotcommensurable with 2rc,thebody willnever return toits initial state. Inthiscasethetrajectory ofthepoint oftangency isdense inanannulus with center 0'intheplane (Figure 123). PROBLEM. Show that theconnected components oftheinvariant two- dimensional manifold I/C(Section 28B) inthesix-dimensional space TSO(3) 146 29:Euler‘s equations. Poinsot’s description ofthemotion ‘Y l Figure 123 Trajectory ofthepoint ofcontact ontheinvariable plane aretori,andthatonecanchoose coordinates (pland(p2mod 21:onthem so that (/31=w1(C) and (P2=w2(C)- Hint. Take thephase oftheperiodic variation ofMasrp1. Wenow look attheimportant special case when theinertia ellipsoid is anellipsoid ofrevolution: I2 = I3 ¢ I1. Inthiscase theaxisoftheellipsoid B,e1, theinstantaneous axisofrotation 0),andthevector malways lieinoneplane. Theangles between them andthe length ofthevector toarepreserved; theaxes ofrotation (om)andsymmetry (B,e,) sweep outcones around theangular momentum vector mwith the same angular velocity (Figure 124). This motion around miscalled pre- cesston. PROBLEM. Find theangular velocity ofprecession. ANSWER. Decompose theangular velocity vector tointocomponents inthedirections ofthe angular momentum vector mandtheaxisofthebody B,e,. Theﬁrstcomponent gives theangular velocity ofprecession, top,=M/I2. m Ah;/4' rsé,Tl’ Figure 124 Rolling ofanellipsoid ofrevolution ontheinvariable plane 147:1.11 1 n st 6:Rigid bodies Hint. Represent themotion ofthebody astheproduct ofarotation around theaxisof momentum andasubsequent rotation around theaxisofthebody. Thesum oftheangular velocity vectors ofthese rotations isequal totheangular velocity vector oftheproduct. Remark. Intheabsence ofoutside forces, arigid body ﬁxed atapoint 0isrepresented bya lagrangian system whose conﬁguration space isagroup, namely S0(3), andthelagrangian function isinvariant under lefttranslations. Onecanshow thatasigniﬁcant partofEuler‘s theory ofrigid body motion usesonly thisproperty andtherefore holds foranarbitrary left-invariant lagrangian system onanarbitrary Liegroup. Inparticular, byapplying thistheory tothegroup ofvolume-preserving diffeomorphisms ofadomain Dinariemannian manifold, onecanobtain thebasic theorems ofthehydrodynamics ofanideal ﬂuid. (SeeAppendix 2.) 30Lagrange’s top Weconsider herethemotion ofanaxially symmetric rigid body ﬁxed atastationary point ina uniform force ﬁeld. This motion iscomposed ofthree periodic processes: rotation, precession, andnutation. AEuler angles Consider arigid body ﬁxed atastationary point 0andsubject totheaction ofthegravitational force mg.Theproblem ofthemotion ofsuch a“heavy rigid body” hasnotyetbeen solved inthegeneral case andinsome sense is unsolvable. Inthisproblem with three degrees offreedom, only twoﬁrstintegrals areknown: thetotal energy E=T+U,and theprojection M,ofthe angular momentum onthevertical. There isanimportant special case in which theproblem canbecompletely solved*the caseofasymmetric top.A symmetric orlagrangian topisarigid body ﬁxed atastationary point 0 whose inertia ellipsoid at0isanellipsoid ofrevolution andwhose center of gravity liesontheaxisofsymmetry e3(Figure 125). Inthiscase, arotation ea 6 Q~. ©§ zo=Icos6 mg 0 / Figure 125 Lagrangian top around thee3axisdoes notchange thelagrangian function, andbyNoether’s theorem there must exist aﬁrstintegral inaddition toEandM,(aswewill see,itturns outtobetheprojection M3oftheangular momentum vector on thee3axis). Ifwecanintroduce three coordinates sothattheangles ofrotation around thezaxisandaround theaxisofthetopareamong them, then these co- 148 .'.".i'-W5‘l'l'ft‘Zf'?>°'1‘ .- 3'30:Lagrange’s top ordinates willbecyclic, andtheproblem with three degrees offreedom will reduce toaproblem with onedegree offreedom (forthethird coordinate). Such achoice ofcoordinates ontheconﬁguration space S0(3) ispossible; these coordinates tp,I//.49arecalled theEuler angles andform alocal co- ordinate system inS0(3) similar togeographical coordinates onthesphere: theyexclude thepoles andaremultiple-valued ononemeridian. ez P3 0 Axis of Vertical thetop “P /:t§:{\\"Projection ofthetop’s.. ‘.1 Nodal line Figure 126 Euler angles Weintroduce thefollowing notation (Figure 126): ex,ey,ande,aretheunit vectors ofaright-handed cartesian stationary coordinate system atthestationary point O; el,e2,ande3aretheunit vectors ofaright moving coordinate system connected tothebody, directed along theprincipal axes atO; I,=I2aéI3arethemoments ofinertia ofthebody at0; eN istheunitvector oftheaxis[e2,es],called the“line ofnodes” (allvectors areinthe“stationary space” k). Inorder tocarry thestationary frame (ex,ey,e2)into themoving frame (e2,e2,e3),wemust perform three rotations: 1.Through anangle (paround thee,axis. Under thisrotation, e,remains ﬁxed, andexgoes toeN. 2.Through anangle 6around theeNaxis. Under thisrotation, e2goes to es,ande,,,remains ﬁxed. 3.Through anangle 11/around thee3axis. Under thisrotation, e,.,goes to e1,ande3stays ﬁxed. After allthree rotations, exhasgone toel,ande,toe2,;therefore, ey goestoe2. 149 6:Rigid bodies Theangles (p,I//,and6arecalled theEuler angles. Itiseasytoprove: Theorem. Toevery triple ofnumbers (p,9,11/theconstruction above associates arotation ofthree-dimensional space, B((,0, 9,ti)eS0(3), taking the frame (ex,ey,e2)into theframe (e1,e2,e3).Inaddition, the-mapping ((p,6,up)—>B((p, 6,t//)gives local coordinates 0<<p<2rc O<t//<21: 0<0<rc onS0(3), theconﬁguration space ofthetop.Like geographical longitude, (pandupcanbeconsidered asangles mod 2rc;for0=Oor6=rtthemap (go,ti,lb)->Bhasapole-type singularity. BCalculation ofthelagrangian function Wewillexpress thelagrangian function interms ofthecoordinates go,6,1/1 andtheir derivatives. Thepotential energy, clearly, isequal to U=ffjzg dm=mgzo =mglcos0, where zoistheheight ofthecenter ofgravity above O(Figure 125). Wenow calculate thekinetic energy. Asmall trick isuseful here: we consider theparticular casewhen (p=1/1=O. Lemma. Theangular velocity ofatopisexpressed interms ofthederivatives oftheEuler angles bytheformula to=tie,+(rpsin0)e2 +(ll;+cpcos 6)e2, r¢=w=o PROOF. Welook atthevelocity ofapoint ofthetopoccupying theposition rattime t.After time dtthispoint takes theposition (within (dt)2) B(¢+do9+d9,W+dtl)B"(<t>, 9»//)r, where dtp=q'>dt,d0=tldtanddt//=dt. Consequently, tothesame accuracy thedisplacement vector isthesum ofthethree terms Bo»+do,H.=/ormp.0.or—r=tw...r1dt.Bo.6+det/»>B*<<r.t1.t//it -I=[<»i.r1dt.Bo».H.it+dl//)B“‘(</1.19.1//)r -r=rm...an (theangular velocities co,,,,(1)0,and0),,aredeﬁned bythese formulas). Therefore, thevelocity ofthepoint risv=[(04,+tn,+u),,,,r],sothe angular velocity ofthebody is om=0),,+we+0),), where theterms aredeﬁned bytheformulas above. 150 30:Lagrange’s top Itremains todecompose thevectors a),,,,0),,and 0),,with respect to e1,e2,ande3.Wehave notyetused thefactthatrp=1/1=O.Iltp=1/1=0, then B(¢>+d</1.9.1l1)B“(¢. 9.11) issimply arotation around theaxise,through anangle dzp,so (1),,=qbez. Furthermore, B(<p, 0+d6,1/1)B_1(<p, 6,1/1) issimply arotation around the axise,,,=ex=e,through anangle d0inthecase(p=1/1=0,so Finally, B((p, 6,1ﬁ+d1/1)B"(g0,0,1b) isarotation through anangle d1/1 around theaxise3,so...,=I,b€3. Inshort, for(p=1/1=Owehave to=(be,+6le1+ 1/1e3. But,clearly, fortp=1/1=O e,=e2cos9+e2sin19. Sothecomponents oftheangular velocity along theprincipal axes e2,e2, ande2are w,=0 co2=¢sin6 o)3=I//+q'>cosl9. 1:] Since T=§(I,cof +I2012+I,w§), thekinetic energy for(p=1/1=Ois given bytheformula I I-T=?1(ti2 +¢12sin20)+ 330/1 +q'>cos6)2. Butthekinetic energy cannot depend ontpand 1/1:these arecyclic co- ordinates, andbyachoice oforigin ofreference forzpand1pwhich does not change Twecanalways make (,0=Oand1/1=0.Thus theformula wegot forthekinetic energy istrueforallrpand1/1. Inthiswayweobtain thelagrangian function L=-I—£((i2 +¢2sin26) + +q'>cos6)2 -mglcosli. CInvestigation ofthemotion Tothecyclic coordinates (,0and1/1there correspond theﬁrstintegrals 5L -6—¢=M,=qi>(I, sinzli +I3cos2 0)+1/1I3cos0 5L - ‘W=M3=(/‘)l3COS9+l//I3. 151 6:Rigid bodies Theorem. Theinclination 6oftheaxisofthetoptothevertical changes with time inthesame wayasintheone-dimensional system withenergy E=%W+vRn where theeﬂective potential energy isgiven bytheformula (M, —Mcos(9)2 Ueff = +mglCOS PROOF. Following thegeneral theory, weexpress ¢1and interms ofM3 andM2.Wegetthetotal energy ofthesystem as I,- M§ (M,— M3cos6)2E=—62 _ 0W412+21,+"'g’°°S +2I1sin2t9 and (b_M,—M3cos6 U I1sin29 ' The number M§/213 =E—E’,independent of6,does notaffect the equation for0. I1 Inorder tostudy theone-dimensional system above itisconvenient to make thesubstitution cos0=u(-1 3u£1). Wealsowrite M2 M3 2E’ Zmgl 4 I a 4 Z b 4 Z Q 4 I B> 0_ I1 I1 I1 I1 Then wecanrewrite thelawofconservation ofenergy E’as 512=f(M), where f(u)=(a—/.'iu)(1 —uz)—(a—bu)2, and thelawofvariation of theazimuth (pas _a—bu ‘P=for Wenotice that f(u) isapolynomial ofdegree 3,f(+oo) =+00, and f(i1) =—(aTb)2<0ifa75ib.Ontheother hand, actual motions correspond toconstants a,b,(X,and /5'forwhich f(u)20forsome -15u31.Thus f(u)hasexactly tworealroots u,andu2ontheinterval —1gu31(and oneforu>1,Figure 127). Therefore, theinclination 6 oftheaxisofthetopchanges periodically between twolimit values 0,and02 (Figure 128). This periodic change ininclination iscalled nutation. 152 30:Lagrange’s top /. —l 1 J —-—~—— l >11 111 143 Figure 127 Graph ofthefunction f(u) Wenow consider themotion oftheazimuth oftheaxisofthetop.The point ofintersection oftheaxiswiththeunitsphere moves intheringbetween theparallels 19,and02.Thevariation oftheazimuth oftheaxisisdetermined bytheequation _a—bu ‘P=1—-IF‘ Iftheroot u’oftheequation a=buliesoutside of(ul,u2),then theangle (p varies monotonically andtheaxistraces acurve likeasinusoid ontheunit sphere (Figure l28(a)). Iftheroot u’oftheequation a=buliesinside (u1,u2), thentherateofchange ofzpisinopposite directions ontheparallels 61and02,andtheaxistraces alooping curve inthesphere (Figure l28(b)). Iftheroot u’ofa=buliesontheboundary (e.g., u’=u2),then theaxis traces acurve with cusps (Figure 128(c)). The lastcase, although exceptional, isobserved every time werelease theaxisofatoplaunched atinclination 62without initial velocity; thetop ﬁrstfalls, butthen rises again. The azimuthal motion ofthetopiscalled precession. The complete motion ofthetopconsists ofrotation around itsown axis, nutation, and precession. Each ofthethree motions hasitsownfrequency. Ifthefrequencies areincommensurable, thetopnever returns toitsinitial position, although itapproaches itarbitrarily closely. I 2 2 61 ‘An? “is,-aﬁ 6. ta) (bl (cl Figure 128 Path ofthetop’s axisontheunitsphere 1531 1 1 1 l | 1 I l 6:Rigid bodies 31Sleeping topsandfasttops The formulas obtained inSection 30reduce thesolution oftheequations ofmotion ofatopto elliptic integrals. However, qualitative information about themotion isusually easy toobtain without turning toquadrature. Inthisparagraph weinvestigate thestability ofavertical topandgiveapproximate formulas forthemotion ofarapidly spinning top. ASleeping tops Weconsider ﬁrst theparticular solution oftheequations ofmotion in which theaxisofthetopisalways vertical (6=0)andtheangular velocity isconstant (a“sleeping” top). Inthis case, clearly, M,=M3=13003 (Figure 129). Z / Figure 129 Sleeping top PROBLEM. Show thatastationary rotation around thevertical axisisalways Liapunov unstable. Wewilllook atthemotion oftheaxisofthetop,andnotofthetopitself. Will theaxisofthetopstably remain close tothevertical, i.e.,will0remain small‘? Expressing theeffective potential energy ofthesystem (M—M3cos(9)2 Uefr = +mgl COS9 asapower series in8,weﬁnd 1§w§(9“/4) 92 U: ..._ _ ...=C A92 ...,55 2I102 ‘l’ 2'1' ‘l’ ‘l’ A:cogI§_Ll 81, 2' IfA>O,theequilibrium position 19=0oftheone-dimensional system isstable, andifA<Oitisunstable. Thus, thecondition forstability hasthe form ls 154 31:Sleeping topsandfasttops When friction reduces thevelocity ofasleeping toptobelow thislimit, the topwakes up. PROBLEM. Show that, for(1)2>4mgII,/I2, theaxisofasleeping topisstable with respect to perturbations which change thevalues ofM,andM3,aswellas6. BFast tops Atopiscalled fastifthekinetic energy ofitsrotation islarge incomparison with itspotential energy: 213tog>mgl. Itisclear from asimilarity argument thatmultiplying theangular velocity byNisexactly equivalent todividing theweight byN2. Theorem. If,while theinitial position ofatopispreserved, theangular velocity ismultiplied byN,thenthetrajectory ofthetopwillbeexactly thesame as iftheangular velocity remained asitwasandtheacceleration ofgravity gwere divided byN2.Inthecase oflarge angular velocity thetrajectory clearly goes Ntimes faster.“ Inthisway wecanstudy thecase g—>0andapply theresults tostudy thecaseco—>oo. Tobegin, weconsider thecase g=0,i.e.,themotion ofasymmetric topintheabsence ofgravity. Wecompare twodescriptions ofthismotion: Lagrange’s (Section 30C) andPoinsot’s (Section 29C). Weﬁrstconsider Lagrange’s equation forthevariation oftheangle of inclination 6ofthetop’s axis. Lemma. Intheabsence ofgravity, theangle 00satisfying M2=M3cos00 isastable equilibrium position oftheequation ofmotion ofthetop’s axis. Thefrequency ofsmall oscillations of0near thisequilibrium position is equal to I3(03 wnut : 1 PROOF. Intheabsence ofgravity theeffective potential energy reduces to (M2 ~M3cos(9)2 Um=4.42118111 6 Thisnonnegative function hastheminimum value ofzero fortheangle 6=00determined by thecondition M,=M3cos90(Figure 130). Thus, theangle ofinclination 60ofthetop’s axis 5‘Denote by<pg(t, Q)theposition ofthetopattime twith initial condition ﬁeTSO(3) and gravitational acceleration g.Then thetheorem saysthat . q2g(tr :(PN—1,;(Nt> 155 6:Rigid bodies Um- 0,, 9 Figure 130 Effective potential energy ofatop tothevertical isstably stationary: forsmall deviations ofthe initial angle 6from 60,there will beperiodic oscillations of6near 60(nutation). Thefrequency ofthese oscillations iseasily determined bythefollowing general formula: thefrequency o)ofsmall oscillations inaone- dimensional system with energy -2ax _E=T+U(x), U(x0) =mmU(x) isgiven (Section 22D) bytheformula 2U"(X.)to=4. £1 Theenergy oftheone-dimensional system describing oscillations oftheinclination ofthe top’s axisis I5'62 +Um. For6=60+xweﬁndM,—M3cos6=M3(cos 60—cos(60 +x))=M3x sin60+0(x2) M§‘x2-sin2 60 I§co§U,=4 +o(x2)=4x2+"',H 21,s|n260 21, from which weobtain theexpression forthefrequency ofnutation I wnui = D 1 From theformula q)=(M2 —M3cos6)/I1 sin26itisclear that, for 6=60,theazimuth oftheaxis does notchange with time: theaxis is stationary. The azimuthal motion oftheaxis under small deviations of6 from 60could alsobestudied with thehelp ofthisformula, butwewilldeal with itdifferently. The motion ofatopintheabsence ofgravity canbeconsidered in Poinsot’s description. Then theaxisofthetoprotates uniformly around the angular momentum vector, preserving itsposition inspace. Thus, theaxis ofthetopdescribes acircle onthesphere whose center corresponds tothe angular momentum vector (Figure 131). Remark. Now themotion ofthetop’s axis, which according toLagrange wascalled nutation, iscalled precession inPoinsot’s description ofmotion. 156 31:Sleeping topsandfasttops I71 Z sl- Figure 131 Comparison ofthedescriptions ofthemotion ofatopaccording to Lagrange andPoinsot This means thattheformula obtained above forthefrequency ofasmall nutation, eon“, =I3w3/I1, agrees with theformula forthefrequency of precession to=M/I1inPoinsot’s description: when theamplitude of nutation approaches zero, I3co3—>M. CAtopinaweak ﬁeld Wegonowtothecasewhen theforce ofgravity isnotabsent, butisvery small (the values ofM,andM3areﬁxed). Inthiscase aterm mglcos 6, small together with itsderivatives, isadded totheeffective potential energy. Wewillshow thatthisterm slightly changes thefrequency ofnutation. Lemma. Suppose thatthefunction f(x) hasaminimum atx=0andTaylor expansion f(x) = Ax2/2 +...,A>0.Suppose thatthefunction h(x)hasTaylor expansion h(x)=B+Cx+---. Then, forsuﬁiciently small 8,thefunction f,(x) =f(x) +£h(x) hasaminimum atthepoint (Figure 132) Cs O(2) xi Z __T‘ + 8 aA wMdHsdmewzwoInmMnmmfKx)=A-+O@) PROOF. Wehave j;’(x) =Ax+Ce+O(x2) +O(ex), andtheresult isobtained byapplying the immmhﬁmmmnﬂmommtoﬂ(m. U f f(X)ft(X) eh(x) X Xe Figure 132 Displacement oftheminimum under asmall change ofthefunction 157 6:Rigid bodies Bythelemma, theeffective potential energy forsmall ghasaminimum 09close to60,andatthispoint U”differs slightly from U"(90). Therefore, the frequency ofasmall nutation near 00isclose tothatobtained forg=0: . I3hmcon", =1-C03. g-*0 1 DArapidly thrown top Wenow consider thespecial initial conditions when werelease theaxisof thetopwithout aninitial push from aposition with inclination 60tothe vertical. Theorem. Iftheaxisofthetopisstationary attheinitial moment (gb=9=0) andthetopisrotating rapidly around itsaxis(co;—>00),which isinclined from thevertical with angle 00(M, =M3cos60),then asymptotically, as cos—>oo, 1.thenutation frequency isproportional totheangular velocity; 2.theamplitude ofnutation isinversely proportional tothesquare ofthe angular velocity; 3.thefrequency ofprecession isinversely proportional totheangular velocity; 4.thefollowing asymptoticformulas hold(as003—>00): I Ilmgl _ mgl 3to ~—to a~-—— sin6 to ~Znut I1 3 nut 1%mg O prec I3(D3 (het@f(w3) ~y(w3) if1imm»w(f/Q) =1)- Fortheproof, welook atthecase when theinitial angular velocity is ﬁxed, butg—>0.Then byinterpreting theformulas with theaidofasimilarity argument (cf.Section B),weobtain thetheorem. Wealready know from Section 30Cthatunder ourinitial conditions theaxisofthetoptraces acurve with cusps onthesphere. Um E’ — 60,, as Figure 133 Deﬁnition oftheamplitude ofnutation 158 3|:Sleeping tops andfasttops Weapply thelemma tolocate theminimum point Hgoftheeffective potential energy. We set(Figure I33) 9=9O+x c0s6=c0s90—xsinH0+---. Then weobtain, asabove, theTaylor expansion inxat60 I2 2 i U¢rrlg=o =€%x2 + mglcos!) =mglcos 60—XmglS1Il60 +-~~. 1 Applying thelemma tof=U,,,|g=0, g=t-;,h=mlcos(60 +x),weﬁndthattheminimum ofthe effective potential energy U,,-,-isattained atangle ofinclination Imlsin0(lg=60+xy X9=~',——-2-99 +O(g2). I3(U3 Thus theinclination 6ofthe top’s axiswilloscillate near Hg(Figure 134). But,attheinitial moment, _\;< 50 >\ 9, Figure 134 Motion ofatop’s axis 6=60andll=O.This means that60corresponds tothehighest position ofthe axisofthetop. Thus, forsmall g,theamplitude ofnutation isasymptotically equal to Imlsinllan~X,~Q?-ofg (9~0). Wenowﬁndtheprecessional motion oftheaxis. From thegeneral formula _M,—M3cos6 ‘*0_1,sin’0 forM, =M3cos60and6=60+x,we ﬁndthatM,—M3cos6 =Max sinB0+---zso . Ma += ‘M X Q)llsin60 Butxoscillates harmonically between 0andZxg(uptoO(g2)). Therefore, theaverage value of thevelocity ofprecession over theperiod ofnutation isasymptotically equal to ? M3 mgl‘Q°’" PROBLEM. Show that .. —0lll'I1llI1‘lL(t) (p()=1. g—*0t—~0o ""91/laws 159 PART III HAMILTONIAN MECHANICS Hamiltonian mechanics isgeometry inphase space. Phase space hasthe structure ofasymplectic manifold. Thegroup ofsymplectic diﬂeomorphisms acts onphase space. The basic concepts and theorems ofhamiltonian mechanics (even when formulated interms oflocal symplectic coordinates) areinvariant under thisgroup (and under thelarger group oftransformations which alsotransform time). Ahamiltonian mechanical system isgiven byaneven-dimensional mani- fold(the“phase space ”),asymplectic structure onit(the“Poincare integral invariant”) andafunction onit(the“hamiltonian function ”).Every one- parameter group ofsymplectic diffeomorphisms ofthephase space pre- serving thehamiltonian function isassociated toaﬁrst integral ofthe equations ofmotion. Lagrangian mechanics iscontained inhamiltonian mechanics asaspecial case(thephase space inthiscaseisthecotangent bundle oftheconﬁguration space, andthehamiltonian function istheLegendre transform ofthelagrang- ianfunction). Thehamiltonian point ofview allows ustosolve completely aseries of mechanical problems which donotyield solutions byother means (for example, theproblem ofattraction bytwostationary centers andtheproblem ofgeodesics onthetriaxial ellipsoid). The hamiltonian point ofview has even greater value fortheapproximate methods ofperturbation theory (celestial mechanics), forunderstanding thegeneral character ofmotion incomplicated mechanical systems (ergodic theory, statistical mechanics) andinconnection with other areas ofmathematical physics (optics, quantum mechanics, etc.). Differential forms Exterior differential forms arise when concepts such asthework ofaﬁeld along apath andtheﬂuxofaﬂuid through asurface aregeneralized tohigher dimensions. Hamiltonian mechanics cannot beunderstood without differential forms. Theinformation weneed about differential forms involves exterior multi- plication, exterior differentiation, integration, andStokes’ formula. 32Exterior forms Here wedeﬁne exterior algebraic forms AI-forms LetR"beann-dimensional realvector space.“ Wewilldenote vectors inthis space by§,r|,.... Deﬁnition. Aform ofdegree 1(ora1-form) isalinear function oi:R"->R,i.e., @('l~i§i +'l~2g2)='l~1w(§i) ‘l’/l2w(E.»2), A1,'l~25Rand altQ2ER"- Werecall thebasic facts about 1-forms from linear algebra. Thesetofall 1-forms becomes arealvector space ifwedeﬁne thesumoftwoforms by (wt ‘l’ =wl(€) ‘l’a)2(&)9 andscalar multiplication by (lw)(§) =101(5)- “Itisessential tonotethatwedonotﬁxanyspecial euclidean structure onR".Insome examples weusesuch astructure; inthese cases thiswillbespeciﬁcally stated (“euclidean |R"”). 163 7:Differential forms Thespace of1-forms onR"isitself n-dimensional, andisalsocalled thedual space (lR")*. Suppose thatwehavechosen alinear coordinate system x1,...,x,,onR". Each coordinate xiisitselfa1-form. These n1-forms arelinearly independent. Therefore, every 1-form tohastheform w=a1x1+---+a,,x,,, a,eIR. Thevalue ofoionavector §isequal to w(§)=aix1(§) ++a..x..(§), where x,(§), ...,x,,(§) arethecomponents ofQinthechosen coordinate system. EXAMPLE. Ifauniform force ﬁeldFisgiven oneuclidean [R3,itswork Aonthedisplacement 2; isal-form acting onQ(Figure 135). F(force) w(£)=(F,f) f(displacement) Figure 135 Thework ofaforce isaI-form acting onthedisplacement. B2-forms Deﬁnition. Anexterior form ofdegree 2(ora2-form) isafunction onpairs of vectors co’:R"xIR"->R,which isbilinear andskew symmetric: w2(A'1§1 +A-2&.»2»Eta)=A'1w2(§1» gs)'l'A-20-l2(§z, gs) w2(&.m §z)="'w2(§z> gr), Wm A2ER,gt»ézias5Rn- EXAMPLE l.LetS(§,, Q2)betheoriented area oftheparallelogram constructed onthevectors 5,,andQ2oftheoriented euclidean plane [R2,i.e., S(€lI€2) =in 612IWhere gr=éllel +€i2e2~§2 =éllel +£2292-éll 522 with e,,ezabasis giving theorientation onR2. ItiseasytoseethatS(§,, §2)isa2-form (Figure I36). EXAMPLE 2.Letvbeauniform velocity vector ﬁeld foraﬂuid inthree-dimensional oriented euclidean space (Figure 137). Then theﬂuxofthefluid over thearea oftheparallelogram §,,§2isabilinear skew symmetric function ofQ,andQ2,i.e.,a2-form deﬁned bythetriple scalar product w2(§i Q2)=(Vigt,5,2)- 164 IL Figure 136 Oriented area isa2-form.A32:Exterior forms Figure 137 Flux ofaﬁuid through asurface isa2-form. EXAMPLE 3.Theoriented area oftheprojection oftheparallelogram with sides ii,andQ2on thex1,xz-plane ineuclidean R3isa2-form. PROBLEM 1.Show thatforevery 2-form ofonR"wehave o)2(§, Q)=0, VQeR". Solution. Byskew symmetry, w2(§, Q)=—w2(§, Q). Thesetofall2-forms onR"becomes arealvector space ifwedeﬁne the addition offorms bytheformula (W1'l'@2)(§1, Q2)=¢9i(§i» §2)+w2(§i, §2) andmultiplication byscalars bytheformula ()~¢°)(§i> g2)=)*(°(§i» PROBLEM 2.Show thatthisspace isﬁnite-dimensional, andﬁnditsdimension. ANSWER. n(n—I)/2: abasis isshown below. Ck-forms Deﬁnition. Anexterior form ofdegree k,orak-form, isafunction ofkvectors which isk-linear andantisymmetric: w(l~i§’1 +)'2§’i1§2"">gk) =}'1a)(g/1’ 52»---9git)+)~2(9(§i> gzi~-~>lit) where(0(§i|, -''1€ik)=(—1)vw(€l> ''‘7girls Oifthepermutation i1,...,ikiseven;\'= . ._ ,.1ifthepermutation 11,...,1,,isodd. l65 7:Differential forms £3 52 $1 Figure 138 Oriented volume isa3-form. EXAMPLE 1.Theoriented volume oftheparallelepiped withedges §,,...,Q,inoriented euclidean space R”isann-form (Figure 138). in-~ ii» in! '''gum where E,,=5,-,e1 + +§,~,,e,, ande,,...,e,,areabasis ofR“. EXAMPLE 2,Let113.?"beanoriented k-plane inn-dimensional euclidean space R”.Then the k-dimensional oriented volume oftheprojection oftheparallelepiped with edges E,,,Q2, Q,eR"onto R‘isak-form onR". The setofallk-forms inR"form arealvector space ifweintroduce operations ofaddition (wt+w2)(§) =w1(E.)+ 012(5), E,={K1,---,§r}»§; ER", andmultiplication byscalars (l~w)(€) =l~w(€)- PROBLEM 3.Show thatthisvector space isﬁnite-dimensional andﬁnditsdimension. ANSWER. Cf:abasis isshown below. DTheexterior product oftwo1-forms Wenow introduce onemore operation: exterior multiplication offorms. Ifco“isak-form andofisanI-form onIR",then their exterior product w"Aoi‘ willbeak+l-form. Weﬁrstdeﬁne theexterior product of1-forms, which associates toevery pairof1-forms col,co,onR"a2-form (01/\co,onR". LetQbeavector inR".Given two1-forms co,and(U2,wecandeﬁne a mapping ofIR"totheplane IRxIRbyassociating to§eR"thevector co(§) with components co,(§) andw2(§) intheplane with coordinates col,(02 (Figure 139). Deﬁnition. Thevalue oftheexterior product col/\(02onthepairofvectors §1,§2eIR”istheoriented area oftheimage oftheparallelogram with sides oJ(§,) andw(§2) ontheC01,(1)2-plﬂnﬂl _w1(§i) w2(§i) (ml Aw2)(§11€2) _l w2(€2) 166 32:Exterior forms Rn 52 51 00 “Zw(E2) w(£,) wt Figure 139 Deﬁnition oftheexterior product oftwo1-forms PROBLEM 4.Show thatmlAto,really isa2-form. PRQBLEM 5.Show thatthemapping (1/J1» 932)_’wl/\W2 isbilinear andskew symmetric: (D1 /\(U2 =-0): /\(U1, (,l'm’, +,l”w’{) Awz=lo)‘, A(U2+/l"w'{ A(1)2. Hint. Thedeterminant isbilinear andskew-symmetric notonly with respect torows, but alsowith respect tocolumns. Now suppose wehave chosen asystem oflinear coordinates onIR",i.e.,we aregiven nindependent 1-forms xl,...,x,,.Wewillcallthese forms basic. Theexterior products ofthebasic forms arethe2-forms x,-AxJ-.Byskew- symmetry, x,-Ax,-=Oandx,-AxJ=—xj Ax,-.Thegeometric meaning of theform xiAx1-isverysimple: itsvalue onthepairofvectors Q1,Q2isequal totheoriented areaoftheimage oftheparallelogram §1,Q2onthecoordinate plane x,-,xjunder theprojection parallel totheremaining coordinate directions. PROBLEM 6.Show thattheCf=n(n—1)/2forms x,-Ax,(i<j)arelinearly independent. Inparticular, inthree-dimensional euclidean space (xl,x2,x3),thearea oftheprojection onthe(xl,x2)-plane isx1Ax2,onthe(xl,x3)-plane itis x2Ax3,andonthe(x3,x1)-plane itisx3Axi. PROBLEM 7.Show thatevery 2-form inthethree-dimensional space (x,,xl,x3)isoftheform P.\‘2 /\x3+Qxa /\x1+ RX1/\ X2. 167 7:Differential forms PROBLEM 8.Show that every 2-form onthen-dimensional space with coordinates x,,...,x,, canbeuniquely represented intheform (1)2 :Zaijx; /\Xi. ‘ l<_] Hint. Lete,bethei-thbasis vector, i.e.,x,~(e.) =1,x,(e,-) =0fori#1‘.Look atthevalue of theform wzonthepaire,~,ej.Then a,-I=w2(e,~,e1-). EExterior monomials Suppose that wearegiven k1-forms wl,...,wk.Wedeﬁne their exterior product w,A Awk. Deﬁnition. Set @1(€1) wk(g1) (0)1/\"'/\(Uk)(é1,"'>gk): E Z- wi(§k) wu(§k) Inother words, thevalue ofaproduct of1-forms ontheparallelepiped §,,...,Q,isequal totheoriented volume oftheimage oftheparallelepiped intheoriented euclidean coordinate space R"under themapping §—> (w1(g)s ''-a PRQBLEM 9.Show thatw,A---Awkisak-form. PROBLEM 10.Show thattheoperation ofexterior product of1-forms gives amulti-linear skew- symmetric mapping (w1,...,w,,)—>w1 A Awk. Inother words, (/l’w’, +,l"w'{) A(U2A Awk=/l'w’1 A(U2A /\wk+l"w’,’ A(.02AA wk and U),-1A Awik=(—1)‘w1 A Awk, where {Oifthe permutation i1,...,itiseven, vI . . _ _.1ifthepermutation 11,...,lkisodd. Now consider acoordinate sstem onR"ivenbthebasic forms x,..., Y g Y 1 x,,.Theexterior product ofkbasic forms x,~‘A---Axik, lsimsn, istheoriented volume oftheimage ofak-parallelepiped onthek-plane (xii, x,-k)under theprojection parallel totheremaining coordinate directions. 168 32:Exterior forms PROBLEM 11.Showthat, iftwo ofthe indicesi,,...,ik arethe same, then theform x,-1A Axkk 1SZBIO. PROBLEM 12.Show thattheforms x,-lA Ax,~k, where15i,<i2 <'-- <ik$n, arelinearly independent. Thenumber ofsuch forms isclearly C,f.Wewillcallthem basic k-forms. PRQBLEM 13.Show thatevery k-form onR"canbeuniquely represented asalinear combination ofbasic forms: to"= Z a,-______kkxkl A Ax,-X. 15i'|< <i;,5n Hint.a,h__,_,-k =w"(e,-I, ...,ekk). Itfollows asaresult ofthisproblem thatthedimension ofthevector space ofk-forms onR"isequal toCi.Inparticular, fork=n,C’;=1,from which follows Corollary. Every n-form onR”iseither theoriented volume ofaparallelepiped withsome choice ofunitvolume, orzero: w"=a-xk A /\x,,. PROBLEM 14.Show thatevery k-form onR"with k>niszero. Wenow consider theproduct ofak-form wkandanl-form w‘.First, suppose thatwearegiven twomonomials k_ l_w_wk A Awk and w-wk“ A Awkkk, where wk,...,wkkk are1-forms. Wedeﬁne their product w"Aw‘tobethe monomial (C91 A Awk)A(@k+1 A Awt+1) =-(01 /\ "' A A CUk+1 A "‘ /\ (0k+;. PROBLEM l5.Show thattheproduct ofmonomials isassociative: (w"Aw‘)Aw'"=w"A(w'Aw"') andskew-commutative: wkAw‘=(—l)"'w' Aw" Hint. Inorder tomove each ofthelfactors ofw’forward, weneed kinversions with the kfactors ofw". Remark. Itisuseful toremember that skew-commutativity means commutativity only if oneofthe degrees kandliseven, andanti-commutativity ifboth degrees kandIareodd. 169 7:Differential forms 33Exterior multiplication Wedeﬁne here theoperation ofexterior multiplication offorms andshow that itisskew- commutative, distributive, andassociative. ADeﬁnition ofexterior multiplication Wenow deﬁne theexterior multiplication ofanarbitrary k-form w"byan arbitrary l-form w‘.Theresult wkAto’willbeak+l-form. Theoperation of multiplication turns outtobe: l.skew-commutative: w"Aw‘=(—1)"'w' Aw"; 2.distributive: ().kw'{ +3.2w§)Aw'=,l,w'{ Aw'+Alto‘; Aw'; 3.associative: (wkAw‘)Aw"'=w"A(w'Aw"'). Deﬁnition. The exterior product w"Aw'ofak-form w"onR"with an l-form w‘onIR"isthek+l-form onR"whose value onthek+lvectors E1,---,git»E.Ik+lv --->§k+lE R"issqual to (wk Awl)(§1v' '*’€k+l) :§:,("_1)v(0k(€i,1- -'>€ik)a)l(€j|!"'>€j|la whereik < <ikandjk < <j,;(i,,...,ik,jk,...,j,)isapermutation ofthe numbers (1,2,..., k+I);and lifthispermutation isodd;v= .. ..0ifthispermutation iseven. Inother words, every partition ofthek+Ivectors Q1,...,Qk+,intotwo groups (ofkandoflvectors) gives oneterminoursum(1).Thistermisequal totheproduct ofthevalue ofthek-form w"onthekvectors oftheﬁrstgroup withthevalue ofthel-form w‘onthelvectors ofthesecond group, withsign +or—depending onhowthevectors areordered inthegroups. Iftheyare ordered insuch awaythatthekvectors oftheﬁrstgroup andthelvectors of thesecond group written insuccession form aneven permutation ofthe vectors Qk,Q2,...,Qkkk, then wetake thesigntobe+,andifthey form an oddpermutation wetakethesigntobe—. EXAMPLE. Ifk=I=1,thenthere arejusttwopartitions: Q1,Q2andQ2,Q1. Therefore, (wt Awzlfgii E2)=¢°i(§i)¢°2@2) —¢02(§1)‘91(§2)» which agrees with thedeﬁnition ofmultiplication of1-forms inSection 32. PROBLEM 1.Show thatthedeﬁnition above actually deﬁnes ak+l-form (i.e.,thatthevalue of (w"Aw')(Qk, ...,Qkkk) depends linearly andskew-symmetrically onthevectors Q). 170 33:Exterior multiplication BProperties oftheexterior product Theorem. The exterior multiplication offorms deﬁned above isskew-com- mutative, distributive, andassociative. Formonomials itcoincides with the multiplication deﬁned inSection 32. Theproof ofskew-commutativity isbased onthesimplest properties of even andoddpermutations (cf.theproblem attheendofSection 32)andwill belefttothereader. Distributivity follows from thefactthat every term in(1)islinear with respect tow"andw‘. Theproof ofassociativity requires alittle more combinatorics. Since the corresponding arguments arecustomarily carried outinalgebra courses for theproof ofLaplace’s theorem ontheexpansion ofadeterminant bycolumn minors, wemay usethistheorem.” Webegin with thefollowing observation: ifassociativity isproved forthe terms ofasum, then itisalsotrueforthesum, i.e., (w:kAC02)Aw3=w:kA((1)2Aw3)} implies (wkA(1)2)/\ w3=(01/\(a)2 Aw3) ((011+wi)Awz)A(vs=(£01+wl)A(wzAW3)- For,bydistributivity, which hasalready been proved, wehave 'l'mi) A(92) AW3= AW2) A(93)+ AW2) A(93), (wl+W1’)A(wzAwt)=(wiA((92A(D3))+(wlA(wzA013))- Wealready know from Section 32(Problem 13)thatevery form onIR"isa sum ofmonomials; therefore, itisenough toshow associativity formulti- plication ofmonomials. Since wehave notyetproved theequivalence ofthedeﬁnition inSection 32ofmultiplication ofk1-forms with thegeneral deﬁnition (1),wewill temporarily denote themultiplication ofk1-forms bythesymbol K,sothat ourmonomials have theform k_ —- — l__ -— —w-wkA---Awk and w—(0k+1/\---/\(Uk+k, where wk,...,wkkk are1-forms. 53Adirect proof ofassociativity (also containing aproof ofLaplace’s theorem) consists of checking thesigns intheidentity (((0k A(U!) Au)m)(§|, ---,§|k+|+m) :Z i‘ a)k(gi|a *'-1éi|()u)l(gjp '‘'1€j|)u)m(§|l|1 '''ah;-,1)? where i,< <ik,j, < <j,,h, < <h,,,;(i,,...,h,,,) isapermutation ofthenumbers (l,...,k+l+m). l7l 7:Differential forms Lemma. Theexterior product oftwomonomials isamonomial: (wt /T"' Kalli)/\(w|t+1K Xwk+l) —(01/\"‘/\(1)k/\(Uk+1/\‘ '/\(l]k+j. PROOF. Wecalculate thevalues oftheleftandright sides onk+Ivectors Qk,...,Qkkk. Thevalue oftheleftside, byformula (1),isequal tothesumof theproducts Zatdet|[email protected]...)|- det|w.(é,...)I 1gigk k<igh+l oftheminors oftheﬁrstkcolumns ofthedeterminant oforder k+Iandthe remaining minors. Laplace’s theorem ontheexpansion byminors ofthe ﬁrstkcolumns asserts exactly thatthissum, with thesame ruleofsignchoice asinDeﬁnition (1),isequal tothedeterminant det|wk(Qk-)|. Cl Itfollows from thelemma thattheoperations Aand Acoincide: weget, inturn, (1Jk7\(.t)2=(1)k/\(.02, wkAw2Kw3=(wk Aw2)Aw3=(wk /\(i)2)/\C!)3, wkAw2A---7\wk=(--~((wk Aw2)Aw3)A---Awk). Theassociativity ofA-multiplication ofmonomials therefore follows from theobvious associativity ofA-multiplication of1-forms. Thus, inview ofthe observation made above, associativity isproved inthegeneral case. PROBLEM 2.Show thattheexterior square ofal-form, or,ingeneral. ofaform ofoddorder, is equal tozero: w“Aw‘=0ifkisodd. EXAMPLE I.Consider acoordinate system pk....,p,,_qk,....q,,onW"and the2-form (oz=Z:'=k pkAqk. Lﬁeometrically, thisform signiﬁes thesumoftheoriented areas oftheprojection ofaparal- lelogram onthentwo-dimensional coordinate planes (pk,qk)_....([1,.q,.).Later, wewillsee thatthe2-form ofhasaspecial meaning forhamiltonian mechanics. ltcanbeshown thatevery nondegenerate“ 2-form onR“hastheform wzinsome coordinate system (pk,...,q,,).] PROBLEM 3.Find theexterior square ofthe2-form w’. ANSWER. wzAw’=~ZXp.~ AP]AqrAq!‘ i>j PROBLEM 4.Find theexterior k-thpower ofwz. Ai~iswER. (U2/\U)1A"'/\U‘)2=i'k! Z p|iA'UAp||<Aq4i/\"'/\q|u' v H‘. __In k 5*‘Abilinear form ofisnondegenerate ifVQabO,311:w2(Q, I])¢0.SeeSection 41B. 172 33:Exterior multiplication lnparticular, wzA /\(1)2 =inlpk A ApkAqkA /\q,, L‘-vi-' V1‘ is,uptoafactor, thevolume ofa2n-dimensional parallelepiped inR2”. EXAMPLE 2.Consider theoriented euclidean space R3.Every vector AeR3determines al-form wlk,byw}k(Q) =(A,Q)(scalar product) anda2-form wiby w§(Qk, Q2)=(A,Qk,Q2) (triple scalar product). PROBLEM 5.Show thatthemaps A—>wlkandA—>wfkestablish isomorphisms ofthe linear space R5ofvectors Awith thelinear spaces of1-forms onR‘and2-forms onR3.Ifwechoose an orthonormal oriented coordinate system (xk,x2,x_k)onR3,then (U; = /1kXk + AZXZ + A3X3 and wfk=Akx2 Ax2,+/12x3 Axk+Akxk Ax2. Remark. Thus theisomorphisms donotdepend onthechoice oftheorthonormal oriented coordinate system (xk,x2,x3).Butthey dodepend onthechoice oftheeuclidean structure onR3,andtheisomorphism A—>wfkalsodepends ontheorientation (coming implicitly inthe deﬁnition oftriple scalar product). PROBLEM 6.Show that, under theisomorphisms established above, theexterior product of 1-forms becomes thevector product inR3,i.e.,that rujkAwk‘,=wk2,k_k,k foranyA.BeR3. lnthiswaytheexterior product ofl-forms canbeconsidered asanextension ofthevector product inR’tohigher dimensions. However, inthen-dimensional case, theproduct isnota vector inthesame space: thespace of2-forms onR"isisomorphic toR"only forn=3. PROBLEM 7.Show that, under theisomorphisms established above, theexterior product ofa l-form anda2-form becomes thescalar product ofvectors inR3: (UriA@)f!=lA.B)-Y1 AX2AX3- CBehavior under mappings Letf:Rm—>R"bealinear map, andwkanexterior k-form onIR".Then there isak-form f*w"onR“,whose value onthekvectors Qk,...,Qke[Rm isequal tothevalue ofw"ontheir images: (f*wh)(€1> '~'1git) =wk(f§1> '''vf€k)- PROBLEM 8.Verify that] “w"isanexterior form. PROBLEM 9.Verify that/* isalinear operator from thespace ofk-forms onR"tothespace of k-forms onR“(thestarsuperscript means that] *actsintheopposite direction from /). PROBLEM I0.Letf: R”—>R"andg:R"—>R".Verify that(gcf)‘ =f*@g*. PROBLEM ll.Verify thatf*preserves exterior multiplication:f*(w" Aw‘)=(f*w") A(f*w‘). 173l li 3 ¢_...__.____ 7:Differential forms 34Differential forms Wegivehere thedeﬁnition ofdifferential forms ondifferentiable manifolds. ADiﬂerential I-forms Thesimplest example ofadifferential form isthedifferential ofafunction. EXAMPLE. Consider thefunction y=f(x) =x2.Itsdifferential df=2xdxdepends onthe point xandonthe“increment oftheargument,” i.e.,onthetangent vector Qtothexaxis. We ﬁxthepoint x.Then thedifferential ofthefunction atx,dflxsdepends linearly onQ.So,ifx=1 andthecoordinate ofthetangent vector Qisequal to1,then df=2,andifthecoordinate of Qisequal to10,then df=20(Figure 140). f. (If E X Figure 140 Differential ofafunction Letf:M—>IRbeadifferentiable function onthemanifold M(wecan imagine a“function ofmany variables” f:I.R"—>1R).The differential dflk offatXisalinear map dfxtTM, —+[R ofthetangent space toMatXintotherealline.Werecall from Section 18Fthe deﬁnition ofthismap: LetQeTM, bethevelocity vector ofthecurve x(t): IR—>M;x(0) =X andX(0) =Q.Then, bydeﬁnition, are=%=0f(X(r))- PROBLEM 1,LetQbethevelocity vector oftheplane curve x(t)=cost, y(t)=sintatt=0. Calculate thevalues ofthedifferentials dxanddyofthefunctions xandyenthevector Q (Figure l4l). ANSWER. dXlti,o;(§) =0,dYlu.o>(§l =l Note thatthedifferential ofafunction fatapoint xeMisa1-form df,on thetangent space TM,2. 174 34:Diﬂerential forms J’ € x Figure 141 Problem 1 Thedifferential dfoffonthemanifold Misasmooth map ofthetangent bundle TMtotheline df:TM->IR (TM =UTMX). This map isdifferentiable andislinear oneach tangent space TMkcTM. Deﬁnition. Adiﬂerential form ofdegree 1(ora1-form) onamanifold Misa smooth map w:TM—>[R ofthetangent bundle ofMtotheline,linear oneach tangent space TMX. One could saythat adiﬁferential l-form onMisanalgebraic l-form on TM, which is“diﬂerentiable with respect toX.” PROBLEM 2.Show thatevery differential l-form onthelineisthedifferential ofsome function. PROBLEM 3.Find differential l-forms onthecircle andtheplane which arenotthedifferential ofanyfunction. BThegeneral form ofadiﬂerential I-form onlR" Wetake asourmanifold Mavector space with coordinates xk,...,x,k. Recall that thecomponents Qk,...,Q,ofatangent vector QeTIRQ arethe values ofthedifferentials dxk,...,dx,onthevector Q.These n1-forms on TlR§arelinearly independent. Thus the1-forms dxk,...,dx,,form abasis for then-dimensional space of1-forms onTIRQ, andevery l-form onTlR',1 can beuniquely written intheform akdxk+---+a,,dxk,where theakarereal coefficients. Now letwbeanarbitrary differential 1-form onlR".Atevery point Xitcanbeexpanded uniquely inthebasis dxk,...,dx,,.From thisweget: Theorem. Every diﬂerential 1-form onthespace lR"with agiven coordinate system xk,...,x,,canbewritten uniquely intheform w=ak(x)dxk + +a,,(x)dx,,, where thecoeﬁicients ak(x) aresmooth functions. 175i 1 li i 7:Differential forms X2 E E 2 2 3 I E1 0 1 2 3 X’ Figure 142 Problem 4 PROBLEM 4.Calcu1atethe value ofthe formsw, =dx,,[U2=x1dx2,andw3 =dr2(r2 =xf+xi) onthevectors §,,Q2,and5,5(Figure 142). ANSWER. 515,1is (U1 0 *1 1 (1)2 O-2 -2 PROBLEM 5.Letx,,...,x,,befunctions onamanifold Mforming alocal coordinate system in some region. Show that every l-form onthisregion canbeuniquely written intheform w=a,(x) dx,+ +a,,(x) dx,,. CDifferential k-forms Deﬁnition. Adifferential k-form 00"I,atapoint Xofamanifold Misanexterior k-form onthetangent space TM‘ toMatx,i.e.,ak-linear skew-symmetric function ofkvectors §1,...,§,,tangent toMatx. Ifsuch aform co"|,‘ isgiven atevery point xofthemanifold Mandifitis differentiable, then wesaythatwearegiven ak-form 0)"onthemanifold M. PROBLEM 6.Putanatural differentiable manifold structure onthesetwhose elements arek-tuples ofvectors tangent toMatsome point x. Adifferential k-form isasmooth map from themanifold ofProblem 6to theline. PROBLEM 7.Show thatthek-forms onMform avector space (inﬁnite-dimensional ifkdoes not exceed thedimension ofM). Differential forms canbemultiplied byfunctions aswellasbynumbers. Therefore, thesetofC°°differential k-forms hasanatural structure asa module over theringofinﬁnitely differentiable realfunctions onM. 176 34:Differential forms DThegeneral form ofadifferential k-form onR" Take asthemanifold Mthevector space R"with ﬁxed coordinate functions x1,.. .,x,,:IR"—>R.Fixapoint x.Wesawabove thatthen1-forms dxl,..., dx,,form abasis ofthespace of1-forms onthetangent space TR}. Consider exterior products ofthebasic forms: dxil/\‘““/\dx|'k, i1<"'<ik. InSection 32wesawthatthese C’;k-forms form abasis ofthespace ofexterior k-forms onTIRQ. Therefore, every exterior k-form onTIRQ canbewritten uniquely intheform Z t1;1____,,-k dxil /\'''/\dx,-k. i1<...<ik Now letcobeanarbitrary differential k-form onR".Atevery point xit canbeuniquely expressed interms ofthebasis above. From thisfollows: Theorem. Every differential k-form onthespace R"with agiven coordinate system xl,...,x,,canbewritten uniquely intheform oi"= Za,-I ik(x)dxi1 A Adxik, ix< <ik ’ ’ where theail, I-k(x)aresmooth functions onR". PROBLEM 8.Calculate thevalue ofthe forms tn,=dxl/\dxz, (1)2=x,dx,Adx,-x2dxzA dxl,andto,=rdrAdtp(where x,=rcos(pandX2=rsingo)onthepairs ofvectors (Q1,111), (lip'11).and(Q.W3)(Figure 143)- Answsn. (gt,Th) (En,TI2) (gs,'l3) (01 l l —l 0); 2 l -3 (03 l l -l X2 3 2 111V I‘E' 0 1 2 X‘ Figure 143 Problem 8 l77 7:Differential forms PROBLEM 9.Calculate thevalue oftheforms to,=dxzAdxl, co,=x,dx3Adxz, and (1)3=dx3Adrz(r2=xi+x§+xi).onthepairofvectors E_,=(l,l,l),I|=(1,2,3)atthe point x=(2,0,O). ANSWER. to,=1,(U2=-2,(1)3=V8. PROBLEM l0.Letx,,....x,,:M—>[Rbefunctions onamanifold which form alocal coordinate system onsome region. Show thatevery differential form onthisregion canbewritten uniquely in theform 0)"= Z a'l__,__,k(x) dxr-I A Adx,-k. ||<~~<.rk EXAMPLE. Change ofvariables inaform. Suppose that wearegiven two coordinate systems on[R3:xl,x2,x3andy,,yz,y3.Letcobea2-form on{R3. Then, bythetheorem above, cocanbewritten inthesystem ofx-coordinates asw=X,dxzAdx3+X2dx3Adx1+ X3dx,Adxz, where X1, X2, andX3arefunctions ofxl,x2,andx3,andinthesystem ofy-coordinates as co=Y,dyzAdyg+Y2dy3Ady,+Y3dy,Adyz, where Y1,Y2,and Y3 arefunctions ofyl,yz,andy3. PROBLEM ll.Given theform written inthex-coordinates (i.e.,theX,-)andthechange ofvariables formulas x=x(y), write theform iny-coordinates, i.e.,ﬁndY. Solution. Wehave dx,=(8x,-/ay,) dy,+(Ex,/ayz) dyz+(8x,-,/6y3) dy,Therefore, ° a a a = vdxzAdxt=(indyi+fld)‘2+15db)A3dyi+id)‘: +§d.v3).vyi an 51'; @y1 5y; Fr; from which weget 0 D 0,y3=Xl>@ ,X, H3 ,,,C_ D(y1.I2) D01.Y2) D()’1»}’2) EAppendix. Differential forms inthree-dimensional spaces LetMbeathree-dimensional oriented riemannian manifold (inallfuture examples Mwillbeeuclidean three~space R3).Letxl,xl,andx3belocal coordinates, andletthesquare ofthelength element have theform dsz=E1dxf+ E2dx§+E3dx§ (i.e.,thecoordinate system istriply orthogonal). PROBLEM l2.Find E1,E2,andE3forcartesian coordinates x,y,z,forcylindrical coordinates r,tp,:andforspherical coordinates R,tp,6intheeuclidean space R3(Figure 144). Awswsn. dsz=dxz+dyz+dzz=drz+r2dqaz+dzz=dR2+Rzcosz Bdtpz +R2d6Z. Welete,,e2,ande3denote theunitvectors inthecoordinate directions. These three vectors form abasis ofthetangent space. 178 34:Differential forms z R Q ’ X w Figure 144 Problem I2 PROBLEM l3.Find thevalues oftheforms dx,,dxz, anddx3onthevectors e,,e2,ande3. ANSWER. dx,-(e,-) =l/\/E, therestarezero. Inparticular, forcartesian coordinates dx(e_,.) = dy(e,.) =dz(e:) =l;forcylindrical coordinates dr(e,) =d:(e,) =land dq>(e,,,) =l/r(Figure 145), forspherical coordinates dR(eR) =1,d<p(e,,) =1/RcosHanddH(e,) =l/R. Themetric andorientation onthemanifold Mfurnish thetangent space toMatevery point with thestructure ofanoriented euclidean three-dimen- sional space. Interms ofthisstructure, wecantalkabout scalar, vector, and triple scalar products. PROBLEM I4.Calculate [e,,ez],(eR,e,,),and(cg,ex,ey). Answen. e3,0_1, Inanoriented euclidean three-space every vector Acorresponds toa 1-form colanda2-form co},,deﬁned bytheconditions w§(§) =(A,€) wi(€,'1)=(A,Q11), @116R3- Thecorrespondence between vector ﬁelds andforms does notdepend on thesystem ofcoordinates, butonly ontheeuclidean structure andorienta- tion. Therefore, every vector ﬁeld Aonourmanifold Mcorresponds toa differential 1-form co}onMandadifferential 2-form wﬁonM. z ez eg er s/1} Y S9 r Figure 145 Problem 13 179 7:Differential forms Theformulas forchanging from ﬁelds toforms andback have adifferent form ineach coordinate system. Suppose thatinthecoordinates xl,x2,and x3described above, thevector ﬁeld hastheform A = /1181+ A28; + A393 (thecomponents A,aresmooth functions onM).Thecorresponding 1-form to},decomposes over thebasis dxi,andthecorresponding 2-form over the basis dx,Aax,-. PROBLEM 15.Given thecomponents ofthevector ﬁeldA,ﬁndthedecompositions ofthel-form ta},andthe2-form cu}. Solution. We have co},(e,) =(A,el)=A,. Also, (a,dx,+azdxz+a3dx3)(e,) = a,a'x,(e,) =a,/,/E1. From thiswegetthata,=A1,/E1,sothat mi,=A“/E dx,+Ah/E; dx;+143\/E3dx3. Inthesame way, wehave cof,(e2, ea)=(A,ez,ea)=A1.Also, l(oz,dxzAdx3+012dxsAdx,+<13dx,Adx2)(e2, e3)=oz,—i../E25, Hence,ot, =A“/Q“ i.e., mi=A1,/E21; dx;Adx3+AA/E5 dxsAdx,+AM/EEi1dx, Adxz. Inparticular, incartesian, cylindrical, andspherical coordinates on[R23thevector ﬁeld A=A,,e, +Aye, +A,e, =A,e, +Awe‘, +A,e, =Age, +Awe“, +A,,e,, corresponds tothel-form ml.=Axdx +Aydy +/4,dz =A,dr +rA¢,d(,0 +Azdz =ARdR+Rcos (;lA¢d(p +R/49110 andthe2-form w§=A,dyAdz+A,dz Adx+A,dxAdy =rA,drpAdz+AdzAdr+rA,drAdIR.Recall that thegradient ofafunction isthevector ﬁeld gradf corresponding tothedifferential: wéradf =dﬁi-<1,df(€) =(gradf,€) Vii- PROBLEM 16.Find thecomponents ofthegradient ofafunction inthebasis e,.e2,e3. Solution. Wehave df=(of/8x,)dx, +(éf/6x2) dxz+(ﬁf/(3x3)dx3. Bytheproblem above 1of 1af 1ofgradf= e+ e+ e. ,/Elaxi 1,/Ezaxz 2t/E35)‘; 3 180 i 35:Integration ofdifferential forms Inparticular, incartesian, cylindrical, andspherical coordinates 6 ii 6 8 16 6grad_f=a£e,,+a€e,+a£e,=;e,+;a£e,,+6£e, of 1af 15/“wttﬁmwi +12an 35Integration ofdifferential forms Wedeﬁne heretheconcepts ofachain, theboundary ofachain, andtheintegration ofaform overachain. Theintegral ofadifferential form isahigher-dimensional generalization ofsuch ideas asthe fluxofaﬂuid across asurface orthework ofaforce along apath. ATheintegral ofaI-form along apath Webegin byintegrating a1-form mlonamanifold M.Let y:[03ts1]—>M beasmooth map (the “path ofintegration”). The integral oftheform ofonthepath yisdeﬁned asalimit ofRiemann sums. Every Riemann sum consists ofthevalues oftheform co‘onsome tangent vectors Q,(Figure 146): fl fa)‘ =limzco1(§,-). r A—>Oi=1 Thetangent vectors §,areconstructed inthefollowing way. Theinterval 03t31isdivided intoparts A,-:t,-3t5t,-+1bythepoints t,-.Theinterval A,canbelooked atasatangent vector A,tothetaxisatthepoint t,-.Its image inthetangent space toMatthepoint y(t,-)is gt=dVlr,-(A1) ETMy(1,)- Thesum hasalimit asthelargest oftheintervals A,tends tozero. Itis called theintegral ofthel-form ax‘along thepath y. Thedeﬁnition oftheintegral ofak-form along ak-dimensional surface follows ananalogous pattern. Thesurface ofintegration ispartitioned into vf Ar l-———i———+-—->l-—| fr Figure 146 Integrating al-form along apath 181 7:Differential forms Figure 147 Integrating a2-form over asurface small curvilinear k-dimensional parallelepipeds (Figure 147); these paral- lelepipeds arereplaced byparallelepipeds inthetangent space. Thesumofthe values oftheform ontheparallelepipeds inthetangent space approaches theintegral asthepartition isreﬁned. Wewillﬁrstconsider aparticular case. BTheintegral ofak-form onoriented euclidean space R" Letxl,...,xkbeanoriented coordinate system onIR“.Then every k-form onIR“isproportional totheform dxlA Adxk, i.e.,ithastheform co"=<p(x)dx, A Adxk,where (p(x) isasmooth function. LetDbeabounded convex polyhedron inR“(Figure 148). Bydeﬁnition, theintegral oftheform cu“onDistheintegral ofthefunction rp: fad‘ =J<p(x)dx1, ...,dxk, D D where theintegral ontheright isunderstood tobetheusual limit ofRiemann sums. Such adeﬁnition follows thepattern outlined above, since inthiscasethe tangent space tothemanifold isidentiﬁed with themanifold. PRoBLEM l.Show thatIDwkdepends linearly onw". PROBLEM 2.Show thatifwedivide Dintotwodistinct polyhedra D,andD2,then ya;"=f w"+_[ to“. D D; D1 Inthegeneral case (ak-form onann-dimensional space) itisnotsoeasy toidentify theelements ofthepartition with tangent parallelepipeds; wewill consider thiscase below. Figure 148 Integrating ak-form ink-dimensional space 182 35:Integration ofdifferential forms CThebehavior ofdifferential forms under maps Letf:M—>Nbeadifferentiable map ofasmooth manifold Mtoasmooth manifold N,andletcobeadifferential k-form onN(Figure 149). Then, a well-deﬁned k-form arises alsoonM:itisdenoted byf*wandisdeﬁned by therelation (f*¢9)(§1>--- »Sh)=w(f.|.§1»---»f*Sit) foranytangent vectors §,,...,§,,eTMX. Here f*isthedifferential ofthe mapfInother words, thevalue oftheform f*coonthevectors §1,...,Q,‘is equal tothevalue ofcuontheimages ofthese vectors. R M Al --?L i f*w Figure 149 Aform onNinduces aform onM. EXAMPLE. Ify=f(x,, X2)=xf+x§andto=dy,then f*ﬂ) = 2x1dX|+ 2X2 dxz. PROBLEM 3.Show thatf*w isak-form onM. PRoBLEM 4.Show thatthemapf*preserves operations onforms: f*('l1w1 +42(1):) ='l1f*(w1) +/lzf*((/J2), f*(w1 /\W2) =(f*w1) /\(f*w2)- PROBLEM S.Letg:L—>Mbeadifferentiable map. Show that (fg)* =g*f*. PROBLEM 6.LetD,andD2betwocompact, convex polyhedra intheoriented k-dimensional space R"andf:D,—>D2adifferentiable map which isanorientation-preserving diffeomor- phism” ofthe interior ofD, onto theinterior ofD,. Then, foranydifferential k-form cu"onD2, J‘f*a2" =J‘to". D1 D; Hint. This isthechange ofvariables theorem foramultiple integral: f—(Miy—) ¢(y(x))d><1 ~-dxn =f¢(y)dy1 ---dy.- 1|) DzDI @(Xl,...,X 55i.e.,one-to-one with adifferentiable inverse. 183 7:Differential forms DIntegration ofak-form onann-dimensional manifold Letcobeadifferential k-form onann-dimensional manifold M.LetDbea bounded convex k-dimensional polyhedron ink-dimensional euclidean space ER"(Figure 150). Theroleof“path ofintegration” willbeplayed bya \ M 0 0 R" Figure 150 Singular k-dimensional polyhedron k-dimensional cell560ofMrepresented byatriple 0=(D,f,Or)consisting of 1.aconvex polyhedron DcIR“, 2.adifferentiable map f:D—->M,and 3.anorientation onIR“,denoted byOr. Deﬁnition. Theintegral ofthek-form (Doverthek-dimensional cellaisthe integral ofthecorresponding form over thepolyhedron D Leo=Lf*0). PROBLEM 7.Show thattheintegral depends linearly ontheform: J‘111601 "l"A-2(1): =A1 -{(1)1 “l"rf-2J-CD2. Thek-dimensional cellwhich differs from aonlybythechoice oforienta- tioniscalled thenegative ofaandisdenoted by—aor-1-cr(Figure 151). Q *0 V V Figure 151 Problem 8 PROBLEM 8.Show that, under achange oforientation, theintegral changes sign: 5‘Thecellaisusually called asingular k-dimensional polyhedron. 184 35:Integration ofdifferential forms EChains Thesetf(D)isnotnecessarily asmooth submanifold ofM.Itcould have “self~intersections” or“folds” andcould even bereduced toapoint. How- ever, even intheone-dimensional case, itisclear that itisinconvenient to restrict ourselves tocontours ofintegration consisting ofonepiece: itis useful tobeabletoconsider contours consisting ofseveral pieces which can betraversed ineither direction, perhaps more than once. The analogous concept inhigher dimensions iscalled achain. Deﬁnition. Achain ofdimension konamanifold Mconsists ofaﬁnite collection ofk-dimensional oriented cells 0,,...,0,inMandintegers ml,...,m,, called multiplicities (themultiplicities canbepositive, negative, orzero). Achain isdenoted by ck=mlol + +m,a,. Weintroduce thenatural identiﬁcations mlo -1-m_,_o' =(m1 +m2)o mlal +m2a2=m2a2+m,a, 0a=0 c,,+0=c,,. PROBLEM 9.Show thatthesetofallk-chains onMforms acommutative group ifwedeﬁne the addition ofchains bytheformula (mlo, + +m,a,) +(m',a', + +m},a},) =m,a, + +m,a, +m’,a', + +mflafl. FExample: theboundary ofapolyhedron LetDbeaconvex oriented k-dimensional polyhedron ink-dimensional euclidean space IR".Theboundary ofDisthe(k—1)-chain 6DonIR"deﬁned inthefollowing way(Figure 152). Thecells o,-ofthechain 8Darethe(k—1)-dimensional faces D,ofthe polyhedron D,together with maps f,:D,—->IR"embedding thefaces inIR“and orientations Or,deﬁned below; themultiplicities areequal to1: 6D=Za, oi=(D,-,f,, Or,-). Rule oforientation oftheboundary. Lete1,...,ckbeanoriented frame in IR".LetD,beoneofthefaces ofD.Wechoose aninterior ointofD-andthere P 1 f Figure 152 Oriented boundary 185 7:Differential forms construct avector noutwardly normal tothepolyhedron D.Anorienting frame forthefaceD,willbeaframe fl,...,fk_, onD,such thattheframe (n,f,,...,fk_1)isoriented correctly (i.e.,thesame wayastheframe e,,...,ck). Theboundary ofachain isdeﬁned inananalogous way. Leto=(D,f,Or) beak-dimensional cellinthemanifold M.Itsboundary doisthe(k—1) chain: 00=Z0,consisting ofthecells 0,=(D,-,f,,Or,), where theD,are the(k—1)-dimensional faces ofD,Or,areorientations chosen bytherule above, andf,aretherestrictions ofthemapping f:D—->MtothefaceD,. The boundary 8ckofthek-dimensional chain ckinMisthesum ofthe boundaries ofthecells ofckwith multiplicities (Figure 153): tick=5(m,o', + +m,o,) =mldo‘,+ +m,5o',. Obviously, tickisa(k—1)-chain onM.57 80;RWQ Figure 153 Boundary ofachain PROBLEM 10.Show thattheboundary oftheboundary ofanychain iszero: 560,, =0. Hint. Bythelinearity ofaitisenough toshow that 88D =0foraconvex polyhedron D.It remains toverify that every (k—2)-dimensional face ofDappears in85D twice, with opposite signs. Itisenough toprove thisfork=2(planar cross-sections). GTheintegral ofaform over achain Letco"beak-form onM,andckak-chain onM,ck=Zm,a,. Theintegral oftheform wkoverthechain ckisthesumoftheintegrals onthecells, counting multiplicities: J‘ (Uh = 2 m,- J (Dk. Ck Gt PROBLEM 11.Show thattheintegral depends linearly ontheform: Jw'{+w'§=Jw'§+Jw'§. Ck Ck 5|: PROBLEM 12.Show thatintegration ofaﬁxed form w"onchains ckdeﬁnes ahomomorphism from thegroup ofchains totheline. 57Wearetaking k>1here. One-dimensional chains areincluded inthegeneral scheme ifwe make thefollowing deﬁnitions: azero-dimensional Qain consists ofacollection ofpoints with multiplicities; theboundary ofanoriented interval ABisB—A(thepoint Bwithmultiplicity 1 andAwith multiplicity -1); theboundary ofapoint isempty. 186 35:Integration ofdifferential forms EXAMPLE l.Let Mbetheplane {(p,q)},oi‘theform pdq,andc,thechain consisting ofonecello withmultiplicity 1: [Ogts2rr]i>(p=cost.q=sinr). Then Inpdq=rt.Ingeneral, ifachain ckrepresents theboundary ofaregion G(Figure 154), then Inpdqisequal tothearea ofGwith sign +or—depending onwhether thepairofvectors (outward normal, oriented boundary vector) hasthesame oropposite orientation asthepair (paxis, qaxis). 1/ V//71W//I///A pdqI, ,, Figure 154 Theintegral oftheform pdqovertheboundary ofaregion isequal tothe areaoftheregion. EXAMPLE 2.LetMbetheoriented three-dimensional euclidean space IR3.Then every l-form on Mcorresponds tosome vector ﬁeld A(of=(of),where w,f(€) =(A,€)- Theintegral of01,1onachain c,representing acurve Iiscalled thecirculation oftheﬁeld A over thecurve I: J‘(of=JTA, dl). 0| I Every 2-form onMalsocorresponds tosome ﬁeldA(0)2=raj,where o),{(E_,, 1])=(A,Q,11)). Theintegral oftheform wionachain c2representing anoriented surface Siscalled the ﬂuxoftheﬁeld Athrough thesurface S: ‘Loaf =L(A, dn). PROBLEM 13.Find theﬂuxofthe ﬁeldA=(1/R2)eR overthesurface ofthe sphere x2+yz+22= 1,oriented bythevectors ek,e,atthepoint z=1.Find theﬂuxofthe same ﬁeldoverthesurface ofthe ellipsoid (x2/a2) +(yz/b2) +22=1oriented thesame way. Hint. Cf.Section 36H. PROBLEM 14.Suppose that, inthe2n-dimensional space 1R"={(p,, ...,p,,; q,,_..,q,,)}, weare given a2-chain ckrepresenting atwo-dimensional oriented surface Swith boundary I.Find fdP1’\ dlli‘I' ‘I'dPn/\dllnand Jlllidfli ‘I' +Pad?»- £1 I ANSWER. Thesumoftheoriented areas oftheprojection ofSonthetwo-dimensional coordinate planes p,,q,. 187 7:Differential forms 36Exterior differentiation Wedeﬁne hereexterior differentiation ofk-forms andprove Stokes’ theorem: theintegral ofthe derivative ofaform over achain isequal totheintegral oftheform itself over theboundary of thechain. AExample: thedivergence ofavector ﬁeld Theexterior derivative ofak-form toonamanifold Misa(k+1)-form do) onthesame manifold. Going from aform toitsexterior derivative isanalo- gous toforming thedifferential ofafunction orthedivergence ofavector ﬁeld. Werecall thedeﬁnition ofdivergence. E; "'611 2, Figure 155 Deﬁnition ofdivergence ofavector ﬁeld LetAbeavector ﬁeld ontheoriented euclidean three-space IR3,andletS betheboundary ofaparallelepiped 1'1with edges §1,§2,and<23atthevertex x (Figure 155). Consider the(“outward ”)flux oftheﬁeld Athrough the surface S: F(l'l)=Lox,dn). Iftheparallelepiped flisvery small, theﬂuxFisapproximately propor- tional totheproduct ofthevolume oftheparallelepiped, V=(§,,5,2,5,3), andthe“source density” atthepoint x.This isthelimit .F(sfl) 21$ 23V where sflistheparallelepiped with edges £15,,s§2,e§3.This limit does not depend onthechoice oftheparallelepiped Hbutonly onthepoint x,andis called thedivergence, divA,oftheﬁeld Aatx. Togotohigher-dimensional cases, wenote thatthe“ﬂux ofAthrough a surface element” isthe2-form which wecalled 0)}.Thedivergence, then, isthedensity intheexpression forthe3-form 093=divAdx AdyAdz, w3(§1,§z,§s) =divA‘V(§1,§z,§s), characterizing the“sources inanelementary parallelepiped.” 188 36:Exterior differentiation Theexterior derivative dw"ofak-form w"onann-dimensional manifold Mmay bedeﬁned astheprincipal multilinear partoftheintegral ofw"over theboundaries of(k+1)-dimensional parallelepipeds. BDeﬁnition oftheexterior derivative Wedeﬁne thevalue oftheform dwonk+1vectors §,,...,Qk,.1tangent toM atx.Todothis,wechoose some coordinate system inaneighborhood ofx onM,i.e.,adifferentiable mapfofaneighborhood ofthepoint 0ineuclidean space IR"toaneighborhood ofxinM(Figure 156). ek+I k+1 Figure 156 Thecurvilinear parallelepiped II. Thepre-images ofthevectors Q1,...,Qk,1eTM, under thedifferential offlieinthetangent space toIR"at0.This tangent space canbenaturally identiﬁed with IR",sowemay consider thepre-images tobevectors §i‘,---,§i2‘+1ER"- Wetake theparallelepiped lT*inIR"spanned bythese vectors (strictly speaking, wemust look atthestandard oriented cube inIR“1anditslinear map onto fI*,taking theedges ck,...,ekkk toQ1‘,...,§,‘{‘+,, asa(k+1)- dimensional cellinIR").Themap ftakes theparallelepiped H*toa(k+1)- dimensional cellonM(a“curvilinear parallelepiped ”).Theboundary ofthe cellTIisak-chain, 011Consider theintegral oftheform w"ontheboundary OHoffl: Fe...-..t..,> =limo“. EXAMPLE. Wewillcallasmooth function (p:M—>Ra0-form onM.Theintegral ofthe 0-form (p onthe0-chain co=Xm,-A, (where them,areintegers andtheA,points ofM)is Lo=Zm,</>(A.-)- Then thedeﬁnition above gives the“increment” F(§,) =tp(x,) —(p(x) (Figure 157)ofthe function (p,andtheprincipal linear partofF(§,) at0issimply thedifferential of(p. PROBLEM 1.Show thatthefunction F(§,, ...,§k.,,)isskew-symmetric with respect to§. Itturns outthat theprincipal (k+1)-linear part ofthe“increment” F(l?,,,...,§k,.1)isanexterior (k+1)-form onthetangent space TM, toM 189 7:Differential forms Figure 157 Theintegral overtheboundary ofaone-dimensional parallelepiped isthe change inthefunction. atx.This form does notdepend onthecoordinate system thatwasused to deﬁne thecurvilinear parallelepiped I'I.Itiscalled theexterior derivative, or differential, oftheform w"(atthepoint x)andisdenoted bydw". CAtheorem onexterior derivatives Theorem. There isaunique (k+1)-form QonTM, which istheprincipal (k+1)-linear part at0oftheintegral over theboundary ofacurvilinear parallelepiped, F(§1, ...,Qk,1);i.e., (1) F(£§1,--»,ﬁ€i+1) =8"+‘9(§1,---,§i+1) +0(8"+1) (8—>0)- Theform Qdoes notdepend onthechoice ofcoordinates involved inthe deﬁnition ofF.If,inthelocal coordinate system x1,...,x,,onM,theform w"iswritten as (‘uh = Z ai|,...,i|< dxl1 A 'D'Adxiks then Qiswritten as (2) Q=dw"=Zda,,____,,,, Adx,, A Adxk,. Wewillcarry outtheproof ofthistheorem forthecase ofaform w‘= a(x1, x2)dx1 onthex,,x2plane. The proof inthegeneral case isentirely analogous, butthecalculations aresomewhat longer. Wecalculate F(§,1]),i.e.,theintegral ofwlontheboundary oftheparal- lelogram Hwith sides §and1|andvertex at0(Figure 158). Thechain 611is X2 'q+fI "1 E 15: X1 Figure 158 Theorem onexterior derivatives 190 36:Exterior differentiation given bythemappings oftheinterval 05tg1totheplane t—>§t,t—> Q+r|t,t—~>t|t,andt —>I]+§twithmultiplicities 1,1,—1,and—1.Therefore, Lnw‘ =folfafét) —aflét+'t)]€1 —[a('1t) —a(t]!+€)ln1dl where 61=d><1(§), n1=dX1('t), 6».=dX2(€), and'12=dx2(n) arethe components ofthevectors §and1].But 6 8ate+~11-ate)=5"->1,+in,+0&1,112) 1 (thederivatives aretaken atx,=x2=0).Inthesame way am»+o-a<~1o= t,+65;‘;c,+018.112). Byusing these expressions intheintegral, weﬁndthat 1 aa 22F(é,1t) =6w=5— (§2'l1—inn)+oté,n)- I1 X2 Theprincipal bilinear partofF,aspromised in(1),turns outtobethevalue oftheexterior 2-form Q=Edxz/\dx1 5X2 onthepairofvectors Q,1].Thus theform obtained isgiven byformula (2), since a a ada/\11,<,=5_;'1.i><,A,1x,+a—;’2.1x, Adx1=5x(i2dx1 /\dx1. Finally, ifthecoordinate system x1,x2ischanged toanother (Figure 159), theparallelogram TIischanged toanearby curvilinear parallelogram IT,so that thedifference inthevalues oftheintegrals, 1,11w‘—I,,11,w‘ willbe small ofmore thansecond order (prove itI). El \1 ll '2 Figure 159 Independence oftheexterior derivative from thecoordinate system.\‘/ I91 7:Differential forms PROBLEM 2.Carry outtheproof ofthetheorem inthegeneral case. PROBLEM 3.Prove theformulas fordifferentiating asumandaproduct: d(w1+ wk)=dw1+ dwk. and d(w" Aw‘)=do)" Aw’+(~l)"w" Ado)‘. PROBLEM 4.Show thatthedifferential ofadifferential isequal tozero: dd=0. PROBLEM 5.Letf:M—>Nbeasmooth mapandwak-form onN.Show thatf*(dw) =d(1“‘w). DStokes’ formula One ofthemost important corollaries ofthetheorem onexterior derivatives isthe Newton-Leibniz-Gauss-Green-Ostrogradskii-Stokes-Poincare for- mula: (3) La,=dw, where cisany(k+1)-chain onamanifold Mandwisanyk-form onM. Toprove thisformula itissufficient toprove itforthecasewhen thechain consists ofonecell0.Weassume ﬁrstthatthiscellaisgiven byanoriented parallelepiped Hc:lR""‘ (Figure 160). I;-~i Figure l60 Proof ofStokes’ formula foraparallelepiped Wepartition IIinto N"*‘ small equal parallelepipeds 1'1,similar toTI. Then, clearly, Nk+| J\(.t)=ZF,-, WI18I'€F,-=Jl CU. 511 II on, Byformula (1)wehave Ff = '~'a§lt+1)+ O(N_(k+l))> k*1where §'1,...,§k,1 aretheedges ofII,-.ButZZZ, dw(E,‘1,...,§k,.1) isa Riemann sumfor1,1dw.Itiseasy toverify thato(N""*1’)isuniform, so ~k+l ~k+l limZF,=limZdw(§*,,...,§;',..,)= fdw. N—~oo i=1 lV~+oc~ i=1 Tl 192 36:Exterior differentiation Finally, weobtain Jw=ZF,=1im ZF,= Jdw. EH N—*oo l'I Formula (3)follows automatically from thisforanychain whose polyhedra areparallelepipeds. Toprove formula (3)foranyconvex polyhedron D,itisenough toprove itforasimplex,“ since Dcanalways bepartitioned into simplices (Figure 161): D=ZD, 6D=ZOD,. Figure 161 Division ofaconvex polyhedron intosimplices Figure 162 Proof ofStokes‘ formula forasimplex Wewillprove formula (3)forasimplex. Notice that ak-dimensional oriented cube canbemapped onto ak-dimensional simplex sothat: 1.The interior ofthecube goes diﬂeomorphically, with itsorientation preserved, onto theinterior ofthesimplex; 2.The interiors ofsome (k—1)-dimensional faces ofthecube godiffeo- morphically, with their orientations preserved, onto theinteriors ofthe faces ofthesimplex; theimages oftheremaining (k—1)-dimensional faces ofthecube lieinthe(k—2)-dimensional faces ofthesimplex. Forexample, fork=2such amap ofthecube 05xl,x251onto the triangle isgiven bytheformula yl=xl,yz=xlxz (Figure 162). Then, 58Atwo-dimensional simplex isatriangle, athree-dimensional simplex isatetrahedron, a k-dimensional simplex istheconvex hullofk+1points inR"which donotlieinanyk—1- dimensional plane. EXAMPLEZ {xe R":x,- ZOandXL, x,-31}. 193 7:Differential forms formula (3)forthesimplex follows from formula (3)forthecube andthe change ofvariables theorem (cf.Section 35C). EXAMPLE 1.Consider thel-form 031=P1dqi + +pndqn= Pdq onR2"with coordinates p1,..., p,,,ql,...,q,,.Then dw‘ =dpl/\dq,+--~ +dp,,/\dq,,=dp/\dq,so Ifdp/\dq=f pdq. C; dc; Inparticular, ifc2isaclosed surface (dc; =0),then U62dp/\dq=0. EExample 2-Vector analysis Inathree-dimensional oriented riemannian space M,every vector ﬁeld A corresponds toal-form 0),}anda2-form mi.Therefore, exterior differentia- tioncanbeconsidered asanoperation onvectors. Exterior differentiation of0-forms (functions), 1-forms, and2-forms cor- respond totheoperations ofgradient, curl, anddivergence deﬁned bythe relations df= wéradf dwi =COZMIA dwft =(div A)‘/33 (theform 003isthevolume element onM).Thus, itfollows from (3)that f(y) —-f(x) =Jlgradfdl if5l= y—x LAdl=fLcurlA~dn if5S =1 JJA dn= (divA)a>3 if5D=S. s 0 diV[A, B]=(curl A,B)—(curl B,A),PROBLEM 6.Show that curlaA=[grad a,A]+acurlA, divaA=(grad a,A)+adivA. Hint. Bytheformula fordifferentiating theproduct offorms, d(w[2A_,]) =d(w,‘\ /\wﬂ)=do); A0),‘;—tn},/\dw}, PROBLEM 7.Show thatcurlgrad =divcurl=0. Hint. dd=0. 194 36:Exterior differentiation FAppendix I."Vector operations intriply orthogonal systems Letxl,x2,x3beatriply orthogonal coordinate system onM,dsz= E1dxf+E2dx§+E3dx§ande,thecoordinate unit vectors (cf.Section 34E). PROBLEM 8.Given thecomponents ofavector ﬁeldA=Ale, +A2e2 +A2423, ﬁndthecompo- nents ofitscurl. Solution. According toSection 34E Therefore.co;=A“//IT, dxl+A2\/[T2 dX2+A3\/Q (t'.\'_,. PAJ? 5/1jf (3% 3 2“ 2)dx2 Adx3+ =wfw“. .1w;= é -A According toSection 34E,wehave \/E91 \/E92 \//E393 0/1,/E, aA2\/E2 IA 1 ( ) l 6 6 5 cuf : __ e+ _ __ y/EZEJ 6X2 6x3 1 -\/EIEZES ax] 0x2 OX3 Aix/ET AA/E AA/E Inparticular, incartesian, cylindrical, andspherical coordinates onR3, a/4,a/1,. a/1, 0,4, a/1,. 5,4,curlA= -2 e,,+ — e,.+ ;—7e, 5y 62 dz 5x ﬁx dy 1a/1, ar/1,) +a/1, 5,4, 1at/1,, a/1, Z— ﬁ-w— — -—-- Q 7 — — M _ —/2 rdo) 62 ' dz ﬁr8°+rdr dtpez "A 5A cost) ldAR ERA‘, 1(iRA lGA=; Q_~=i,,+_ ___*e+,?@___JeRcosfl a¢ at) "Ras an“’R012 costldtp "' PROBLEM 9.Find thedivergence oftheﬁeldA=A,e, +A2e2 +A2e3. Solution. mi=A“/E2 E3dx2Adx3+~--.Therefore, '1aw:=it/1,./E2E,)d>t, /\dx2Adx3+ l Bythedeﬁnition ofdivergence, This meansdeoi=divA,/E,E2E3 dx,Adx2Adxa. 1 5 5 5 =Y Al,/EZE3 +5"'*/12,/E3E1 +T143,/EIEZ).\/E152 E,X1 X2 X3 195 7:Differential forms Inparticular, incartesian, cylindrical, andspherical coordinates onR3: d_A0,4,+0,4,+é‘A, 1(am,+6,4,)+6,1, \l = 2 =— e - 1 ﬁx 5y dz rﬁr dzp 52 _1(@111COS0/4,5+0R,4,,+0Rcos0/4,) _R1cos0 0R 5(1) an ‘ PROBLEM 10.TheLaplace operator onMistheoperator A=divgrad. Find itsexpression inthe coordinates x,-. ANSWER. Af: 1_[a (/5215, 0f)+___] \/E1E2E3 axl El axl Inparticular, onR3 01/elf01;0*;1of101;elfA= = _ _. fﬁx’+0y’+622 or’+rdr+r2@(p2+022'r<~~1>~<'‘owen=l — os .R2cost) 17R C 8R dtpcosﬁldtp 66co 80 GAppendix 2:Closed forms andcycles Theﬂuxofanincompressible ﬂuid (without sources) across theboundary ofaregion Disequal tozero. Wewillformulate ahigher-dimensional analogue tothisobvious assertion. Thehigher-dimensional analogue ofan incompressible ﬂuid iscalled aclosed form. Theﬁeld Ahasnosources if divA=0. Deﬁnition. Adifferential form soonamanifold Misclosed ifitsexterior derivative iszero: do)=0. Inparticular, the2-form wﬁcorresponding toaﬁeld Awithout sources isclosed. Also, wehave, byStokes’ formula (3): Theorem. Theintegral ofaclosed form wkover theboundary ofany(k+1)- dimensional chain c,,+1 isequal tozero: J w"=0 tfdto"=0. 5¢‘k+1 PROBLEM 11.Show thatthedifferential ofaform isalways closed. Ontheother hand, there areclosed forms which arenotdifferentials. For example, take forMthethree-dimensional euclidean space R3without 0: M=R3-0,with the2-form being theﬂux oftheﬁeld A=(l/R2)eR (Figure 163). Itiseasy toconvince oneself thatdivA=0,sothatour2-form I96 36:Exterior differentiation Figure 163 TheﬁeldA wf,isclosed. Atthesame time, theﬂuxover anysphere with center 0isequal to41:.Wewillshow that theintegral ofthedifferential ofaform over the sphere must bezero. Deﬁnition. Acycle onamanifold Misachain whose boundary isequal to zero. The oriented surface ofoursphere canbeconsidered tobeacycle. It immediately follows from Stokes’ formula (3)that Theorem. Theintegral ofadifferential over anycycle isequal tozero: J‘ d(Dk=0 l_.faCk+1=0. I-'I<+ 1 Thus, our2-form co}isnotthedifferential ofany1-form. Theexistence ofclosed forms onMwhich arenotdifferentials isrelated tothetopological properties ofM.Onecanshow thatevery closed k-form onavector space isthedifferential ofsome (k—1)-form (Poincaré’s lemma). PROBLEM 12.Prove P0incaré’s lemma for1-forms. Hint. Consider Ii;tn‘=tp(x,). PROBLEM 13.Show thatinavector space theintegral ofaclosed form over anycycle iszero. Hint. Construct a(k+1)-chain whose boundary isthegiven cycle (Figure 164). Figure 164 Cone over acycle 197 7:Differential forms Namely, foranychain cconsider the“cone overcwith vertex 0."Ifwedenote theoperation ofconstructing acone byp,then 60p+p=>6=1 (theidentity map). Therefore, ifthechain cisclosed, d(pc) =c. PROBLEM. Show thatevery closed form onavector space isanexterior derivative. Hint. Usethecone construction. Letw‘beadifferential k-form onIR".Wedeﬁne a(k—1)- form (the“co-cone over co”)pm“inthefollowing way: foranychain c,,_, ll pw"='l at". ¢|<—i Pfk-1 Itiseasy toseethat the(k—1)-form poi“exists andisunique: itsvalue onthevectors 5,,...,§,,L1, tangent toR"atx,isequal to (Pw)x(&.m -~-,§|t-1)=lﬁwt,,(X, [En~-~=f§i-i)df~ ltiseasy toseethat dOp+pQd=1 (theidentity map). Therefore, iftheform tn"isclosed, d(pw") =oi". PROBLEM. LetXbeavector ﬁeld onMandwadifferential k-form. Wedeﬁne adifferential (k—1)-form ixw(theinterior derivative ofwbyX)bytherelation flxwlfgii -~-45.,i~i) =('u(xs gt»-~-4§i-1)- Prove thehomotopy formula I-Xd "l' = Lx, where Lxisthedifferentiation operator inthedirection oftheﬁeldX. [The action ofL,‘onaform isdeﬁned, using thephase ﬁow{g’}oftheﬁeldX,bytherelation (Lxw)(§) =5;w(t1LE)- |=0 L,iscalled theLiederivative orﬁsherman’s derivative: theﬁowcarries allpossible differential- geometric objects pasttheﬁsherman, andtheﬁsherman sitsthere anddifferentiates them.] Hint. Wedenote byHthe“homotopy operator” associating toak-chain y:a—>Mthe (k+1)-chain Hy:(Ix0)-»Maccording totheformula (Hy)(t, x)=g'y(x) (where I=[0,1]). Then y‘v~Y=5(Hv) +Hwy). PROBLEM. Prove theformula fordifferentiating avector product onthree-dimensional euclidean space (oronariemannian manifold): curl[a, b]={a,ll}+adivIi—bdiva (where {a,b}=L,bisthePoisson bracket ofthevector ﬁelds, cf.Section 39). Hint. Ifristhevolume element, then i,,,|[,_,|r =di,i,t diva=di,r and {a,b} =L,b; byusing these relations andthefactthatdr=0,itiseasytoderive theformula forcurl[a, Ii]from thehomotopy formula. 198 36:Exterior differentiation HAppendix 3:Cohomology andhomology The setofallk-forms onMisavector space, theclosed k-forms asub- space andthedifferentials of(k—1)-forms asubspace ofthesubspace of closed forms. Thequotient space (closed forms) __R Rr”‘M’> iscalled thek-thcohomology group ofthemanifold M.Anelement ofthis group isaclass ofclosed forms differing from oneanother only byadiffer- ential. PROBLEM l4.Show thatforthecircle S‘wehave H'(S‘, R)=R. Thedimension ofthespace H"(M, IR)iscalled thek-thBetti number ofM. PROBLEM I5.Find theﬁrstBetti number ofthe torus T2=S‘xS‘. Theﬂuxofanincompressible ﬂuid (without sources) over thesurfaces of twoconcentric spheres isthesame. Ingeneral, when integrating aclosed form as Figure I65 Homologous cycles over ak-dimensional cycle, wecanreplace thecycle with another onepro- vided thattheir difference istheboundary ofa(k+1)-chain (Figure 165): fa)": Ito“ a b ifa—b=t3c,,+1anddw"=0. Poincare called twosuch cycles aandbhomologous. With asuitable deﬁnition” ofthe group ofchains onamanifold Mandits 5°Forthisourgroup {c,,}must bemade smaller byidentifying pieces which differ only bythe choice ofparametrization forthechoice ofpolyhedron D.Inparticular, wemayassume that Disalways oneandthesame simplex orcube. Furthermore. wemust take every degenerate k—cell (D,f,Or)to bezero, i.e.,(D, f.Or)=Oiff =f2-f,,wheref, :D—>D’andD’hasdimension smaller than k. 199 7:Differential forms subgroups ofcycles andboundaries (i.e., cycles homologous tozero), the quotient group (cycles)=Hf”) iscalled thek-thhomology group ofM. Anelement ofthisgroup isaclass ofcycles homologous tooneanother. Therank ofthisgroup isalso equal tothek-thBetti number ofM(“De Rham’s Theorem”). 200 Symplectic manifolds Asymplectic structure onamanifold isaclosed nondegenerate differential 2-form. Thephase space ofamechanical system hasanatural symplectic structure. Onasymplectic manifold, asonariemannian manifold, there isanatural isomorphism between vector ﬁelds and 1-forms. Avector ﬁeld onasym- plectic manifold corresponding tothedifferential ofafunction iscalled a hamiltonian vector ﬁeld. Avector ﬁeld onamanifold determines aphase ﬂow, i.e.,aone-parameter group ofdiffeomorphisms. The phase ﬂow ofa hamiltonian vector ﬁeld onasymplectic manifold preserves thesymplectic structure ofphase space. The vector ﬁelds onamanifold form aLiealgebra. The hamiltonian vector ﬁelds onasymplectic manifold alsoform aLiealgebra. Theoperation inthisalgebra iscalled thePoisson bracket. 37Symplectic structures onmanifolds Wedeﬁne here symplectic manifolds, hamiltonian vector ﬁelds, andthestandard symplectic structure onthecotangent bundle. ADeﬁnition LetM2"beaneven-dimensional differentiable manifold. Asymplectic structure onM2"isaclosed nondegenerate differential 2-form wzonM2": dwz=0and Vi;aé03q:w2(§, 1])aé0 (§,1|e TM,,). Thepair(M2",(oz)iscalled asymplectic manifold. 201 8:Symplectic manifolds EXAMPLE. Consider thevector space R2"with coordinates p,-,q,-andlet(oz=Zdp,Adq,. PROBLEM. Verify that(Rh, (oz)isasymplectic manifold. Forn=1thepair(R2,(oz)isthepair (theplane, area). Thefollowing example explains theappearance ofsymplectic manifolds indynamics. Along with thetangent bundle ofadifferentiable manifold, itis often useful tolook atitsdual—the cotangent bundle. BThecotangent bundle anditssymplectic structure LetVbeann-dimensional differentiable manifold. Al-form onthetangent space toVatapoint xiscalled acotangent vector toVatx.Thesetofall cotangent vectors toVatxforms ann-dimensional vector space, dual to thetangent space TVx.Wewilldenote thisvector space ofcotangent vectors byT*V, andcallitthecotangent space toVatx. The union ofthecotangent spaces tothemanifold atallofitspoints is called thecotangent bundle ofVandisdenoted byT*V. ThesetT*V hasa natural structure ofadifferentiable manifold ofdimension 2n.Apoint of T*V isa1-form onthetangent space toVatsome point ofV.Ifqisachoice ofnlocal coordinates forpoints inV,then such aform isgiven byitsncom- ponents p.Together, the2nnumbers p,qform acollection oflocal coordinates forpoints inT*V. There isanatural projection f:T*V —>V(sending every 1-form onTV,to thepoint x).Theprojection fisdifferentiable andsurjective. Thepre-image ofapoint XeVunder fisthecotangent space T*V, Theorem. Thecotangent bundle T*Vhasanatural symplectic structure. Inthe local coordinates described above, thissymplectic structure isgiven bythe formula w2=d|)/\dq=tlp1/\dq1+-'--l-dpn/\dq,,. PROOF. First, wedeﬁne adistinguished 1-form onT*V.LetEeT(T* V),be avector tangent tothecotangent bundle atthepoint peT*V, (Figure 166). Thederivative f*:T(T*V) —>TVof thenatural projection f:T*V —>Vtakes §toavector f,,,§tangent toVatx.Wedeﬁne a1-form to‘onT*V bythe relation co‘(§) =p(f*lg).Inthelocal coordinates described above, thisform isto‘=pdq. Bytheexample inA,theclosed 2-form (oz=dto‘ isnon- degenerate. El Remark. Consider alagrangian mechanical system with conﬁguration manifold Vand lagrangian function L.Itiseasy toseethatthelagrangian “generalized velocity" t'|isatan- gent vector totheconﬁguration manifold V,andthe“generalized momentum" p=dl,/01'] isacotangent vector. Therefore, the“p,q”phase space ofthe lagrangian system isthecotangent bundle oftheconﬁguration manifold. Thetheorem above shows thatthephase space ofa mechanical problem hasanatural symplectic manifold structure. 202L\_s4It/...A\f.hiaMu7a~n.. 37:Symplectic structures onmanifoldsill11it W V Figure 166 The1-form pdqonthecotangent bundle PROBLFM. Show that theLegendre transform does notdepend onthecoordinate system: it takes afunction L:Tl’—+ll\’onthetangent bundle toafunction H:T*V —>Ronthecotangent bundle. CHamiltonian vector ﬁelds Ariemannian structure onamanifold establishes anisomorphism between thespaces oftangent vectors and1-forms. Asymplectic structure establishes asimilar isomorphism. Deﬁnition. Toeach vector §,tangent toasymplectic manifold (Mz",(oz)at thepoint X,weassociate al-form cogonTM, bytheformula @501)=wz(1t,é)V116TM,- PROBLEM. Show thatthecorrespondence l’;—>(ogisanisomorphism between the2n-dimensional vector spaces ofvectors andof1-forms. EXAMPLE. InRz”={(p,q)}wewillidentify vectors andl-forms byusing theeuclidean structure (x,x)=pz+qz.Then thecorrespondence E,—>tog‘determines atransformation R2"—>R2", PROBLEM. Calculate thematrix ofthis transformation inthebasis p,q. ANSWER.( 0E —E O Wewilldenote byItheisomorphism I:T*M,, ->TM, constructed above. Now letHbeafunction onasymplectic manifold Mz".Then dHisadiffer- ential 1-form onM,andatevery point there isatangent vector toMas- sociated toit.Inthiswayweobtain avector ﬁeld IdHonM. Deﬁnition. The vector ﬁeld IdHiscalled ahamiltonian vector ﬁeld; His called thehamiltonianfunction. 203 8:Symplectic manifolds EXAMPLE. IfMz”=Rz"={(p.q)},then weobtain thephase velocity vector ﬁeldofHamilton's canonical equations: _ _ 6H _6Hx=ldH(x)~==-p=—E and q=?3—p. 38Hamiltonian phase ﬂows andtheir integral invariants Liouville's theorem asserts that thephase flow preserves volume. Poincare found awhole series ofdifferential forms which arepreserved bythehamiltonian phase ﬂow. AHamiltonian phase ﬂows preserve thesymplectic structure Let(Mz",(oz)beasymplectic manifold andH:Mz"—>IRafunction. Assume thatthevector ﬁeldIdHcorresponding toHgives a1-parameter group of diffeomorphisms g‘:Mz"—>Mz": d I EMgr_1anon. Thegroup g‘iscalled thehamiltonian phase ﬂowwithhamiltonian function H. Theorem. Ahamiltonian phase ﬂow preserves thesymplectic structure: (gt)=lI(02 =(02. Inthecase n=1,Mz"=Rz,thistheorem saysthatthephase ﬂowg‘ preserves area(Liouville’s theorem). Fortheproof ofthistheorem, itisuseful tointroduce thefollowing nota- tion(Figure 167). LetMbeanarbitrary manifold, cak-chain onMandg‘:M—>Maone- parameter family ofdifferentiable mappings. Wewillconstruct a(k+1)- chain JconM,which wewillcallthetrack ofthechain cunder thehomotopy g‘,O$t31'. Let(D,f,Or)beoneofthecellsinthechain c.Tothiscellwillbeassociated acell(D’,f',Or’)inthechain Jc,where D’=IxDisthedirect product of theinterval 03tgrand D;themapping f’:D’—>M isobtained from f:D—>Mbytheformula f’(t,x)=g'f(x); andtheorientation Or’ofthe 1 glcnth/@144 k=2 k=I Figure 167 Track ofacycle under homotopy 204 4-‘ 38:Hamiltonian phase ﬂows andtheir integral invariants space El\"‘*‘ containing D’isgiven bytheframe co,el,...,e,,,where coisthe unitvector ofthetaxis,andel,...,e,,isanoriented frame forD. Wecould saythatJcisthechain swept outbycunder thehomotopy g’, 05t5r.Theboundary ofthechain Jcconsists of“end-walls” made up oftheinitial andﬁnal positions ofc,and“side surfaces” ﬁlled inbythe boundary ofc. Itiseasytoverify thatunder thechoice oforientation made above, (1) (3(Jc,,) =g’c,,—0,,—‘J0c,,. Lemma. Let)1bea1-chain inthesymplectic manifold (Mz",cuz). Letg’bea phase ﬂow onMwithhamiltonian function H.Then if (oz=f dH. dtJr a’v PROOF. Itissufficient toconsider achain itwith onecellf:[0,1]—>M.We introduce thenotation 0 5f'(s, t)=g'f(s), Q=K]; and 1]=Ffte TMf,(,,,,. Bythedeﬁnition oftheintegral if”oz=L1cuz(§,I])dtds. Butbythedeﬁnition ofthephase ﬂow, 1|isavector (atthepoint f'(s, t))of thehamiltonian ﬁeld with hamiltonian function H.Bydeﬁnition ofahamil- tonian ﬁeld, coz(§, 1])=dH(§). Thus in0,1=UmdH)dt. El Corollary. Ifthechain yisclosed (dy=0),thenIn(oz=0. Pttoor. j,an=j,,,H=0. El PROOF orTHETHEOREM. Weconsider any2~chairi c.Wehave Oijldrozéj coz%(f -I-J. )(oz§fcuz—fcuz J6 51¢ g'e c Joe g'c c (1since cuzisclosed, 2byStokes’ formula, 3byformula (1),4bythecorollary above with y=dc).Thus theintegrals oftheform cuzonanychain candon itsimage g'carethesame. [:1 PROBLEM. Isevery one-parameter group ofdilfeomorphisms ofMz"which preserves thesym- plectic structure ahamiltonian phase ﬂow? Hint. Cf.Section 40. 205 8:Symplectic manifolds BIntegral invariants Letg:M—>Mbeadifferentiable map. Deﬁnition. Adifferential k-form (oiscalled anintegral invariant ofthemap g iftheintegrals oftoonanyk-chain candonitsimage under garethesame:it EXAMPLE. IfM=Rzand(oz=dpAdqisthearea element, then (ozisanintegral invariant of anymap gwithjacobian 1. PROBLEM. Show thataform to"isanintegral invariant ofamapgifandonly ifg*(o" =(o". PROBLEM. Show thatiftheforms to‘andto’areintegral invariants ofthemap g,then theform (o"/\to‘isalsoanintegral invariant ofg. Thetheorem insubsection Acanbeformulated asfollows: Theorem. Theform (ozgiving thesymplectic structure isanintegral invariant ofahamiltonian phase ﬂow. Wenow consider theexterior powers of(oz, ((oz)z =coz/\coz ((oz)3 =cozA(ozA(oz,... 2 23 24 ICorollary. Each oftheforms (coz) ,((o),((o),...isanintegra invariant ofa hamiltonian phase ﬂow. PROBLEM. Suppose thatthedimension ofthesymplectic manifold (Mz", (oz)is2n.Show that ((oz)" =0fork>n,andthat((oz)" isanondegenerate 2n-form onMz". Wedeﬁne avolume element onMz"using ((oz)". Then, ahamiltonian phase ﬂow preserves volume, andweobtain Liouville’s theorem from the corollary above. EXAMPLE. Consider thesymplectic coordinate space Mz"=Rz"={(p,q)}, (oz=dp/\dq=Zdp,Adqi.Inthiscase theform ((oz)" isproportional to theform w2h=. 2‘dp,-, A AdpikAdqilA Adqir Theintegral of(oz"isequal tothesumoftheoriented volumes ofprojections onto thecoordinate planes (p,,,...,p,-k,q,,,...,q,k). Amap g:Rz"—>Rz“iscalled canonical ifithas(ozasanintegral invariant. Acanonical map isgenerally called acanonical transformation. Each ofthe 206 38:Hamiltonian phase ﬂows andtheir integral invariants forms co‘,(o6,...,coz"isanintegral invariant ofevery canonical transforma- tion. Therefore, under acanonical transformation, thesumoftheoriented areas ofprojections onto thecoordinate planes (p,-1,...,p,-k,qil,...,q,,,),15k5n, ispreserved. Inparticular, canonical transformations preserve volume. Thehamiltonian phase ﬂow given bytheequations it=—6H/dq, ('1= 6H/dp consists ofcanonical transformations g’. Theintegral invariants considered above arealsocalled absolute integral invariants. Deﬁnition. Adifferential k-form (oiscalled arelative integral invariant ofthe map g:M—>Miflg,co=I,coforevery closed k-chain c. Theorem. Let(obearelative integral invariant ofamapg.Then dtoisanab- solute integral invariant ofg. PROOF. Letcbeak+1-chain. Then Jdcuéjcoif wéfméj dco. c dc gdc dgc gc (1and4arebyStokes’ formula, 2bythedeﬁnition ofrelative invariant, and 3bythedeﬁnition ofboundary). El EXAMPLE. Acanonical mapg:Rz”—>Rz"hasthel-form II to‘=pdq=Zp,-dq, asarelative integral invariant. l-:1 Infact,every closed chain conRz"istheboundary ofsome chain (1,andweﬁnd ftoléf toli-J‘ cu‘-%J‘d(o'§J~d(i)1§J_w1§J‘w1; gc gfo (‘go go (1 60 c (1and6arebydeﬁnition ofa, 2bydeﬁnition of5,3and5byStokes’ formula, and4since g iscanonical anddco‘=d(pdq)=dqAdq=(oz), PROBLEM. Letdo)"beanabsolute integral invariant ofthemapg:M->M.Does itfollow that to"isarelative integral invariant‘? ANSWER. No,ifthere isaclosed k-chain onMwhich isnotaboundary. CThelawofconservation ofenergy Theorem. Thefunction Hisaﬁrst integral ofthehamiltonian phase ﬂow with hamiltonian function H. PROOF. Thederivative ofHinthedirection ofavector I]isequal tothevalue ofdHonI].Bydeﬁnition ofthehamiltonian ﬁeld 11=IdHweﬁnd dH(n)=wz(n,I dH)=wztnnt) =0- El PROBLEM. Show thattheI-form dHisanintegral invariant ofthephase flow with hamiltonian function H. 207 8:Symplectic manifolds 39TheLiealgebra ofvector ﬁelds Every pairofvector ﬁelds onamanifold determines anew vector ﬁeld, called their Poisson bracket.“ ThePoisson bracket operation makes thevector space ofinﬁnitely differentiable vector ﬁelds onamanifold intoaLiealgebra. ALiealgebras One example ofaLiealgebra isathree-dimensional oriented euclidean vector space equipped with theoperation ofvector multiplication. The vector product isbilinear, skew-symmetric, andsatisﬁes theJacobi identity [[.4,B],c]+[[B,c],A]+[[c,A],B]=0. Deﬁnition. ALiealgebra isavector space L,together withabilinear skew- symmetric operation LxL—>Lwhich satisﬁes theJacobi identity. The operation isusually denoted bysquare brackets andcalled the commutator. PROBLEM. Show thatthesetofnXnmatrices becomes aLiealgebra ifwedeﬁne thecommutator by[A,B]=AB-BA. BVector ﬁelds anddtﬂerential operators LetMbeasmooth manifold andAasmooth vector ﬁeld onM:atevery point xeM wearegiven atangent vector A(x)e TM,,. With every such vector ﬁeldweassociate thefollowing twoobjects: 1.Theone-parameter group ofdiﬂeomorphisms orﬂow A‘:M—>Mforwhich Aisthevelocity vector ﬁeld (Figure l68):6‘ d— A'x=A(x). dti=0 2.Theﬁrst-order differential operator LA.Werefer heretothedifferentiation offunctions inthedirection oftheﬁeldA:foranyfunction (p:M—>R thederivative inthedirection ofAisanew function LA(p,whose value atapoint xis tuow=% awn 1=0 6°OrLiebracket [Trans note]. 6‘Bytheorems ofexistence, uniqueness, anddifferentiability inthetheory ofordinary dif- ferential equations, thegroup A’isdeﬁned ifthemanifold Miscompact. Inthegeneral case themaps A’aredeﬁned only inaneighborhood ofxandonly forsmall t;thisisenough forthe following constructions. 2084..n...u4m.ﬂ ».. 39:The Liealgebra ofvector ﬁelds M A Figure 168 Thegroup ofdiffeomorphisms given byavector ﬁeld PROBLEM. Show thattheoperator LAislinear: Li\('li‘Pi 'l''12‘#2)=}~1LA¢'i +dzL-MP2 (Ah/l2ER)- Also, prove Leibniz's formula L,\(¢1|(Pz) =4011., (p,+(plL,\qp1. EXAMPLE. Let(x,,.-..,x,,)belocal coordinates onM.Inthiscoordinate system thevector A(x) isgiven byitscomponents (A,(x), ...,A,,(x)); theﬂow A’isgiven bythesystem ofdiﬂerential equauons )2,=A,(x),...,>Z,, =A,,(x) and,therefore, thederivative Of(p =<p(x,, ....x,,)inthedirection Ais L,\zp=/4,3-+ +Ania6x1 0x,, Wecould saythatinthecoordinates (x,,....x,,)theoperator LAhastheform = 4_+... 1_ L A 5 A 8A '<°\', +"ﬁr, thisisthegeneral form ofaﬁrst-order linear differential operator oncoordinate space. PROBLEM. Show thatthecorrespondences between vector ﬁelds A,ﬂows A‘,anddifferentiations L,areone-to-one. CThePoisson bracket ofvector ﬁelds Suppose thatwearegiven twovector ﬁelds AandBonamanifold M.The corresponding ﬂows A’andB‘donot,ingeneral, commute: A’B‘;éBSA‘ (Figure 169). PROBLEM. Find anexample. Solution. Theﬁelds A=e,,B=x,e2 onthe(xl,X2)plain, Bsx A'B‘x B‘lA'x B A x A'x Figure 169 Non-commutative ﬂows 209I i E El 4-.~_@=_=<-v=,.T_-av"'"I1*i l P iI 8:Symplectic manifolds Tomeasure thedegree ofnoncommutativity ofthetwoﬂows A‘andB‘we consider thepoints A‘B‘x and B‘A'x. Inorder toestimate thedifference between these points, wecompare thevalue atthem ofsome smooth function (ponthemanifold M.Thedifference A(t;s;x)=(p(A'B‘x) —(p(B‘A'x) isclearly adifferentiable function which iszero fors=0and fort=0. Therefore, theﬁrstterm different from 0intheTaylor series insandtofA at0contains st,andtheother terms ofsecond order vanish. Wewillcalculate thisprincipal bilinear term ofAatO. Lemma 1.Themixed partial derivative 52A/55 6tat0isequal tothecom- mutator ofdiﬂerentiation inthedirections AandB: 62 —'— ‘l(P(A'Bsx) _(P(Bs/ltxll =(LBLA(P _LALB(P)lX)-6s6ts=,=0 PROOF. Bythedeﬁnition ofLA, 5T <P(A‘B‘><) =(LA<0)(B’><)- (Il=O Ifwedenote thefunction LA(pbyI//,then bythedeﬁnition ofLB 6 " l//(BSX) =(LB5ss=0 Thus, 62 ISE ¢(/1BX)=(LBLA <11)» U s=t=0 Wenow consider thecommutator ofdifferentiation operators LBLA — LALB.Atﬁrstglance thisisasecond-order differential operator. Lemma 2.The operator LBLA —LALBisaﬁrst-order linear diﬂerential operator. PROOF. Let(A1,...,A,,) and(B1,...,B,,)bethecomponents oftheﬁelds AandBinthelocal coordinate system (xl,...,x,,)onM.Then n 5 n 6 n 6A’ F n 62¢ = .__ ._ = B._1_ B..}a_ .LBl\¢, .g%1% 5x;;Z%64]5xjq, K2: Ia}Q aXj¢,4_L;;1 H4J5xi5xj@-R4 I-I Ifwesubtract LAL|;(p, theterm with thesecond derivatives of(pvanishes, andweobtain " 5A1 6Bj otp (L51-A -L.\Ls)(P —Z(Baa: "/11-3?!) U -.k-.i—l 210 39:The Liealgebra ofvector ﬁelds Since every ﬁrst-order linear differential operator isgiven byavector ﬁeld, ouroperator LBLA—LAL,alsocorresponds tosome vector ﬁeld C. Deﬁnition. The Poisson bracket orcommutator oftwovector ﬁelds Aand Bonamanifold M62isthevector ﬁeld Cforwhich LC =LnLA —LALB. ThePoisson bracket oftwovector ﬁelds isdenoted by C=[A,B]. PROBLEM, Suppose thatthevector ﬁelds AandBaregiven bytheir components /ti,Biincoor- dinates x,-,Find thecomponents ofthePoisson bracket. Solution. Intheproof ofLemma 2weproved theformula " 314- 3B‘ A, B Z B‘ 4 * l" [ ]l .21 axe Aext PROBLEM. LetA1bethelinear vector ﬁeld ofvelocities ofarigid body rotating with angular velocity um,around 0,andA2thesame thing withangular velocity ml.Find thePoisson bracket [AnA1]- DTheJacobi identity Theorem. ThePoisson bracket makes thevector space ofvector ﬁelds ona manifold MintoaLiealgebra. PROOF. Linearity andskew—symmetry ofthePoisson bracket areclear. We willprove theJacobi identity. Bydeﬁnition ofPoisson bracket, wehave L[[A,B],C] =LCL[A,B] —L[.\,|t]Lc =LCLBLA —LCLALB ‘l’LALBLC —LBLALC. There willbe12terms inallinthesum LH,,,|,}_¢] +Lmtq, A]+L[[¢_AH]. Each term appears inthesumtwice, with opposite signs. El EAcondition forthecommutativity ofﬂows LetAandBbevector ﬁelds onamanifold M. Theorem. Thetwoﬂows A‘andB‘commute ifandonly ifthePoisson bracket ofthecorresponding vector ﬁelds [A,B]isequal tozero. PROOF. IfA'B‘ EB‘/1‘, then [A,B]=0byLemma 1.If[A,B]=0,then, byLemma 1, <p(A‘B’x) —<p(B‘A'x) =o(s2 +t2), s—>0andt->0 62Inmany books thebracket isgiven theopposite sign. Oursignagrees with thesignofthe commutator inthetheory ofLiegroups (cf.subsection F). 211 8:Symplectic manifolds foranyfunction (patanypoint x.Wewillshow thatthisimplies (p(A'B‘x) = <p(B‘A‘x) forsufﬁciently small sandt.Ifweapply thistothelocal coordinates (rp=xl,...,rp==x,,),weobtain A'B‘ =B‘A'. Consider therectangle 05t5to,05s5so(Figure 170)inthet.s-plane. Toevery path going from (0,0)to(to,so)andconsisting ofaﬁnite number ofintervals inthecoordinate direc- tions, weassociate aproduct oftransformations oftheflows A‘andB‘.Namely, toeach interval t,5t5toweassociate /4'1"‘, andtoeach interval s,5s5soweassociate B“"';the trans- formations areapplied intheorder inwhich theintervals occur inthepath, beginning at(0,0). Forexample, thesides (05t5to,s=0)and(t=to,05s5so)corresponds totheproduct B‘°A'°, andthesides (1=0,05sso)and(s=so,05t5to)totheproduct A'°B‘°. < s 10--Y0 S0 0 T0 l Figure 170 Proof ofthecommutativity ofﬂows A’~B‘"x B-Y-ix Q, 6! a BRA [l\.\. €5 5 7 Mx _ e A'°x Figure 171 Curvilinear quadrilateral /nan Inaddition, weassociate toeachsuchpath inthe(t,s)-plane apathonthemanifold M starting atthepoint xandcomposed oftrajectories oftheflows A’andB‘(Figure 171). Ifa path inthe(1,s)-plane corresponds totheproduct A"B“ --~A'"B‘", then onthemanifold M thecorresponding path ends atthepoint A“B" A'"B‘"x. Ourgoal willbetoshow thatall these paths actually terminate attheonepoint A'°B‘"x =B‘°A“‘x. Wepartition theintervals 05t5toand05s5sointoNequal parts, sothatthewhole rectangle isdivided intoN2small rectangles. Thepassage from thesides (0,O)-(to,O)-(to,so) tothesides (0,0)—(0,so)—(to,so)canbeaccomplished inN2steps, ineach ofwhich apair ofneighboring sides ofasmall rectangle isexchanged fortheother pair(Figure 172). Ingeneral, 212 39TheLiealgebra ofvector ﬁelds Figure I72 Going from onepairofsides totheother thissmall rectangle corresponds toanon-closed curvilinear quadrilateral ﬁyotzu onthemanifold M(Figure I'll).Consider thedistance“ between itsvertices atand[3corresponding tothelargest values ofsand1.Aswesawearlier. p(ai,ft)5C,N" (where theconstant C,>0does not depend onN).Using thetheorem ofthedilferentiability ofsolutions ofdifferential equations with respect totheinitial data, itisnotdifficult toderive from thisabound onthedistance between theends oz’andB’ofthepaths x6"/ﬁB' andx6:-zeta’ onM:p(a’, ll’)<C,N",where the constant C,>0again does notdepend onN.Butwebroke upthewholejourney from B‘°A’°x toA'°B“’x into N2such pieces. Thus, p(A'°B‘°x, B'°A'°x) 5N1C2N‘3 VN. Therefore. A'°B‘°x =B’°A'°x. Cl FAppendix; Liealgebras andLiegroups ALiegroup isagroup Gwhich isadifferentiable manifold, andforwhich the operations (product andinverse) aredifferentiable maps GxG—>Gand G—»G. Thetangent space, TG,, toaLiegroup Gattheidentity hasanatural Liealgebra structure; itisdeﬁned asfollows: Foreach tangent vectorA 6TG,.thereisaone-parameter subgroup A’CG with velocity vector A=(d/dt)|,=oA'. Thedegree ofnon-commutativity oftwosubgroups A‘andB‘ismeasured bytheproduct A'B‘A"B". Itturns outthatthere isoneandonly one subgroup C’forwhich p(A‘B‘A_'B“, C")=o(s2 +t2) assandt—>0. The corresponding vector C=(d/dr)|,=oC' iscalled theLie bracket C=[A,B]ofthevectors AandB.Itcanbeveriﬁed that theoperation of Liebracket introduced inthiswaymakes thespace TG,intoaLiealgebra (i.e., theoperation isbilinear, skew-symmetric, andsatisﬁes theJacobi identity). This algebra iscalled theLiealgebra oftheLiegroup G. Pnoausm. Compute thebracket operation intheLiealgebra ofthe group SO(3) ofrotations in three-dimensional euclidean space. Lemma 1shows thatthePoisson bracket ofvector ﬁelds canbedeﬁned astheLiebracket forthe“inﬁnite-dimensional Liegroup” ofalldiffeo- morphisms“ ofthemanifold M. "3Insome riemannian metric onM. °‘Ourchoice ofsign inthedeﬁnition ofPoisson bracket wasdetermined bythiscorrespondence. 213 8:Symplectic manifolds Ontheother hand, theLiebracket canbedeﬁned using thePoisson bracket ofvector ﬁelds onaLiegroup G.LetgeG.Right translation R,is themap R9:G—>G,Rgh=hg.The differential ofRaatthepoint emaps TG,. into TGQ. Inthisway, every vector AeTGQ corresponds toavector ﬁeld onthegroup: itconsists oftheright translations (Rg)*A andiscalled a right-invariant vector ﬁeld. Clearly, aright-invariant vector ﬁeld onagroup isuniquely determined byitsvalue attheidentity. PROBLEM. Show thatthePoisson bracket ofright-invariant vector ﬁelds ona Liegroup Gisaright-invariant vector ﬁeld, anditsvalue attheidentity of thegroup isequal totheLiebracket ofthevalues oftheoriginal vector ﬁelds attheidentity. 40TheLiealgebra ofhamiltonian functions Thehamiltonian vector ﬁelds onasymplectic manifold form asubalgebra ofthe Liealgebra of allﬁelds. Thehamiltonian functions alsoform aLiealgebra: theoperation inthisalgebra is called thePoisson bracket offunctions. Theﬁrstintegrals ofahamiltonian phase flowform a subalgebra ofthe Liealgebra ofhamiltonian functions. AThePoisson bracket oftwofunctions Let(M2",cu’)beasymplectic manifold. Toagiven function H:M2“—>[R onthesymplectic manifold there corresponds aone-parameter group g},:M2" —>M2" ofcanonical transformations ofM2"—the phase ﬂow ofthe hamiltonian function equal toH.LetF:M2"—>Rbeanother function onM2". Deﬁnition. The Poisson bracket (F,H)offunctions Fand Hgiven ona symplectic manifold (M2",012)isthederivative ofthefunction Finthe direction ofthephase ﬂow with hamiltonian function H: (F.H)(><)=%Foam). t=O Thus, thePoisson bracket oftwofunctions onMisagain afunction onM. Corollary 1.Afunction Fisaﬁrst integral ofthephase ﬂow withhamiltonian function Hifandonly ifitsPoisson bracket with Hisidentically zero: (F,H)EO. Wecangivethedeﬁnition ofPoisson bracket inaslightly different form ifweusetheisomorphism Ibetween 1-forms andvector ﬁelds onasymplectic manifold (M2",(oz).This isomorphism isdeﬁned bytherelation (cf.Section 37) w’(n.Iw‘)=w‘(n)~ Thevelocity vector ofthephase ﬂow g},isIdH.This implies 214 40:TheLiealgebra ofhamiltonian functions Corollary 2.ThePoisson bracket ofthefunctions FandHisequal tothe value ofthel-form dFonthevelocity vector IdHofthephase ﬂow with hamiltonian function H: (F,H)=dF(IdH). Using thepreceding formula again, weobtain Corollary 3.ThePoisson bracket ofthefunctions FandHisequal tothe “skew scalar product” ofthevelocity vectors ofthephase ﬂows withhamil- tonian functions HandF: (F,H)=cull]dH,IdF). Itisnow clear that Corollary 4.ThePoisson bracket ofthefunctions FandHisaskew-symmetric bilinear function ofFandH: and (Hi/liFi +)~zF2)=/lilH,F1)'l"l2(H,F2) l)~t6R)- Although thearguments above areobvious, they lead tonontrivial deductions, including thefollowing generalization ofatheorem ofE.Noether. Theorem. Ifahamiltonian function Honasymplectic manifold (M2",coz) admits theone-parameter group ofcanonical transformations given bya hamiltonian F,then Fisaﬁrst integral ofthesystem with hamiltonian function H. PROOF. Since Hisaﬁrst integral oftheﬂow g},(H,F)=0(Corollary 1). Therefore, (F,H)=0(Corollary 4)andFisaﬁrstintegral (Corollary 1).[:1 PROBLEM l.Compute thePoisson bracket oftwo functions FandHinthecanonical coordinate SpaceR“={(M1)}. w’(é.t|) =(15.11). Solution. ByCorollary 3wehave ""HEF 6H"F (RH): ZliT’TT_ .g‘ tel cploql (lqlcpl (weusethefactthatIissymplectic andhastheform 0-EI=<> E 0 PROBLEM 2.Compute thePoisson brackets ofthebasic functions p,andq,. Solution. Thegradients ofthebasic functions form a"symplectic basis“: their skew-scalar products areinthebasis (p,q)). (pit11,)=(p..q,) =(q.-4,) =0(irt#1) (q,.p.) =—(p,.q,-)=t. 215 8:Symplectic manifolds PROBLEM 3.Show thatthemapA:R2"—>R2"sending (p,q)—>(P(p, 1|),Q(p,q))iscanonical if andonly ifthe Poisson brackets ofanytwofunctions inthevariables (p,q)and(P,Q)coincide: 5H13F 5H3F 5H5F (‘ll-IBF (RH)""' Z0pdq—ﬁqopZGPFQ—6Q6P=(F’H)?“ Solution. LetAbecanonical. Then thesymplectic structures dpAdqanddP/\dQcoincide. Butthedeﬁnition ofthe Poisson bracket (F,H)wasgiven invariantly interms ofthe symplectic structure; itdidnotinvolve thecoordinates. Therefore, (F,H)“ =(F.H)=(PkH)r.Q- Conversely, suppose thatthePoisson brackets (P,-,Q,-)|,_,, have thestandard form ofProblem 2. Then, clearly, dP/\dQ=dp/\dq,i.e.,themap Aiscanonical. PROBLEM 4.Show thatthePoisson bracket ofaproduct canbecalculated byLeibniz’s rule: (F1F2- H)=F1(F2,H) +F2(Fi, H)- Hint. ThePoisson bracket (F,FZ. H)isthederivative oftheproduct F,F2 inthedirection ofthe ﬁeld IdH. BTheJacobi identity Theorem. ThePoisson bracket ofthree functions A,B,andCsatisﬁes the Jacobi identity: ((/1,B),C)+((13,C),A)+((C,A).B)=0- Corollary (Poisson’s theorem). The Poisson bracket oftwoﬁrst integrals F1,F2ofasystem with hamiltonian function Hisagain aﬁrst integral. PROOF orTHECOROLLARY. BytheJacobi identity, ((F19F2)1H) =(F1s(F2sH)) +(F2>(HsF1)) =0+0’ aswastobeshown. U Inthisway, byknowing twoﬁrstintegrals wecanﬁndathird, fourth, etc. byasimple computation. Ofcourse, notalltheintegrals wegetwillbe essentially new, since there cannot bemore than 2nindependent functions onM2". Sometimes wemay getfunctions ofoldintegrals orconstants, which maybezero. Butsometimes wedoobtain newintegrals. PROBLEM. Calculate thePoisson brackets ofthecomponents p,,p2,p3, M1,M2,M3ofthe linear andangular momentum vectors ofamechanical system. ANSWER~(M1v M2) :M31(Ml!p1) =0’(M19 p2) :P3’ (M19 P3) =—pZ' Theorem. Iftwocomponents, M1andM2,oftheangular momentum ofsome mechanical problem areconserved, thenthethird component isalsoconserved. Pnoor orTHEJACOBI IDENTITY. Consider thesum ((A,B),C)+((B.C),A)+((0.A),B)- 216 40:TheLiealgebra ofhamiltonian functions This sum isa“linear combination ofsecond partial derivatives” ofthe functions A,B,andC.Wewillcompute theterms inthesecond derivatives ofA: ((/4.B),C)+((C,A).B)=(LcLn —Lnl-c)/1. where L;isdifferentiation inthedirection ofQandFisthehamiltonian ﬁeldwithhamiltonian function F. But, byLemma 2,Section 39,thecommutator ofthedifferentiations LoL,—LBLC isaﬁrst-order differential operator. This means that none ofthesecond derivatives ofAarecontained inoursum. Thesame thing is trueforthesecond derivatives ofBandC.Therefore, thesumiszero. Cl Corollary 5.LetBandCbehamiltonian ﬁelds with hamiltonian functions BandC.Consider thePoisson bracket [B,C]ofthevector ﬁelds. This vector ﬁeld ishamiltonian, anditshamiltonian function isequal tothe Poisson bracket ofthehamiltonian functions (B,C). PROOF. Set(B,C)=D.TheJacobi identity canberewritten intheform (A.D)=((/1.B).C)—((4.C).B). Ln=LcL|| _L51-c Ln=L[||,c], aswastobeshown. E] CTheLiealgebras ofhamiltonian ﬁelds, hamiltonian functions, andﬁrst integrals Alinear subspace ofaLiealgebra iscalled asubalgebra ifthecommutator ofanytwoelements ofthesubspace belongs toit.Asubalgebra ofaLie algebra isitself aLiealgebra. Thepreceding corollary implies, inparticular, Corollary 6.Thehamiltonian vector ﬁelds onasymplectic manifold form a subalgebra oftheLiealgebra ofallvector ﬁelds. Poisson’s theorem onﬁrstintegrals canbere-formulated as Corollary 7.Theﬁrst integrals ofahamiltonian phase ﬂowform asubalgebra oftheLiealgebra ofallfunctions. TheLiealgebra ofhamiltonian functions canbemapped naturally onto theLiealgebra ofhamiltonian vector ﬁelds. Todothis, toevery function H weassociate thehamiltonian vector ﬁeld Hwith hamiltonian function H. Corollary 8.ThemapoftheLiealgebra offunctions onto theLiealgebra of hamiltonian ﬁelds isanalgebra homomorphism. Itskernel consists ofthe locally constant functions. IfM2"isconnected, thekernel isone-dimensional andconsists ofconstants. 217 8:Symplectic manifolds PRooF. Ourmap islinear. Corollary 5saysthatourmap carries thePoisson bracket offunctions into thePoisson bracket ofvector ﬁelds. The kernel consists offunctions Hforwhich IdHE0.Since Iisanisomorphism, dHE0andH=const. Cl Corollary 9.Thephase ﬂows with hamiltonian functions H1andH2commute ifandonly ifthePoisson bracket ofthefunctions H1andH2is(locally) constant. PROOF. Bythetheorem inSection 39,E,itisnecessary andsufficient that [H1, H2] E0,andbyCorollary 8thiscondition isequivalent tod(H1, H2) E0. l:l Weobtain yetanother generalization ofE.Noether’s theorem: given a flowwhich commutes withtheoneunder consideration, onecanconstruct aﬁrstintegral. DLocally hamiltonian vector ﬁelds Let(M2", wz)beasymplectic manifold andg’:M2"->M2"aone-parameter group ofdiffeo- morphisms preserving thesymplectic structure. Willg‘beahamiltonian flow? EXAMPLE. LetM2" beatwo-dimensional torus T2,apoint ofwhich isgiven byapairofco- ordinates (p,q)rn0d l.Letofbetheusual area element dp/\dq.Consider thefamily oftrans- lations g'(p,q)=(p+t,q)(Figure 173). Themaps g’preserve thesymplectic structure (i.e., area). Can weﬁndahamiltonian function corresponding tothevector ﬁeld (p=1,4=0)’? Ifp'=—5H/dq andq =5H/dp, wewould have 6H/op =Oand 6H/dq =—l,i.e.,H =—q+C. Butqisonly alocal coordinate onT2;there isnomap H:T2—>Rforwhich 5H/6p =0and 5H/dq =l.Thus g’isnotahamiltonian phase ﬂow. [I (I Figure 173 Alocally hamiltonial ﬁeldonthetorus Deﬁnition. Alocally hamiltonian vector ﬁeld onasymplectic manifold (M2", (oz)isthevector ﬁeldlo)‘, where co‘isaclosed l-form onM“. Locally, aclosed l-form isthedifferential ofafunction, of=dH.However, inattempting toextend thefunction Htothewhole manifold M2"wemayobtain a“many-valued hamiltonian function,” since aclosed 1-form onanon-simply-connected manifold maynotbeadifferential (forexample, theform dqonT2).Aphase flowgiven byalocally hamiltonian vector ﬁeldiscalled alocally hamiltonian ﬁow. PROBLEM. Show thataone-parameter group ofdiffeomorphisms ofasymplectic manifold pre- serves thesymplectic structure ifand only ifitisalocally hamiltonian phase ﬂow. Hint. Cf.Section 38A. 218 41:Symplectic geometry PRonLEM. Show that inthesymplectic space R2", every one-parameter group ofcanonical dilfeomorphisms (preserving dpAdq)isahamiltonian flow. Hint. Every closed l-form onIR“isthedifferential ofafunction. PROBLEM. Show thatthelocally hamiltonian vector ﬁelds form asub-algebra ofthe Liealgebra ofallvector ﬁelds. Inaddition, thePoisson bracket oftwolocally hamiltonian ﬁelds isactually ahamiltonian ﬁeld, with ahamiltonian function uniquely“ determined bythegiven ﬁelds Q andItbytheformula H=w2(§, 1|).Thus, thehamiltonian ﬁelds form anideal intheLiealgebra oflocally hamiltonian ﬁelds. 41Symplectic geometry Aeuclidean structure onavector space isgiven byasymmetric bilinear form, andasymplectic structure byaskew-symmetric one. The geometry ofasymplectic space isdifferent from that of aeuclidean space, although there aremany similarities. ASymplectic vector spaces LetR2"beaneven-dimensional vector space. Deﬁnition. Asymplectic linear structure onR2"isanondegenerate“ bi- linear skew-symmetric 2-form given inR2". This form iscalled the skew-scalar product andisdenoted by[§,1|]=-"[1], §].Thespace R2”, together with thesymplectic structure [,],iscalled asymplectic vector space. EXAMPLE. Let(p,,...,p,,,ql,...,q,,)becoordinate functions onR2", and roztheform a)2=p1 /\¢11'l""‘l'P»/\¢1..- Since thisform isnondegenerate andskew-symmetric, itcanbetaken fora skew-scalar product: [Q11] =oJ2(§, 1]).Inthisway thecoordinate space R2"={(p,q)} receives asymplectic structure. This structure iscalled the standard symplectic structure. Inthestandard symplectic structure the skew-scalar product oftwovectors §and1]isequal tothesumoftheoriented areas oftheparallelogram (§,t1) onthertcoordinate planes (p,-,q,). Two vectors Qand1|inasymplectic space arecalled skew-orthogonal (E41])iftheir skew-scalar product isequal tozero. PROBLEM. Show thatQ4Q:every vector isskew-orthogonal toitself. Thesetofallvectors skew-orthogonal toagiven vector 1|iscalled the skew-orthogonal complement to1|. 65Notjustuptoaconstant. 6°A2-form [,]onR2"isnondegenerate if([§, 11]=0,Vq)=>(Q=0). 219 8:Symplectic manifolds PROBLEM. Show thattheskew-orthogonal complement to1|isa2n—l-dimensional hyperplane containing 1|. Hint. Ifallvectors were skew-orthogonal toI],then theform [,]would bedegenerate. BThesymplectic basis Aeuclidean structure under asuitable choice ofbasis (itmust beortho- normal) isgiven byascalar product inaparticular standard form. Inexactly thesame way, asymplectic structure takes thestandard form indicated above inasuitable basis. PROBLEM. Find theskew-scalar product ofthe basis vectors emandem(i=l...,n)intheexample presented above. Solution. Therelations (1) fey.»em]=[eatea]=[eat°-1.-1=0[¢.=.»%.~1 =1 follow from thedeﬁnition Ofpl /‘\q,+ +p,,Aq,,. Wenow return tothegeneral symplectic space. Deﬁnition. Asymplectic basis isasetof2nvectors, em,eq,(i=1,...,n) whose scalar products have theform (1). Inother words, every basis vector isskew-orthogonal toallthebasis vectors except one, associated toit;itsproduct with theassociated vector isequal toi1. Theorem. Every symplectic space hasasymplectic basis. Furthermore, wecan takeanynonzero vector 6fortheﬁrst basis vector. PROOF. This theorem isentirely analogous tothecorresponding theorem in euclidean geometry andisproved inalmost thesame way. Since thevector eisnotzero, there isavector fnotskew-orthogonal toit (theform [,]isnondegenerate). Bychoosing thelength ofthisvector, we caninsure thatitsskew-scalar product with eisequal to1.Inthecasen=1, thetheorem isproved. Ifn>1,consider theskew-orthogonal complement D(Figure 174) to thepairofvectors e,f.Distheintersection oftheskew-orthogonal comple- ments toeandf.These two2n—1-dimensional spaces donotcoincide, f € Figure 174 Skew-orthogonal complement 220 41:Symplectic geometry since eisnotintheskew-orthogonal complement tof.Therefore, their inter- section haseven dimension 2n-—2. Wewillshow thatDisasymplectic subspace ofR2",i.e.,thattheskew- scalar product [,]restricted toDisnondegenerate. Ifavector §eD were skew-orthogonal tothewhole subspace D,then since itwould alsobe skew-orthogonal toeandtof,Ewould beskew-orthogonal toR2",which contradicts thenondegeneracy of[,]onR2”.Thus D2“ 2issymplectic. Now ifweadjoin thevectors eandftoasymplectic basis forD2"'2 we getasympletic basis forR2",andthetheorem isproved byinduction onn. III Corollary. Allsymplectic spaces ofthesame dimension areisomorphic. Ifwetake thevectors ofasymplectic basis ascoordinate unit vectors, weobtain acoordinate system p,-,q,inwhich [,]takes thestandard form p,/\q,+ +p,,/\q,,.Such acoordinate system iscalled sym- plectic. CThesymplectic group Toaeuclidean structure weassociated theorthogonal group oflinear map- pings which preserved theeuclidean structure. Inasymplectic space the symplectic group plays ananalogous role. Deﬁnition. Alinear transformation S:R2”-—>R2"ofthesymplectic space R2"toitself iscalled symplectic ifitpreserves theskew-scalar product: [$§.5'1]=[lint]. I/5.116 R2"- Thesetofallsymplectic transformations ofR2"iscalled thesymplectic group andisdenoted bySp(2n). Itisclear that thecomposition oftwo symplectic transformations is symplectic. Tojustify theterm symplectic group, wemust only show thata symplectic transformation isnonsingular; itisthenclear thattheinverse is alsosymplectic. PROBLEM. Show thatthegroup Sp(2) isisomorphic tothegroup ofrealtwo-by-two matrices withdeterminant landishomeomorphic totheinterior ofasolid three-dimensional torus. Theorem. Atransformation S:R2"—>R2"ofthestandard symplectic space (p,q)issymplectic ifandonly ifitislinear andcanonical, i.e.,preserves the dtﬂerential 2-form w’=dinAdq.++dt..Adc..- PROOF. Under thenatural identiﬁcation ofthetangent space toR2"with R2",the2-form co’goes to[,]. [1 221 8:Symplectic manifolds Corollary. Thedeterminant ofanysymplectic transformation isequal to1. PROOF. Wealready know (Section 38B) that canonical maps preserve the exterior powers oftheform (02.Butitsn-thexterior power is(uptoaconstant multiple) thevolume element onR2". This means that symplectic trans- formations Softhestandard R2"={(p,q)}preserve thevolume element, sodetS=1.Butsince every symplectic linear structure canbewritten down instandard form inasymplectic coordinate system, thedeterminant ofa symplectic transformation ofanysymplectic space isequal to1. El Theorem. Alinear transformation S:R2"—>R2”issymplectic ifandonly ifit takes some (and therefore any) symplectic basis intoasymplectic basis. PROOF. Theskew-scalar product ofanytwolinear combinations ofbasis vec- torscanbeexpressed interms ofskew-scalar products ofbasis vectors. Ifthe transformation does notchange theskew-scalar products ofbasis vectors, then itdoes notchange theskew-scalar products ofanyvectors. El DPlanes insymplectic space Inaeuclidean space allplanes areequivalent: each ofthem canbecarried into anyother onebyamotion. Wewillnow look atasymplectic vector space from thispoint ofview. PROBLEM. Show thatanonzero vector inasymplectic space canbecarried intoanyother non- zero vector byasymplectic transformation. PROBLEM. Show that notevery two-dimensional plane ofthesymplectic space IRZ"canbe obtained from agiven 2-plane byasymplectic transformation. Hint. Consider theplanes (p,,p2)and(pl,ql). Deﬁnition. Ak-dimensional plane (i.e., subspace) ofasymplectic space is called null“ ifitisskew-orthogonal toitself, i.e.,iftheskew-scalar product ofanytwovectors oftheplane isequal tozero. EXAMPLE. Thecoordinate plane (p,,...,pk)inthesymplectic coordinate system p.qisnull. (Prove it!) PROBLEM. Show thatanynon-null two-dimensional plane canbecarried intoanyother non- nulltwo-plane byasymplectic transformation. Forcalculations insymplectic geometry itmay beuseful toimpose some euclidean structure onthesymplectic space. Weﬁxasymplectic coordinate system p,qandintroduce aeuclidean structure using thecoordinate scalar product (x,x)=Zpiz+61?, where x=Zp,-e,,_. +q,-em. 67Null planes arealsocalled isotropic, andfork=n,lagrangian. 222 411Symplectic geometry Thesymplectic basis ep,eqisorthonormal inthiseuclidean structure. The skew-scalar product, likeevery bilinear form, canbeexpressed interms of thescalar product by (2) [5,'1]=(1%,It) where 1:R2“~>R2"issome operator. Itfollows from theskew-symmetry of theskew-scalar product thattheoperator Iisskew-symmetric. PROBLEM. Compute thematrix oftheoperator Iinthesymplectic basis cpl,em. (°'E)E0‘ANSWER. where Eisthen>1nidentity matrix. Thus, forn=1(inthep,q-plane), Iissimply rotation by90°,andinthe general case Iisrotation by90°ineach ofthenplanes p,-,q,-. PROBLEM. Show thattheoperator Iissymplectic andthatI’=—E2,. Although theeuclidean structures andtheoperator Iarenotinvariantly associated toasymplectic space, they areoften convenient. Thefollowing theorem follows directly from (2). Theorem. Aplane rtofasymplectic space isnullifand only ifthe plane In:is orthogonal tort. Notice thatthedimensions oftheplanes rtandIrtarethesame, since Iis nonsingular. Hence Corollary. Thedimension ofanullplane inR2"islessthan orequal ton. This follows since thetwo k-dimensional planes rtand Incannot be orthogonal ifk>n. Weconsider more carefully then-dimensional nullplanes inthesymplectic coordinate space R2".Anexample ofsuch aplane isthecoordinate p-plane. There areinallC5,,n-dimensional coordinate planes inR2"={(p,q)}. PROBLEM. Show thatthere are2"nullplanes among theCZ,n-dimensional coordinate planes: toeach ofthe2"partitions oftheset(I.....n)intotwoparts (i,....,i,_).(j,....__j,,_k) weasso- ciate thenullcoordinate plane pm,...,p,_,qh.....q,,__,. Inorder tostudy thegenerating functions ofcanonical transformations weneed 223 8:Symplectic manifolds & Figure 175 Construction ofacoordinate plane trtransversal toagiven plane rt. Theorem. Every n-dimensional nullplane 1tinthesymplectic coordinate space R2"istransverse“ toatleast oneofthe2"coordinate nullplanes. PROOF. LetPbethenullplane p,,...,p,,(Figure 175). Consider theinter- section t=rtrwP.Suppose thatthedimension ofrisequal tok,0sk3n. Like every k-dimensional subspace ofthen-dimensional space, theplane tis transverse toatleast one(n—k)-dimensional coordinate plane inP,letus saytheplane l1=(p,,,...,p,~"_k); t+r1=P,rn11=0. Wenow consider thenulln-dimensional coordinate plane 5:(Pm---,Pi,..,,»‘l'j,»~~»>qj,.)» 'l=°'f')P, andshow thatourplane rtistransverse tooz rtr'\o=O. Wehave tc1t,rt4rt=t41tncmazaanza}=(t+n)4(rtr\o)=>P4(1tr\o). ButPisann-dimensional nullplane. Therefore, every vector skew-orthogonal toPbelongs toP(cf.thecorollary above). Thus (rtn0')cP.Finally, 1tr\o=(rtnP)n(or\P)= tn11=(), aswastobeshown. El PROBLEM, Letrt,andrt,betwok-dimensional planes insymplectic R2".Isitalways possible to carry rt,tort;byasymplectic transformation? How many classes ofplanes arethere which cannot becarried oneintoanother? ANSWER. [t/2]+1,ifkgn;[(2n-to/2]+1ilk2n. ESymplectic structure andcomplex structure Since I2=—Ewecanintroduce into ourspace R2"notonly asymplectic structure [,]andeuclidean structure (,),butalsoacomplex structure, bydeﬁning multiplication byi=./—1tobetheaction ofI.Thespace R2” 68Two subspaces L,andL2ofavector space Laretransverse ifL,+L2=L.Two n-dimen- sional planes inR2"aretransverse ifandonly iftheyintersect only in0. 224 42:Parametric resonance insystems with many degrees offreedom isidentiﬁed inthiswaywith acomplex space C"(thecoordinate space with coordinates zk=pk+iqk).Thelinear transformations ofR2"which preserve theeuclidean structure form theorthogonal group O(2n); those preserving thecomplex structure form thecomplex linear group GL(n, C). PROBLEM. Show that transformations which areboth orthogonal andsymplectic arecomplex, thatthose which areboth complex andorthogonal aresymplectic. andthatthose which are both symplectic andcomplex areorthogonal: thus that theintersection oftwoofthethree groups isequal totheintersection ofallthree: 0(2):) OSp(2n) =Sp(2n) mGL(n, C)=GL(n_ C)OO(2n). This intersection iscalled theunitary group U(n). Unitary transformations preserve thehermitian scalar product (§,1])+ i[§,11];thescalar andskew-scalar products onR2"areitsrealandimaginary parts. 42Parametric resonance insystems with many degrees offreedom During ourinvestigation ofoscillating systems withperiodically varying parameters (cf.Section 25),\veexplained thatparametric resonance depends onthebehavior oftheeigenvalues ofa certain linear transformation (“the mapping ataperiod ").Thedependence consists ofthefact thatanequilibrium position ofasystem with periodically varying parameters isstable ifthe eigenvalues ofthemapping ataperiod have modulus lessthan l,andunstable ifatleast oneof theeigenvalues hasmodulus greater than l. Themapping ataperiod obtained from asystem ofHamilton’s equations with periodic coefficients issymplectic. Theinvestigation inSection 25ofparametric resonance inasystem with onedegree offreedom relied onouranalysis ofthebehavior ofthe eigenvalues ofsymplectic transformations ofthe plane. Inthisparagraph wewillanalyze. inananalogous way,thebehavior ofthe eigenvalues ofsymplectic transformations inaphase space ofanydimension. Theresults oflhis analysis (due toM.G.Krein) canbeapplied tothestudy ofconditions fortheappearance ofparametric resonance inmechanical systems with many degrees offreedom. ASymplectic matrices Consider alinear transformation ofasymplectic space, S:R2"—>R2". Let p,,...,p,,;q,,...,q,,beasymplectic coordinate system. Inthiscoordinate system, thetransformation isgiven byamatrix S. Theorem. Atransformation issymplectic ifandonlyifitsmatrix Sinthesym- plectic coordinate system (p,q)satisﬁes therelation s'1s=1, 0—EI=(E0)where andS’isthetranspose ofS. 225 8:Symplectic manifolds PROOF. Thecondition forbeing symplectic ([S§, S11]=[§,1|]forall§and1|) canbewritten interms ofthescalar product byusing theoperator I,as follows: (13%,$11)=(IQII), V5,'1 or ($755.11) =(1%,Ii). va.Ii, aswastobeshown. III BSymmetry ofthespectrum ofasymplectic transformation Theorem. Thecharacteristic polynomial ofasymplectic transformation p(/I) =det(S —/IE) isreflexive,” i.e.,p(,l) =,l.2"p( 1//1). PROOF. Wewillusethefacts thatdetS=detI=1,I2=—E,anddetA’= detA.Bythetheorem above, S=—IS" ‘I.Therefore, p(/l.)=det(S-AE)=det(—IS"‘I -»1E)= det(—S"1 +ts) =det(—E +/ts) l 1_ 2n __ = 2n _-/1det(S AE) /l.p(x). l] Corollary. If/lisaneigenvalue ofasymplectic transformation, then l/Zisalso aneigenvalue. Ontheother hand, thecharacteristic polynomial isreal; therefore, ifA isacomplex eigenvalue, then Iisaneigenvalue different from /1.Itfollows that theroots /lofthecharacteristic polynomial liesymmetrically with respect totherealaxis andtotheunit circle (Figure 176). They come in 4-tuples, _11,1,}.,I,j (|/ll9*1,Iml¢0), andpairs lying ontherealaxis, 11 Z 2 —2—A i1’ 69Areﬂexive polynomial isapolynomial a0x"' +a,x"‘_' + +amwhich hassymmetric coefﬁcients ao=a,,,,a,=a,,,_,, . 226 42:Parametric resonance insystems with many degrees offreedom A 1A:K O _ >\=>\ I O _1 1 A1‘X0 . -1 Figure 176 Distribution oftheeigenvalues ofasymplectic transformation orontheunitcircle, 1 1 1:1 1:1. Itisnothard toverify thatthemultiplicities ofallfourpoints ofa4-tuple (or both points ofapair) arethesame. cStability Deﬁnition. Atransformation Siscalled stable if ‘v’s>0,35>0:|x|<5=>|S”x|<s, \iN>0. PROBLEM. Show thatifatleast oneoftheeigenvalues ofasymplectic transformation Sdoes not lieontheunitcircle, then Sisunstable. Hint. Inview ofthedemonstrated symmetry, ifoneoftheeigenvalues does notlieonthe unitcircle, then there exists aneigenvalue outside theunitcircle l/ll>1;inthecorresponding invariant subspace, Sisan“expansion with arotation.” PROBLEM. Show thatifalltheeigenvalues ofalinear transformation aredistinct andlieonthe unitcircle, then thetransformation isstable. Hint. Change toabasis ofeigenvectors. Deﬁnition. Asymplectic transformation Siscalled strongly stable ifevery symplectic transformation suﬂiciently close7° toSisstable. InSection 25weestablished thatS:R2—>R2isstrongly stable if»l,_2= eh“and1,aé12. Theorem. Ifall2neigenvalues ofasymplectic transformation Saredistinct andlieontheunitcircle, thenSisstrongly stable. PROOF. Weenclose the2neigenvalues Ain2nnon-intersecting neighborhoods, symmetric with respect totheunitcircle andtherealaxis(Figure 177). The 2nroots ofthecharacteristic polynomial depend continuously ontheele- ments ofthematrix ofS.Therefore, ifthematrix S,issuﬁiciently close toS, 7°S,is“sufficiently close” toSiftheelements ofthe matrix ofS, inaﬁxed basis differ from the elements ofthe matrix ofSinthesame basis bylessthan asufficiently small number s. 227 8:Symplectic manifolds new I ea Figure 177 Behavior ofsimple eigenvalues under asmall change ofthesymplectic transformation exactly oneeigenvalue /1,ofthematrix ofS,willlieineach ofthe2nneigh- borhoods ofthe2npoints of/l.Butifoneofthepoints ,1,didnotlieonthe unitcircle, forexample, ifitlayoutside theunitcircle, then bythetheorem in subsection B,there would beanother point 2.2,|.l.2|<1inthesame neighbor- hood, andthetotal number ofroots would begreater than 2n,which isnot possible. Thus alltheroots ofS,lieontheunit circle andaredistinct, soS,is stable. El Wemight saythat aneigenvalue Aofasymplectic transformation can leave theunitcircle only bycolliding with another eigenvalue (Figure 178); atthesame time, thecomplex-conjugate eigenvalues willcollide, andfrom thetwopairs ofroots ontheunit circle weobtain one4-tuple (orpairof real2). 0 I,, Figure 178 Behavior ofmultiple eigenvalues under asmall change ofthesymplectic transformation Itfollows from theresults ofSection 25thatthecondition forparametric resonance toarise inalinear canonical system with aperiodically changing hamilton function isprecisely thatthecorresponding symplectic transforma- tion ofphase space should cease tobestable. Itisclear from thetheorem above thatthiscanhappen only after acollision ofeigenvalues ontheunit circle. Infact,asM.G.Krein noticed, notevery such collision isdangerous. Itturns outthattheeigenvalues ,1with |}.|=1aredivided intotwoclasses: positive andnegative. When tworoots with thesame signcollide, theroots “gothrough oneanother,” andcannot leave theunit circle. Ontheother 228 43:Asymplectic atlas hand, when tworoots with different signs collide, they generally leave the unitcircle. M.G. l(rein’s theory goes beyond thelimits ofthisbook; wewillformulate thebasic results here intheform ofproblems. PROBLEM. LetZandIbesimple (multiplicity 1)eigenvalues ofasymplectic transformation S with Ill=1.Show thatthetwo-dimensional invariant plane rt,corresponding to,1,1,isnon- null. Hint. LetQ,andQ2becomplex eigenvectors ofSwitheigenvalues /'.,andZ2.Then if).,i.2 aé1, thevectors Q,andQ2areskew-orthogonal: [Q,,Q2]=0. LetQbearealvector oftheplane rt,,,where Im1>0and|).|=1.The eigenvalue itiscalled positive if[SQ,Q]>0. PROBLEM. Show thatthisdeﬁnition iscorrect, i.e.,itdoes notdepend onthechoice ofQaé0in theplane 1:2. Hint. Ifthe plane rt,contained twonon-collinear skew-orthogonal vectors, itwould benull. Inthesame way, aneigenvalue Zofmultiplicity kwith |/ll=lisofdeﬁnite sign ifthe quad- ratic form [S§,Q]is(positive ornegative) deﬁnite ontheinvariant 2k-dimensional subspace corresponding to/1,I. PROBLEM. Show thatSisstrongly stable ifandonly ifalltheeigenvalues J.lieontheunitcircle andareofdeﬁnite sign. Hint. Thequadratic form [SE_,,Q]isinvariant with respect toS. 43Asymplectic atlas Inthisparagraph weprove Darboux‘s theorem, according towhich every symplectic manifold haslocal coordinates p,qinwhich thesymplectic structure canbewritten inthesimplest way: of=dp/\dq. ASymplectic coordinates Recall thatthedeﬁnition ofmanifold includes acompatibility condition for thecharts ofanatlas. This isacondition onthemaps (pf1(1)jgoing from one chart toanother. Themaps (pf1(p,aremaps ofaregion ofcoordinate space. Deﬁnition. Anatlas ofamanifold M2" iscalled symplectic ifthestandard symplectic structure of=dp/\dqisintroduced into thecoordinate space R2"={(p,q)},andthetransfer from onechart toanother isrealized byacanonical (i.e.,(02-preserving) transformation“ (pf‘rp,-. PROBLEM. Show thatasymplectic atlas deﬁnes asymplectic structure onM2". The converse isalso true: every symplectic manifold hasasymplectic atlas. This follows from thefollowing theorem. 71Complex-analytic manifolds, forexample, aredeﬁned analogously; there must beacomplex- analytic structure oncoordinate space, andthetransfer from Onechart toanother must be complex analytic. 229 8:Symplectic manifolds BDarboux’s theorem Theorem. Letofbeaclosed nondegenerate differential 2-form inaneighbor- hood ofapoint xinthespace R2".Then insome neighborhood ofxonecan choose acoordinate system (p,,...,p,,;q,,...,q,,)such thattheform hasthe standard form : (1)2=Zdp, /\dqi. i=1 This theorem allows ustoextend toallsymplectic manifolds anyassertion ofalocal character which isinvariant with respect tocanonical transforma- tions andisproven forthestandard phase space (lR2", (02=dp/\dq). CConstruction ofthecoordinates p,andq, Fortheﬁrstcoordinate p,wetake anon-constant linear function (wecould have taken anydifferentiable function whose differential isnotzero atthe point x).Forsimplicity wewillassume thatp,(X) =0. LetP,=Idp,denote thehamiltonian ﬁeldcorresponding tothefunction p,(Figure 179). Note thatP,(x) at0;therefore, wecandraw ahyperplane NM" through thepoint xwhich does notcontain thevector P,(x) (we could have taken anysurface transverse toP,(x) asl\/2"‘ ‘). M211-'2 02 lllllllrg.llllmini4-in-aw - -_vP1 Figure 179 Construction ofsymplectic coordinates Consider thehamiltonian ﬂow P’,with hamiltonian function p,.We consider thetime tnecessary togofrom Ntothepoint z=P‘,(y) (yeN) under theaction ofP’,asafunction ofthepoint z.Bytheusual theorems in thetheory ofordinary differential equations, thisfunction isdeﬁned and differentiable inaneighborhood ofthepoint xeR2“.Denote itbyq,.Note thatq,=0onNandthatthederivative ofq,inthedirection oftheﬁeld P, isequal to1.Thus thePoisson bracket ofthefunctions q,andp,wecon- structed isequal to1: (‘InPi)E1- 230 43:Asymplectic atlas DConstruction ofsymplectic coordinates by induction onn Ifn=1,theconstruction isﬁnished. Letn>1.Wewillassume that Dar- boux’s theorem isalready proved forR2“ 2.Consider thesetMgiven bythe equations p,=q,=0.Thedifferentials dp,anddq,arelinearly independent atxsince w2(I dp,, ldq,) =(q,,p,)E1.Thus, bytheimplicit function theorem, thesetMisamanifold ofdimension 2n—2inaneighborhood of x;wewilldenote itbyM2“ 2. Lemma. Thesymplectic structure ofonR2"induces asymplectic structure on some neighborhood ofthe point xonM2“ 2. PROOF. Fortheproof weneed only thenondegeneracy ofofonTM‘. Consider thesymplectic vector space TlR,f". The vectors P,(x) andQ,(x) ofthe hamiltonian vector ﬁelds with hamiltonian functions p,andq,belong toTlltf". LetQ6TM,,. The derivatives ofp,andq,inthedirection Qare equal tozero. This means thatdp,(Q) =w-’"(Q, P,)=0anddq,(Q) =o)2(Q, Q,) =0.Thus TM, istheskew-orthogonal complement toP,(x), Q,(x). By Section 41B, theform to’onTM, isnondegenerate. 1:1 Bytheinduction hypothesis there aresymplectic coordinates inaneigh- borhood ofthepoint xonthesymplectic manifold (Mz"‘2, to’IM).Denote them byp,.q, (i=2,...,n).Weextend thefunctions p2,...,q,,toaneighbor- hood ofxinR2"inthefollowing way. Every point zinaneighborhood of xinR2"can beuniquely represented intheform z=P‘,Q‘,w, where weM2"'2, andsandraresmall numbers. Wesetthevalues ofthecoor- dinates p2,...,q,,atzequal totheir values atthepoint w(Figure 179). The 2nfunctions p,,...,p,,,q,,...,q,,thus constructed form alocal coordinate system inaneighborhood ofxinR2". EProof that thecoordinates constructed are symplectic Denote byPfandQ}(i=l,...,n)thehamiltonian ﬂows with hamiltonian functions p,andq,-,andbyP,andQ,thecorresponding vector ﬁelds. Wewill compute thePoisson brackets ofthefunctions p,,...,q,,.Wealready sawin Cthat(q,,p,)21.Therefore, theﬂows P‘,andQ‘,commute: P',Q‘, =Q‘,P‘,. Recalling thedeﬁnitions ofp2, ...,q,,weseethateach ofthese functions is invariant with respect totheﬂows P’,andQ',.Thus thePoisson brackets of p,andq,with all2n-2functions p,,q,(i>I)areequal tozero. Themap P',Q’, therefore commutes with all2n—2ﬂows P1,Q?(i>1). Consequently, itleaves each ofthe2n—2vector ﬁelds P,,Q,(i>1)ﬁxed. P',Q"‘, preserves thesymplectic structure ofsince theﬂows P‘,andQ1are hamiltonian; therefore, thevalues oftheform (1)2onthevectors ofanytwo 231 8:Symplectic manifolds ofthe2n—2ﬁelds Pi,Q,(i>1)arethesame atthepoints z=P‘,Q‘,WeR2” andweM2"‘2.Butthese values areequal tothevalues ofthePoisson brack- etsofthecorresponding hamiltonian functions. Thus, thevalues ofthe Poisson bracket ofanytwoofthe2n—-2coordinates p,,q,(i>1)atthe points zandwarethesame ifz=P'1Q‘,w. The functions p,andqlareﬁrst integrals ofeach ofthe2n—2ﬂows Pj-,Qf(i>1).Therefore, each ofthe2n—2ﬁelds Pi,Q,istangent tothe level manifold pl=q,=0.Butthismanifold isM2""2.Therefore, each of the2n—2ﬁelds P,,Q,(i>1)istangent toM2“? Consequently, these ﬁelds arehamiltonian ﬁelds onthesymplectic manifold (M2"‘2,wz|M),and thecorresponding hamiltonian functions arep,|M, q,|M(i>l).Thus, inthe whole space (lR2", wz), thePoisson bracket ofanytwoofthe2n—2co- ordinates p,,q,-(i>1)considered onM2"'2 isthesame asthePoisson bracket ofthese coordinates inthesymplectic space (M2""2,ofIM). But, byourinduction hypothesis, thecoordinates onM2"'2 (p,-IM, q,-|M; i>l)aresymplectic. Therefore, inthewhole space R2",thePoisson brackets oftheconstructed coordinates have thestandard values (PoP1“); (Pi,qj)E(qr,qj)EOand (qt,Pi)E1- The Poisson brackets ofthecoordinates p,qonR2"have thesame form if wz=Zdp,/\dq,-.Butabilinear form ofisdetermined byitsvalues on pairs ofbasis vectors. Therefore, thePoisson brackets ofthecoordinate functions determine theshape ofofuniquely. Thus wz=dp1/\ dq,+ +dp,, /\dq,,, andDarboux’s theorem isproved. El 232 Canonical formalism Thecoordinate point ofview willpredominate inthischapter. Thetechnique ofgenerating functions forcanonical transformations, developed by Hamilton andJacobi, isthemost powerful method available forintegrating thedifferential equations ofdynamics. Inaddition tothistechnique, the chapter contains an“odd-dimensional” approach tohamiltonian phase ﬂows. This chapter isindependent oftheprevious one. Itcontains new proofs ofseveral oftheresults inChapter 8,aswellasanexplanation oftheorigin ofthetheory ofsymplectic manifolds. 44Theintegral invariant ofPoincaré—Cartan Inthissection welook atthegeometry of1-forms inanodd-dimensional space. AAhydrodynamical lemma Letvbeavector ﬁeld inthree-dimensional oriented euclidean space [R3, andr=curlvitscurl. Theintegral curves ofrarecalled vortex lines. Ifyl isanyclosed curve in[R3(Figure 180), thevortex lines passing through the points ofylform atube called avortex tube. Let7/2beanother curve encircling thesame vortex tube, sothaty,—yz= 80,where oisa2-cycle representing apart ofthevortex tube. Then: Stokes’ lemma. Theﬁeld vhasequal circulation along thecurves yland322: ifvdl=ifvdl. VI Y2 233 9:Canonical formalism 72 \ 51 '71 Figure 180 Vortex tube PROOF. ByStokes’ formula, [,1vdl—[hvdl=H,curlvdn=0,since curlv istangent tothevortex tube. El BThemulti-dimensional Stokes’ lemma Itturns outthatStokes’ lemma generalizes tothecaseofanyodd-dimensional manifold M2'”1(inplace ofR3).Toformulate thisgeneralization wereplace ourvector ﬁeld byadifferential form. The circulation ofavector ﬁeld vistheintegral ofthel-form w‘ (w‘(§) =(v,§)).Tothecurl ofvthere corresponds the2-form of=dw‘ (dw‘(§, 1])=(r,§,11)).Itisclear from these formulas thatthere isadirection I‘ T7 Figure 181 Axis invariantly connected with a2-form inanodd-dimensional space atevery point (namely, thedirection ofr,Figure 181), having theproperty thatthecirculation ofvalong theboundary ofevery “inﬁnitesimal square” containing risequal tozero: dw1(r, 1])=0, VI]. Infact,dw1(r, 1|)=(r,r,1])=0. Remark. Passing from the2-form wz=dw‘tothevector ﬁeld r=curlv isnotaninvariant operation: itdepends ontheeuclidean structure of[R3. Only thedirection” ofrisinvariantly associated with wz(and, therefore, with the1-form 0)‘).Itiseasy toverify that, ifraé0,then thedirection ofr isuniquely determined bythecondition thatw2(r, 1|)=OforallI]. 72l.e.,theunoriented lineinTR3 with direction vector r. 234 44:Theintegral invariant ofPoincaré~Cartan The algebraic basis forthemulti-dimensional Stokes’ lemma isthe existence ofanaxisforevery rotation ofanodd-dimensional space. Lemma. Letofbeanexterior algebraic 2-form ontheodd-dimensional vector space R2” 1.Then there isavector §atOsuch that w2(§’ TI)=0’ V"6R2n+1_ PROOF. Askew-symmetric form wzisgiven byaskew-symmetric matrix A W293,I1)=(Ali,11) ofoddorder 2n+1.Thedeterminant ofsuch amatrix isequal tozero, since A’=—A detA =detA’=det(—A) =(-1)’"*‘ detA =—detA. Thus thedeterminant ofAiszero. This means Ahasaneigenvector Qaé0 with eigenvalue 0,aswastobeshown. Cl Avector Qforwhich w2(§, 1|)=0,Vqiscalled anullvector fortheform wz. Thenullvectors ofwzclearly form alinear subspace. Theform £02iscalled nonsingular ifthedimension ofthisspace istheminimal possible (i.e., 1 foranodd-dimensional space lR2"*‘ or0foraneven-dimensional space). PROBLEM. Consider the2-form to’=dp,/\dq,+ +dp,,/\dq,,Onaneven-dimensional space R2”with coordinates pf,...,p,,;q1,.. .,q,,.Show thatofisnonsingular. PROBLEM. Onanodd-dimensional space R2"*‘ with coordinates p,.....p,,: q,. q..:t,con~ sider the2-form of=Zdp,/\dq,—oi‘/\dt,where to‘isanyl-form onR2“ 1.Show that(02is nonsingular. Ifofisanonsingular form onanodd-dimensional space lR2"*‘, then thenullvectors Qofofalllieonaline.This lineisinvariantly associated to theform (oz. Now letM2"*‘ beanodd-dimensional differentiable manifold andat‘ a1-form onM.Bythelemma above, atevery point xeMthere isadirection (i.e., astraight line{ctﬁ} inthetangent space TMX) having theproperty that theintegral ofto‘along theboundary ofan“inﬁnitesimal square containing thisdirection” isequal tozero: dw'(§, 1])=0, Vqe TM, Suppose further that the2-form dw‘ isnonsingular. Then thedirection § isuniquely determined. Wecallitthe“vortex direction” oftheform oi‘. Theintegral curves oftheﬁeld ofvortex directions arecalled thevortex lines (orcharacteristic lines) oftheform cu‘. Let",1,beaclosed curve onM.The vortex lines going outfrom points of"/1form a“vortex tube.” Wehave 235 9:Canonical formalism Themulti-dimensional Stokes’ lemma. Theintegrals ofa1-form w‘along any twocurves encircling thesame vortex tubearethesame :to‘=fHm‘, if7/1—"/2=60,where aisapiece ofthevortex tube. PROOF. ByStokes’ formula §w‘—t]€w‘=foJ‘=Jdw‘. vi vz 13¢ v Butthevalue ofdw‘onanypairofvectors tangent tothevortex tube isequal tozero. (These twovectors lieina2-plane containing thevortex direction, anddo)‘vanishes onthisplane.) Thus, L,do)‘=0. El CHamilton’s equations Allthebasic propositions ofhamiltonian mechanics follow directly from Stokes’ lemma. For M2"*1 wewilltake the“extended phase space lR2"*1” with co- ordinates P1,...,p,,;q1,..., q,,;t.Suppose wearegiven afunction H= H(p,q,t).Then wecanconstruct” the1-form w‘=ndq—Hdt (pdq=indqi++ride.)- Weapply Stokes’ lemma toco‘(Figure 182). 72 71/ \\ - P (—Hq,H,,,l) 1 >1ll Figure 182 Hamiltonian ﬁeld andvortex lines ofthe form pdq—Hdt. Theorem. The vortex lines oftheform cu‘=pdq—Hdt onthe2n+1- dimensional extended phase space p,q,thave aone-to-one projection onto thetaxis, i.e.,theyaregiven byfunctions p=p(t),q=q(t).These functions satisfy thesystem ofcanonical differential equations with hamiltonian function H: (1) dp__6H dq_5H at" aq’ dt_5p' Inother words, thevortex lines oftheform pdq—Hdtarethetrajectories ofthephase ﬂow intheextended phase space, i.e.,theintegral curves ofthe canonical equations (1). 7’Theform oi‘seems heretoappear outofthin air.Inthefollowing paragraph wewillseehow theideaofusing thisform arose from optics. 236 44:Theintegral invariant ofPoincaré—Cartan PROOF. Thedifferential oftheform pdq—Hdtisequal to " an 6H d 1= d1' dI"——_d ' —% ' . w (pAq apt p,/\dt aqidq,Adt) +- Itisclear from thisexpression that thematrix ofthe2-form dwlinthe coordinates p,q,thastheform 0—Ein A: .5 0Hq, -H,-an 0 where E 1, H_8HH_aH= -ii"? ""a_ 1 P (I mm (verify this!). Therank ofthismatrix is2n(theupper left2n-corner isnon-degenerate); therefore, dw‘ isnonsingular. Itcanbeveriﬁed directly that thevector (—H,,, Hp,1)isaneigenvector ofAwith eigenvalue 0(doitl).This means thatitgives thedirection ofthevortex lines oftheform pdq—Hdt.Butthe vector (—Hq, Hp,1)isalsothevelocity vector ofthephase ﬂow of(1).Thus theintegral curves of(1)arethevortex lines oftheform pdq—Hdt,aswas tobeshown. El DAtheorem ontheintegral invariant of Poincaré—Cartan Wenow apply Stokes’ lemma. Weobtain thefundamental Theorem. Suppose that thetwocurves yland112encircle thesame tube of phase trajectories of(1). Then theintegrals oftheform pdq—Hdtalong them arethesame: f£pdq—Hdt= dipdq-Hdt. Yi Y2 Theform pdq—Hdtiscalled theintegral invariant ofPoincaré—Cartan.7“ PROOF. Thephase trajectories arethevortex lines oftheform pdq—Hdt, andtheintegrals along closed curves contained inthesame vortex tube are thesame byStokes’ lemma. Cl "4lnthecalculus ofvariations pdq~Hdtiscalled Hilbert’s invariant integral. Z37 9:Canonical formalism ll sf;*1 P [0 I [1 >t Figure 183 Poincaré’s integral invariant Wewillconsider, inparticular, curves consisting ofsimultaneous states, i.e.,lying intheplanes t=const (Figure 183). Along such curves, dt=0 and §pdq—Hdt=§pdq.From thepreceding theorem weobtain the important: Corollary 1.The phase ﬂow preserves theintegral oftheform pdq = p,dql+ +p,,dq,,onclosed curves. PROOF. Letgig:R2"—>R2"bethetransformation ofthephase space (p,q) realized bythephase ﬂow from time tototl(i.e.,gj0(p0, qo)isthesolution tothecanonical equations (1)with initial conditions p(t0) =po,q(t0) =qo). Let7:beanyclosed curve inthespace R2"clR2"*‘ (t=to).Then gay isaclosed curve inthespace R2"(t=t1),contained inthesame tube of phase trajectories inlR2"*‘. Since dt=0onyandongig):weﬁnd bythe preceding theorem that§,pdq=f,,_pdq,aswastobeshown. El.ll,Or The form pdqiscalled Poincaré’s relative integral invariant. Ithasa simple geometric meaning. Let0beatwo—dimensional oriented chain and y=50.Then, byStokes’ formula, weﬁnd §pdq=JT dpAdq. )1 0’ Thus wehave proved theimportant: Corollary 2.Thephase ﬂow preserves thesumoftheoriented areas ofthe projections ofasurface onto thencoordinate planes (pi,q,-): H_dpAdq=~U dpAdq. a 9,2,0 Inother words, the2-form wz=dpAdqisanabsolute integral invariant ofthephase ﬂow. EXAMPLE. Forn=1,ofisarea, andweobtain Liouville’s theorem: the phase ﬂow preserves area. 238 44:Theintegral invariant ofPoincaré—Cartan ECanonical transformations Letgbeadifferentiable mapping ofthephase space R2"={(p,q)}toR2". Deﬁnition. Themapping giscalled canonical, oracanonical transformation, ifgpreserves the2-form col=2dp,Adqi. Itisclear from theargument above that thisdeﬁnition canbewritten inanyofthree equivalent forms: 1.g*a)2 =col(gpreserves the2-form Zdp,Adq,-); 2.H,of=Hg,of,Va(gpreserves thesum oftheareas oftheprojections ofanysurface); 3.§,pdq=§,,pdq(theform pdqisarelative integral invariant ofg). PROBLEM. Show thatdeﬁnitions (1)and(2)areequivalent to(3)ifthedomain ofthemap in question isasimply connected region inthephase space R2"; inthegeneral case 3=>2<=~l. Thecorollaries above cannowbeformulated as: Theorem. Thetransformation ofphase space induced bythephase ﬂow is canonical.-'5 Letg:R2"—>R2"beacanonical transformation: gpreserves theform 012. Then galsopreserves theexterior square ofwzz g*(w2 Awz)=wz/\wz and g*(co2)" =(w2)". Theexterior powers oftheform Zdp,Adq,-areproportional totheforms co‘=Zdp,/\dpjAdq,/\dqj, t<; 0)“: Z dPiiA /\dplkAdqi.A Adqiiv il<"'<ik Thus wehave proved Theorem. Canonical transformations preserve the integral invariants 4 2ncu . Geometrically, theintegral oftheform oi“isthesum oftheoriented volumes oftheprojections onto thecoordinate planes (p,-I,...,p,-k,q,-1,...,qjk). Inparticular, 012"isproportional tothevolume element, andweobtain: Corollary. Canonical transformations preserve thevolume element inphase space: thevolume ofgDisequal tothevolume ofD,foranyregion D. 75Theproof ofthistheorem which ispresented intheexcellent book byLandau andLifshitz (Mechanics, Pergamon, Oxford, 1960) isincorrect. 239 9:Canonical formalism Inparticular, applying thistothephase ﬂow weobtain Corollary. The phase ﬂow (1)has asintegral invariants theforms w2,o)“,...,w2". Thelastofthese invariants isthephase volume, sowehave again proved Liouville’s theorem. 45Applications oftheintegral invariant of Poincaré—Cartan Inthisparagraph weprove thatcanonical transformations preserve theform ofHamilton’s equations. thataﬁrstintegral ofHamilton's equations allows ustoreduce immediately theorder ofthesystem bytwoandthatmotion inanatural lagrangian system proceeds along geodesics oftheconﬁguration space provided with acertain riemannian metric. AChanges ofvariables inthecanonical equations Theinvariant nature oftheconnection between theform pdq—Hdtand itscurl lines gives risetoaway ofwriting theequations ofmotion inany system of2n+1coordinates inextended phase space {(p,q,t)}. p,q,t X1.---X2»+1 /”\ Figure 184 Change ofvariables inHamilton’s equations\\\\ //// Let(x1,...,x2,,, 1)becoordinate functions insome chart ofextended phase space (considered asamanifold M2'”1,Figure 184). Thecoordinates (p,q,t)canbeconsidered asgiving another chart onM.The form wl= pdq—Hdtcanbeconsidered asadifferential 1-form onM.Invariantly associated (notdepending onthechart) tothisform isafamily oflines onM— thevortex lines. Inthechart (p,q,t),these lines arerepresented asthetra- jectories ofthephase ﬂow (1) dp__8H dq_6H at_aq at'ap with hamiltonian function H(p, q,t). Suppose thatinthecoordinates (x1,...,xl,, 1)theform w‘iswritten as + "' + X2"+1dX2n+1. 240.....i._u-ct.a.a..:.‘i 45:Applications oftheintegral invariant ofPoincaré—Cartan Theorem. Inthechart (xi).thetrajectories of(1)arerepresented bythevortex lines oftheform ZX1dx1. PROOF. Thecurllines oftheforms ZX1dx,-andpdq—Hdtaretheimages intwodifferent charts ofthevortex lines ofthesame form onM.Butthe integral curves of(1)arethevortex lines ofpdq—Hdt.Thus, their images inthechart (x1)arethevortex lines oftheform ZX1dxi. l:| Corollary. Let(P1,...,P,,; Q1,..., Q,,;T)beacoordinate system onthe extended phase space (p,q,t)andK(P,Q T)andS(P,Q, T)functions such that pdq—Hdt=PdQ—KdT+dS (theleft-andright-hand sides areforms onextended phase space). Then thetrajectories ofthephase ﬂow (I)arerepresented inthechart (P,Q,T)bytheintegral curves ofthecanonical equations (2) g£__?l<_ Q-6541- aQ dT_6P' PROOF. Bythetheorem above, thetrajectories of(1)arerepresented bythe vortex lines oftheform PdQ —KdT+dS.ButdShasnoinﬂuence on thevortex lines (since ddS=0).Therefore, theimages ofthetrajectories of(1) arethevortex lines oftheform PdQ—KdTAccording toSection 44,C, thevortex lines ofsuch aform areintegral curves ofthecanonical equations (2) I] Inparticular, letg:R2"—>R2"beacanonical transformation ofphase space taking apoint with coordinates (p,q)toapoint with coordinates (P,Q).Thefunctions P(p,q)andQ(p,q)canbeconsidered asnewco- ordinates onphase space. Theorem. lnthenew coordinates (P,Q)thecanonical equations (l)have thecanonical form“ 1,, t2__@1 Q15at‘ ao dt_6P withthesame hamiltonian function: K(P, Q,t)=H(p, q,t). 7°Insome textbooks theproperty ofpreserving thecanonical form ofHamilton's equations is taken asthedeﬁnition ofacanonical transformation. This deﬁnition isnotequivalent tothe generally accepted onementioned above. Forexample, thetransformation P=2p,Q=q, which isnotcanonical byourdeﬁnition, preserves thehamiltonian form oftheequations of motion. Thisconfusion appears evenintheexcellent textbook byLandau andLifshitz (Mechanics, Oxford, Pergamon, 1960); inSection 45ofthisbook theyshow thatevery transformation which preserves thecanonical equations iscanonical inoursense. 241 9:Canonical formalism 4 W P1.qt I pUv q0 P Figure 185 Closedness oftheform pdq—PdQ PROOF. Consider the1-form pdq-—PdQonR2". Foranyclosed curve *1» wehave (Figure 185) dipdq—PdQ= t£pdq— t£PdQ=0 since giscanonical. Therefore, f:(',1:1', pdq—PdQ=Sdoes notdepend on thepath ofintegration butonly ontheendpoint (p1,q1)(foraﬁxed initial point (p11,q1,)). Thus dS=pdq —PdQ. Consequently, intheextended phase space, wehave pdq—Hdt=PdQ-Hdt+dS. Thus, thetheorem above isapplicable, and(2)istransformed to(3). Cl PROBLEM. Letg(!)I R“—>R2"beacanonical transformation ofphase space depending onthe parameter t,g(t)(p, q)=(P(p, q,t),Q(p, q,t)).Show thatinthevariables P,Q,tthecanonical equations (l)have thecanonical form with newhamiltonian function OS . K(P1Q1t)= qt where Phil S(Pi.qi.t) =f pdq—PdQ P0410 BReduction oforder using theenergy integral Suppose nowthatthehamiltonian function H(p, q)does notdepend ontime. Then thecanonical equations (1)have aﬁrstintegral: H(p(t), q(t)) =const. Itturns outthatbyusing thisintegral wecanreduce thedimension (2n+1) oftheextended phase space bytwo, thereby reducing theproblem toin- tegration ofasystem ofcanonical equations ina(2n—1)-dimensional space. Weassume that(insome region) theequation h=H(p1,...,p,,;q1,...,q,,) canbesolved forp1: pl = Q9 h)! 242 45:Applications oftheintegral invariant ofPoincare-Cartan whereP =(p2,...,p,,);Q =(q2,...,q,,); T= —q1.Then weﬁnd pdq— Hdt= PdQ— KdT—d(Ht)+ tdH. Now letybeanintegral curve ofthecanonical equations (1)lying onthe 2n-dimensional surface H(p, q)=hinR2” 1.Then yisavortex lineofthe formp dq—Hdt(Figure 186). Weproject theextended phase space lR2"+‘ = {(p,q,t)}onto thephase space R2"={(p,q)}.The surface H=hispro- jected onto a(2n—1)-dimensional manifold M2"_‘:H(p, q)=hinR2", andyisprojected toacurve )7lying onthissubmanifold. The variables P,Q,Tform local coordinates onM2"_1. Q AMZn—l "F ' H=h P >1‘ Figure 186 Lowering theorder ofahamiltonian system PROBLEM. Show thatthecurve 9isavortex lineofthe form pdq=PdQ—KdTonM2“ 1. Hint. d(Ht) does notaffect thevortex lines, anddHiszeroonM. Butthevortex lines ofPdQ——KdTsatisfy Hamilton’s equations (2). Thus wehave proved Theorem. Thephase trajectories oftheequations (1)onthesurface M2"", H=h,satisfy thecanonical equations dp1_r'iK dq1_ 5K (i_2 n) dqi—@q. dq1— 611.’ f ’ where thefunction K(p2, ...,p,,;ql,...,q,,;T,h)isdeﬁned bytheequation H(K,p2,...,p,,;—-'l:q2,...,q,1)= h. CTheprinciple ofleast action inphase space Intheextended phase space {(p,q,t)},weconsider anintegral curve ofthe canonical equations (1)connecting thepoints (p1,,q0,to)and(p1,q1,t1). Theorem. Theintegral Ipdq—Hdthasyasanextremal under variations ofyforwhich theends ofthecurve remain inthen-dimensional subspaces (I=t0.q=q0)<md (I=ti.1|=Q1)- PROOF. Thecurve yisavortex lineoftheform pdq~Hdt(Figure 187). Therefore, theintegral ofpdq—Hdtover an“inﬁnitely small parallelogram 243 9:Canonical formalism P 7! It Iq— 1»1 /10.00 7 >I Figure 187 Principle ofleast action inphase space passing through thevortex direction” isequal tozero. Inother words, the increment 1,.-I,pdq—Hdtissmall toahigher order incomparison with thedifference ofthecurves yandy’,aswastobeshown. Ifthisargument does notseem rigorous enough, itcanbereplaced bythe computation @f<p<i—H>di= j(<i@p+i><t<i-ii-,§<§p-‘;%@q)di 1 5H OH = 5 I '———5—' ——5:|dt.""@+ .i(q ﬁplp (p+@q)q Weseethat theintegral curves ofHamilton’s equations aretheonly extremals oftheintegral Ipdq—Hdtintheclass ofcurves ywhose ends lieinthen-dimensional subspaces (t=to,q=qo)and (t=t1,q=q1) ofextended phase space. El Remark. Theprinciple ofleast action inHamilton’s form isaparticular caseoftheprinciple considered above. Along extremals, wehave 11.0: 1| It J.pdq—Hdt=J.(pq—H)dt=J‘Ldt lo.qo lo lo (since thelagrangian Landthehamiltonian HareLegendre transforms ofoneanother). Now let7‘(Figure I88)betheprojection oftheextremal yonto theq.tplane. Toanynearby curve ?' connecting thesame points (to.qo)and((1,q1)intheq.tplane weassociate acurve 7’inthe Pl if//ll///4 ///////.. q 0 7 ljq] !0~q0 ._.. »r Figure 188 Comparison curves fortheprinciples ofleast action intheconﬁguration andphase spaces 244 45:Applications oftheintegral invariant ofPoincaré—Cartan phase space (p,q,t)bysetting p=at/aq. Then, along ;",too, pdq—Hdt= Ldt.But bythetheorem above, (3prlq*Htlt=0foranyvariation curve 7(with boundary conditions (t=t1,.q :qo)and(t=t1,q=q1).Inparticular. thisistrueforvariations ofthe special form taking ~,-to7’.Thus 7isanextremal offLdt,aswastobeshown. Inthetheorem above weareallowed tocompare )1with asigniﬁcantly wider class ofcurves y’than inHamilton’s principle: there arenorestrictions placed ontherelation ofpwith q.Surprisingly, onecanshow thatthetwo principles arenevertheless equivalent: anextremal inthenarrower class of variations (p=6L/0(1) isanextremal under allvariations. The explana- tionisthat, forﬁxed q,thevalue p=at/aq isanextremal ofpq—H(cf.the deﬁnition oftheLegendre transform, Section 14). DTheprinciple ofleast action inthe Maupertuis—Euler—Lagrange~Jac0bi form Suppose now thatthehamiltonian function H(p,q)does notdepend ontime. Then H(p, q)isaﬁrstintegral ofHamilton’s equations (1).Weproject the surface H(p, q)=hfrom theextended phase space {(p,q,t)}tothespace {(p,q)}.Weobtain a(2n—1)-dimensional surface H(p,q)=hinR2”, which wealready studied insubsection Bandwhich wedenoted byM2"‘1. Thephase trajectories ofthecanonical equations (1)beginning onthe surface M2"'1 lieentirely inM2” 1.They arethevortex lines oftheform pdq=PdQ~KdT(inthenotation ofB)onM2"'1. Bythetheorem in subsection C,thecurves (1)onM2"'1 areextremals forthevariational principle corresponding tothisform. Therefore, wehave proved Theorem. Ifthehamiltonian function H=H(p, q)does notdepend ontime, thenthephase trajectories ofthecanonical equations (1)lying onthesurface M2"‘1:H(p, q)=hareextremals oftheintegral jpdqintheclass of curves lying onM2"‘1andconnecting thesubspaces q=qoandq=q1. Wenow consider theprojection onto theq-space ofanextremal lying onthesurface M2"“1:H(p, q)=h.This curve connects thepoints qoand q1.Letybeanother curve connecting thepoints qoandq1(Figure 189). Thecurve yistheprojection ofsome curve itonM2"“1. Speciﬁcally, we P H(P.q) =/1 T ql q iﬁaqo 4 4’ Figure 189 Maupertuis‘ principle 245 9:Canonical formalism parametrize ybyr,a3r3b,y(a)=qo,y(b)=q1.Then atevery point q of)1there isavelocity vector q=dy(r)/dt, andthecorresponding momentum p=at/aq. Iftheparameter Tischosen sothatH(p, q)=h,then weobtain acurve $1:q=y(t), p=at/aq onthesurface M2"‘1.Applying thetheorem above tothecurve $1onM2"‘1,weobtain Corollary. Among allcurves q=y(t)connecting thetwopoints qoandq1on theplane qandparametrized sothatthehamiltonian function hasaﬁxed value H(0L/dq, q)=h,thetrajectory oftheequations ofdynamics (1)is anextremal oftheintegral of“reduced action” Ipdq =Ipq dr=I1%(r)q(r)dt. )1 7' Yq This isalsotheprinciple ofleast action ofMaupertuis (Euler—Lagrange— Jacobi)."'7 Itisimportant tonote thattheinterval a3T3bparametrizing thecurve yisnotﬁxed andcanbedifferent fordifferent curves being com- pared. Ontheother hand, theenergy (the hamiltonian function) must be thesame. Wenote alsothattheprinciple determines theshape ofatrajectory butnotthetime: inorder todetermine thetime wemust usetheenergy constant. Theprinciple above takes aparticularly simple form inthecasewhen the system represents inertial motion onasmooth manifold. Theorem. Apoint mass conﬁned toasmooth riemannian manifold moves along geodesic lines (i.e.,along extremals ofthelength Ids). PROOF. Inthiscase, ldsz dL_ dS2 Therefore, inorder toguarantee aﬁxed value ofH=h,theparameter must bechosen proportional tothelength dr=ds/,/ 2h.The reduced action integral isthen equal to I61: qdr =I,/2hds =,/2h>I ds; I’ 7 I’ therefore, extremals aregeodesics ofourmanifold. El Inthecase when there isapotential energy, thetrajectories oftheequa- tions ofdynamics arealsogeodesics inacertain riemannian metric. 77“Inalmost alltextbooks, even thebest, thisprinciple ispresented sothatitisimpossible to understand.” (K.Jacobi, Lectures onDynamics, 1842-1843). Idonotchoose tobreak with tradition. Averyinteresting “proof ”ofMaupertuis’ principle isinSection 44ofthemechanics textbook ofLandau andLifshitz (Mechanics, Oxford, Pergamon, 1960). 246 45:Applications oftheintegral invariant ofPoincare-Cartan Letdszbeariemannian metric onconﬁguration space which gives the kinetic energy (sothat T=%(ds/dr)2). Lethbeaconstant. Theorem. Intheregion ofconﬁguration space where U(q) <hwedeﬁne ariemannian metric bytheformula dp=,/h—U(q)ds. Then thetrajectories ofthesystem with kinetic energy T=§(ds/dr)2, potential energy U(q),andtotal energy hwillbegeodesic lines ofthemetric dp. PROOF. Inthis case L=T—U,H=T+U,and (at/aqiq =2T= (ds/dr)2 =2(h— U).Therefore, inorder toguarantee aﬁxed value of H=h,theparameter rmust bechosen proportional tolength: dr= ds/,/2(h —U).Thereduced action integral willthen beequal to Iy‘;_§q.i.=Iy,/r<7.T>.i.=ﬂIy.i,.. ByMaupertuis’ principle, thetrajectories aregeodesics inthemetric dp, aswastobeshown. E] Remark 1.Themetric dpisobtained from dsbya“stretching” depending onthepoint qbutnotdepending onthedirection. Therefore, angles inthe metric dparethesame asangles inthemetric ds.Ontheboundary ofthe region U$hthemetric dphasasingularity: thecloser wecome tothe boundary, thesmaller thep-length becomes. Inparticular, thelength ofany curve lying intheboundary (U=h)isequal tozero. Remark 2.Iftheinitial andendpoints ofageodesic yaresufficiently close, then theextremum oflength isaminimum. This justiﬁes thename “principle ofleast action.” Ingeneral, anextremum oftheaction isnotnecessarily a minimum, asweseebyconsidering geodesics ontheunitsphere (Figure 190). Every arcofagreat circle isageodesic, butonly those with length lessthan 1: areminimal: thearcNS'M isshorter than thegreat circle areNSM. 3s Figure 190 Non-minimal geodesic 247 9:Canonical formalism Remark 3.Ifhislarger than themaximum value ofUontheconﬁguration space, then themetric dphasnosingularities; therefore, wecanapply topological theorems about geodesics onriemannian manifolds tothestudy ofmechanical systems. Forexample, weconsider thetorus T2with some riemannian metric. Among allclosed curves onT2making mrotations Figure 191 Periodic motion ofadouble pendulum around theparallel andnaround themeridian, there exists acurve ofshortest length (Figure 191). This curve isaclosed geodesic (foraproof seebooks onthecalculus ofvariations or“Morse theory”). Ontheother hand, the torus T2istheconﬁguration space ofaplanar double pendulum. Therefore, Theorem. Foranyintegers mandnthere isaperiodic motion ofthedouble pendulum under which onesegment makes mrotations while theother segment makes nrotations. Furthermore, such periodic motions exist foranysufficiently large values oftheconstant h(hmust belarger than thepotential energy atthehighest position). Asalastexample weconsider arigid body fastened atastationary point andlocated inanarbitrary potential ﬁeld. Theconﬁguration space (SO(3)) isnotsimply connected: there exist non-contractible curves init.Theabove arguments imply Theorem. Inanypotential force ﬁeld, there exists atleast oneperiodic motion ofthebody. Furthermore, there exist periodic motions forwhich thetotal energy hisarbitrarily large. 46Huygens’ principle Thefundamental notions ofhamiltonian mechanics (momenta, thehamiltonian function H, theform pdq—HdtandtheHamilton-Jacobi equations, allofwhich wewillbeconcerned with below) arose bythetransforming ofseveral verysimple andnatural notions ofgeometric optics. guided byaparticular variational principle—that ofFermat, intogeneral variational principles (and inparticular intoHamilton’s principle ofstationary action, 6Ldt=O). 248 46:Huygens’ principle AWave fronts Weconsider brieﬁy7 8thefundamental notions ofgeometric optics. According totheextremal principle ofFermat, light travels from apoint qotoapoint q,intheshortest possible time. Thespeed ofthelight candepend both onthe point q(an“inhomogeneous medium”) and onthedirection oftheray (inan“anisotropic medium,” such asacrystal). The characteristics ofa medium canbedescribed bygiving asurface (the“indicatrix”) inthetangent space ateach point q.Todothis,wetakeinevery direction thevelocity vector ofthepropagation oflight atthegiven point inthegiven direction (Figure Q $5%?$48? Figure 192 Ananisotropic, inhomogeneous medium I T’ $- 1 \. /¢q0(,, ',,(i> \ to / \ I’ 4- 4. I’ ¢‘q0(t+s) <l>,,0(r) } r—'— q, "0 Figure 193 Envelope ofwave fronts‘1’,,,(s) Now lett>0.Welook atthesetofallpoints qtowhich light from agiven point qocantravel intime lessthan orequal tot.Theboundary ofthisset, (I>q0(t), iscalled thewavefront ofthepoint qoafter time tandconsists ofpoints towhich light cantravel intime tandnotfaster. There isaremarkable relation, discovered byHuygens, between thewave fronts corresponding todiﬂerent values oft.(Figure 193) 78Wewillnotpursue rigor here, andwillassume thatalldeterminants aredifferent from zero, etc.Theproofs ofthesubsequent theorems donotdepend onthesemi-heuristic arguments of thisparagraph. 249 91Canonical formalism Huygens’ theorem. Let<Dq0(t) bethewave front ofthepoint qoafter time t. Forevery point qofthisfront, consider thewave front after time s,(Dq(s). Then thewavefront ofthepoint qoafter time s+t,<D%(s +t),willbethe envelope ofthefronts <Dq(s), qe%(t). Pnoor. Letq,,,eq0(t +s).Then there exists apath from qotoq,,,along which thetime oftravel oflight equals t+s,andthere isnone shorter. We look atthepoint q,onthispath, towhich light travels intime t.Noshorter path from qotoq,canexist; otherwise, thepath qOq,,, would notbethe shortest. Therefore, thepoint q,liesonthefront @q0(t)o Inexactly thesame waylight travels thepath q,q,,, intime s,andthere isnoshorter path from q,toq,+,. Therefore, thepoint q,,,liesonthefront ofthepoint q,attime s, (D_“(s). Wewillshow that thefronts <Dq:(s) and<Dqo(t +s)aretangent. In fact, ifthey crossed each other (Figure 194), then itwould bepossible to reach some points of<D,|0(t +s)from q,intime lessthan s,andtherefore from qointime lessthan s+t.This contradicts thedeﬁnition of<D,,0(t +s); andsothefronts @qt(S) and<Dq0(t +s)aretangent atthepoint q,,,, aswas tobeproved. E] O W q, qo qr rs (bq(S) ¢qU(s +1) I Figure 194 Proof ofHuygens‘ theorem The theorem which hasbeen proved iscalled Huygens’ principle. Itis clear that thepoint qocould bereplaced byacurve, surface, or,ingeneral, byaclosed set,thethree-dimensional space {q}byanysmooth manifold, andpropagation oflight bythepropagation ofanydisturbance transmitting itself “locally.” Huygens’ principle reduces totwodescriptions oftheprocess ofprop- agation. First, wecantrace therays, i.e.,theshortest paths ofthepropagation oflight. Inthiscase thelocal character ofthepropagation isgiven bya velocity vector q.Ifthedirection oftherayisknown, then themagnitude ofthevelocity vector isgiven bythecharacteristics ofthemedium (the indicatrix). Ontheother hand, wecantrace thewave fronts. Assuming thatweare given ariemannian metric onthespace {q},wecantalkabout thevelocity ofmotion ofthewave front. Welook, forexample, atthepropagation of light inamedium ﬁlling ordinary euclidean space. Then onecancharacterize themotion ofthewave front byavector pperpendicular tothefront, which willbeconstructed inthefollowing manner. 250 46:Huygens" principle Direction oftheray Q Ra fl p=gradS (I0 y Direction ofmotion ofthefront Front Sqtfql If Figure 195 Direction ofarayanddirection ofmotion ofthewave front Forevery point qowedeﬁne thefunction S,,O(q) astheoptical length of thepath from qotoq,i.e.,theleast time ofthepropagation oflight from qo toq.Thelevel set{qzSq0(q) =t}isnothing other than thewave front <D,,O(t) (Figure 195). The gradient ofthefunction S(inthesense ofthemetric mentioned above) isperpendicular tothewave front andcharacterizes the motion ofthewave front. Inthisconnection, thebigger thegradient, the slower thefront moves. Therefore, Hamilton called thevector _as thevector ofnormal slowness ofthefront. Thedirection ofthe rayqandthedirection ofmotion ofthe front pdonot coincide inananisotropic medium. However, theyarerelated tooneanother byasimple relationship, easily derived from Huygens’ principle. Recall that thecharacteristics ofthemedium areatevery point described bya surface ofvelocity vectors oflight—the indicatrix. Deﬁnition. Thedirection ofthehyperplane tangent totheindicatrix atthe point viscalled conjugate tothedirection v(Figure 196). Theorem. Thedirection ofthewave front <Dq0(t) atthepoint q,isconjugate tothedirection oftherayq. PROOF. Welook (Figure 197)atpoints q,oftherayq0q,, 0g1.’3t.Take 8 very small. Then thefront (Dq!_£(s) differs byquantities oforder O(s2) from theindicatrix atthepoint q,,contracted bys.ByHuygens’ principle, this front <Dq[_c(a)istangent tothefront (Dq0(t) atthepoint q,.Passing tothelimit ass—>0,weobtain thetheorem. El U D Conjugate direction Figure 196 Conjugate hyperplane 251 9:Canonical formalism ¢’q.(') lndicatrix of thepoint ll; Direction oftheray WI Direction ofmotion . P ofthefront~@ an @111‘-—e (G) Front tlnmfll Figure 197 Conjugacy ofthedirection ofawave andofthefront Iftheauxiliary metric used todeﬁne thevector pischanged, thenatural velocity ofthemotion ofthefront, i.e.both themagnitude anddirection of thevector p,will bechanged. However, thediﬁerential form pdq=dS onthespace {q}=R3isdeﬁned inaway which isindependent ofthe auxiliary metric; itsvalue depends only onthechosen fronts (orrays). Onthe hyperplane conjugate tothevelocity vector ofaray,thisform isequal to zero, anditsvalue onthevelocity vector isequal to1.79 BTheoptical-mechanical analogy Wereturn now tomechanics. Here thetrajectories ofmotion arealso extremals ofavariational principle, andonecanconstruct mechanics as thegeometric optics ofamany-dimensional space, asHamilton did;wewill notdevelop thisconstruction infulldetail, butwillonlyenumerate those optical concepts which ledHamilton tobasic mechanical concepts. Optics Mechanics Optical medium Extended conﬁguration space {(q,t)} Fermat’s principle Hamilton’s principle 5jLdt=0 Rays Trajectories q(t) Indicatrices Lagrangian L Normal slowness vector p Momentum p ofthefront Expression ofpinterms of Legendre transformation thevelocity oftheray,q 1-form pdq 1-form pdq—Hdt 79Inthisway,thevectors pcorresponding tovarious fronts passing through agiven point arenot arbitrary, butaresubject toonecondition: thepermissible values ofpﬁllahypersurface in {p}-space which isdual totheindicatrix ofvelocities. 252 46:Huygens’ principle Theoptical length ofthepath Sqn(q) andHuygens’ principle have notyet been used. Their mechanical analogues aretheaction function and the Hamilton—J acobi equation, towhich wenow turn. CAction asafunction ofcoordinates andtime Deﬁnition. Theaction function S(q,t)istheintegral SqQ,tQ(q7 t)= V along theextremal yconnecting thepoints (qo,to)and(q,t). Inorder forthisdeﬁnition tobecorrect, wemust takeseveral precautions: wemust require thattheextremals going from thepoint (q0,to)donotinter- sectelsewhere, butinstead form aso-called “central ﬁeld ofextremals” (Figure 198). More precisely, weassociate toevery pair(q,,,t)apoint (q,t) which istheendoftheextremal with initial condition q(0)=qo,q(0) =qo. Wesaythat anextremal yiscontained inacentral ﬁeld ifthemapping (q,,t)—>(q,t)isnondegenerate (atthepoint corresponding totheextremal yunder consideration, andtherefore insome neighborhood ofit). 4 [.1] !0.lI0 t Figure 198 Acentral ﬁeldofextremals Itcanbeshown thatforit—toIsmall enough theextremal yiscontained in 80acentral ﬁeld. Wenow look atasufficiently small neighborhood oftheendpoint (q,t) ofourextremal. Every point ofthisneighborhood isconnected to(qo,to) byaunique extremal ofthecentral ﬁeld under consideration. This extremal depends differentiably ontheendpoint (q,t).Therefore, intheindicated neighborhood theaction function iscorrectly deﬁned S,,O,,0(q, t)=ILdt. Y Ingeometric optics wewere looking atthedifferential oftheoptical length ofapath. Itisnatural here tolook atthedifferential oftheaction function. 8°PROBLEM. Show thatthisisnottrueforlarger —to.Hint. ii:—q(Figure 199). 253 9:Canonical formalism ‘I I Figure 199 Extremal withafocal point which isnotcontained inanycentral ﬁeld Theorem. Thediﬂerential oftheaction function (for aﬁxed initial point) is equal to dS=pdq—Hdt where p=at/aq andH=pq~—Laredeﬁned withthehelpoftheterminal velocity qofthetrajectory y. PROOF. Weliftevery extremal from (q,t)-space totheextended phase space {(p,q,t)},setting p=at/aq, i.e.,replacing theextremal byaphase trajectory. Wethen getann+1-dimensional manifold intheextended phase space consisting ofphase trajectories, i.e.,characteristic curves oftheform pdq—Hdt.Wenow givetheendpoint (q,t)anincrement (Aq,At),and consider thesetofextremals connecting (qo,to)with points ofthesegment q+0Aq,t +0At,0 sl931(Figure 200). Inphase space wegetaquadrangle acomposed ofcharacteristic curves oftheform pdq—Hdt,theboundary ofwhich consists oftwophase trajectories ylandyz,asegment ofacurve at lying inthespace (q=qo,t=to),andasegment ofacurve [3projecting tothesegment (Aq,At). Since aconsists ofcharacteristic curves ofthe form pdq—Hdt,wehave 0=Hd(pdq—Hdt)=jpdq—Hdt a 60 =j —j +j—jpdq—Hdt. vi 72 B 1 But,onthesegment at,wehave dq=0,dt=0.Onthephase trajectories yland yz,pdq —Hdt =Ldt(Section 45C). So,thedifference jn—[,1pdq —Hdt P 72 . At,Aq q.2%; to»qo "‘7,_ ,1. Figure 200 Calculation ofthedifferential oftheaction function 254 46:Huygens’ principle isequal totheincrease oftheaction function, andweﬁnd Lpdq—Hdt=S(q+Aq,t+At)—S(q,t). Ifnow Aq~>O,At—>0,then Lpdq—Hdt =pAq—HAt +o(At, Aq) which proves thetheorem. U Theform pdq—Hdtwasformerly introduced tousartiﬁcially. Wesee now, bycarrying outtheoptical-mechanical analogue, that itarises from examining theaction function corresponding totheoptical length ofapath. DTheHamilton—Jacobi equation Recall thatthe“ vector ofnormal slowness p”cannot bealtogether arbitrary: itissubject toonecondition, pq=1,following from Huygens’ principle. Ananalogous condition restricts thegradient oftheaction function S. Theorem. Theaction function satisﬁes theequation ES GS(1) €t+H(a—q,q,t) =0. This nonlinear ﬁrst-order partial differential equation iscalled the Hamilton—Jacobi equation. PROOF. Itissuﬁicient tonotice that, bytheprevious theorem, 6S 6S_=-11 =___ at (P,q.t)paq E1 Therelation justestablished between trajectories ofmechanical systems (“rays”) andpartial differential equations (“wave fronts”) canbeused in twodirections. First, solutions ofEquation (1)canbeused forintegrating theordinary differential equations ofdynamics. Jacobi’s method ofintegrating Hamilton’s canonical equations, presented inthenext section, consists ofjustthis. Second, therelation oftherayandwave points ofview allows oneto reduce integration ofthepartial diﬁerential equations (1)tointegration ofahamiltonian system ofordinary differential equations. Letusgointo thisinalittle more detail. FortheHamilton—Jacobi equation (1),theCauchy problem is as as(2) S(q.I0)=$0(q) 5+H(56.q,t)=0- 255 9:Canonical formalism Inorder toconstruct asolution tothisproblem, welook atthehamiltonian system _ 6H __6H P=-EH ‘I-55- Weconsider theinitial conditions (Figure 201): 5S0 ‘I(fo) —‘I0 Pftol '-T] qu- Thesolution corresponding tothese equations isrepresented in(q,t)-space bythecurve q=q(t), which istheextremal oftheprinciple 6jLdt=0 (where thelagrangian L(q,q,t)istheLegendre transformation with respect topofthehamiltonian function H(p, q,t)).This extremal iscalled the characteristic ofproblem (2),emanating from thepoint qo. Ifthevalue t1issufficiently close tota,then thecharacteristics emanating from points close toqodonotintersect forto3t3t1,lq—qol<R. Furthermore, thevalues ofqoandtcanbetaken ascoordinates forpoints intheregion |q—q0,,,| <R,to3t3t1(Figure 201). q 4-‘Q ‘lollllllll\\\w||‘$25!’/3"§\ —— . > to I][2 [3 Figure 201 Characteristics forasolution ofCauchy’s problem fortheHamilton- Jacobi equation Wenow construct the“action function with initial condition S0”: A<3) son=sod.)+jLo.q.oat Q0-lo (integrating along thecharacteristic leading toA). Theorem. Thefunction (3)isasolution ofproblem (2). PROOF. Theinitial condition isclearly fulﬁlled. ThefactthattheHamilton- Jacobi equation issatisﬁed isveriﬁed justasinthetheorem ondifferentials ofaction functions (Figure 202). ByStokes’ lemma, jn—jvl+jg—j,pdq —Hdt=0.Butonoz,Hdt =0andp=550/dq, SO jndq—Hdt=jvdq =jd$@=$0010+Aq)—$0010)- 256 P ll\\*..46:Huygens’ principle '72O‘. i A-l-AA Otl 10,470 IvI Figure 202 Theaction function asasolution oftheHamilton-Jacobi equation Further, 7,and"/2arephase trajectories, so Sojpdq—Hdr=j Ldt. 3.‘: ..‘2 jpdq —Hdr=[S0(q0 +Aq)+ILtit]—[S0(q0) +jLdt:| H 2 ,2t. =S(A +AA) —S(/1). ForAt,Aq—+0,wegetas/at =—H, 68/0q =p,which proves thetheorem. E] PROBLEM. Show theuniqueness ofthe solution toproblem (2). Hint. Differentiate Salong thecharacteristics. PROBLEM. Solve theCauchy problem (2)for pl q2 n=- s=-.2 °2 PROBLEM. Draw agraph ofthe multiple-valued “functions” S(q)andp(q)fort =:3(Figure 201). ANSWER. Cf.Figure 203. S ___ ,1, P >q Figure 203 Atypical singularity ofasolution oftheHamilton—Jacobi equation 257l<1 9:Canonical formalism Thepoint ofself-intersection ofthegraph ofScorresponds onthegraph ofptotheMaxwell line:theshaded areas areequal. Thegraph ofS(q,t)hasasingularity called aswallows tailatthe point (0,:2). 47TheHami1ton~Jacobi method forintegrating Hamilton’s canonical equations Inthisparagraph wedeﬁne thegenerating function ofafreecanonical transformation. TheideaoftheHamilton—Jacobi method consists ofthefollowing. Under canonical changes ofcoordinates, thecanonical form oftheequations of motion ispreserved, asisthehamiltonian function (Section 45A). Therefore, ifwesucceed inﬁnding acanonical transformation which reduces the hamiltonian function toaform such that thecanonical equations canbe integrated, then wecanalso integrate theoriginal canonical equations. It turns outthattheproblem ofconstructing such acanonical transformation reduces tothedetermination ofasuﬂfciently large number ofsolutions to theHamilton—Jacobi partial differential equation. Thegenerating function ofthedesired canonical transformation must satisfy thisequation. Before turning totheapparatus ofgenerating functions, weremark thatitisunfortunately noninvariant andituses, inanessential way, theco- ordinate structure inphase space {(p,q)}.Itisnecessary tousetheapparatus ofpartial derivatives, inwhich even thenotation isambiguous.“ AGenerating functions Suppose thatthe2nfunctions P(p,q)andQ(p,q)ofthe2nvariables pandq giveacanonical transformation g:R2"—>R2".Then the1-form pdq—PtlQ isanexact differential (Section 45A): (1) Pdq—PdQ=d$(P,ll)- PROBLEM. Show theconverse: ifthisform isanexact differential, then thetransformation is canonical. Wenow assume that, inaneighborhood ofsome point (po,qo),wecan take (Q,q)asindependent coordinates. Inother words, weassume that thefollowing jacobian isnotzero at(po,qo): detm=detQ9—#0.80>.<1) 01> 81Itisimportant tonote thatthequantity ou/ﬁx onthex,y-plane depends notonly onthe function which istaken forx,butalsoonthechoice ofthefunction y:innewvariables (x,:) thevalue ofﬁu/Ex willbedifferent. One should write -\ CM _l":.vQ;KZ\'=COHRI Z:COHSI 258 47:TheHamilton—Jacobi method Such canonical transformations willbecalled free. Inthiscase, thefunction S canbeexpressed locally inthese coordinates: 8(P.<1)=S1(Qsq)- Deﬁnition. The function S1(Q,q) iscalled agenerating function ofour canonical transformation g. Weemphasize that S1isnotafunction onthephase space R2“: itisa function onaregion inthedirect product R;xR3oftwon-dimensional coordinate spaces, whose points aredenoted byqandQ.Itfollows from (1) thatthe“partial derivatives” ofS1are Conversely, every function S1gives acanonical transformation gby formulas (2). Theorem. LetS,(Q, q)beafunction given onaneighborhood ofsome point (Q0, qo)ofthedirect product oftwon-dimensional euclidean spaces. If 432$dt7‘ at0, e Q0-‘Io thenS1isagenerating function ofsome freecanonical transformation. PROOF. Consider theequation fortheQcoordinates: wnzozp dq ' Bytheimplicit function theorem thisequation canbesolved todetermine a function Q(p, q)inaneighborhood ofthepoint fl,P=(i (O 0 Qosqo (with Q(p0, qo)=Q0). Infact, thedeterminant weneed here is 02$(Q11)), 6Q Q0410 andthisisdifferent from zero byhypothesis. Wenow consider thefunction m@o=—%&@m. andset P(p, = ql, 259 9:Canonical formalism Then thelocal map g:R2"—>R2"sending thepoint (p,q)tothepoint (P(p, q),Q(p,q)) willbecanonical with generating function S1,since by construction pdq—PdQ= dq+ dQ. Itisfree, since det(6Q/dp) =det(62S1(Q, q)/6Q dq)” ¢0. l:| The transformation g:R2"—>R2"isgiven ingeneral by2nfunctions of 2nvariables. Weseethat acanonical transformation isgiven entirely by onefunction of2nvariables—its generating function. Itiseasy toseehow useful generating functions areinallcalculations related tocanonical trans- formations. This becomes even more soasthenumber ofvariables, 2n, becomes large. BTheHamilton-Jacobi equation forgenerating functions Wenotice that canonical equations inwhich thehamiltonian function depends only onthevariable Qareeasy tointegrate. IfH=K(Q, t),then the canonical equations have theform . . GK (3) Q=0 P=% from which wehave immediately ’8KQo=Qo mn=mo+j- a9QQ(0) Wewillnow look foracanonical transformation reducing thehamiltonian H(p, q)totheform K(Q). Tothisendwewilllook foragenerating function ofsuch atransformation, S(Q, q).From (2)weobtain thecondition to H@%%QmJ=K@m where after differentiation wemust substitute q(P,Q)forq.Wenotice that forﬁxed Q,Equation (4)hastheform oftheHamilton~Jacobi equation. Jacobi’s theorem. Ifasolution S(Q, q)isfound totheHamilton—Jacobi equa- tion(4),depending onnparameters“ Q,andsuch thatdet(52S/6Q5q) 760, then thecanonical equations , an _an(5) 1>——E and(1-5 canbesolved explicitly byquadratures. Thefunctions Q(p, q)determined bytheequations dS(Q, q)/6q =pareﬁrst integrals oftheequation (5). 82Ann-parameter family ofsolutions of(4) iscalled acomplete integral oftheequation. 260 47:TheHamilton—Jacobi method PROOF. Consider thecanonical transformation with generating function S(Q, q).By(2)wehave p=(GS/6q)(Q, q),from which wecandetermine Q(p, q).Wecalculate thefunction H(p, q)inthenew coordinates P,Q. Wehave H(p, q)=H((6S/6q)(Q, q),q).Inorder toﬁnd thehamiltonian function inthenew coordinates wemust substitute into thisexpression (after differentiation) forqitsexpression interms ofPandQ.However, by(4),thisexpression does notdepend onqatall,sowehave simply H(p,q)=K(Q). Thus, inthenewvariables, Equation (5)hastheform (3),from which Jacobi’s theorem follows directly. D Jacobi’s theorem reduces solving thesystem ofordinary differential equations (5)toﬁnding acomplete integral ofthepartial differential equation (4).Itmay appear surprising that this“reduction” from thesimple tothe complicated provides aneffective method forsolving concrete problems. Nevertheless, itturns outthatthisisthemost powerful method known for exact integration, andmany problems which were solved byJacobi cannot be solved byother methods. CExamples Weconsider theproblem ofattraction bytwoﬁxed centers. Interest inthis problem hasgrown recently inconnection with thestudy ofthemotion of artiﬁcial earth satellites. Itisfairly clear thattwoclose centers ofattraction onthez-axis approximate attraction byanellipsoid slightly extended along thez-axis. Unfortunately, theearth isnotprolate, butoblate. Toovercome thisdifficulty, onemust place thecenters atimaginary points atdistances iis from theorigin along thez-axis. Analytic formulas forthesolution aretrue, ofcourse, inthecomplex region. Inthisway weobtain anapproximation totheearth’s ﬁeldofgravity, inwhich theequations ofmotion canbeexactly integrated andwhich iscloser toreality than thekeplerian approximation inwhich theearth isapoint. Forsimplicity wewillconsider only theplanar problem ofattraction by twoﬁxed points with equal masses. Thesuccess ofJacobi’s method isbased ontheadoption ofasuitable coordinate system, called elliptic coordinates. Suppose thatthedistance between theﬁxed points O1andO2is2c(Figure I Figure 204 Elliptical coordinates 261 9:Canonical formalism ‘C_~.'.?t&iif,:?*.“\'v;':lIl\;¢';$323’ Figure 205 Confocal ellipses andhyperbolas 204), andthatthedistances ofamoving mass from them arer1andr2,re- spectively. Theelliptic coordinates tf,11aredeﬁned asthesumanddifference ofthedistances tothepoints O1andO2:Q‘=r1+r2,11=r,—r2. PROBLEM. Express thehamiltonian function inelliptic coordinates. Solution. Thelinesﬁ =const areellipses withfociat O,andO2:the linesn =const arehyper- bolas with thesame foci(Figure 205). They aremutually orthogonal; therefore, dsz=a2dﬁz+bld112. Wewillﬁndthecoefficients aandb.Formotion along anellipse wehave drl=dscosatand drz=—dscosat,sodn=2cosaids. Formotion along ahyperbola wehave dr,=dssinat anddrz=dssinat,sodtf=2sinatds.Thus a=(2sinat)" andb=(2cosat)’'.Furthermore, from thetriangle O,MO2 weﬁndrf+r§+2r|rZ cos20:=4c2,which implies 4cZ—rz—rz2 -2 1 Zcos at—sm at=4-H,2r,r2 21")";cosz at+Slﬂzat=-7,2r,r2 2 4t-2—(r1—r2)2 ,2 (r,+r2)2—402cos 1:?--4- sm0t=—i-————i. 4l'1l'2 4r1r2 Butifdsz=2ofdq,-2,then Z 2r=za§i,.».=ei..H=£,—j;+ u. Thus, _2(r,+r2)2—41:2 24c2—(r,—r2)2 k k H—P: *2 "l-P» ———'-rlrz 2r1r2 rl rz Butr,+r2=5,r1—rz=11,4r,r2 =if-r12.Therefore, ﬁnally, 51-48 4t-1-111 4/<5 Wewillnow solve theHamilton—Jacobi equation. Deﬁnition. If,intheequation 8S 6S(D»~ —' =9 aaqna ql, 9qn) Os 1 262 47:TheHamilton~Jacobi method thevariable q,andderivative 6S/6q1 appear only intheform ofacombina- tion<p(6S/dql, ql),then wesaythatthevariable qlisseparable. Inthiscaseitisuseful tolook forasolution oftheequation oftheform 5=$1(q1) +$'(q2, ---iqt.)- Bysetting (p(@S1 /6q,, ql)=clinthisequation, weobtain anequation forS’ with asmaller number ofvariables 6S’ 6S’(D— ...,—, ,..., ,,; =0.2(aq29 aqn q2 q cl) LetS’=S’(q2,...,q,,;c,,c) beafamily ofsolutions tothis equation depending ontheparameters c,-.The functions S,(q1, cl)+S’willsatisfy thedesired equation ifS,satisﬁes theordinary differential equation <p(6S1/6q1,q,) =cl.This equation iseasy tosolve; weexpress 681/dql interms ofq,and cltoobtain 6S1/dq, =I//(q,,c1), from which S1= lm'//(qt, C1)dq1- Ifoneofthevariables, sayqz,isseparable inthenewequation (with (D2) wecanrepeat thisprocedure and(inthemost favorable case) wecanﬁnd asolution oftheoriginal equation depending onnconstants S1(qi§C1)+ Sz(¢I2lC1t C2)+ +Sn(q»§ Ci,---,¢'n)- Inthiscasewesaythatthevariables arecompletely separable. Ifthevariables arecompletely separable, then asolution depending onn parameters oftheHamilton—Jacobi equation, <D,(6S/ﬁq, q)=0,isfound by quadratures. Butthen thecorresponding system ofcanonical equations can alsobeintegrated byquadratures (Jacobi’s theorem). Weapply theabove totheproblem oftwoﬁxed centers. TheHamilton- Jacobi equation (4)hastheform (:;2)2(€2 —462)+(%3)2(4¢2 —'12)=K(§2—'12)+4l<€- Wecanseparate variables by,forinstance, setting (%2<c’ -4c’)-4k:-Kc’=ct and (gg)2(4c2 —112)+Knz =—c1. Then weﬁndthecomplete integral ofEquation (4)intheform 2 — —c2s(t,»1;c1,c2>=l / dc+l /%_—,72Tndn- 263 9:Canonical formalism Jacobi’s theorem now gives anexplicit expression, interms ofelliptic integrals, formotion intheproblem oftwoﬁxed centers. Amore detailed investigation ofthismotion canbefound inCharlier’s book “Die Mechanik desHimmels,” Berlin, Leipzig, W.deGruyter &Co., 1927. Another application oftheproblem oftheattraction oftwoﬁxed centers is thestudy ofmotion withﬁxed pullinaﬁeld withoneattracting center. This isaquestion ofthemotion ofapoint mass under theaction ofa newtonian attraction ofaﬁxed center andonemore force (“pull”) ofcon- stant magnitude anddirection. This problem canbelooked atasthelimiting case oftheproblem ofattraction bytwoﬁxed centers. Inthepassage to thelimit, onecenter goes olTtoinﬁnity inthedirection ofthethrust force (during which itsmass must grow proportionally tothesquare ofthedistance moved inorder toguarantee constant pull). This limiting case oftheproblem oftheattraction oftwoﬁxed centers canbeintegrated explicitly (inelliptic functions). Wecanconvince ourselves ofthisbypassing toalimit orbydirectly separating variables intheproblem ofmotion with constant pull inaﬁeld with onecenter. The coordinates inwhich thevariables areseparated inthisproblem areobtained asthe limit ofelliptic coordinates asoneofthecenters approaches inﬁnity. They arecalled parabolic coordinates andaregiven bytheformulas u=r~—x v=r+x (thepullisdirected along thex-axis). Adescription ofthetrajectories ofamotion with constant pull(many ofwhich arevery intricate) canbefound inV.V.Beletzkii’s book “Sketches ofmotions ofcelestial bodies,” Nauka, 1972. Asonemore example weconsider theproblem ofgeodesics onatriaxial ellipsoid.“ Here Jacobi’s elliptical coordinates /l,,/12,and13arehelpful, where the1,aretheroots oftheequation Z 2 2 X1 +X2 +X3 =1, 2.,>).2>l3; 411+)» a2+/l (13+). xl,x2,andx3arecartesian coordinates. Wewillnotcarry outthecomputa- tions showing thatthevariables areseparable (they canbefound, forexample, inJacobi’s “Lectures ondynamics”), butwillmention only theresult: we willdescribe thebehavior ofthegeodesics. The surfaces /ll=const, 2.2=const, and /l3=const aresurfaces of second degree, called confocal quadrics. Theﬁrstofthese isanellipsoid, the second ahyperboloid ofonesheet, andthethird ahyperboloid oftwosheets. Theellipsoid candegenerate intotheinterior ofanellipse, theone-sheeted hyperboloid either intotheexterior ofanellipse orintothepart ofaplane 83The problem ofgeodesics onanellipsoid andtheclosely related problem ofellipsoidal billiards have found application inaseries ofrecent results inphysics connected with laser devices. 264 47:The Hamilton—Jacobi method between thebranches ofahyperbola, and thetwo-sheeted hyperboloid either intothepartofaplane outside thebranches ofahyperbola orintoa plane. Suppose that theellipsoid under consideration isoneoftheellipsoids inthefamily with semiaxes a>b>c.Each ofthethree ellipses xl=0, x2=0,andx3=0isaclosed geodesic. Ageodesic starting from apoint ofthelargest ellipse (with semiaxes aand b)inadirection close tothe direction oftheellipse (Figure 206), isalternately tangent tothetwoclosed lines ofintersection oftheellipsoid with theone-sheeted hyperboloid ofour family 1=const.“ This geodesic iseither closed orisdense inthearea \I--It‘tiI .\\ Figure 206 Geodesic onatriaxial ellipsoid (‘ C Figure 207 Geodesics emanating from anumbilical point between thetwolines ofintersection. Astheslope ofthegeodesic increases, thehyperboloids collapse down totheregion “inside” thehyperbola which intersects ourellipsoid initsfour “umbilical points.” Inthelimiting case weobtain geodesics passing through theumbilical points (Figure 207). Itisinteresting tonote that allthegeodesics starting atanumbilical point again converge attheopposite umbilical point, andallhave thesame length between thetwoumbilical points. Only oneofthese geodesics isclosed, namely, themiddle ellipse with semiaxes aandc.Ifwetravel along any other geodesic passing through anumbilical point inanydirection, wewill approach thisellipse asymptotically. Finally, geodesics which intersect thelargest ellipse even more “steeply” (Figure 208) arealternately tangent tothetwolines ofintersection ofour s‘These lines ofintersection oftheconfocal surfaces arealsolines ofcurvature oftheellipsoid. 265 9:Canonical formalism .-/I 1 ‘ / ' L Figure 208 Geodesics ofanellipsoid which aretangent toatwo-sheeted hyperboloid./'_\ ellipsoid with atwo-sheeted hyperboloid.“ Ingeneral, they aredense inthe region between these lines. Thesmall ellipse with semiaxes bandcisamong these geodesics. “The main diﬂiculty inintegrating agiven differential equation liesin introducing convenient variables, which there isnoruleforﬁnding. There- fore, wemust travel thereverse path and after ﬁnding some noticeable substitution, look forproblems towhich itcanbesuccessfully applied.” (Jacobi, “Lectures ondynamics”). Alistofproblems admitting separation ofvariables inspherical, elliptical, andparabolic coordinates isgiven inSection 48ofLandau andLifshitz’s “Mechanics” (Oxford, Pergamon, 1960). 48Generating functions Inthisparagraph weconstruct theapparatus ofgenerating functions fornon-free canonical transformations. AThegenerating function S2(P,q) Letf:R2"—>R2"beacanonical transformation with g(p,q)=(P,Q).By thedeﬁnition ofcanonical transformation thedifferential form onR2" pdq—PdQ=dS isthetotal differential ofsome function S(p,q).Acanonical transformation is freeifwecantake q,Qas2nindependent coordinates. Inthiscase the function Sexpressed inthecoordinates qandQiscalled agenerating function S1(q, Q).Knowing thisfunction alone, wecanﬁndall2nfunctions giving the transformation from therelations (1) P=0815;:Q)andP:_@$1(g¢5Q)_ Itisfarfrom thecase that allcanonical transformations arefree. For example, inthecase oftheidentity transformation qandQ=qaredepen- dent. Therefore, theidentity transformation cannot begiven byagenerating 85These arealsolines ofcurvature. 266 48:Generating functions function S,(q,Q).Wecan,however, obtain generating functions ofanother form bymeans oftheLegendre transformation. Suppose, forinstance, that wecantake P,qasindependent local coordinates onR2"(i.e., the determinant det(6(P,q)/6(p, q))=det(0P/0p) isnotzero). Then wehave pdq—PdQ=dS and pdq+QdP=d(PQ+S). Thequantity PQ+S,expressed interms of(P,q),isalsocalled agenerating function $2(P,q)=PQ+8(1).q). Forthisfunction, weﬁnd as(P, as(P,)(2) p= andQ=%q. Conversely, ifS2(P,q)isanyfunction forwhich thedeterminant a2S2(P1atm°(8q6P ,.o__o isnotzero, then inaneighborhood ofthepoint wecansolve theﬁrstgroup ofequations (2)forPandobtain afunction P(p,q)(where P(p0, qo)=P0).After this, thesecond group ofequations (2) determine Q(p, q),andthemap (p,q)—~>(P,Q)iscanonical (prove thisl). PROBLEM. Find agenerating function S,fortheidentity map P=p,Q=q. Auswsa. Pq. Remark. Thegenerating function S2(P, q)isconvenient also because there arenominus signs intheformulas (2).andthey areeasy toremember ifweremember thatthegenerating function oftheidentity transformation isPq. B2"generating functions Unfortunately, thevariables P,qcannot always bechosen forlocal co- ordinates either; however, wecanalways choose some setofnnew co- ordinates Pt=(Pt,,---»Pt,,) Qj=(Qj,,---=Qj.,-,,) sothattogether with theoldqweobtain 2nindependent coordinates. Here (i1,..., i,,)(j1,...,j,,_,,) isanypartition oftheset(l,...,n) into twonon-intersecting parts; sothere areinall2"cases. 267 9:Canonical formalism Theorem. Letg:R2"—>R2"beacanonical transformation given bythe functions P(p,q)andQ(p, q).Inaneighborhood ofevery point (po,qo)at least oneofthe2"setsoffunctions (P,-,Q1-,q)canbetaken asindependent coordinates onR2": a(Pi> Q": Qd J =d___'i1 ggQ_°@<p..p,-.in6‘ant.p,) Inaneighborhood ofsuch apoint, thecanonical transformation gcanbe reconstructed from thefunction S3(Pi’Q,-.qt=(P.-Qt)+lpdq—Pdo bytherelations 3 ii .="S andP.=_%.3 () p 9 Qt apil J Conversely, ifS3(P,, QI-,q)isanyfunction forwhich thedeterminant det(('i2S3/0P 6q)|Po_.0 (P=P,-,Qj)isnotzero, then therelations (3)givea canonical transformation inaneighborhood ofthepoint p0,qo. PROOF. Theproof ofthistheorem isalmost thesame astheonecarried out above intheparticular casek=n.Weneed only verify thatthedeterminant det[(0(P,, QJ-)/8(1),, p,~))] isnotzero foroneofthe2".sets (P,,Q1-,q). Weconsider thedillerential ofourtransformation gatthepoint (po,qo).Byidentifying the tangent space toR2"with R2",wecanconsider dgasasymplectic transformation S:R2"—>R2". Consider thecoordinate p-plane PinR2"(Figure 209). This isanulln-plane, anditsimage SP isalsoanullplane. Weproject theplane SPonto thecoordinate plane a={(p,,q,»)}parallel to theremaining coordinate axes, i.e.,inthedirection ofthen-dimensional nullcoordinate plane 6={(pj,q,)}.Wedenote theprojection operator byTS:P—>a. Thecondition det(6(P,, Q,-)/t3(p,, p,-))9*0means thatT:SP—>0isnonsingular. Theoperator Sisnonsingular. Therefore, TSisnonsingular ifandonly ifT:SP—>oisnonsingular. Inother words, thenullplane SPmust betransverse tothenullcoordinate plane 6.Butweshowed in P $1 SP -T 6 Figure 209 Checking non-degeneracy 268'.i...t..a§Enﬂ tut.-, 48:Generating functions Section 41thatatleast oneofthe2"nullcoordinate planes istransverse toSP.This means that oneofour2"determinants isnonzero, aswastobeshown. El PROBLEM. Show thatthissystem of2"types ofgenerating functions isminimal: given anyoneof the2"determinants, there exists acanonical transformation forwhich only thisdeterminant is nonzero.“ CInﬁnitesimal canonical transformations Wenow consider acanonical transformation which isclose totheidentity. Itsgenerating function canbetaken close tothegenerating function Pq oftheidentity. Welook atafamily ofcanonical transformations g,depending differentiably ontheparameter e,such that thegenerating functions have theform <3 6S(4) Pq+sS(P,q;a) p=P+a—é€ Q=q+t-2&3. Aninﬁnitesimal canonical transformation isanequivalence class offamilies g,,twofamilies g,andh,,being equivalent iftheir diﬁerence issmall ofhigher than ﬁrstorder, lg,—h,|=O(s2), s->0. Theorem. Aninﬁnitesimal canonical transformation satisﬁes Hamilton’s diﬂerential equations av __@ dQ _@ ds.-..“ atd»=..<.'@p withhamiltonian function H(p,q)=S(p,q,0). PROOF. Theresult follows from formula (4):P—->pass—~>0. Cl Corollary. Aone-parameter group oftransformations ofphase space R2" satisﬁes Hamilton’s canonical equations ifandonly ifthetransformations arecanonical. Q l. ‘. I7 Figure 210 Geometric meaning ofHamilton’s function 8°Thenumber ofkinds ofgenerating functions indifferent textbooks ranges from 4to4". 269 9:Canonical formalism The hamiltonian function Hiscalled the“generating function ofthe inﬁnitesimal canonical transformation.” Wenotice thatunlike thegenerating function S,thefunction Hisafunction ofpoints ofphase space, invariantly associated tothetransformation. Thefunction Hhasasimple geometric meaning. Letxandybetwopoints inR2"(Figure 210), yacurve connecting them, andat=y—x.Consider theimages ofthecurve yunder thetransformations g,,03Tga;they form aband 0(2). Now consider theintegral oftheform of=Zdp,/\dq, over the2-chain a,using thefactthat80=gay—it+g,x-g,y. PROBLEM. Show that .1 2l1m— 0)=H(x)-H(y) 8to e—~0 0 exists anddoes notdepend ontherepresentative oftheclass g,. From thisresult weonce more obtain thewell-known Corollary. Under canonical transformations thecanonical equations retain their form, with thesame hamiltonian function. PROOF. Wecomputed thevariation ofthehamiltonian function using only aninﬁnitesimal canonical transformation andthesymplectic structure of lR2"—the form wz. Cl 270 Introduction toperturbation theory 10 Perturbation theory consists ofaveryuseful collection ofmethods forﬁnding approximate solutions of“perturbed” problems which areclose tocom- pletely solvable “non-perturbed” problems. These methods canbeeasily justiﬁed ifweareinvestigating motion over asmall interval oftime. Relatively little isknown about how farwecantrust theconclusions ofperturbation theory ininvestigating motion over large orinﬁnite intervals oftime. Wewillseethatthemotion inmany “non-perturbed ”integrable problems turns outtobeconditionally periodic. Inthestudy ofunperturbed problems, andeven more sointhestudy oftheperturbed problems, special symplectic coordinates, called “action-angle” variables, areuseful. Inconclusion, we willprove atheorem justifying perturbation theory forsingle-frequency systems andwillprove theadiabatic invariance ofaction variables insuch systems. 49Integrable systems Inorder tointegrate asystem of2nordinary differential equations, wemust know 2nﬁrst integrals. Itturns outthatifwearegiven acanonical system ofdifferential equations, itisoften sufficient toknow onlynﬁrstintegrals—each ofthem allows ustoreduce theorder ofthe system notjustbyone,butbytwo. ALiouville’s theorem onintegrable systems Recall that afunction Fisaﬁrst integral ofasystem with hamiltonian function Hifandonly ifthePoisson bracket (H,F)50 isidentically equal tozero. 271 10:Introduction toperturbation theory Deﬁnition. Two functions F1andF2onasymplectic manifold areininvolution iftheir Poisson bracket isequal tozero. Liouville proved thatif,inasystem with ndegrees offreedom (i.e.,with a2n-dimensional phase space), nindependent ﬁrst integrals ininvolution areknown, then thesystem isintegrable byquadratures. Here istheexact formulation ofthistheorem: Suppose thatwearegiven n functions ininvolution onasymplectic 2n-dimensional manifold F1’.-.,Fn i,j:l,2,...,n. Consider alevel setofthefunctions F,- Mf Z = Z 1,---,n}~ Assume that thenfunctions F,areindependent onM,(i.e., then1-forms dF,arelinearly independent ateach point ofM,). Then 1.M,isasmooth manifold, invariant under thephase ﬂowwith hamiltonian function H=F1. 2.Ifthemanifold M,iscompact andconnected, then itisdiffeomorphic tothen-dimensional torus T"={(<p1, ...,(p,,)mod 2rc}. 3.Thephase ﬂow with hamiltonian function Hdetermines aconditionally periodic motion onMf, i.e.,inangular coordinates tp=(tpl,...,<p,,) wehave f-19=to, to=to(f).dt 4.Thecanonical equations with hamiltonian function Hcanbeintegrated byquadratures. Before proving thistheorem, wenote afewofitscorollaries. Corollary 1.If,inacanonical system with twodegrees offreedom, aﬁrst integral Fisknown which does notdepend onthehamiltonian H,then the system isintegrable byquadratures; acompact connected two-dimensional submanifold ofthephase space H=h,F=fisaninvariant torus, and motion onitisconditionally periodic. PRooF. FandHareininvolution since Fisaﬁrstintegral ofasystem with hamiltonian function H. El Asanexample with three degrees offreedom, weconsider aheavy sym- metric Lagrange topﬁxed atapoint onitsaxis. Three ﬁrst integrals are immediately obvious: H,M2,andM3.Itiseasy toverify thattheintegrals 272 49:Integrable systems M,andM3areininvolution. Furthermore, themanifold H=hinthephase space iscompact. Therefore, wecanimmediately say,without anycalcula- tions, that forthemajority ofinitial conditions“ themotion ofthetopis conditionally periodic: thephase trajectories ﬁllupthethree-dimensional torus H=c,,M,=c2,M3=c3.Thecorresponding three frequencies are called frequencies offundamental rotation, precession, andnutation. Other examples arise from thefollowing observation: ifacanonical system canbeintegrated bythemethod ofHamilton—Jacobi, then ithasn ﬁrstintegrals ininvolution. Themethod consists ofacanonical transformation (p,q)—>(P,Q)such that theQ,areﬁrst integrals. Butthefunctions Q, andQ,»areclearly ininvolution. Inparticular, theobservation above applies totheproblem ofattraction bytwoﬁxed centers. Other examples areeasily found. Infact, thetheorem ofLiouville formulated above covers alltheproblems ofdynamics which have been integrated tothepresent day. BBeginning oftheproof ofLiouville’s theorem Weturn now totheproof ofthetheorem. Consider thelevel setofthe integrals: Mf={XIF,=ﬁ,l=l,...,H}. Byhypothesis, then1-forms dF,arelinearly independent ateach point of M,; therefore, bytheimplicit function theorem, M,isann-dimensional submanifold ofthe2n-dimensional phase space. Lemma 1.Onthen-dimensional manifold M,there exist ntangent vector ﬁelds which commute withoneanother andwhich arelinearly independent atevery point. PROOF. Thesymplectic structure ofphase space deﬁnes anoperator Itaking 1-forms tovector ﬁelds. This operator 1carries the1-form dF,totheﬁeld IdF,-ofphase velocities ofthesystem with hamiltonian function F,-.We willshow that thenﬁelds IdF,-aretangent toM,,commute, andareinde- pendent. Theindependence oftheIdF,atevery point ofM,follows from theinde- pendence ofthedF,~and thenonsingularity oftheisomorphism I.The ﬁelds IdF,-commute with oneanother, since thePoisson brackets oftheir hamiltonian functions (F,-,F1-)areidentically 0.Forthesame reason, the derivative ofthefunction F,inthedirection oftheﬁeld IdF,-isequal tozero foranyi,j=1,...,n.Thus theﬁelds IdF,-aretangent toM,,andLemma 1 isproved. El 8"Thesingular level sets.where theintegrals arenotfunctionally independent, constitute the exception. 273 10:Introduction toperturbation theory Wenotice thatwehave proved even more than Lemma 1: 1'.Themanifold M,isinvariant with respect toeach ofthencommuting phase ﬂows glwith hamiltonian functions F,-:gf-gﬁ=gjgl. 1”.Themanifold M,isnull(i.e.,the2-form (oziszero onTM,|,,). This istruesince thenvectors IdF,IXareskew-orthogonal tooneanother ((F,, FJ-)E0)andform abasis ofthetangent plane tothemanifold M,at thepoint x. CManifolds onwhich theaction ofthegroup IR"istransitive Wewillnowusethefollowing topological proposition (theproof iscompleted inSection D). Lemma 2.LetM"beacompact connected diﬂerentiable n-dimensional mani- fold, onwhich wearegiven npairwise commutative andlinearly independent ateach point vector ﬁelds. Then M“isdiﬂeomorphic toann-dimensional torus. PROOF. Wedenote bygf,i=1,...,n,theone-parameter groups ofdiffeo- morphisms ofMcorresponding tothengiven vector ﬁelds. Since theﬁelds commute, thegroups gﬁandgjcommute. Therefore, wecandeﬁne anaction g ofthecommutative group R"={t}onthemanifold Mbysetting g‘:M—>M g'=g','---g§;', (t=(t,,...,t,,)elRi"). Clearly, g‘*’ =g‘g’, t,seR".Now ﬁxapoint xoeM.Then wehave amap giR"—+M g(t) =gtxo. (The point x0moves along thetrajectory oftheﬁrstﬂow fortime t,,along thesecond ﬂowfortimet2,etc.) PROBLEM 1.Show thatthemap g(Figure 211) ofasufﬁciently small neighborhood Vofthe point 0eR"gives achart inaneighborhood ofxo:every point xoeMhasaneighborhood U(xo6UCM)such that gmaps Vdiﬁeomorphically onto U. Hint. Apply theimplicit function theorem andusethelinear independence ofthe ﬁelds atxo. PROBLEM 2.Show thatg:R"—>Misonto. R” M lllllli llllU'llllllil|||ItIl7‘llllilll’ Figure 211 Problem 1 274 49:Integrable systems Figure 212 Problem 2 Hint. Connect apoint x6Mwith x,,byacurve (Figure 212), cover thecurve byaﬁnite number oftheneighborhoods Uofthepreceding problem anddeﬁne tasthesumofshifts t,- corresponding topieces ofthecurve. Wenote that themap g:R"—>M"cannot beone-to-one since M"is compact andR"isnot.Wewillexamine thesetofpre-images ofxoeM". Deﬁnition. Thestationary group ofthepoint xoisthesetFofpoints t6R" forwhich g'x,, =xo. PROBLEM 3.Show thatFisasubgroup ofthe group R",independent ofthepoint x,,. Solution. Ifg'x0=xoand g'x0 =xo,then _q“'x0 =g‘g'x0 =gsxo =xoand g"x,, = g7‘g‘xO =xo.Therefore, Fisasubgroup ofIR".lfx=g'x,, andteF,then g‘x=g""x,, = o'a'><0 =9'X<>=X- lnthisway thestationary group Fisawell-deﬁned subgroup ofR" independent ofthepoint xo.Inparticular, thepoint t=0clearly belongs toF. PROBLEM 4.Show that, inasuﬁiciently small neighborhood Vofthepoint 0eIR",there isno point ofthestationary group other than t=0. Hint. Themapg: V—>Uisadiffeomorphism. PROBLEM 5.Show that, intheneighborhood t+Vofanypoint teFcIR",there isnopoint of thestationary group Fother than t.(Figure 213) Thus thepoints ofthestationary group FlieinR"discretely. Such sub- groups arecalled discrete subgroups. W V Figure 213 Problem 5 275 10:Introduction toperturbation theory F, 0 P2 P1'§- . 0000 Figure 214 Adiscrete subgroup oftheplane EXAMPLE. Lete,,...,e,,beklinearly independent vectors inR",03k3n. Thesetofalltheir integral linear combinations (Figure 214) m,e1+---+m,,e,,, m,-eZ=(...,—2,—1,0,l,...) forms adiscrete subgroup ofIR".Forexample, thesetofallintegral points intheplane isadiscrete subgroup oftheplane. DDiscrete subgroups inR" Wewillnowusethealgebraic factthattheexample above includes alldiscrete subgroups ofR”.More precisely, wewillprove Lemma 3.LetFbeadiscrete subgroup ofR".Then there exist k(03k3n) linearly independent vectors e,,-...,ekEFsuch thatFisexactly thesetof alltheir integral linear combinations. PROOF. Wewill consider R"with some euclidean structure. Wealways have 0eF.IfF={0}thelemma isproved. Ifnot,there isapoint eoeF, e,,aé0(Figure 215). Consider thelinelR€e,,. Wewillshow that among the elements ofFonthisline, there isapoint e,which isclosest to0.Infact, inthediskofradius |e,,|with center 0,there areonly aﬁnite number ofpoints ofF(aswesawabove, every point xofFhasaneighborhood Vofstandard sizewhich does notcontain anyother point ofF).Among theﬁnite number ofpoints ofFinside thisdiscandlying onthelineRea, thepoint closest to0 willbetheclosest point to0onthewhole line.Theintegral multiples ofthis point e,(me,, meZ)constitute theintersection ofthelineReowith F. Figure 215 Proof ofthelemma ondiscrete subgroups 276 49:Integrable systems Infact,thepoints meldivide thelineintopieces oflength |elI.Ifthere were apoint eeFinside oneofthese pieces (mel, (m+l)e,), then thepoint e—meleFwould becloser to0than el. Ifthere arenopoints ofFoffthelineRel, thelemma isproved. Suppose there isapoint eeF,e¢Rel.Wewillshow thatthere isapoint elleFclosest tothelineRel(butnotlying ontheline). Weproject eorthogonally onto Rel. The projection liesinexactly one interval A={lel},m32.<m+1. Consider theright circular cylinder Cwith axisAandradius equal tothe distance from Atoe.Inthiscylinder lieaﬁnite (nonempty) number ofpoints ofthegroup F.Letezbetheclosest onetotheaxisRelnotlying ontheaxis. PROBLEM 6.Show thatthedistance from thisaxistoanypoint eofFnotlying onRelisgreater than orequal tothedistance ofelfrom Rel. Hint. Byashiftofmelwecanmove theprojection ofeonto theaxisinterval A. Theintegral linear combinations ofelandezform alattice intheplane Rel+Rez. PROBLEM 7.Show thatthere arenopoints ofFontheplane Rel+Relother than integral linear combinations ofel ande2. Hint. Partition the plane into parallelograms (Figure 216) A={Alel +22cl}, ml32.,<m,-+l.lfthere wereanee Awithe aémlel +m2BZ,[l1€ll[l'1€p0ll1IE —mlel —mlel would becloser toRelthan ez. A F2 P1 Figure 216 Problem 7 Ifthere arenopoints ofFoutside theplane Rel+Rel, thelemma is proved. Suppose thatthere isapoint eeFoutside thisplane. Then there exists apoint e3eFclosest toRel+Rez; thepoints mlel +mzez +m3e3 exhaust Finthethree-dimensional-space Rel+Rel+Re3. IfFisnot exhausted bythese, wetake theclosest point tothis three-dimensional space, etc. PROBLEM 8.Show thatthisclosest point always exists. Hint. Take theclosest oftheﬁnite number ofpoints ina“cylinder” ('. Note thatthevectors el,ez,ell,...arelinearly independent. Since they all lieinR",there arek3nofthem. 277 l0:Introduction toperturbation theory PROBLEM 9.Show thatFisexhausted bytheintegral linear combinations ofel. ...,ell. Hint. Partition theplane Rel+---+Relintoparallelepipeds Aandshow thatthere cannot beapoint ofFinanyA.lfthere isaneeFoutside theplane Rel+---+Re,,theconstruction isnotﬁnished. Thus Lemma 3isproved. El Itisnow easy toprove Lemma 2:M,isdifleomorphic toatorus T". Consider thedirect product ofkcircles andn-kstraight lines: T“><RH‘={(<t>1.---.<t>t;y1.---.y..-t)}. 't>m0d21r. together with thenatural map p:R2”—>T“xR""", p(<t>.r)=(<0mod21¢.y)- The points fl,...,fl,eR"(flhascoordinates (pl=21:,(pl=0,y=0)are mapped to0under thismap. Letel,...,eleFcR"bethegenerators ofthegroup F(cf.Lemma 3). Wemap thevector space R"={((p,y)}onto thespace R"={t}sothatthe vectors flgotoel.LetA:R"—+R"besuch anisomorphism. Wenow note thatR"={(cp,y)}gives charts forT"xR""‘, andR"={t} gives charts forourmanifold M,. PROBLEM 10.Show that the map ofcharts /1:R"—+R" gives adiffeomorphism Z;T‘XR""‘-+ M,, R"=ttwii—<'—» R"-ltl Tk XRn-it _________1€l_______, Mr But, since themanifold M,iscompact byhypothesis, k=nandM,isan n-dimensional torus. Lemma 2isproved. El Inview ofLemma l,theﬁrsttwostatements ofthetheorem areproved. Atthesame time, wehave constructed angular coordinates (pl,...,cp,lmod 21: onM,. PROBLEM ll.Show that under theaction ofthephase ﬂow with hamiltonian Htheangular coordinates cpvary uniformly with time (lb,=oi, U)l-=to,-(f) tp(t) ==tp(0) +tot. Inother words, motion ontheinvariant torus M,isconditionally periodic. Hint. up=A'‘t. Ofalltheassertions ofthetheorem, only thelastremains tobeproved: that thesystem canbeintegrated byquadratures. 278 50:Action-angle variables 50Action-angle variables Weshow here that, under thehypotheses ofLiouville’s theorem, wecanﬁndsymplectic co- ordinates (I,tp)such that theﬁrst integrals Fdepend only onI,andrpareangular coordinates onthetorus M,. ADescription ofaction-angle variables InSection 49westudied oneparticular compact connected level manifold oftheintegrals: M,={xzF(x) =f};itturned outthat M,wasann-di- mensional torus, invariant with respect tothephase ﬂow. Wechose angular coordinates (pl-onMsothatthephase ﬂowwith hamiltonian function H=Fl takes anespecially simple form: do5=wtf) ¢(r)=<t>(0)+wr- Wewillnow look ataneighborhood ofthen-dimensional manifold M, in2n-dimensional phase space. PROBLEM. Show thatthemanifold M,hasaneighborhood diffeomorphic tothedirect product ofthen-dimensional torus T"andthediscD"inn-dimensional euclidean space. Hint. Take thefunctions F,»andtheangles (plconstructed above ascoordinates. Inview of thelinear independence ofthedFl-,thefunctions Fland(pl(i=I,...,rt)giveadiffeomorphism ofaneighborhood ofM,onto thedirect product T“xD". Inthecoordinates (F,tp)thephase ﬂowwith hamiltonian function H=Fl canbewritten intheform ofthesimple system of2nordinary differential equations dF dq)1 _= _= <> d,Odzwtr). which iseasily integrated: F(t)=F(0), q>(t)=q>(0) +co(F(0))t. Thus, inorder tointegrate explicitly theoriginal canonical system of differential equations, itissufﬁcient toﬁndthevariables tpinexplicit form. Itturns outthatthiscanbedone using only quadratures. Aconstruction of thevariables (pisgiven below. Wenote that thevariables (F,qr)arenot, ingeneral, symplectic co- ordinates. Itturns outthat there arefunctions ofF,which wewilldenote byI—I(F), I=(1l,..., 1,),such that thevariables (I,tp)aresymplectic coordinates: theoriginal symplectic structure wzisexpressed inthem by theusual formula (U2 = /\ 279 10:Introduction toperturbation theory Thevariables Iarecalled action variables;88 together withtheangle variables tpthey form theaction-angle system ofcanonical coordinates inaneighbor- hood ofM,. Thequantities Ilareﬁrstintegrals ofthesystem with hamiltonian function H=Fl,since theyarefunctions oftheﬁrstintegrals F,-.Inturn, thevariables Flcanbeexpressed interms ofIand, inparticular, H=Fl=H(I).In action-angle variables thedifferential equations ofourﬂow(1)have theform dI_ @_ I(2) E_0 dl_tt>(). PROBLEM. Canthefunctions co(I)in(2)bearbitrary? Solution. Inthevariables (I.rp),theequations oftheﬁow (2)have thecanonical form with hamiltonian function H(l). Therefore, (o(I) =an/at; thus ifthenumber ofdegrees offreedom isnZ2,thefunctions co(I) arenotarbitrary, butsatisfy thesymmetry condition dwl/61, = 5(1),(71,. Action-angle variables areespecially important forperturbation theory; inSection 52wewilldemonstrate their application tothetheory ofadiabatic invariants. BConstruction ofaction-angle variables inthe case ofonedegree offreedom Asystem with onedegree offreedom inthephase plane (p,q)isgiven bythe hamiltonian function H(p, q). EXAMPLE 1.The harmonic oscillator H=%p2+%q2; or,more generally, H=§a2p2 +%b2q2./ EXAMPLE 2.The mathematical pendulum H=%p2—cosq.Inboth cases wehave acompact closed curve M,,(H =h),and theconditions ofthe theorem ofSection 49forn=1aresatisﬁed. Inorder toconstruct theaction-angle variables, wewill look fora canonical transformation (p,q)—+(I,(p)satisfying thetwoconditions: L1=nn (3)2.li11¢=21¢. PROBLEM. Find theaction-angle variables inthecase ofthesimple harmonic oscillator H=if+idl- Solution. Ifr.(,0arepolar coordinates. then dpAdq=rdrAdtp=d(i-2/2) /\dq).There- fore, I=HI(pl+qz)/2. 88Itisnothard toseethatIhasthedimensions ofaction. 280.>..u=n1..~_~¢..A.s;.s3maIi1 50:Action-angle variables Inorder toconstruct thecanonical transformation p,q->I,cpinthe general case, wewilllook foritsgenerating function S(I,q): (4) p=5%’3 <t>=-‘l-§§,ll’—") H(@i§,!,;_‘”..)=i. Weﬁrstassume thatthefunction h(I)isknown andinvertible, sothatevery curve M,,isdetermined bythevalue ofI(Mll=Mllll). Then foraﬁxed value ofIwehave from (4) dSlI=const =p This relation determines awell-deﬁned differential 1-form dSonthecurve MM’). Integrating thisl-form onthecurve M,,ll, weobtain (inaneighborhood ofapoint ql,)afunction ll $(I.q)=lpdq- qo This function willbethegenerating function ofthetransformation (4)in aneighborhood ofthepoint (I,ql,).Theﬁrstoftheconditions (3)issatisﬁed automatically: I=I(h). Toverify thesecond condition, weconsider the behavior ofS(I,q)“inthelarge.” After acircuit oftheclosed curve M,,l,lthe integral ofpdqincreases by Asa)=35pdq. Mm!) equal tothearea Flenclosed bythecurve M,,l,l.Therefore, thefunction S isa“multiple-valued function” onMlllll:itisdetermined uptoaddition ofintegral multiples ofII.This term hasnoeffect onthederivative 0S(I, q)/dq; butitleads tothemulti-valuedness ofgo=0S/81. This derivative turns out tobedeﬁned only uptomultiples ofdAS(I)/dI. More precisely, theformulas (4)deﬁne a1-form drponthecurve M,,lll,andtheintegral ofthisform on M,,,” isequal todAS(I)/dl. Inorder tofulﬁll thesecond condition, llM,dcp=21:,weneed that d AS II I--52-? where II=§M,pdqisthearea bounded bythephase curve H=h. Deﬁnition. The action variable intheone-dimensional problem with hamiltonian function H(p,q)isthequantity I(h)=(1/21t)II(h). Finally, wearrive atthefollowing conclusion. Letdl'l/dh abO.Then the inverse I(h)ofthefunction h(I)isdeﬁned. 281 10:Introduction toperturbation theory Theorem. SetS(I,q)=[gopdq|H=,,(,,. Then formulas (4)give acanonical transformation p,q—>I,(psatisfying conditions (3). Thus, theaction-angle variables intheone-dimensional case arecon- structed. PROBLEM. Find SandIforaharmonic oscillator. Answen. IfH=%a2p2+%b2q2 (Figure 217),thenM,,istheellipse bounding the area Fl(h) =1r(,/2h/a)(, /2h/b) =21th/ab =21th/w. Thus foraharmonic oscillator theaction variable istheratio ofenergy tofrequency. Theangle variable tpis,ofcourse, thephase of oscillation. [I \/.7 £7 Figure 217 Action variable forahamonic oscillator PROBLEM. Show thattheperiod Tofmotion along theclosed curve H=honthephase plane p,qisequal tothederivative with respect tohofthearea bounded bythiscurve: dI‘IT=4L.dh Solution. Inaction-angle variables theequations ofmotion (2)give _EH d1'1 dfl'1 21: dll (PZ2 1 T 1 21[ -i T = I=—_ 61 0h dh tp dh 2nCConstruction ofaction-angle variables in[R Weturn now tosystems with ndegrees offreedom given inR2"={(p,q)} byahamiltonian function H(p,q)andhaving nﬁrstintegrals ininvolution F1=H,F2,...,F,,.Wewillnotrepeat thereasoning which brought usto thechoice of21:1=§pdqintheone-dimensional case, butwillimmediately deﬁne naction variables I. Letyl,...,11,,beabasis fortheone-dimensional cycles onthetorus M, (theincrease ofthecoordinate (pionthecycle y1-isequal to21:ifi=jand Oifi aéj). Weset <5) 1.-to=3;;ffpdq. 282...i./..~;-uu. 50:Action-angle variables , "'1\\ "'///////» 3‘ Figure 218 Independence ofthecurve ofintegration fortheaction variable PROBLEM. Show thatthisintegral does notdepend onthechoice ofthecurve ",3representing thecycle (Figure 218). Hint. InSection 49weshowed thatthe2-form of=Xdp, /\dq,onthemanifold M,is equal_to zero. ByStokes‘ formula, §—§pdq=J-Jldp/\dq=0, where Pa=','—~,~'. Deﬁnition. Thenquantities I,(f)given byformula (5)arecalled theaction variables. Weassume now that, forthegiven values f-ofthenintegrals Fi,then quantities Iiareindependent: det(<3I/0f)|, aé0.Then inaneighborhood ofthetorus M,wecantake thevariables I,cpascoordinates. Theorem. Thetransformation p,q—>I,cpiscanonical, i.e., /\ = /\ Weoutline theproof ofthistheorem. Consider thedifferential 1-form pdqonM,.Since themanifold M,isnull(Section 49)thisl-form onM, isclosed: itsexterior derivative wz=dp/\dqisidentically equal tozero onM,.Therefore (Figure 219), Xao=fran, P > x X0 M’ q4,,W (I Figure 219 Independence ofthepath fortheintegral ofpdqonMI 283 10:Introduction toperturbation theory does notchange under deformations ofthepath ofintegration (Stokes’ formula). Thus S(x) isa“multiple-valued function” onM,,with periods equal to 7|‘ Now letxobeapoint onM,,inaneighborhood ofwhich thenvariables qarecoordinates onMI,such that thesubmanifold MIc:R2"isgiven byn equations oftheform p=p(I,q),q(x0) =qo.Inasimply connected neighbor- hood ofthepoint qoasingle-valued function isdeﬁned, stl,o=Fi>(Lq)dq, andwecanuseitasthegenerating function ofacanonical transformation nq~L¢ _§§ Flp_m ¢_a' Itisnotdiﬁicult toverify that these formulas actually give acanonical transformation, notonlyinaneighborhood ofthepoint under consideration, butalso“inthelarge” inaneighborhood ofM,.Thecoordinates mpwillbe multiple-valued with periods 5S 6 6 » -1 A. i i i . 1 L tZ 2 -- A,qoJ ,all allA,S 61}21:1, 1:5,], aswastobeshown. Cl We now note that allour constructions involve only “algebraic” operations (inverting functions) and “quadrature”—calculation ofthe integrals ofknown functions. Inthis way theproblem ofintegrating a canonical system with 2nequations, ofwhich nﬁrstintegrals ininvolution areknown, issolved byquadratures, which proves thelastassertion of Liouville’s theorem (Section 49). Cl Remark 1.Even intheone-dimensional case theaction-angle variables arenotuniquely deﬁned bytheconditions (3).Wecould have taken I’=I+const fortheaction variable and go’-go+c(I) fortheangle variable. Remark 2.Weconstructed action-angle variables forsystems with phase space R2".Wecould alsohave introduced action-angle variables forasystem onanarbitrary symplectic manifold. Werestrict outselves heretoonesimple example (Figure 220). 284 51:Averaging talkA, > <'&/I .‘“ Figure 220 Action-angle variables onasymplectic manifold Wecould have taken thephase space ofapendulum (H=%p2—cosq) tobe,instead oftheplane {(p,q)},thesurface ofthecylinder IR‘><S‘ obtained byidentifying angles qdiﬁering byanintegral multiple of21:. The critical level lines H=ildivide thecylinder into three parts, A,B,andC,each ofwhich isdiffeomorphic tothedirect product IR‘xS1. Wecanintroduce action-angle variables intoeach part. Inthebounded part (B)theclosed trajectories represent theoscillation ofthependulum; in theunbounded parts they represent rotation. Remark 3.Inthegeneral case, asintheexample analyzed above, the equations F,-=f,-cease tobeindependent forsome values offi,andM,ceases tobeamanifold. Such critical values offcorrespond toseparatrices dividing thephase space oftheintegrable problem into parts corresponding tothe parts A,B,andCabove. Insome ofthese parts themanifolds M,canbe unbounded (parts AandCintheplane {(p,q)}); others arestratiﬁed into n-dimensional invariant toriM,,inaneighborhood ofsuch atorus we canintroduce action-angle variables. 51Averaging Inthisparagraph weshow thattime averages andspace averages areequal forsystems under- going conditionally-periodic motion. AConditionally-periodic motion Intheearlier sections ofthisbook. wehave frequently encountered con- ditionally-periodic motion: Lissajous ﬁgures, precession, nutation, rotation ofatop,etc. Deﬁnition. LetT"bethen-dimensional torus andqa=((p1,...,<p,,)mod 21: angular coordinates. Then byaconditionally-periodic motion wemean a one-parameter group ofdiffeomorphisms T"—>T"given bythedif- ferential equations (Figure 221): ti:=to, to=(col, ...,con)="const. 285 10:Introduction toperturbation theory $2 2. I ‘ 0 J1! $1 Figure 221 Conditionally-periodic motion These differential equations areeasily integrated: 41(1)=<P(0) +"JI- Thus thetrajectories inthechart {q)}arestraight lines. Atrajectory onthe torus iscalled awinding ofthetorus. EXAMPLE. Letn=2.Ifto,/co; =kl/kg. thetrajectories areclosed: ifiv,/wz isirrational. then trajectories onthetorus aredense (cf.Section I6). Thequantities co1,...,conarecalled thefrequencies oftheconditionally- periodic motion. Thefrequencies arecalled independent ifthey arelinearly independent over theﬁeld ofrational numbers: ifkeZ"B9and(k,co)=O, then k=0. BSpace average andtime average Letf(q))beanintegrable function onthetorus T”. Deﬁnition. Thespace average ofafunction fonthetorus T"isthenumber f=<2”)-"L2"---f:”r<¢>d¢. ---d<i».- Consider thevalue ofthefunction f(tp)onthetrajectory (p(t)=qio+cot. This isafunction oftime, f((|)0+cot).Weconsider itsaverage. Deﬁnition. Thetimeaverage ofthefunction fonthetorus T"isthefunction .1T f*((Po) =hm?Jf((Po +mild! 0 T-'00 (deﬁned where thelimit exists). Theorem ontheaverages. Thetime average exists everywhere, andcoincides with thespace average iffiscontinuous (ormerely Riemann integrable) andthefrequencies co,areindependent. "Qk=(k,,...,/<,,)withintegral /<,.. 286 51:Averaging PROBLEM. Show thatifthefrequencies aredependent, then thetime average candiffer from the space average. Corollary 1.Ifthefrequencies areindependent, then every trajectory {q)(t)} isdense onthetorus T". PROOF. Assume thecontrary. Then insome neighborhood Dofsome point ofthetorus, there isnopoint ofthetrajectory q>(t). Itiseasy toconstruct a continuous function fequal tozero outside Dandwith space average equal to1.The time average f*(q>0) onthetrajectory q)(t) isequal to0aé1. This contradicts theassertion ofthetheorem. El Corollary 2.Ifthefrequencies areindependent, then every trajectory is uniformly distributed onthetorus T". This means that thetime thetrajectory spends inaneighborhood Dis proportional tothemeasure ofD. More precisely, letDbea(Jordan) measurable region ofT".Wedenote bytD(T) theamount oftime that theinterval 03t5Tofthetrajectory q>(t)isinside ofD.Then r,,(T) mesDl'i‘3.T=w=r~ PROOF. Weapply thetheorem tothecharacteristic function fofthesetD (fisRiemann integrable since DisJordan measurable). Then [5f(q>(t))dt = rD(T), andf=(21z)"' mesD,andthecorollary follows immediately from thetheorem. Cl Corollary. Inthesequence 1,2.4,8, 1,3,6, 1,2,5, 1,2,... o_/‘ﬁrst digits ofthenumbers 2",thenumber 7appears (log8—log7)/(log 9—log8)times as often as8. Thetheorem onaverages may befound implicitly inthework ofLaplace, Lagrange, andGauss oncelestial mechanics; itisoneoftheﬁrst“ergodic theorems.” Arigorous proof wasgiven only in1909 byP.Bohl, W.Sierpinski, andH.Weyl inconnection with aproblem ofLagrange onthemean motion oftheearth’s perihelion. Below wereproduce H.Weyl’s proof. CProof ofthetheorem onaverages Lemma 1.Thetheorem istrueforexponentials f=e“"""’, keZ”. PROOF. Ifk=0,thenf=f=f*=1andthetheorem isobvious. Ifk¢O, then f=O.Ontheother hand, T ei(k,m)T _1 Iei(k.o.i+wt) dt=eitluoot M 0 i(k.w) 287 10:Introduction toperturbation theory Therefore, thetimeaverage is I ei(|l.lP0) ei(k,m)T _1 lim _§—i-— =O. [:1Tau, zﬂt,co) T Lemma 2.Thetheorem istruefortrigonometric polynomials f:Zfkei(k.o>_ fit]<N PROOF. Both thetime andspace averages depend linearly onf,andtherefore agree byLemma 1. El Lemma 3.Letfbearealcontinuous (oratleast Riemann integrable) function. Then, forany8>O,there exist twotrigonometric polynomials P1andP2 such thatP1<f<P2and(l/(21t)")fT.. (P2—P1)d(p 5s. PROOF. Suppose ﬁrstthatfiscontinuous. BytheWeierstrass theorem, we canapproximate fbyatrigonometric polynomial Pwith If—PI<2.2. Thepolynomials P1=P—21;andP2=P+28aretheones wearelooking for. Iffisnotcontinuous butRiemann integrable, then there aretwocontinu- ousfunctionsfl andf2such thatfl<f<f2and(21r)_" I(f2——f,)dq> <':1.§8 (Figure 222corresponds tothecharacteristic function ofaninterval). By approximating fland f2bypolynomials P,<fl<f2<P2, (2rt)‘" I(P2—f2)dq) <§s,(21t)'" I(fl—P,)dtp <§e,weobtain what we need. Lemma 3isproved. U 1':1'1 _LLiii Pl Figure 222 Approximation ofthefunction fbytrigonometric polynomials P,andP2 Itisnow easy toﬁnish theproof ofthetheorem. Let8>0.Then, byLemma 3,there aretrigonometric polynomials P,<f<P2with (2n:)'" j(P2—P1)dq> <s. ForanyT,wethen have %fTP.<¢<v>dt <2-Tf(¢(t))dr <§fTP.<<i»<t>>di. O 0 ByLemma 2,forT>T0(s), iP,-%VP,-(q>(t))dt <.9(i=1,2). 0 288 51:Averaging Furthermore, P,<f< P2andP2—P,<.2.Therefore, P2—f< sand f—P,<a;therefore, forT>T,,(s), I}fTf(¢(t))dt -fl<28. 0 aswastobeproved. [:1 PROBLEM. Atwo-dimensional oscillator with kinetic energy T=2&2+2}”and potential energy U=2x2+yzperforms anoscillation with amplitudes a,=landa,=l.Find the timeaverage ofthekinetic energy. PROBLEM.” Letwkbeindependent, a,>0.Calculate _I 3 .lim—argZa,e'°"".tI-m lc=l ANSWER. (amt, +(1)2012 +(e313)/rt, where 01,,0:2,and13aretheangles ofthetriangle with sides ak(Figure 223). OJ) 003 /\ (X1 . O ll] U3 (*3! . (X3 al (X3 ? . _ Figure 223 Problem onmean motion ofperihelia DDegeneracies Sofarwehave considered thecasewhen thefrequencies toareindependent. Anintegral vector kEZ"iscalled arelation among thefrequencies if (k,0))=O. PROBLEM. Show thatthesetofallrelations between agiven setoffrequencies 0)isasubgroup Fofthelattice Z". WesawinSection 49that such asubgroup consists entirely oflinear combinations ofrindependent vectors k,,1srgn.Wesaythatthere are r(independent) relations among thefrequencies.” 9°Lagrange showed thattheinvestigation oftheaverage motion oftheperihelion ofaplanet reduces toasimilar problem. Thesolution ofthisproblem canbefound inthework ofI-I.Weyl. Theeccentricity ofthe earth’s orbit varies asthemodulus ofananalogous sum. Iceagesappear toberelated tothese changes ineccentricity. "‘Show thatthenumber rdoes notdepend onthechoice ofindependent vectors k,-. 289 10:Introduction toperturbation theory PROBLEM. Show thattheclosure ofatrajectory {¢p(t) =oo+cot}(onT")isatorus ofdimen- sionn—rifthere arerindependent relations among thefrequencies co;inthiscasethemotion onT"" isconditionally-periodic with n—rindependent frequencies. Weturn now totheintegrable hamiltonian system given inaction-angle variables I,tpbytheequations . , 0HI=0 tp=10(1), where o)(l) =—aT. Every n-dimensional torus I=const inthe2n-dimensional phase space is invariant, andmotion onitisconditionally-periodic. Deﬁnition. Asystem iscalled nondegenerate ifthedeterminant 60) 62H d€t H =det W isnotzero. PROBLEM. Show that. ifasystem isnondegenerate. then inanyneighborhood ofany point there isaconditionally-periodic motion with nfrequencies, andalso with anysmaller number of frequencies. Hint. Wecantake thefrequencies tothemselves instead ofthevariables Iaslocal coordinates. Inthespace ofcollections offrequencies. thesetofpoints towith anynumber ofrelations r(0Sr<n)isdense. Corollary. Ifasystem isnondegenerate, then theinvariant toriI=const areuniquely deﬁned, independent ofthechoice ofaction-angle coordinates I,(p,theconstruction ofwhich always involves some arbitrariness.” PROOF. The toriI=const canbedeﬁned astheclosures ofthephase tra- jectories corresponding totheindependent 0). U Wenote incidentally that, forthemajority ofvalues I,thefrequencies towillbeindependent. PROBLEM. Show thatthesetofIforwhich thefrequencies l1)(|) inanondegenerate system are dependent hasLebesgue measure equal tozero. Hint. Show ﬁrstthat mes{oJ:3k ab0,(m, k)=O}=O. Ontheother hand, indegenerate systems wecanconstruct systems of action-angle variables such that thetoriI=const willbedifferent indif- ferent systems. This isthecase because theclosures oftrajectories ina degenerate system aretoriofdimension k<n,andtheycanbecontained indifferent ways inn-dimensional tori. 92For example. wecan always write thesubstitution I’=I,(p'=(p+S,(I). orl,,I2; ‘Pp‘P2“l1'l'I2~ l2¥<Pi~ ‘P2_‘P1- 290 52:Averaging ofperturbations EXAMPLE 1.Theplanar harmonic oscillator x=—x; n=2,k=1.Separa- tionofvariables incartesian andpolar coordinates leads todifferent action- angle variables anddifferent tori. EXAMPLE 2.Keplerian planar motion (U=-1/r), n=2,k=1.Here, too,separation ofvariables inpolar andinelliptical coordinates leads to different I. 52Averaging ofperturbations Here weshow theadiabatic invariance oftheaction variable inasystem with onedegree of freedom. ASystems close tointegrable ones Wehave considered agreat many integrable systems (one-dimensional problems, thetwo-body problem, small oscillations, theEuler andLagrange cases ofthemotion ofarigid body with aﬁxed point, etc.). Westudied the characteristics ofphase trajectories inthese systems: they turned outtobe “windings oftori,” densely ﬁlling uptheinvariant toriinphase space; every trajectory isuniformly distributed onthistorus. One should notconclude from this that integrability isthetypical situation. Actually, theproperties oftrajectories inmany-dimensional systems canbehighly diverse andnotatallsimilar totheproperties of conditionally-periodic motions. Inparticular, theclosure ofatrajectory ofasystem with ndegrees offreedom canﬁllupcomplicated setsofdimension greater than nin2n-dimensional phase space; atrajectory could even be dense anduniformly distributed onawhole (2n—1)-dimensional manifold given bytheequation H=h.”One may callsuch systems “nonintegrable” since they donotadmit single-valued ﬁrst integrals independent ofH. Thestudy ofsuch systems isstillfarfrom complete; itconstitutes aproblem in“ergodic theory.” One approach tononintegrable systems istostudy systems which are close tointegrable ones. Forexample, theproblem ofthemotion ofplanets around thesunisclose totheintegrable problem ofthemotion ofnon- interacting points around astationary center; other examples aretheprob- lemofthemotion ofaslightly nonsymmetric heavy topandtheproblem of nonlinear oscillations close toanequilibrium position (thenearby integrable problem islinear). The following method isespecially fruitful inthein- vestigation ofthese andsimilar problems. BTheaveraging principle LetI,q)beaction-angle variables inanintegrable (“nonperturbed”) system with hamiltonian function H,,(I): i=0 ¢=o)(I) ti>(t)=6(%. 93Forexample, inertial motion onamanifold ofnegative curvature hasthisproperty. 291 IO:Introduction toperturbation theory Asthenearby “perturbed” system wetake thesystem <1) =w(l)+em.<0)i=ago.(P), where e<1. Wewillignore forawhile that thesystem ishamiltonian andconsider anarbitrary system ofdifferential equations intheform (l)given onthedirect product T"><Gofthek-dimensional torus T"={cp=(<p,,...,<p,,)mod 21:} andaregion Ginl-dimensional space GcIR’={I=(I,,...,I,)}. For 1»:=0themotion in(1)isconditionally-periodic with atmost kfrequencies andwithk-dimensional invariant tori. The averaging principle forsystem (1)consists ofitsreplacement by another system, called theaveraged system: _ 212 21:<2)J=@2o> at-I>=(2n"*j jg<J.¢>d¢....-.d<i». 0 0 inthel-dimensional region GcIR‘={J=(J,,...,J,)}. Weclaim thatsystem (2)isa“good approximation” tosystem (1). Wenote thatthisprinciple isneither atheorem, anaxiom, noradeﬁnition, butrather aphysical proposition, i.e.,avaguely formulated and, strictly speaking, untrue assertion. Such assertions areoften fruitful sources of mathematical theorems. This averaging principle may befound explicitly inthework ofGauss (instudying theperturbations ofplanets ononeanother, Gauss proposed todistribute themass ofeach planet around itsorbit proportionally totime andtoreplace theattraction ofeach planet bytheattraction oftheringso obtained). Nevertheless, asatisfactory description oftheconnection between thesolutions ofsystems (1)and(2)inthegeneral casehasnotyetbeen found. Inreplacing system (1)bysystem (2)wediscard theterm eg(I, qr)= eg(I, q))—ag(l) ontheright-hand side. This term hasorder easdoes the remaining term cg.Inorder tounderstand thedifferent roles oftheterms gandging,weconsider thesimplest example. PROBLEM. Consider thecasek=l=1, q'>=wré0 l=8y(</>)- Show thatforO<t<1/s, |I(t)—J(t)| <ca,where J(t)=[(0)+sgt. Solution I I 8 wt 8 I(t)H1(0)=J‘t:g(</2,, +cot)dt =fag dt+—I§(<p)d(p =agt+——h(o.n) 0 0 ‘D0 0) where h((p) =jgg(<p)d\I ll”J11) I I~El I Figure 224 Evolution andoscillation Thus thevariation inIwith time consists oftwoparts: anoscillation of order sdepending ongand asystematic “evolution” with velocity ag (Figure 224). The averaging principle isbased ontheassertion that inthegeneral case themotion ofsystem (1)canbedivided into the“evolution” (2)and small oscillations. Initsgeneral form, thisassertion isinvalid andtheprinciple itself isuntrue. Nevertheless, wewillapply theprinciple tothehamiltonian system (1): ¢l>=—(%(H0(l) +eH,(l, <P)) l=3%(H0(l) +-EH10. (Pl)- Fortheright-hand sideoftheaveraged system (2)wethen obtain 2"a -2_,WI=_gon a¢H1(,¢)dq’ 0 Inother words, there isnoevolution inanondegenerate hamiltonian system. One variant ofthis entirely nonrigorous deduction leads totheso- called Laplace theorem: The semi-major axes ofthekeplerian ellipses of theplanets have nosecular perturbations. The discussion above sufﬁces toconvince usoftheimportance ofthe averaging principle; wenow formulate atheorem justifying thisprinciple inonevery particular case—that ofsingle-frequency oscillations (k=1). Thistheorem shows thattheaveraging principle correctly describes evolution over alarge interval oftime (0<t<1/s). CAveraging inasingle-frequency system Consider thesystem ofl+1differential equations (1) ¢=co(l)+g(t,(p) (0mod21¢Es1, l=8g(l,(p) leGclR', where f(I,tp+21:)Ef(I,go)and g(I,(p+21:)Eg(I,(p),together with the “averaged” system oflequations . 1 21! (2) J=820). Where 2(J)=ﬁj g(J.<t>)d¢- 0 293 IO:Introduction toperturbation theory Gmd Figure 225 Theorem onaveraging Wedenote byI(t), (p(t) thesolution ofsystem (1)with initial conditions 1(0), (p(()), andbyJ(t)thesolution ofsystem (2)with thesame initial con- ditions J(0) =1(0)(Figure 225). Theorem. Suppose that: 1.thefunctions to,f,andgaredeﬁned forIinabounded region G,andin thisregion theyarebounded, together with their derivatives uptosecond order: llwﬁf,gllc1(c><s1)< C1; 2.intheregion G,wehave w(I) >c>0; 3.for05t31/s,aneighborhood ofradius dofthepoint J(t)belongs toG: J(t)eG—d. Then forsuﬂlciently small 8(O<s<£0) 1|l(t)—J(t)|<C98, forallt,0gtsE, where theconstant C9>0depends onc,,c,andd,butnotons. Some applications ofthistheorem willbegiven below (“adiabatic in- variants”). Weremark that thebasic idea oftheproof ofthistheorem (achange ofvariables diminishing theperturbation) ismore important than thetheorem itself; thisisoneofthebasic ideas inthetheory ofordinary differential equations; itisencountered inelementary courses asthe“method ofvariation ofconstants.” DProof ofthetheorem onaveraging Inplace ofthevariables Iwewillintroduce newvariables P (3) P=I+::k(I, tp), where thefunction k,21:-periodic in(p,willbechosen sothat thevector P willsatisfy asimpler differential equation. 294 52:Averaging ofperturbations By(1)and(3),therateofchange ofP(t)is ' v ~ . .. “k. Fk Pk Pk 5kP=I‘_1_"=I _I 1~ 1_. (4) +85, +8a¢<t> 8[g(.¢)+6(pw()j+H0,2+@a¢f Weassume thatthesubstitution (3)canbeinverted, sothat (5) I=P+8h(P,</1,8) (where thefunctions hare21:-periodic in(p). Then (4)and(5)imply thatP(t)satisﬁes thesystem ofequations to P=alga:+57';w(P)j+R. where the“remainder term” Rissmall ofsecond order with respect toa: (7) lRl<C232, C2(¢'1i C3,C4)>O, ifonly (8) llwllcl <C1llfllcl <C1llgllcl<C1llkllcl<C3llhllcl<@4- Wewillnow trytochoose thechange ofvariables (3)sothat theterm involving sin(6)becomes zero. Forkwegettheequation (‘k 1n"‘sg Ingeneral, such anequation isnotsolvable intheclass offunctions k periodic ingo.Infact,theaverage value (with respect to(p)oftheleft-hand side isalways equal to0,andtheaverage value oftheright-hand sidecanbe different from O.Therefore, wecannot choose kinsuch awayastokillthe entire term involving sin(6).However, wecankilltheentire “periodic” partofg, é(P,<t1) =g(P.<0)—2(P), bysetting <9) k(P.=—j:§%;’T‘;’3d<r- Sowedeﬁne thefunction kbyformula (9).Then, byhypotheses 1.and 2.ofthetheorem, thefunction ksatisﬁes theestimate llkllcl <c3,where c3(c,, c)>0.Inorder toestablish theinequality (8),wemust estimate h. Forthiswemust ﬁrstshow thatthesubstitution (3)isinvertible. Fixapositive number Oi. Lemma. Ifsissuﬂlcientl ysmall, thentherestriction ofthemapping (3)94 I—+I+ck, where lk|C=,G, <c3, 9‘Foranyﬁxed value oftheparameter (p. 295 10:Introduction toperturbation theory totheregion G—at(consisting ofpoints whose at-neighborhood iscontained inG)isadijfeomorphism. The inverse diﬂeomorphism (5)intheregion G—2ozsatisﬁes theestimate ||h||,;2 <c4withsome constant c4(a, c3)>0. PROOF. The necessary estimate follows directly from theimplicit function theorem. Theonly difﬁculty isinverifying thatthemap I-+I+skisone- to-one intheregion G—oz.Wenote thatthefunction ksatisﬁes aLipschitz condition (with some constant L(a,c3)) inG—a.Consider twopoints I,,I2inG-oi.Forsufficiently small s(namely, forLa<1)thedistance between cI{(I,) and t2I((I2) will besmaller than II,—I2|. Therefore, I,+t:k(I,) 7':I2+sk(I2). Thus themap (3)isone-to-one onG—oi,and thelemma isproved. l:l Itfollows from thelemma that forssmall enough alltheestimates (8) aresatisﬁed. Thus theestimate (7)isalsotrue. Wenow compare thesystem ofdifferential equations forJ <2) J=ego) andforP;thelatter, inview of(9),takes theform (6') P=8g(P)+R. Since thedifference between theright sides isoforder $s2(cf.(7)),fortime t$1/sthedifference IP—JIbetween thesolutions isoforder s(Figure 226). Ontheother hand, ll—P|=s|k|$s.Thus, fort3l/c,thedifference II—J|isoforder gs,aswastobeproved. El (}—or 1(1) P(t) J(t) o Figure 226 Proof ofthetheorem onaveraging Toﬁndanaccurate estimate. weintroduce thequantity (10) z(t):P(t) —J(t). Then (6')and(9)imply z=i;(g(P) -gm)+R=llii1+R’, I; where |R’|<c2s2+c,slz| ifthesegment (P,J)liesinG—:1.Under thisassumption weﬁnd (ll) Ii]<c,,i:|z| +c2+:2 (Wh€f€L‘6=(‘5 +<',) |z(O)| <('38. 296 S2:Averaging ofperturbations Lemma. If|il 5a|l|+band |z(O)| <dfor a,b,d,t>O,thenlz(r)| 5(d+b.t)e‘", PROOF. lz(t)| isnogreater than thesolutioii y(t)oftheequation _i"=ay+b,y(0)=d.Solving thisequation, weﬁndy=Ce“. Ce”' =b,C=e“"b, (‘(0) =d,C311'+bt. Cl Now from (Il)and theassumption thatthesegment (P,J)liesinG—a(Figure 226), wehave |l(t)l <(c3s+c2s2t)e‘“'. From thisitfollows that,for05t51/z, |z(r)l <c-,s c-,=(c,+c2)e“. Weseethat, ifa=d/3andsissmall enough, theentire segment (P(t), J(t))(t 5l/s)liesinside G—orand, therefore, llP(t) --J(t)l <(:58 forall05I3F. Ontheother hand, |P(r) —I(t)| <|sk|<t-,2.Thus, foralltwith 05t5I/a, lI(l) —Jllll <F93 ('9=cs‘I’V3>0 andthetheorem isproved. El EAdiabatic invariants Consider ahamiltonian system withonedegree offreedom, with hamiltonian function H(p,q;,1)depending onaparameter 1.Asanexample, wecantake apendulum: 2 2_L L.H'212+lg2’ astheparameter /1wecantake thelength lortheacceleration ofgravity g. Suppose that theparameter changes slowly with time. Itturns outthat in thelimit astherateofchange oftheparameter approaches 0,there isa remarkable asymptotic phenomenon: twoquantities, generally independent, become functions ofoneanother. Assume, forexample, that thelength ofthependulum changes slowly (incomparison with itscharacteristic oscillations). Then theamplitude ofitsoscillation becomes afunction ofthelength ofthependulum. Ifwe veryslowly increase byafactor oftwothelength ofthependulum andthen very slowly decrease ittotheoriginal value, then attheendofthisprocess theamplitude oftheoscillation willbethesame asitwasatthestart. Furthermore, itturns outthattheratio oftheenergy Hofthependulum tothefrequency tochanges very little under aslow change oftheparameter, although theenergy andfrequency themselves maychange alot.Quantities such asthisratio, which change little under slow changes ofparameter, arecalled byphysicists adiabatic invariants. Itiseasy toseethat theadiabatic invariance oftheratio oftheenergy ofapendulum toitsfrequency isanassertion ofaphysical character, i.e.,itis untrue without further assumptions. Infact, ifwevary thelength ofa pendulum arbitrarily slowly, butchose thephase ofoscillation under which 297 IO:Introduction toperturbation theory 7777a / / / Figure 227 Adiabatic change inthelength ofapendulum thelength increases and decreases, wecan setthependulum swinging (parametric resonance). Inview ofthis,physicists have suggested formulating thedeﬁnition ofadiabatic invariance asfollows: theperson changing the parameters ofthesystem must notseewhat state thesystem isin(Figure 227). Giving thisdeﬁnition arigorous mathematical meaning isavery delicate andasyetunsolved problem. Fortunately, wecangetalong with asurrogate. Theassumption ofignorance oftheinternal state ofthesystem onthepart oftheperson controlling theparameter may bereplaced bytherequirement that thechange ofparameter must besmooth, i.e.,twice continuously differentiable. More precisely, letH(p, q;2)beaﬁxed, twice continuously differentiable function of,1.SetI=stand consider theresulting system with slowly varying parameter 2=st: , an _an(*) i>=—~— q=—. H=H(i>.q;@t)-511 5:» Deﬁnition. Thequantity I(p,q;/1)isanadiabatic invariant ofthesystem (*) iffor every K>0there isanso>0such thatif0<s<soand0<t<1/s, then H(p(t), q(t);8!)—1(P(0). q(0);0)|<K- Clearly, every ﬁrstintegral isalsoanadiabatic invariant. Itturns outthat every one-dimensional system (*)hasanadiabatic invariant. Namely, the adiabatic invariant istheaction variable inthecorresponding problem with constant coefficients. Assume that thephase trajectories ofthesystem with hamiltonian H(p, q;1)areclosed. Wedeﬁne afunction I(p,q;2.)inthefollowing way. Forﬁxed /Ithere isaphase portrait corresponding tothehamiltonian function H(p,q;2)(Figure 228). Consider theclosed phase trajectory passing through apoint (p,q).Itbounds some region inthephase plane. Wedenote thearea ofthis region by2rtI(p,q;A).I=const onevery phase trajectory (for given /I).Clearly, Iisnothing buttheaction variable (cf.Section 50). Theorem. Ifthefrequency co(I,2)ofthesystem (*)isnowhere zero, then I(p,q;2)isanadiabatic invariant. 298 52:Averaging ofperturbations /J Itfixed ’/am Figure 228 Adiabatic invariant ofaone-dimensional system(I FProof oftheadiabatic invariance ofaction Forﬁxed 2wecanintroduce action-angle variables-1, <pintothesystem (*) byacanonical transformation depending onA:p,q-+I,zp;(p=w(1,/1), 1=0;to(1,2)=6H,,/61, H0=H,,(1, A). Wedenote byS(I,q;/1)the(multiple-valued) generating function ofthis transformation : _§s _as ”‘aq "’*a1' Now letA=st.Since thechange from variables p,qtovariables I,(pisnow performed byatime dependent canonical transformation, theequations of motion inthenew variables I,(phave thehamiltonian form, butwith hamiltonian function (cf.Section 45A) OS OS PROBLEM. Show that0S(1, q;A)/8/I isasingle-valued function onthephase plane. Hint. Sisdetermined uptotheaddition ofmultiples of21:1. Inthiswayweobtain theequations ofmotion intheform _ 52S ‘P=w(I=/l)+*f(1i‘P§/1) ! 2- 5S I=egg, (P;/1) Q=— , i=i-: Since cu¢0,theaveraging theorem (Section 52C) isapplicable. The averaged system hastheform .1=tag ==s. Butg=(5/5(p)(5S/5/I), and8S/8/I isasingle-valued function onthecircle I=const. Therefore, g=(2rt)" jgdtp=0,andintheaveraged system J does notchange atall:J(t)=J(0). 299 10:Introduction toperturbation theory Bytheaveraging theorem, |I(t) ——I(0)| <csforalltwith 0sts1/s, aswastobeproved. El EXAMPLE. Foraharmonic oscillator (cf.Figure 217), (12 bl, 1\/E,/2% h H =Z 2+ Z IZ —— ii i» Z —, Z b 2p 2q 21:na b to w a’ i.e.,theratio ofenergy tofrequency isanadiabatic invariant. U I (I p ~ —~v--i——>IQ, Figure 229 Adiabatic invariant ofanabsolutely elastic ballbetween slowly changing walls PROBLEM. The length ofapendulum isslowly doubled (I=l0(1+st), 05t1;1/s).How does theamplitude qm,oftheoscillations vary? Solution. I=%l3’2g1/Zqﬁm; therefore, qmax(t)=qmax(0) Asasecond example, consider themotion ofaperfectly elastic rigid ball ofmass 1between perfectly elastic walls whose separation Islowly varies (Figure 229). Wemay consider thatapoint ismoving inan“inﬁnitely deep rectangular potential well,” andthat thephase trajectories arerectangles ofarea 2vl,where visthevelocity oftheball. Inthiscase theproduct vl ofthevelocity oftheballandthedistance between thewalls turns outtobe anadiabatic invariant.” Thus ifwemake thewalls twice asclose together, thevelocity oftheballdoubles, andifweseparate thewalls, thevelocity decreases. 95This does notformally follow from thetheorem, since thetheorem concerns smooth systems without shocks. Theproofofthe adiabatic invariance ofvlinthissystem isaninstructive elemen- taryproblem. 300 Appendix l:Riemannian curvature From asheet ofpaper, onecanform acone oracylinder, butitisimpossible toobtain apiece ofasphere without folding, stretching, orcutting. Thereason liesinthedifference between the“intrinsic geometries ”ofthese surfaces: no partofthesphere canbeisometrically mapped onto theplane. Theinvariant which distinguishes riemannian metrics iscalled riemannian curvature. Theriemannian curvature ofaplane iszero, andthecurvature of asphere ofradius Risequal toR”. Ifoneriemannian manifold canbeiso- metrically mapped toanother, then theriemannian curvature atcorrespond- ingpoints isthesame. Forexample, since acone orcylinder islocally iso- metric totheplane, theriemannian curvature ofthecone orcylinder atany point isequal tozero. Therefore, noregion ofacone orcylinder canbemapped isometrically toasphere. Theriemannian curvature ofamanifold hasavery important inﬂuence onthebehavior ofgeodesics onit,i.e.,onmotion inthecorresponding dynamical system. Iftheriemannian curvature ofamanifold ispositive (as onasphere orellipsoid), then nearby geodesics oscillate about oneanother inmost cases, andifthecurvature isnegative (asonthesurface ofahyper- boloid ofonesheet), geodesics rapidly diverge from oneanother. Inthisappendix wedeﬁne riemannian curvature andbrieﬂy discuss the properties ofgeodesics onmanifolds ofnegative curvature. Afurther treat- ment ofriemannian curvature canbefound inthebook, “Morse Theory” byJohn Milnor, Princeton University Press, 1963, and atreatment of geodesics onmanifolds ofnegative curvature inD.V.Anosov’s book, “Geodesic ﬂows onclosed riemannian manifolds with negative curvature,” Proceedings oftheSteklov Institute ofMathematics, No.90(1967), Am. Math. Soc., 1969. AParallel translation onsurfaces Thedeﬁnition ofriemannian curvature isbased ontheconstruction ofparallel translation ofvectors along curves onariemannian manifold. Webegin with thecase when thegiven riemannian manifold istwo- dimensional, i.e.,asurface, andthegiven curve isageodesic onthissurface. [See Carmo, Manfredo Perdigao do,Differential Geometry ofCurves and Surfaces, Prentice-Hall, 1976. (Translator’s note)] Parallel translation ofavector tangent tothesurface along ageodesic on thissurface isdeﬁned asfollows: thepoint oforigin ofthevector moves along thegeodesic, andthevector itself moves continuously sothatitsangle with thegeodesic anditslength remain constant. Bytranslating totheendpoint ofthegeodesic allvectors tangent tothesurface attheinitial point, weobtain amap from thetangent plane attheinitial point tothetangent plane atthe endpoint. This map islinear andisometric. Wenow deﬁne parallel translation ofavector onasurface along abroken lineconsisting ofseveral geodesic arcs(Figure 230). Inorder totranslate a vector along abroken line,wetranslate itfrom theﬁrstvertex tothesecond 301 Appendix l:Riemannian curvature Figure 230 Parallel translation along abroken geodesic along theﬁrstgeodesic arc,then translate thisvector along thesecond arc tothenext vertex, etc. PROBLEM. Given avector tangent tothesphere atonevertex ofaspherical triangle with three right angles, translate thisvector around thetriangle andback tothesame vertex. ANSWER. Asaresult ofthistranslation thetangent plane tothesphere attheinitial vertex will beturned byaright angle. Finally, parallel translation ofavector along anysmooth curve onasurface isdeﬁned byalimiting procedure, inwhich thecurve isapproximated by broken lines consisting ofgeodesic arcs. PROBLEM. Translate avector directed towards theNorth Pole andlocated atLeningrad (latitude A=60°)around the60th parallel andback toLeningrad, moving totheeast. ANswER. Thevector turns through theangle 21:(1-sinA),i.e.,approximately 50°tothewest. Thus thesizeoftheangle ofrotation isproportional tothearea bounded byourparallel, and thedirection ofrotation coincides withthedirection theorigin ofthevector isgoing around the North Pole. Hint. Itissufficient totranslate thevector along thesame circle onthecone formed bythe tangent lines tothemeridian, going through allthepoints ofthe parallel (Figure 231). Thiscone then canbeunrolled onto theplane, after which parallel translation onitssurface becomes ordinary parallel translation ontheplane. Figure 231 Parallel translation onthesphere 302 Appendix 1;Riemannian curvature EXAMPLE, Weconsider theupper half-plane y>Ooftheplane ofcomplex numbers z=x+iy withthemetric dsz=éadxz Edyz. .V Itiseasytocompute thatthegeodesics ofthis two-dimensional riemannian manifold arecircles andstraight lines perpendicular tothex-axis. Linear fractional transformations with real coefficients az+bz->—i cz+d areisometric transformations ofourmanifold, which iscalled theLobachevsky plane. PRQBLEM. Translate avector directed along theimaginary axisatthepoint z=itothepoint z=I+ialong thehorizontal line(dy=0)(Figure 232). ANSWER. Under translation bytthevector turns tradians inthedirection from they-axis towards thex-axis. —'>.\' Figure 232 Parallel translation ontheLobachevsky plane BThecurvature form Wewillnow deﬁne theriemannian curvature ateach point ofatwo-dimen- sional riemannian manifold (i.e.,asurface). Forthispurpose, wechoose an orientation ofoursurface inaneighborhood ofthepoint under consideration andconsider parallel translation ofvectors along theboundary ofasmall region Donoursurface. Itiseasy tocalculate thattheresult ofsuch atrans- lation isrotation byasmall angle. Wedenote thisangle by(,p(D) (thesignofthe angle isﬁxed bythechoice oforientation ofthesurface). Ifwedivide theregion Dintotwoparts D1andD2,theresult ofparallel translation along theboundary ofDcanbeobtained byﬁrstgoing around onepart, andthen theother. Thus, <P(D) =<P(Di) +<P(Dz), i.e.,theangle rpisanadditive function ofregions. When wechange thedirec- tionoftravel along theboundary, theangle gochanges sign. Itisnatural therefore torepresent <p(D) astheintegral over Dofasuitable 2-form. Such 303 Appendix l:Riemannian curvature a2-form infactexists; itiscalled thecurvature form, andwedenote itbyQ. Thus wedeﬁne thecurvature form Qbytherelation <1) <P(D)=IQ. D Thevalue ofQonapairoftangent vectors 6,ninTMxcanbedeﬁned inthe following way. Weidentify aneighborhood ofthepoint Ointhetangent space toMatxwith aneighborhood ofthepoint xonM(using, forexample, some local coordinate system). Wecanthenconstruct onMtheparallelogram 1'1,spanned bythevectors £6,an,atleast forsuﬁiciently small s. Now thevalue ofthecurvature form onourvectors isdeﬁned bythe formula H<2) areto=lim s—>O 8 Inother words, thevalue ofthecurvature form onapairoftangent vectors isequal totheangle ofrotation under translation along theinﬁnitely small parallelogram determined bythese vectors. PROBLEM. Find thecurvature forms ontheplane, onasphere ofradius R,andontheLobachevsky plane. ANSWER. Q=0,Q=R-2dS,Q=-dS, where the2-form dSisthearea element onour oriented surface. PROBLEM. Show thatthefunction deﬁned byformula (2)isreally adifferential 2-form, independent ofthearbitrary choice involved intheconstruction, andthattherotation ofavector under translation along theboundary ofaﬁnite oriented region Disexpressed, interms ofthisform, byformula (l). PRoBLEM. Show thattheintegral ofthecurvature form over anyconvex surface inthree-dimen- sional euclidean space isequal to4n. CTheriemannian curvature ofasurface We note that every differential 2-form onatwo-dimensional oriented riemannian manifold Mcanbewritten intheform pdS, where a’Sisthe oriented area element andpisascalar function uniquely determined bythe choice ofmetric andorientation. Inparticular, thecurvature form canbewritten intheform Q=KdS, where K:M—>Risasmooth function onManddSisthearea element. Thevalue ofthefunction Katapoint xiscalled theriemannian curvature ofthesurface atx. PROBLEM. Calculate theriemannian curvature oftheeuclidean space, thesphere ofradius R, andtheLobachevsky plane. 304......i.<i..imnn-a Appendix 1:Riemannian curvature ANswER.K =0,K=R‘1,K =—1. PROBLEM. Show thattheriemannian curvature does notdepend ontheorientation ofthe mani- fold, butonly onitsmetric. Hint. The2-forms QanddSboth change signunder achange oforientation. PROBLEM. Show that, forsurfaces inordinary three-dimensional euclidean space, theriemannian curvature atevery point isequal totheproduct ofthe inverses ofthe principal radii ofcurvature (with minus signifthecenters ofcurvature lieonopposite sides ofthesurface). Wenote thatthesignofamanifold’s curvature atapoint does notdepend ontheorientation ofthemanifold; thissignmay bedeﬁned without using the orientation atall. Namely, onmanifolds ofpositive curvature, avector parallel translated around theboundary ofasmall region turns around itsorigin inthesame direction asthepoint ontheboundary goes around theregion; onmanifolds ofnegative curvature thedirection ofrotation isopposite. Wenote further thatthevalue ofthecurvature atapoint isdetermined bythemetric inaneighborhood ofthispoint, andtherefore ispreserved under bending: thecurvature isthesame atcorresponding points ofiso- metric surfaces. Hence, riemannian curvature isalsocalled intrinsic curvature. The formulas forcomputing curvature interms ofcomponents ofthe metric insome coordinate system involve thesecond derivatives ofthemetric andarerather complicated: cf.theproblems inSection Gbelow. DHigher-dimensional parallel translation The construction ofparallel translation onriemannian manifolds ofdi- mension greater than two issomewhat more complicated than thetwo- dimensional construction presented above. The reason isthat inthese dimensions thedirection ofthevector being translated isnolonger determined bythecondition thattheangle withageodesic beinvariant. Infact,thevector could rotate around thedirection ofthegeodesic while preserving itsangle with thegeodesic. Thereﬁnement which wemust introduce intotheconstruction ofparallel translation along ageodesic isthechoice ofatwo-dimensional plane passing through thetangent tothegeodesic, which must contain thetranslated vector. This choice ismade inthefollowing (unfortunately complicated) way. Attheinitial point ofageodesic theneeded plane istheplane spanned by thevector tobetranslated andthedirection vector ofthegeodesic. Welook atallgeodesics proceeding from theinitial point, indirections lying inthis plane. Thesetofallsuch geodesics (close totheinitial point) forms a-smooth surface which contains thegeodesic along which weintend totranslate the vector (Figure 233). Consider anewpoint onthegeodesic atasmall distance Afrom theinitial point. Thetangent plane atthenewpoint tothesurface described above contains thedirection ofthegeodesic atthisnew point. Wetake thisnew 305 Appendix l:Riemannian curvature A Figure 233 Parallel translation inspace point astheinitial point anduseitstangent plane toconstruct anewsurface (formed bythebundle ofgeodesics emanating from thenew point). This surface contains theoriginal geodesic. Wemove along theoriginal geodesic again byAandrepeat theconstruction from thebeginning. After aﬁnite number ofsteps wecanreach anypoint oftheoriginal geo- desic. Asaresult ofourwork wehave, atevery point ofthegeodesic, atangent plane containing thedirection ofthegeodesic. This plane depends onthe length Aofthesteps inourconstruction. AsA—>0thefamily oftangent planes obtained converges (ascanbecalculated) toadeﬁnite limit. Asa result wehave aﬁeld oftwo-dimensional tangent planes along ourgeodesic containing thedirection ofthegeodesic and determined inanintrinsic manner bythemetric onthemanifold. Now parallel translation ofourvector along ageodesic isdeﬁned asinthe two-dimensional case: under translation thevector must remain intheplanes described above; itslength anditsangle with thedirection ofthegeodesic must bepreserved. Parallel translation along anycurve isdeﬁned using approximations bygeodesic polygons, asinthetwo-dimensional case. PROBLEM. Show thatparallel translation ofvectors from onepoint ofariemannian manifold toanother along aﬁxed path isalinear isometric operator from thetangent space attheﬁrst point tothetangent space atthesecond point. PROBLEM. Parallel translate anyvector along theline .\',=t x2=O y=l (Ogtgt) inaLobachevsky space with metric dszIdxf+dxg+dyz y2 ANSWER. Vectors inthedirections ofthe x,andyaxesarerotated byangle rintheplane spanned bythem (rotation isinthedirection from they-axis towards thex1-axis)1 vectors inthex2-direc- tionarecarried parallel tothemselves inthesense oftheeuclidean metric. EThecurvature tensor Wenow consider, asinthetwo-dimensional case, parallel translation along small closed paths beginning andending atapoint ofariemannian manifold. Parallel translation along such apath returns vectors totheoriginal tangent 306 Appendix l:Riemannian curvature space. Themap ofthetangent space toitself thus obtained isasmall rotation (anorthogonal transformation close totheidentity). inthetwo-dimensional casewecharacterized thisrotation byonenumber——the angle ofrotation <,/2.lnhigher dimensions askew-symmetric operator plays theroleolrp. Namely, anyorthogonal operator .4which isclose totheidentity canbewritten inanatural wayintheform -@_ if A-e -E+(D+N+ , where (Disasmall skew-symmetric operator. PRoBLEM. Compute (DifAisarotation oftheplane through asmall angle tp. ANSWER. cos sin (J A: p<0 (P (D2 <0. —S1ll rp cosrp -(p O Unlike inthetwo-dimensional case, thefunction (Disnotgenerally additive (since the orthogonal group ofn-space forn>2isnotcommutative). Nevertheless, wecanconstruct a curvature form using (D,describing the“inﬁnitely small rotation caused byparallel translation around aninﬁnitely small parallelogram” inthesame wayasinthetwo-dimensional case, i.e., using formula (2). Thus, let§and11inTM, bevectors tangent totheriemannian manifold Matthepoint x.Construct asmall curvilinear parallelogram 1'1,onM(the sides oftheparallelogram TI,areobtained from thevectors sfandanbya coordinate identiﬁcation ofaneighborhood ofzeroinTM, with aneighbor- hood ofxinM).Wewilllook atparallel translation along thesides ofthe parallelogram IT,(webegin thecircuit atf). Theresult oftranslation willbeanorthogonal transformation ofTMX, close totheidentity. Itdiffers from theidentity transformation byaquantity oforder £2andhastheform A46,11)=E+829+0(9), where Qisaskew-symmetric operator depending on6andn.Therefore, we candeﬁne afunction Qofpairs ofvectors 5,:1inthetangent space atxwith values inthespace ofskew-symmetric operators onTM, bytheformula ea.in=hm—8.T- 0 3-» PROBLEM. Show thatthefunction Qisadifferential 2-form (with values intheskew-symmetric operators onTMX) anddoes notdepend onthechoice ofcoordinates weused toidentify TM, andM. The form Qiscalled thecurvature tensor oftheriemannian manifold. Wecould saythatthecurvature tensor describes theinﬁnitesimal rotation inthetangent space obtained byparallel translation around aninﬁnitely small parallelogram. 307 Appendix l:Riemannian curvature FCurvature inatwo-dimensional direction Consider atwo-dimensional subspace Linthetangent space toariemannian manifold atsome point. Wetake geodesics emanating from thispoint in allthedirections inL.These geodesics form asmooth surface close toour point. Thesurface constructed liesintheriemannian manifold andhasan induced riemannian metric. Bythecurvature ofariemannian manifold Minthedirection ofa2-plane Linthetangent space toMatapoint x,wemean theriemannian curvature at xofthesurface described above. PROBLEM. Find thecurvatures ofathree-dimensional sphere ofradius RandofLobachevsky space inallpossible two-dimensional directions. ANSWER. R-2, —1. Ingeneral, thecurvatures ofariemannian manifold indifferent two- dimensional directions aredifferent. Their dependence onthedirection is described byformula (3)below. Theorem. The curvature ofariemannian manifold inthetwo-dimensional direction determined byapairoforthogonal vectors 5,i1oflength 1canbe expressed interms ofthecurvature tensor Qbytheformula (3) K=<Q(€,t1)€,11>, where thebrackets denote thescalar product giving theriemannian metric. Theproof isobtained bycomparing thedeﬁnitions ofthecurvature tensor andofcurvature inatwo-dimensional direction. Wewillnotgointoitinarigorous way. Itispossible totake formula (3)forthedeﬁnition ofthe curvature K. GCovariant diﬂerentiation Connected with parallel translation along curves inariemannian manifold isaparticular differential calculus—so-called covariant differentiation, or theriemannian connection. Wedeﬁne thisdifferentiation inthefollowing way. Let5beavector tangent toariemannian manifold Matapoint x,andv avector ﬁeld given onMinaneighborhood ofx.Thecovariant derivative oftheﬁeldvinthedirection Eisdeﬁned byusing anycurve passing through x with velocity 5.After moving along thiscurve forasmall interval oftime t, weﬁndourselves atanewpoint x(t). Wetake thevector ﬁeld vatthispoint x(t)andparallel translate itbackwards along thecurve totheoriginal point x.Weobtain avector depending ontinthetangent space toMatx.For t=0thisvector isv(x), andforother titchanges according tothenon- parallelness ofthevector ﬁeld valong ourcurve inthedirection 5. 308 Appendix l:Riemannian curvature Consider thederivative oftheresulting vector with respect tot,evaluated att=0.This derivative isavector inthetangent space TM,,. Itiscalled the covariant derivative oftheﬁeld valong 6and isdenoted byV;v.Itiseasy to verify thatthevector Vgvdoes notdepend onthechoice ofcurve speciﬁed in thedeﬁnition, butonly onCandv. PROBLEM l.Prove thefollowing properties ofcovariant differentiation: 1.Vévisabilinear function oféandv. 2.Véfr =(L¢f)c +f(x)V;u, where fisasmooth function andLgfisthederivative offinthe direction ofthevector 5inTM,. 3.L_5(v, w)=(Viv, w(x)> +<v(x), Vgw). _ 4.Vvmw —Vwmv =[w,v](x) (where L[,,,,,,] =L,L,—LwLu). PROBLEM 2.Show thatthecurvature tensor canbeexpressed interms ofcovariant differentiation inthefollowing way: Qfio» 'lo)§o =—Vzvii +Vrrvég ‘l’Vtmué» where 5,ri,Q’areanyvector ﬁelds whose values atthepoint under consideration areC0,no,and(0. PROBLEM 3.Show thatthecurvature tensor satisﬁes thefollowing identities: Q(€J1)C +Q(Y1.§)€ +Q(C,5)"=O <Q(€,'1)<1,/3) =(901. BK.'1)- PROBLEM 4.Suppose thattheriemannian metric isgiven inlocal coordinates xl,...,x,,bythe symmetric matrix g,jI (182 =Z gijdx,-dxj. Denote bye,,...,e,,thecoordinate vector ﬁelds (sothatdifferentiation inthedirection e,»is 0,=0/dx,). Then covariant derivatives canbecalculated using theformulas inProblem land thefollowing formulas: Vt,‘-’, =Zrhea rfj=Z”l((3tQ,'i ‘l’ajgil —5t9rj)9“~It 1 where (g“‘)istheinverse matrix to(g,,,). Byusing theexpression forthecurvature tensor interms oftheconnection inProblem 2, wealsoobtain anexplicit formula forthecurvature. Thenumbers RU“ =(Q(e,-, e,»)e,,, e,)are called thecomponents ofthecurvature tensor. HTheJacobi equation Theriemannian curvature ofamanifold isclosely connected with thebe- havior ofitsgeodesics. Inparticular, letusconsider ageodesic passing through some point.in some direction, andalter slightly theinitial conditions, i.e.,theinitial point andinitial direction. Thenewinitial conditions determine anewgeodesic. Atﬁrstthisgeodesic differs very little from theoriginal geo- desic. Toinvestigate thedivergence itisuseful tolinearize thedifferential equation ofgeodesics close totheoriginal geodesic. Thesecond-order linear 309 Appendix l:Riemannian curvature differential equation thusobtained (“the variational equation ”fortheequa- tion ofgeodesics) iscalled theJacobi equation; itisconvenient towrite itinterms ofcovariant derivatives andcurvature tensors. Wedenote byx(t)apoint moving along ageodesic inthemanifold M with velocity (ofconstant magnitude) v(t)e TMx(,,. Iftheinitial condition depends smoothly onaparameter oz,then thegeodesic alsodepends smoothly ontheparameter. Consider themotion corresponding toavalue ofoz.We denote theposition ofapoint attime tonthecorresponding geodesic by x(t,oz)eM.Wewillassume thattheinitial geodesic corresponds tothezero value oftheparameter, sothatx(t,0)=x(t). The vector ﬁeld ofgeodesic variation isthederivative ofthefunction x(t,0!)with respect tooz,evaluated ator=0;thevalue ofthisﬁeldatthepoint x(t)isequal to d ‘dz a:oX([, (X)= ETMX(,). Towrite thevariational equation, wedeﬁne thecovariant derivative with respect totofavector ﬁeld Q(t)given onthegeodesic x(t).Todeﬁne this,we take thevector {(t+h),parallel translate itfrom thepoint x(t+h)to x(t)along thegeodesic, differentiate thevector obtained inthetangent space TMW, with respect tohandevaluate ath=0.The result isavector in TMxm,which iscalled thecovariant derivative oftheﬁeld ((r)with respect tot,anddenoted byDC/Dt. Theorem Thevector ﬁeld ofgeodesic variation satisﬁes thesecond-order linear diﬂerential equation D25 W = —Q(vs é)’-J» where Qisthecurvature tensor, andv=v(t)isthevelocity vector ofmotion along theoriginal geodesic. Conversely, every solution ofthedifferential equation (4)isaﬁeld of variation oftheoriginal geodesic. Equation (4)iscalled theJacobi equation. PROBLEM. Prove thetheorem above. PROBLEM. LetMbeasurface, y(t)themagnitude ofthe component ofthevector f(t)inthedirec- tionnormal toagiven geodesic, andletthelength ofthevector v(t)beequal tol.Show that_r satisﬁes thedifferential equation (5) if=—Ky. where K=K(t)istheriemannian curvature atthepoint x(t) PROBLEM. Using Equation (5),compare thebehavior ofgeodesics close toagiven oneonthe sphere (K=+R’2) andontheLobachevsky plane (K=~1). 310 Appendix 1:Riemannian curvature IInvestigation oftheJacobi equation Ininvestigating thevariational equations, itisuseful todisregard thetrivial variations, i.e.,changes ofthetime origin andofthemagnitude oftheinitial velocity ofmotion. Tothisendwedecompose thevariation vector 5into components parallel andperpendicular tothevelocity vector v.Then (since Q(v,v)=0andsince theoperator Q(v,5)isskew-symmetric) forthenormal component weagain gettheJacobi equation, andfortheparallel component wegettheequation D25—— :0.Dtz Wenow note thattheJacobi equation forthenormal component canbe written intheform of“Newton’s equation” D25-D? =-grad U, where thequadratic form Uofthevector 5isexpressed interms ofthecurva- ture tensor andisproportional tothecurvature Kinthedirection ofthe (5,v)plane: U(€)=%<Q(v, €)v,6)=%K<€, 5)(v,v>- Thus thebehavior ofthenormal component ofthevariation vector ofa geodesic with velocity 1canbedescribed bytheequation ofa(non-autono- mous) linear oscillator whose potential energy isequal totheproduct ofthe curvature inthedirection oftheplane ofvelocity vectors andvariations with thesquare ofthelength ofthenormal component ofthevariation. Inparticular weconsider thecase when thecurvature isnegative inall two-dimensional directions containing thevelocity vector ofthegeodesic (Figure 234). Then thedivergence ofnearby geodesics from thegiven onein x>0 K<0 ' / .\\\ Figure 234 Nearby geodesics onmanifolds ofpositive andnegative curvature 31l Appendix 1:Riemannian curvature thenormal direction canbedescribed bytheequation ofanoscillator with negative deﬁnite (and time-dependent) potential energy. Therefore, the normal component ofdivergence fornearby geodesics behaves likethedi- vergence ofaball,located near thetopofahill,from thetop.Theequilibrium position oftheballatthetopisunstable. This means thatgeodesics nearthe given geodesic willdiverge exponentially from it. Ifthe potential energy ofthenewtonian equation weobtained didnotdepend ontime, our conclusion would berigorous. Letusassume further thatthecurvature inthedifferent direc- tions containing visintheinterval —u2 5K5—b2. WhereO <b<a. Then solutions totheJacobi equation fornormal divergence willbelinear combinations of exponential curves with exponent 1,1,, where thepositive numbers /l,arebetween aandb. Therefore, every solution totheJacobi equation grows atleast asfastase""'aseither t—>+acort—>—1:;most solutions grow even faster, with ratee“"'. Theinstability ofanequilibrium position under negative deﬁnite potential energy isintuitively obvious also inthenon-autonomous case. Itcanbe proven bycomparison with acorresponding autonomous system. Asa result ofsuch acomparison wemay convince ourselves thatunder motion along ageodesic, allsolutions oftheJacobi equation fornormal divergence onamanifold ofnegative curvature grow atleast asfastasanexponential function ofthedistance traveled, whose exponent isequal tothesquare root oftheabsolute value ofthecurvature inthetwo-dimensional direction forwhich thisabsolute value isminimal. Infact, most solutions grow even faster, butwecannot now assert thattheexponent ofgrowth formost solu- tions isdetermined bythedirection inwhich theabsolute value ofthenega- tivecurvature islargest. Insummary, wecansaythatthebehavior ofgeodesics onamanifold of negative curvature ischaracterized byexponential instability. Fornumerical estimates ofthisinstability, itisuseful todeﬁne thecharacteristic path length sastheaverage path length onwhich small errors intheinitial conditions areincreased etimes. More precisely, thecharacteristic path length scanbedeﬁned astheinverse oftheexponent /1which characterizes thegrowth ofthesolution totheJacobi equation fornormal divergence from thegeodesic proceeding with velocity 1: _-1 1/l=lim—max max ln|5(t)| s=—.r-toT|»|<r|€(0ll=1 A Ingeneral, theexponent /1andthepath sdepend ontheinitial geodesic. Ifthecurvature ofourmanifold inalltwo-dimensional directions is bounded away from zero bythenumber —b2, then thecharacteristic path length islessthan orequal tob“1.Thus asthecurvature ofamanifold gets more negative, thecharacteristic path length s,onwhich theinstability of 312 Appendix 1:Riemannian curvature geodesics isreduced toe-fold growth oferror, getssmaller. Inview ofthe exponential character ofthegrowth oferror, thecourse ofageodesic ona manifold ofnegative curvature ispractically impossible topredict. Assume, forexample, that thecurvature isnegative andbounded away from zero by—4m'2. Thecharacteristic path length islessthan orequal to halfameter, i.e.,onageodesic areﬁvemeters long theerror grows byapproxi- mately e‘°~104.Therefore, anerror ofatenth ofamillimeter intheinitial conditions shows upintheform ofaone-meter difference attheendofthe geodesic. JGeodesic ﬂows oncompact manifolds of negative curvature LetMbeacompact riemannian manifold whose curvature atevery point inevery two-dimensional direction isnegative. (Such manifolds exist.) Consider theinertial motion ofapoint ofmass 1onM,without anyexternal forces. Thelagrangian function ofthissystem isequal tothekinetic energy, which isequal tothetotal energy andisaﬁrstintegral oftheequations of motion. IfMhasdimension n,then each energy level manifold hasdimension 2n—1.This manifold isasubmanifold ofthetangent bundle ofM.For example, wecanﬁxthevalue oftheenergy at%(which corresponds toinitial velocity 1).Then thevelocity vector ofthepoint haslength constantly equal to1,andourlevel manifold turns outtobetheﬁber bundle T1McTM consisting oftheunitspheres inthetangent spaces toMatevery point. Thus, apoint ofthemanifold T1M isrepresented asavector oflength 1atapoint ofM.BytheMaupertuis—Jacobi principle, wecandescribe the motion ofapoint mass with ﬁxed initial conditions inthefollowing way: thepoint moves with velocity 1along thegeodesic determined bytheindi- cated vector. Bythelawofconservation ofenergy themanifold TIM isaninvariant manifold inthephase space ofoursystem. Therefore, ourphase ﬂow de- termines aone-parameter group ofdiffeomorphisms onthe(2n—1)- dimensional manifold T1M. This group iscalled thegeodesic flow onM. Thegeodesic ﬂow canbedescribed asfollows: thetransformation attime t carries theunit vector 5eTIM located atthepoint x,totheunit velocity vector ofthegeodesic coming from xinthedirection 5,located atthepoint atdistance tfrom x.Wenote thatthere isanaturally deﬁned volume element onT,M andthatthegeodesic ﬂow preserves it(Liouville’s theorem). Uptonow wehave notused thenegative curvature ofthemanifold M. Butifweinvestigate thetrajectories ofthegeodesic ﬂow, itturns outthatthe negative curvature ofMhasastrong impact onthebehavior ofthese tra- jectories (this isrelated totheexponential instability ofgeodesics onM). 313 Appendix 1:Riemannian curvature Here aresome properties ofgeodesic ﬂows onmanifolds ofnegative curvature (forfurther details, seethebook ofD.V.Anosov cited earlier). 1.Almost allphase trajectories aredense intheenergy level manifold (the exceptional non-dense trajectories form asetofmeasure zero). 2.Uniform distribution: theamount oftime which almost every trajectory spends inanyregion ofthephase space TIM isproportional tothevolume oftheregion. 3.Thephase flowg‘hasthemixing property: ifAandBaretworegions, then limmes[(g’A) tnB]=mesAmesB t—>co (where mes denotes thevolume, normalized bythecondition that the whole space have measure 1). From these properties oftrajectories inphase space follow analogous statements about geodesics onthemanifold itself. Physicists callthese properties “stochastic”: asymptotically forlarge tthetrajectories behave as ifthepoint were random. Forexample, themixing property means thatthe probability ofturning upinBatatime tlong after exiting from Aispropor- tional tothevolume ofB. Thus, theexponential instability ofgeodesics onmanifolds ofnegative curvature leads tothestochasticity ofthecorresponding geodesic ﬂow. KOther applications ofexponential instability Theexponential instability property ofgeodesics onmanifolds ofnegative curvature hasbeen studied bymany authors, beginning withHadamard (and, inthecase ofconstant curvature, also byLobachevsky), butespecially by E.Hopf. Anunexpected discovery ofthe1960s inthisareawasthesurprising stability ofexponentially unstable systems with respect toperturbations ofthe systems themselves. Consider, forexample, thevector ﬁeld giving thegeodesic ﬂow onacom- pact surface ofnegative curvature. Asweshowed above, thephase curves ofthisﬂow arearranged inacomplicated way: almost every oneofthem is dense inthethree-dimensional energy level manifold. Theflow hasinﬁnitely many closed trajectories, andthesetofpoints onclosed trajectories isalso dense inthethree-dimensional energy level manifold. Wenow consider anearby vector ﬁeld. Itturns outthat, inspite ofthe complexity ofthepicture ofphase curves, theentire picture with dense phase curves andinﬁnitely many closed trajectories hardly changes atallif wepass tothenearby ﬁeld. Infact, there isahomeomorphism close tothe identity transformation which takes thephase curves oftheunperturbed ﬂow tothephase curves oftheperturbed ﬂow. Thus ourcomplicated phase ﬂow hasthesame property of“structural 314 Appendix 1:Riemannian curvaturc stability” asalimit cycle, orastable focus intheplane. Wenote thatneither acenter intheplane norawinding ofthetorus hasthisproperty ofstructural stability: thetopological type ofthephase portrait inthese cases changes forarbitrarily small changes inthevector ﬁeld. The existence ofstructurally stable systems with complicated motions, each ofwhich isinitself exponentially unstable, isoneofthebasic discoveries ofrecent years inthetheory ofordinary differential equations (the con- jecture thatgeodesic ﬂows onmanifolds ofnegative curvature arestructurally stable wasmade byS.Smale in1961, and theproof wasgiven byD.V. Anosov andpublished in1967; thebasic results onstochasticity ofthese ﬂows were obtained byYa.G.Sinai andD.V.Anosov, alsointhe1960s). Before these works most mathematicians believed that insystems of differential equations in“general form” only thesimplest stable limiting behaviors were possible: equilibrium positions andcycles. Ifasystem was more complicated (forexample, ifitwasconservative), then itwasassumed thatafter asmall change initsequations (forexample, after imposing small non-conservative perturbations) complicated motions are“dispersed” into simple ones. Wenow know thatthisisnotso,andthatinthefunction space ofvector ﬁelds there arewhole regions consisting ofﬁelds with more com- plicated behavior ofphase curves. Theconclusions which follow from thisarerelevant toawide range of phenomena, inwhich “stochastic” behavior ofdeterministic objects is observed. Namely, suppose that inthephase space ofsome (non-conservative) system there isanattracting invariant manifold (orset)inwhich thephase curves have theproperty ofexponential instability. Wenow know that systems with such aproperty arenotexceptional: under small changes ofthe system thisproperty must persist. What isseenbyanexperimenter observing motions ofsuch asystem‘? Theapproach ofphase curves toanattracting setwillbeinterpreted as theestablishment ofsome sortoflimiting conditions. Thefurther motion ofa phase point neartheattracting setwillinvolve chaotic, unpredictable changes of“phase” ofthelimiting behavior, perceptible as“stochasticity” or “turbulence.” Unfortunately, noconvincing analysis from thispoint ofview hasyet been developed forphysical examples ofaturbulent character. Aprimary example isthehydrodynamic instability ofaviscous ﬂuid, described bythe so-called Navier—Stokes equations. The phase space ofthis problem is inﬁnite-dimensional (itisthespace ofvector ﬁelds with divergence 0inthe domain ofﬂuid ﬂow), buttheinﬁnite-dimensionality oftheproblem is apparently notaserious obstacle, since theviscosity extinguishes thehigh harmonics (small vortices) faster andfaster astheharmonics arehigher and higher. Asaresult, thephase curves from theinﬁnite-dimensional space seem toapproach some ﬁnite-dimensional manifold (orset), towhich the limit regime alsobelongs. 315 Appendix 1:Riemannian curvature Forlarge viscosity, wehave astable attracting equilibrium position inthe phase space (“stable stationary ﬂow”). Astheviscosity decreases itloses sta- bility; forexample, astable limit cycle canappear inphase space (“periodic ﬂow”) orastable equilibrium position ofanewtype (“secondary stationary ﬂow”). 96Astheviscosity decreases further, more andmore harmonics come intoplay, andthelimit regime canbecome ever higher indimension. Forsmall viscosity, theapproach toalimit regime with exponentially unstable trajectories seems very likely. Unfortunately, thecorresponding calculations have notyetbeen carried outduetothelimited capacity of existing computers. However, thefollowing general conclusion canbedrawn without anycalculations: turbulent phenomena may appear even ifsolutions exist andareunique; exponential instability, which isencountered even in deterministic systems with aﬁnite number ofdegrees offreedom, issuﬂicient. Asonemore example ofanapplication ofexponential instability wemen- tion theproof announced byYa.G.Sinai ofthe“ergodic hypothesis” of Boltzmann forsystems ofrigid balls. Thehypothesis isthatthephase ﬂow corresponding tothemotion ofidentical absolutely elastic balls inaboxwith elastic walls isergodic onconnected energy level sets.(Ergodicity means that almost every phase curve spends anamount oftime inevery measurable piece ofthelevel setproportional tothemeasure ofthatpiece.) Boltzmann’s hypothesis allows ustoreplace time averages byspace averages, and was foralong time considered tobenecessary tojustify statistical mechanics. Inreality, Boltzmann’s hypothesis (inwhich itisa question ofalimit astime approaches inﬁnity) isnotnecessary forpassing tothestatistical limit (thenumber ofpieces approaches inﬁnity). However, Boltzmann’s hypothesis inspired theentire analysis ofthestochastic proper- tiesofdynamical systems (so-called ergodic theory), anditsproof serves asa measure ofthematurity ofthistheory. Theexponential instability oftrajectories inBoltzmann’s problem arises asaresult ofcollisions oftheballs with oneanother, andcanbeexplained inthefollowing way. Forsimplicity, wewillconsider asystem ofonly two particles intheplane, andwillrepresent asquare boxwith reﬂection oﬂthe walls bytheplanar torus {(x,y)mod 1}.Then wecanconsider oneofthepar- ticles asstationary (using theconservation ofmomentum); theother particle canbeconsidered asapoint. Inthiswaywearrive atthemodel problem ofmotion ofapoint onatoral billiard table with acircular wall inthemiddle from which thepoint isre- ﬂected according tothelaw“the angle ofincidence isequal totheangle of reﬂection ”(Figure 235). Toinvestigate thissystem welook atananalogous billiard table bounded ontheoutside byaplanar convex curve (e.g., themotion ofapoint inside an ellipse). Motion onsuch abilliard table canbeconsidered asthelimiting case ofthegeodesic ﬂow onthesurface ofanellipsoid. Passage tothelimit 9‘Amore detailed account oflossofstability isgiven in“Lectures onbifurcations andversal families,” Russian Math. Surveys 27,no.5(1972), 55-123. 316 Appendix l:Riemannian curvature Figure 235 Torus-shaped billiard table with scattering byacircular wall consists ofdecreasing thesmallest axisoftheellipsoid tozero. Asaresult, geodesics ontheellipsoid become billiard trajectories ontheellipse. We discover from thisthattheellipse canreasonably bethought ofastwo-sided andthat, under every reﬂection, thegeodesic goes from onesideoftheellipse totheother. Wenowreturn toourtoral billiard table. Motion onitcanbelooked atas thelimiting case ofthegeodesic ﬂow onasmooth surface. This surface is obtained from looking atthetorus with ahole asatwo-sided surface, giving itsome thickness andslightly smoothing thesharp edge. Asaresult wehave a surface with thetopology ofapretzel (asphere with twohandles). After blowing uptheellipse into theellipsoid weobtain asurface of positive curvature; after blowing upthetorus with ahole wegetasurface of negative curvature (inboth cases thecurvature isconcentrated close tothe edge, buttheblowing upcanbedone sothatthesignofthecurvature does notchange). Thus motion inourtoral billiard table canbelooked atasthe limiting caseofmotion along geodesics onasurface ofnegative curvature. Now, toprove Boltzmann’s conjecture (inthesimple case under con- sideration) itissufﬁcient toverify that theanalysis ofstochastic properties ofgeodesic ﬂows onsurfaces ofnegative curvature holds intheindicated limiting case. Amore detailed presentation oftheproof turns outtobeverycomplicated; ithasbeen published only forthecase ofsystems oftwoparticles (Ya. G. Sinai, Dynamical systems with elastic reﬂections, Russian Mathematical Surveys, 25,no.2(1970), 137-189). 317 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups andthehydrodynamics ofideal ﬂuids Eulerian motion ofarigid body canbedescribed asmotion along geodesics inthegroup ofrotations ofthree-dimensional euclidean space provided with aleft-invariant riemannian metric. Asigniﬁcant part ofEuler’s theory depends only upon thisinvariance, andtherefore canbeextended toother groups. Among theexamples involving such ageneralized Euler theory aremotion ofarigid body inahigh-dimensional space and, especially interesting, the hydrodynamics ofanideal (incompressible and inviscid) ﬂuid. Inthe latter case, therelevant group isthegroup ofvolume-preserving diffeo- morphisms ofthedomain ofﬂuid ﬂow. Inthisexample, theprinciple ofleast action implies thatthemotion oftheﬂuid isdescribed bythegeodesics inthe metric given bythekinetic energy. (Ifwewish, wecantakethisprinciple tobe themathematical deﬁnition ofanideal ﬂuid.) Itiseasy toverify that this metric is(right) invariant. Ofcourse, extending results obtained forﬁnite-dimensional Liegroups totheinﬁnite-dimensional case should bedone with care. Forexample, in three-dimensional hydrodynamics anexistence anduniqueness theorem for solutions oftheequations ofmotion hasnotyetbeen proved. Nevertheless, itisinteresting toseewhat conclusions canbedrawn byformally carrying over properties ofgeodesics onﬁnite-dimensional Liegroups totheinﬁnite- dimensional case. These conclusions take thecharacter ofapriori statements (identities, inequalities, etc.) which should besatisﬁed byallreasonable solutions. Insome cases, theformal conclusions canthen berigorously justiﬁed directly, without inﬁnite-dimensional analysis. Forexample, theEuler equations ofmotion forarigid body have astheir analogue inhydrodynamics theEuler equations ofmotion ofanideal ﬂuid. Euler’s theorem onthestability ofrotations around thelarge andsmall axes oftheinertia ellipsoid corresponds inhydrodynamics toaslight generaliza- tionofRayleigh’s theorem onthestability ofﬂows without inﬂection points ofthevelocity proﬁle. Itisalsoeasy toextract from Euler’s formulas anexplicit expression for theriemannian curvature ofagroup with aone-sided invariant metric. Applying thistohydrodynamics weﬁndthecurvature ofthegroup ofdif- feomorphisms preserving thevolume element. Itisinteresting tonote thatin suﬂiciently nice two-dimensional directions, thecurvature turns outtobe ﬁnite and, inmany cases, negative. Negative curvature implies exponential instability ofgeodesics (cf.Appendix 1).Inthecase under consideration, the geodesics aremotions ofanideal ﬂuid; therefore thecalculation ofthe curvature ofthegroup ofdiffeomorphisms gives ussome information onthe instability ofideal ﬂuid ﬂow. Infact,thecurvature determines thecharacter- isticpath length onwhich differences between initial conditions grow bye. Negative curvature leads topractical indeterminacy oftheﬂow: onapath only afewtimes longer than thecharacteristic path length, adeviation in initial conditions grows 100times larger. 318 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Inthisappendix, wewillbrieﬂy setouttheresults ofcalculations related togeodesics ongroups with one-sided (right- orleft-) invariant metrics. Proofs andfurther details canbefound inthefollowing places: V.Arnold, Surlagéométrie diiiérentielle desgroupes deLiededimension inﬁnie etsesapplica- tions al’hydr0dynamique desﬂuides parfaits. Annales del’Institut Fourier, XVI, no.l (1966), 319-361. V.I.Arnold, Anapriori estimate inthetheory ofhydrodynamic stability, Izv.Vyssh. Uchebn. Zaved. Matematicka 1966, no.5(54), 3-5. (Russian) V.I.Arnold, TheHamiltonian nature ofthe Euler equations inthedynamics ofarigid body and ofanideal ﬂuid, Uspekhi Matematischeskikh Nauk, 24(1969), no.3(147) 225-226. (Russian) L.A.Dikii, Aremark onHamiltonian systems connected with therotation group, Functional Analysis andItsApplications, 6:4(1972) 326-327. D.G.Ebin, J.Marsden, Groups ofdiffeomorphisms andthemotion ofanincompressible ﬂuid, Annals ofMath. 92,no.1(1970), 102-I63. O.A.Ladyzhenskaya, Onthelocal solvability ofnon-stationary problems forincompressible ideal andviscous ﬂuids andvanishing viscosity, Boundary problems inmathematical physics, v.5(Zapiski nauchnikh seminarov LOMI, v.21),“Nauka,” 1971, 65-78. (Russian) A.S.Mishchenko, Integrals ofgeodesic ﬂows onLiegroups, Functional Analysis andItsAp- plications, 4,no.3(1970), 232-235. A.M.Obukhov, Onintegral invariants insystems ofhydrodynamic type, Doklady Acad. Nauk. I84,no.2(1969). (Russian) L.D.Faddeev, Towards astability theory ofstationary planar-parallel ﬂows ofanideal ﬂuid, Boundary problems inmathematical physics, v.5(Zapiski nauchnikh seminarov LOMI, v.21),“Nauka,“ 1971, 164-I72. (Russian) ANotation :Thearﬁoint andco-aayoint representations LetGbearealLiegroup andgitsLiealgebra, i.e.,thetangent space tothe group attheidentity provided with thecommutator bracket operation [,1 ALiegroup actsonitself byleftandright translation: every element g ofthegroup Gdeﬁnes diﬂeomorphisms ofthegroup onto itself: Lg:G—+G Lgh=gh Rg=G—>G Rgh=hg. Theinduced maps ofthetangent spaces willbedenoted by L91‘: TGh —> TGgh and R91‘: TGh —> TGhg forevery hinG. Thediﬂeomorphism Rg_lLg isaninner automorphism ofthegroup. It leaves thegroup identity element ﬁxed. Itsderivative attheidentity isa linear map from thealgebra (i.e., thetangent space tothegroup atthe identity) toitself. This map isdenoted by Adg: g—>g Adg=(Ry-tLg)*e 319 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups andiscalled theadjoint representation ofthegroup. Itiseasytoverify that Ad,isanalgebra homomorphism, i.e.,that Adg[€a : isAdg '7], éa'7E Itisalsoclear thatAdgh =AdgAd,,. Wecanconsider Adasamap ofthegroup intothespace oflinear operators onthealgebra: Ad(g) =Adg. Themap Adisdiﬂerentiable. Itsderivative attheidentity ofthegroup isa linear map from thealgebra gtothespace oflinear operations onQ.This map isdenoted byad,anditsimage onanelement Zinthealgebra byadé. Thus ad:isanendomorphism ofthealgebra space, andwehave dad=Ad“: Q->Endg ad;=5; Adezc, t=O where e’:istheone-parameter group with tangent vector f.From theformula written above itiseasy todeduce anexpression foradinterms ofthealgebra alone: add? =[5,7!]- Wenow consider thedual vector space g*totheLiealgebra g.This is thespace ofreallinear functionals ontheLiealgebra. Inother words, g* isthecotangent space tothegroup attheidentity, g*=T*G,,. Thevalue ofanelement ifofthecotangent space tothegroup atsome point gonan element 11ofthetangent space atthesame point willbedenoted byround brackets: (f,n)e[Rl, feT*G,,,neTGg. Leftandright translation induce operators onthecotangent space dual toL“andRH. Wedenote them by Lg‘:T*Gg,, —>T*G,, and R;‘:T*G,,g —>T*G,, forevery hinG.These operators aredeﬁned bytheidentities (L55,'1)E(€,L9,,n)and(R56,11)E(5,R9...'1)- Thetranspose operators Adj, where gruns through theLiegroup G,form arepresentation ofthisgroup, i.e.,they satisfy therelations Adj,=Ad,‘§‘Ad;‘. This representation iscalled theco-adjoint representation ofthegroup and plays animportant roleinallquestions related to(left) invariant metrics on thegroup. Consider thederivative oftheoperator Ad;with respect togattheidentity. Thisderivative isalinear map from thealgebra tothespace oflinear operators 320 Appendix 2;Geodesics ofleft-invariant metrics onLiegroups onthedual space tothealgebra. This linear map isdenoted byad*,andits image onanelement 5inthealgebra isdenoted byadg‘.Thus ad*isalinear operator onthedual space tothealgebra, ad‘§: 9*—>g*. Itiseasy toseethatad;istheadjoint ofadg: (adZ‘rl, QE(11,ad¢§) forallneg*,Qeg. Itissometimes convenient todenote theaction ofad*bybraces: adgn ={f,17}, where Ce9,17e9*. Thus braces mean thebilinear function from g><g*to9*,related tocom- mutation inthealgebra bytheidentity ({€.'1},O=('1,[5,C1)- Weconsider now theorbits oftheco-adjoint representation ofthegroup inthedual space ofthealgebra. Ateach point ofanorbit wehave anatural symplectic structure (called theKirillov form since A.A.Kirillov ﬁrstused it toinvestigate representations ofnilpotent Liegroups). Thus, theorbits of theco-adjoint representation arealways even-dimensional. Wealso note thatweobtain aseries ofexamples ofsymplectic manifolds bylooking at different Liegroups andallpossible orbits. Thesymplectic structure ontheorbits oftheco-adjoint representation is deﬁned bythefollowing construction. Letxbeapoint inthedual space to thealgebra and6avector tangent atthispoint toitsorbit. Since g*isa vector space, wecanconsider thevector E,which really belongs tothetangent space tog*atx,aslying ing*. Thevector 6canberepresented (inmany ways) asthevelocity vector of themotion ofthepoint xunder theco-adjoint action oftheone-parameter group e‘"withvelocity vector ae9.Inother words, every vector tangent to theorbit ofxintheco-adjoint representation ofthegroup canbeexpressed interms ofasuitable vector ainthealgebra bytheformula §={a,x}, a6g,xeg*. Now weareready todeﬁne thevalue ofthesymplectic 2-form Qonapair ofvectors §1,{Ztangent totheorbit ofx.Namely, weexpress £1andC2in terms ofalgebra elements a1anda2bytheformula above, andthen obtain thescalar Q(€1v€2) =(X141, (121), XE9*,atEQ- Itiseasytoverify that(1)thebilinear form Qiswelldeﬁned, i.e.,itsvalue does notdepend onthechoice ofa,;(2)Qisskew-symmetric andtherefore gives adiﬂerential 2-form Qontheorbit; and(3)Qisnondegenerate andclosed (theproofs canbefound, forinstance, inAppendix 5).Thus theform Qisa symplectic structure onanorbit oftheco-adjoint representation. 321 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups BLeft-invariant metrics Ariemannian metric onaLiegroup Giscalled left-invariant ifitispreserved byalllefttranslations Lg,i.e.,ifthederivative oflefttranslation carries every vector toavector ofthesame length. Itissuﬂicient togivealeft-invariant metric atonepoint ofthegroup, for instance theidentity; then themetric canbecarried totheremaining points bylefttranslations. Thus there areasmany left-invariant riemannian metrics onagroup asthere areeuclidean structures onthealgebra. Aeuclidean structure onthealgebra isdeﬁned byasymmetric positive deﬁnite operator from thealgebra toitsdual space. Thus, letA:g ->g*be asymmetric positive linear operator: (A6: = 6)» for is7’in (Itisnotvery important thatAbepositive, butinmechanical applications thequadratic form (AC, C)ispositive deﬁnite.) Wedeﬁne asymmetric operator Ag:TGQ—>T*Gg bylefttranslation: A95 Z L;:—1ALg- 1* Wethus obtain thefollowing commutative diagram oflinear operators: Adt] {W 9L,-.. T69 R,-.. 9 g*,___L: T*Gg R; ,g>|= pl} Ad: Wewilldenote byangled brackets thescalar product determined bythe operator Ag: _ <6,i1>.,=(Agé.'1)=(/1911.5) =('1,€>,- This scalar product gives ariemannian metric onthegroup G,invariant under lefttranslations. Thescalar product inthealgebra willbedenoted simply by (,).Wedeﬁne anoperation B:g><Q—>gbytheidentity ([a,b],c)E(B(c, a),b), forallbing. Clearly, thisoperation Bisbilinear, andforﬁxed ﬁrst argument isskew- symmetric inthesecond: (B(c, a).b)+(B(c, b),a)=O. 322 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups CExample LetG=SO(3) bethegroup ofrotations ofthree-dimensional euclidean space, i.e.theconﬁguration space ofarigid body ﬁxed atapoint. Amotion ofthebody isthendescribed byacurve g=g(t)inthegroup. TheLiealgebra ofGisthethree-dimensional space ofangular velocities ofallpossible rotations. Thecommutator inthisalgebra istheusual vector product. Arotation velocity gofthebody isatangent vector tothegroup atthe point g.Togettheangular velocity, wemust carry thisvector tothetangent space ofthegroup attheidentity, i.e.tothealgebra. Butthiscanbedone in twoways: byleftandright translation. Asaresult, weobtain twodifferent vectors inthealgebra: we=Lg-.*g'eg and 0),=R,-nkgeg. These twovectors arenone other than the“angular velocity inthebody” and the“angular velocity inspace.” Anelement gofthe group Gcorresponds toaposition ofthe body obtained bythemotion g from some initial state (corresponding totheidentity element ofthegroup andchosen abritrar- ily).Letcobeanelement ofthealgebra. Lete""beaone-parameter group ofrotations with angular velocity to;toisthetangent vector tothisone-parameter group attheidentity. Now welook atthedisplacement e"’_q. where g=g(t)eG,toeg,andr<1, obtained from thedisplacement gbyarotation with angular velocity toafter asmall time r. Ifthe vector gcoincides with thevector d (DY _ e , dr r=0 g thentoiscalled theangular velocity relative tospace andisdenoted by0),.Thus to,isobtained from gbyright translation. Inananalogous way wecanshow that theangular velocity in thebody isthelefttranslate ofthevector ginthealgebra. Thedual space 9*tothealgebra inourexample isthespace ofangular momenta. Thekinetic energy ofabody isdetermined bythevector ofangular velocity inthebody anddoes notdepend ontheposition ofthebody inspace. There- fore, kinetic energy gives aleft-invariant riemannian metric onthegroup. The symmetric positive deﬁnite operator Ag:TGQ—>T*Gg given bythis metric iscalled themoment ofinertia operator (ortensor). Itisrelated tothe kinetic energy bytheformula T=%<g', g>_,=%<a>,, wc)=§(A(o,, rot)= §(Agg, g),where A:g —>9*isthevalue ofAgforg=e.The image ofthe vector gunder theaction ofthemoment ofinertia operator Agiscalled the angular momentum andisdenoted byM=Agg. The vector Mliesinthe cotangent space tothegroup atthepoint g,anditcanbecarried totheco- tangent space tothegroup attheidentity byboth leftandright translations. 323 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Weobtain twovectors M,=L;"M eg* and M,=R;"Me 9* These vectors inthedual space tothealgebra arenone other than the angular momentum relative tothebody (MC) andtheangular momentum relative tospace (Ms). This follows easily from theexpression forkinetic energy interms ofmomentum andangular velocity: T=%(A1C9wC): i(M»ti)- Bytheprinciple ofleast action, themotion ofarigid body under inertia (with noexternal forces) isageodesic inthegroup ofrotations with theleft- invariant metric described above. Wewillnow look atageodesic ofanarbitrary left-invariant riemannian metric onanarbitrary Liegroup asamotion ofa“generalized rigid body” with conﬁguration space G.Such a“rigid body with group G”isdetermined byitskinetic energy, i.e.,apositive deﬁnite quadratic form ontheLiealgebra. More precisely, wewillconsider geodesics ofaleft-invariant metric ona group Ggiven byaquadratic form <0),oi)onthealgebra asmotions ofa rigid body with group Gandkinetic energy (w,co)/2. Toevery motion t—>g(t)ofourgeneralized rigid body wecanassociate four curves: r—>w.(r)@9 I->w.(t)@9 r—>M.(r)@9* r—>M.(r)E11*. called motions ofthe vectors ofangular velocity andmomentum inthebody andinspace. Thedifferential equations which these curves satisfy were found byEuler foranordinary rigid body. However, theyaretrueinthemost general caseofanarbitrary group G,andwewillcallthem theEuler equations fora generalized rigid body. Remark. Intheordinary theory ofarigid body sixdifferent three-dimen- sional spaces R3,IR“, g,g*,TG,,. andT*Gg areidentiﬁed. Thefactthatthe dimensions ofthe space R3inwhich thebody moves andofthe Liealgebra g ofitsgroup ofmotions arethesame isanaccident related tothedimension 3; inthen-dimensional case, ghasdimension n(n-1)/2. Theidentiﬁcation oftheLiealgebra 9with itsdual space g*hasamore profound basis. Thefactisthatonthegroup ofrotations there exists (and is unique uptomultiplication) atwo-sided invariant riemannian metric. This metric gives once andforallapreferred isomorphism ofthevector spaces g andg*(and also ofTG,, andT*G,,). Itallows ustherefore toconsider the vectors ofangular velocity andmomentum aslying inthesame euclidean space. With thisidentiﬁcation, theoperation {,}issimply thecommutator ofthealgebra, taken with aminus sign. 324 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Atwo-sided invariant metric exists onanycompact Liegroup. Therefore, tostudy motions ofrigid bodies with compact groups wemay identify the spaces ofangular velocities andmomenta. However, wecannot make this identiﬁcation forapplications tonon-compact (orinﬁnite-dimensional) groups ofdiﬂeomorphisms. DEuler’s equation Theresults ofEuler (obtained byhimintheparticular case G=S0(3)) can beformulated asthefollowing theorems onthemotion ofthevectors of angular velocity andmomentum ofageneralized rigid body with group G. Theorem I.Thevector ofangular momentum relative tospace ispreserved under motion: dM—‘=O.dt Theorem 2.The vector ofangular momentum relative tothebody satisﬁes Euler’s equation dM—‘= M.dt {C067 C} These theorems areproved forageneralized rigid body inthesame wayas foranordinary rigid body. Remark 1.Thevector ofangular velocity inthebody, wc,canbeexpressed linearly interms ofthevector ofangular momentum inthebody, M,,by using theinverse oftheinertia operator: we=A'1M,. Therefore, Euler’s equation canbeconsidered asanequation forthevector ofangular mo- mentum inthebody alone; itsright-hand sideisquadratic inM,. Wecanalsoexpress thisresult inthefollowing way. Consider thephase ﬂowofourrigid body. (Itsphase space T*G hasdimension twice thedimen- sionnofthegroup Gorthespace ofangular momenta g*.)Then thisphase ﬂow ina2n-dimensional manifold factors over theﬂow given byEuler’s equation inthen-dimensional vector space g*. Afactorization ofaphase ﬂow g’onamanifold Xover aphase ﬂowf‘onamanifold Y isasmooth mapping rtofXonto Yunder which motions g‘aremapped tomotions f’,sothat thefollowing diagram commutes (i.e.,rig‘=f'rr): .</‘XL» X IYLa Y 325 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Inourcase, X=T*G isthephase space ofthebody, Y=g*isthespace ofangular momenta. Theprojection rt:T*G ->g*isdeﬁned bylefttranslation (rrM =L;‘M forMeT*Gg), g’is thephase ﬂowofthebody under consideration onthe2n-dimensional space T*G, andf’isthe phase ﬂow oftheEuler equation inthen-dimension space ofangular momenta g*. Inother words, amotion ofthevector ofangular momentum relative to thebody depends only ontheinitial position ofthevector ofangular mo- mentum relative tothebody anddoes notdepend ontheposition ofthe body inthespace. Remark 2.The lawofconservation ofthevector ofangular momentum relative tospace canbeexpressed bysaying that every component ofthis vector insome coordinate system onthespace g*isconserved. Wethus obtain asetofﬁrstintegrals oftheequations ofmotion oftherigid body. In particular, toevery element oftheLiealgebra gthere corresponds alinear function onthespace 5*and,therefore, aﬁrstintegral. ThePoisson brackets ofﬁrstintegrals given byfunctions ong*arethemselves functions ong*,as canbeseen easily. Wethus obtain an(inﬁnite-dimensional) extension ofthe Liealgebra g,consisting ofallfunctions ong*.gitself isincluded inthis extension astheLiealgebra oflinear functions ong*.Ofcourse, ofallthese ﬁrstintegrals ofthephase ﬂow ina2n-dimensional space only narefunc- tionally independent. Asthenindependent integrals wecantake, forexample, nlinear functions ong*which form abasis ing. Because ofpossible inﬁnite-dimensional applications, wewould liketo avoid coordinates andformulate statements about ﬁrstintegrals intrinsically. This canbedone byreformulating Theorem 1inthefollowing way. Theorem 3.Theorbits oftheco-adjoint representation ofagroup inthedual space tothealgebra areinvariant manifolds fortheﬂow inthisspace given byEuler’s equation. PROOF. M,(t) isobtained from M_,(t) bytheaction oftheco-adjoint repre- sentation, andM,(t) remains ﬁxed. [:1 EXAMPLE. Inthecaseofanordinary rigid body, theorbits oftheco-adjoint representation ofthegroup inthespace ofmomenta arethespheres Mf+M§+M§=const. Inthiscase Theorem 3isreduced tothelawof conservation ofthelength oftheangular momentum. Itconsists ofthefact that, iftheinitial point M,liesonsome orbit (i.e., inthegiven case onthe sphere M2=const), then allthepoints ofitstrajectory under theaction of Euler’s equation lieonthesame orbit. Wenow return tothegeneral caseofanarbitrary group Gandrecall that each orbit oftheco-adjoint representation hasasymplectic structure (cf. subsection A).Furthermore, thekinetic energy ofthebody canbeexpressed 326 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups interms oftheangular momentum relative tothebody. Asaresult weobtain aquadratic form onthespace ofangular momenta T=%(M,,A-1M,). Letusﬁxsome oneorbit Voftheco-adjoint representation. Weconsider the kinetic energy asafunction onthisorbit: H:V-»R,H(M,) =git/1,,A“M,,). Theorem 4.Onevery orbit Voftheco-adjoint representation, Euler’s equation ishamiltonian withhamiltonian function H. PROOF. Every vector 5tangent toVatapoint Mhastheform hf=(f.M}.where feg. In particular, thevector ﬁeld ontheright sideofEuler‘s equation canbewritten intheform X={dT, M}(here thedifferential ofthefunction Tatapoint Mofthevector space g*is considered asavector ofthedual space tog*,i.e.,asanelement ofthe Liealgebra g).Itfollows from thedeﬁnitions ofthesymplectic structure Qandtheoperation {,}(cf.subsection A) thatforevery vector 5tangent toVatM, Q(§~X)=(M,[f,dT])=(dT,if,Ml)=(dH.€)- U Euler’s equation canbecarried over from thedual space ofthealgebra to thealgebra itself byinversion ofthemoment ofinertia operator. Asaresult weobtain thefollowing formulation ofEuler’s equation interms ofthe operation B(section B). Theorem 5.Themotion ofthevector ofangular velocity inthebody isdeter- mined bytheinitial position ofthisvector anddoes notdepend ontheinitial position ofthebody. Thevector ofangular velocity inthebody satisﬁes an equation withquadratic right-hand side: CDC Z B(wC’ a)C)' Wewillcallthisequation Euler’s equation forangular velocity. We notice that, under theaction oftheoperator A‘1:9*->g,theorbits ofthe co-adjoint representation arecarried toinvariant manifolds ofEuler’s equation forangular velocity; these manifolds have symplectic structure, etc. However, unlike orbits ing*,these invariant manifolds arenotdetermined bytheLiegroup Gitself, butdepend also onthechoice ofrigid body (i.e., moment ofinertia operator). From thelawofconservation ofenergy wehave Theorem 6.Euler’s equations (for momentum andangular velocity) have a quadratic ﬁrst integral, whose value isequal tothekinetic energy T=%(M,,A'1M,) =%(Ao,, 0),). 327 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups EStationary rotations andtheir stability Astationary rotation ofarigid body isarotation forwhich theangular velocity inthebody isconstant (and thus alsotheangular velocity inspace; itiseasy toseethatoneimplies theother). Weknow from thetheory ofan ordinary rigid body inR3thatstationary rotations arerotations around the major axesofthemoment ofinertia ellipsoid. Below, weformulate ageneral- ization ofthistheorem tothecaseofarigid body with anyLiegroup. Wenote thatstationary rotations aregeodesics ofleft-invariant metrics which areone- parameter subgroups. Wenote alsothat thedirections ofthemajor axes of theinertia ellipsoid canbedetermined bylooking atthestationary points of thekinetic energy onthesphere ofvectors ofmomentum ofﬁxed length. Theorem 7.The angular momentum (respectively, angular velocity) ofa stationary rotation withrespect tothebody isacritical point oftheenergy ontheorbit oftheco-adjoint representation (respectively ontheimage ofthe orbit under theaction oftheoperator A"1).Conversely, every critical point oftheenergy onanorbit determines astationary rotation. Theproof isastraightforward computation orapplication ofTheorem 4. Wenote thatthepartition ofthespace ofmomenta intoorbits oftheco- adjoint representation cannot besoeasily constructed inthecase ofan arbitrary group asitwasinthesimple caseofanordinary rigid body; inthat case itwasthepartition ofthree-dimensional space intospheres with center Oandthepoint Oitself. Inthegeneral case, theorbits canhave different dimensions, and thepartition into orbits atsome points may notbea ﬁbering; such asingularity already appeared inthethree-dimensional case atthepoint 0. Wecallapoint Mofthespace ofangular momenta aregular point ifthe partition ofaneighborhood ofMintoorbits isdiﬁeomorphic toapartition ofeuclidean space intoparallel planes (inparticular, allorbits near thepoint Mhave thesame dimension). Forexample, forthegroup ofrotations of three-dimensional space allpoints ofthespace ofangular momenta are regular except theorigin. Theorem 8.Suppose thataregular point Mofthespace ofangular momenta is acritical point oftheenergy onanorbit oftheco-adjoint representation, andthatthesecond diﬂerential oftheenergy d2Hatthispoint isa(positive ornegative) deﬁniteform. Then Misa(Liapunov) stable equilibrium position ofEuler’s equations. PROOF. Itfollows from theregularity oftheorbits near thispoint that on every neighboring orbit there exists near Mapoint which isaconditional maximum orminimum ofenergy. E] 328 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Theorem 9.Thesecond diﬂerential ofthekinetic energy, restricted totheimage ofanorbit oftheco-adjoint representation inthealgebra, isgiven ata critical point toegbytheformula 34211|w(§)=<B(w,f), B(w,f)> +(U,W],B(w,f)>, where §isatangent vector tothisimage, expressed interms offbythe formula ¢§=B(<v,f), fEQ- FRiemannian curvature ofagroup with left-invariant metric LetGbeaLiegroup provided with theleft-invariant metric given bya scalar product <,)inthealgebra. Wenote thattheriemannian curvature ofthegroup Gatanypoint isdetermined bythecurvature attheidentity (since lefttranslation maps thegroup toitself isometrically). Therefore, itis sufficient tocalculate thecurvature fortwo-dimensional planes lying inthe Liealgebra. Theorem 10.The curvature ofagroup inthedirection determined byan orthonormal pairofvectors QI1inthealgebra isgiven bytheformula K6,” =<6» +2<a¢ —3<a> a> —4<B{s Br|>a where 25=B(€,11)+B('1,O,ZB=B(€,'1)—B01,<5),201=[Q11], 2B;= B(§,§),2B,,=B(t1,11),andwhere Bistheoperation deﬁned insection B. Theproof isatedious butstraightforward calculation. Itisbased onthe easily veriﬁed formula forcovariant derivative (V§")e = _ _ where fand:1ontheleftareleft-invariant vector ﬁelds andontheright are their values attheidentity. Remark 1.Inthecase ofatwo-sided invariant metric, theformula for curvature hastheparticularly simple form Kat,=%<[€,'1],[5,t1]>- Remark 2.Theformula forthecurvature ofagroup with aright-invariant riemannian metric coincides with theformula fortheleft-invariant case. In fact, aright-invariant metric onagroup isaleft-invariant metric onthe group with thereverse multiplication law(gl=l=g2=gzgl). Passage tothe reverse group changes thesigns ofboth thecommutator andtheoperation B inthealgebra. But, inevery term oftheformula forcurvature, there isa product oftwooperations changing thesign. Therefore, theformula for curvature isthesame intheright-invariant case. 329 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups InEuler’s equation theright-hand sidechanges signunder passage tothe right-invariant case. GApplication togroups ofdzﬂeomorphisms LetDbeabounded region inariemannian manifold. Consider thegroup of diﬂeomorphisms ofDwhich preserve thevolume element. Wewilldenote thisgroup bySDiﬂD. TheLiealgebra corresponding tothegroup SDiﬂ Dconsists ofallvector ﬁelds with divergence 0onD,tangent totheboundary (ifitisnotempty). We deﬁne thescalar product oftwoelements ofthisLiealgebra (i.e.,twovector ﬁelds) as <v1,v2>=_lD(v1~v2)dx, where (-)isthescalar product giving theriemannian metric onD,anddx istheriemannian volume element. Wenow consider theﬂow ofauniform ideal (incompressible, non- viscous) ﬂuid ontheregion D.Such aﬂow isdescribed byacurve t—>g,in thegroup SDiﬂD. Namely, thediffeomorphism g,isthemap which carries every particle oftheﬂuid from theplace itwasattime 0totheplace itisat time t.Itturns outthat thekinetic energy ofthemoving ﬂuid isaright- invariant riemannian metric onthegroup ofdiﬂeomorphisms SDiﬂD. Indeed, suppose thatafter time ttheﬂow oftheﬂuid gives adiffeomorphism g,,andthat thevelocity atthismoment oftime isgiven bythevector ﬁeld v.Then thediﬂeomorphism realized bytheﬂow after time t+T(where tissmall) willbee"'g, uptoaquantity small in comparison with 1(here e'”istheone-parameter group with vector v,i.e.,thephase ﬂowofthe differential equation given bytheﬁeld v).Therefore, theﬁeldofvelocities visobtained from the vector gtangent tothegroup atthepoint gbyright translation. This alsoimplies theright- invariance ofthekinetic energy, which isbydeﬁnition equal to T=%<v,v> (weassume thedensity oftheﬂuid tobe1). Theprinciple ofleast action (which inmathematical terms isthedeﬁnition ofanideal ﬂuid) asserts thatﬂows ofanideal ﬂuid aregeodesics intheright- invariant metric justdescribed onthegroup ofdiﬂeomorphisms. Strictly speaking, aninﬁnite-dimensional group ofdilieomorphisms isnotamanifold. Therefore theexact formulation ofthedeﬁnition above requires additional work: wemust choose suitable functional spaces, provc atheorem onexistence anduniqueness ofsolutions, etc.Uptonowthishasbeen done onlyinthecasewhen thedimension olthe region ofthe ﬂowD isequal to2.However, wewillproceed asifthese difﬁculties connected with inﬁnite dimensions didnotexist. Thus thefollowing arguments areheuristic incharacter. Itturns outthatmany oftheresults canbeproved rigorously, independently ofthetheory ofinﬁnite-dimensional manifolds. Wewillnow indicate theform thatthegeneral formulas introduced above take inthecase G=SDiﬂ'D, where Disaconnected region with ﬁnite 330‘i 4 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups volume inathree-dimensional riemannian manifold. Todothiswemust ﬁrst describe explicitly thebilinear operation B:g ><g—>g deﬁned in section Bbytheformula ([61,blC)E(B(c, H),11>- Itiseasy toverify that inthethree-dimensional case thevector ﬁeld B(c,a)canbeexpressed interms ofthevector ﬁelds aandcofourLiealgebra bytheformula B(c,a)=(curl c)/\a+grad oz, where /\denotes thevector product, andatthesingle-valued function onD which isuniquely (uptoaconstant summand) determined bythecondition Beg(i.e.,theconditions divB=0andBistangent totheboundary ofD). Wenote thattheoperation Bdoes notdepend onthechoice oforientation, since thevector product andcurlboth change signwith achange oforienta- tion. Stationary ﬂows. Euler’s equation for“angular velocity” inthecase G=SDiﬁ'D hastheform vI—B(v, v),since themetric isright-invariant. Therefore, inthecaseofthegroup ofdiffeomorphisms ofthree-dimensional space, ittakes theform of“the equations ofmotion inBernoulli’s form” 5€f=v/\curlv+gradoz, divv=0. Euler’s equation formomentum iswritten intheform ofthe“vorticity equation” dcurlv [CI]—-— =v,urv.at Inparticular, thevorticity ofastationary ﬂow commutes with theﬁeld of velocities. This remark leads quickly toatopological classiﬁcation ofstationary ﬂows ofanideal ﬂuid inthree-dimensional space. Theorem ll.Assume thattheregion Disbounded byacompact analytic surface, andthattheﬁeld ofvelocities isanalytic andnoteverywhere collinear with itscurl. Then theregion oftheﬂow canbepartitioned byananalytic sub- manifold intoaﬁnite number ofcells, ineach ofwhich theﬂow isconstructed inastandard way. Namely, thecells areoftwotypes: those ﬁbered intotori invariant under theﬂow andthose ﬁbered intosurfaces invariant under the ﬂow, diﬂeomorphic totheannulus RXS1.Oneach ofthese toritheﬂow lines areeither allclosed oralldense, andoneach annulus alltheﬂow lines areclosed. Toprove thistheorem welook atthe“Bernoulli surfaces,” i.e.,thelevel surfaces ofthefunction oz.Itfollows from thecondition foraﬂow tobe stationary (v/\curlv=—grad oz)that both theﬂow lines andthevortex 331 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups lines lieontheBernoulli surface. Since theﬁelds ofvelocity andvorticity commute, thegroup R2actsontheclosed Bernoulli surface, anditmust bea torus (cf.theproof ofLiouville’s theorem inSection 49).Ananalogous calculation fortheboundary conditions ontheboundary ofDshows thatthe non-closed Bernoulli surfaces consist ofannuli with closed ﬂow lines. Remark. Theanalyticity oftheﬁeld ofvelocities isnotvery essential, but itisimportant that theﬁelds ofvelocity andvorticity notbecollinear. Computer experiments conducted byM.Henon show more complicated behavior than described inthetheorem fortheﬂow lines ofastationary ﬂow onthethree-dimensional torus; thisﬁeld isgiven bytheformulas v,=Asinz+Ccosy v,=Bsinx+Acosz, v,=Csiny+Bcosx. Theformulas areselected sothatthevectors vandcurlvarecollinear. The results ofHenon’s calculations suggest thatsome ﬂow lines densely ﬁllupa three-dimensional region. IIsovorticial ﬁelds Two-dimensional hydrodynamics diﬂers sharply from three-dimensional hydrodynamics. Theessence ofthisdifference iscontained inthedifference inthegeometries oftheorbits oftheco-adjoint representation inthetwo- andthree-dimensional cases. Inthetwo-dimensional case theorbits arein some sense closed andbehave, forexample, likeafamily oflevel setsofa function (more precisely ofseveral functions: actually even aninﬁnite number offunctions). Inthethree-dimensional casetheorbits aremore complicated; inparticular, they areunbounded (and perhaps dense). Theorbits oftheco- adjoint representation ofthegroup ofdiffeomorphisms ofathree-dimensional riemannian manifold canbedescribed inthefollowing way. Letv,andv2be twovector ﬁelds ofvelocities ofanon-compressible ﬂuid intheregion D. Wesaythattheﬁelds v1andv2areisovorticial ifthere isvolume-preserving diﬂeomorphism g:D—>Dwhich carries every closed contour yinDtoanew contour such thatthecirculation oftheﬁrstﬁeld along theoriginal contour isequal tothecirculation ofthesecond ﬁeld along thenewcontour: 7 Q? Itiseasy toverify thattheimage ofanorbit oftheco-adjoint representation inthealgebra (under theaction oftheinverse oftheinertia operator, A-1)is none other than thesetofﬁelds isovorticial tothegiven ﬁeld. Inparticular, Theorem 3now takes theform ofthefollowing lawofcon- servation ofcirculation: Theorem 12.Thecirculation ofaﬁeld ofvelocities ofanidealﬂuid overaclosed ﬂuid contour does notchange when thecontour iscarried bytheﬂow toa newposition. 332 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Wenote thatiftwoﬁelds ofvelocities ofathree-dimensional ideal ﬂuid onDareisovorticial, then thecorresponding diffeomorphism carries thecurl oftheﬁrstﬁeld intothecurlofthesecond: g*curlv1=curl02. Furthermore, theisovorticity oftwoﬁelds canbedeﬁned astheequivalence oftheﬁelds ofvorticity, iftheregion oftheﬂow issimply-connected. Therefore, theproblem oftheoribits oftheco-adjoint representation inthethree- dimensional case includes theproblem ofclassifying vector ﬁelds with divergence zero uptovolume-preserving diffeomorphisms. This lastproblem inthree dimensions ishopelessly diﬂicult. Wenow consider thetwo-dimensional case. First, wetranslate thebasic formulas intonotation convenient forconsidering thetwo-dimensional case. Weassume that theregion Doftheﬂow istwo-dimensional andoriented. Themetric andorientation giveasymplectic structure onD;thevector ﬁeld ofvelocities hasdivergence zero andistherefore hamiltonian. Therefore, this ﬁeldisgiven byahamiltonian function (many-valued, ingeneral, iftheregion Disnotsimply-connected). Thehamiltonian function ofaﬁeld ofvelocities iscalled thestream function inhydrodynamics, andisdenoted byI,//.Thus v=Igrad tb, where Iistheoperator ofclockwise rotation by90°. Thestream function ofthecommutator oftwoﬁelds turns outtobethe jacobian (orthePoisson bracket ofhamiltonian formalism) ofthestream functions oftheoriginal ﬁelds ‘ll/[v1,vg] =J(¢1, Thevector ﬁeld B(c,a)isgiven, inthetwo-dimensional case, bytheformula B=—(At//c)grad t,0,,+grad oz, where 111,,andthearethestream functions oftheﬁelds aandc,andA= divgrad isthelaplacian. Intheparticular case oftheeuclidean plane with cartesian coordinates x andy,theformulas forstream function, commutator andlaplacian take the particularly simple form 51/1 13¢ Ux=$ Uy=—g ,, ["""’]— 6x 6y 6y 6x 82 82 333 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Thevorticity (orcurl) ofatwo-dimensional ﬁeld ofvelocities isthescalar function rsuch that theintegral around anyoriented region 0inDofthe product ofrwith theoriented area element isequal tothecirculation ofthe ﬁeld ofvelocities around theboundary of0: frdS= 3gv. er 50 Itiseasy tocompute anexpression forthevorticity interms ofthestream function: r=——Al,D. Inthetwo-dimensional simply-connected case, isovorticity ofﬁelds v1 andv2means simply that thefunctions r1andr2(the vorticities ofthese ﬁelds) arecarried tooneanother under asuitable volume-preserving dif- feomorphism. Under such conditions thetwofunctions r1andr2have thesame distribu- tionfunction, i.e., mes{xe D:r1(x) 5c}=mes{x eD:r2(x) 5c}, foranynumber c.Therefore, iftwoﬁelds areintheimage ofthesame orbit oftheco-adjoint representation, then awhole series offunctionals areequal; forexample, theintegrals ofallpowers ofthevorticity fr';ds=fr;ds. D D Inparticular, Euler’s equations ofmotion ofatwo-dimensional ideal ﬂuid 6 £+vVv= —gradp divv=O, have aninﬁnite collection ofﬁrstintegrals. Forexample, theintegral ofany power ofthevorticity oftheﬁeld ofvelocities _ @122 501 k Ik—- {ID dx/\ issuch aﬁrstintegral. Theexistence ofthese ﬁrstintegrals (i.e.,therelatively simple structure of orbits oftheco-adjoint representation) allows ustoprove theorems on existence anduniqueness, etc.inthetwo-dimensional hydrodynamics ofar ideal (and alsoofaviscous) ﬂuid; thecomplicated geometry oforbits ofthe co-adjoint representation inthethree-dimensional case (or,perhaps, in- sufﬁcient information about these orbits) makes thefoundations ofthree- dimensional hydrodynamics avery hard problem. 334 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups JStability ofplanar stationary ﬂows Here weformulate general theorems about stationary rotations (Theorems 7,8,and9above) forthecase ofagroup ofdiffeomorphisms. Weobtain in thiswaythefollowing assertions: 1.Astationary ﬂow ofanideal ﬂuid isdistinguished from allﬂows iso- vorticial toitbythefactthatitisaconditional extremum (orcritical point) ofthekinetic energy. 2.If(i)theindicated critical point isactually anextremum, i.e.,alocal con- ditional maximum orminimum, (ii)itsatisﬁes certain (generally satisﬁed) regularity conditions, and (iii) theextremum isnon-degenerate (the second differential ispositive- ornegative-deﬁnite), then thestationary ﬂow isstable (i.e., isaLiapunov stable equilibrium position ofEuler’s equation). 3.Theformula forthesecond differential ofthekinetic energy, onthetangent space tothemanifold ofﬁelds which areisovorticial toagiven one,hasthe following form inthetwo-dimensional case. LetDbearegion inthe euclidean plane with cartesian coordinates xandy.Consider astationary ﬂow with stream function tb=1//(x, y).Then 2d2H =ffD (6v)2 + (A¢NAt,b)(6r)2 dxdy,where 6visthevariation oftheﬁeld ofvelocities (i.e.,avector ofthetangent space indicated above), andor=curl5v. Wenote that forastationary ﬂow, thegradient vectors ofthestream function anditslaplacian arecollinear. Therefore theratio Vtb/VA|,b makes sense. Furthermore, inaneighborhood ofevery point where thegradient of thevorticity isnotzero, thestream function isafunction ofthevorticity function. Theassertions introduced above lead totheconclusion thatthepositive ornegative deﬁniteness ofthequadratic form d2H isasuﬂicient Condition forstability ofthestationary ﬂow under consideration. This conclusion does notformally follow from Theorems 7,8,and9since theapplication ofanyof ourformulas intheinﬁnite-dimensional case requires justiﬁcation. Fortu- nately, wecanjustify theﬁnal conclusion about stability without justifying theintermediate constructions. Thus wecanrigorously prove thefollowing apriori bounds (expressing thestability ofastationary ﬂow interms ofsmall perturbations oftheinitial velocity ﬁeld). Theorem 13.Suppose thatthestream function ofastationary ﬂow, Ill=¢(x,y), inaregion Disafunction ofthevorticit yfunction (i.e.,ofthefunction At//)not onlylocally, butglobally. Suppose thatthederivative ofthestream function with respect tothevorticity satisﬁes theinequality Vc$F|ﬂ$C, where0<c$C<oo. 335 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Let1,0+<p(x, y,t)bethestream function ofanother ﬂow, notnecessarily stationary. Assume that, attheinitial moment, thecirculation ofthevelocity ﬁeld oftheperturbed ﬂow (with ﬂowfunction 1/1+tp)around every boundary component oftheregionD isequal tothecirculation oftheoriginal ﬂow (with stream function I,//).Then theperturbation cp=cp(x, y,t)atevery moment oftimeisbounded interms oftheinitial perturbation (p0=g0(x,y,O)bythe formula If(V<p)2 +C(A(p)2 dxdy3 (Vrp0)2 +C(A(p0)2 dxdy. D D Ifthestationary ﬂow satisﬁes theinequality V c§—VA£ll/5C, O<c<C<o0, then theperturbation (pisbounded interms of(p0bytheformula ifc<A1-(voldxdysifC(A<o0)’ -(vimaxdy.0 1) This theorem implies thestability ofastationary ﬂow inthecase ofa positive-deﬁnite quadratic form HD(V)’ +%(Awdxdy with respect toVtp(where (pisaconstant function onevery component ofthe boundary ofDwhose gradient ﬂow iszeroover every boundary component), andalsointhecase ofanegative deﬁnite form fJD(V<p)2 +(max AViV‘,l:l/)(A<p)2 dxdy. EXAMPLE 1.Consider aplanar parallel ﬂowinthestrip Y,sy3Y2inthe (x,y)-plane with velocity proﬁle v(y)(i.e.,with velocity ﬁeld (v(y), 0)).Such aﬂow isstationary foranyvelocity proﬁle. Tomake theregion oftheﬂow compact, weimpose thecondition thatthevelocity ﬁelds ofallﬂows under consideration beperiodic with period Xinthex-coordinate. Theconditions ofTheorem 13arefulﬁlled ifthevelocity proﬁle hasno points ofinﬂection (i.e., ifdzv/dyz 960).Wecome totheconclusion that planar parallel ﬂows ofanideal ﬂuid with noinﬂection points inthevelocity proﬁle arestable. Theanalogous proposition inthelinearized problem iscalled Rayleigh’s theorem. 336 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Weemphasize thatinTheorem l3itisnotaquestion ofstability “inalinear approxima- tion,” butofactual strict Liapunov stability (i.e., with respect toﬁnite perturbations inthe nonlinear problem). Thedifference between these twoforms ofstability issubstantial inthis case, since ourproblem hasahamiltonian character (cf.Theorem 4);forhamiltonian systems asymptotic stability isimpossible, sostability inalinear approximation isalways neutral and insutﬁcient foraconclusion about thestability ofanequilibrium position ofthenonlinear problem. EXAMPLE 2.Consider theplanar-parallel ﬂowonthetorus {(x,y),xmod X,ymod 21:} with velocity ﬁeld v=(siny,O),parallel tothex-axis. This ﬁeld isdeter- mined bythestream function tb=—cos yandhasvorticity r=—cos y. Thevelocity proﬁle hastwoinﬂection points, butthestream function can beexpressed asafunction ofthevorticity. Theratio Vt///VAtl isequal to minus one. Byapplying Theorem 13wecanconvince ourselves ofthe stability ofourstationary ﬂow inthecase when 21: X 21: X JJ(A(p)2 dxdy2JJ(V<p)2 dxdy O 0 0 O forallfunctions (pofperiod Xinxand21:iny.Itiseasy tocalculate thatthe lastinequality issatisﬁed forX321:andviolated forX>21:. Thus Theorem 13implies thestability ofasinusoidal stationary ﬂow ona short torus, when theperiod inthedirection ofthebasic ﬂow(X)islessthan thewidth oftheﬂow(21t). Ontheother hand, wecandirectly verify thatona long torus (forX>21:)oursinusoidal ﬂow isunstable.” Thus, inthis example, thesuﬁicient condition forstability from Theorem 13turns outto benecessary. Weshould note thatingeneral anindeﬁnite quadratic form d2Hdoes notimply instability ofthe corresponding ﬂow. Ingeneral, anequilibrium position ofahamiltonian system canbe stable even though thehamiltonian function atthisposition isneither amaximum noramini- mum. Thequadratic hamiltonian H=pf+qf—p§—q§isthesimplest example ofthis kind. KRiemannian curvature ofagroup ofdiﬂeomorphisms The expression forthecurvature ofaLiegroup provided with aone- sided-invariant metric, introduced insubsection E,makes sense alsoforthe group SDiﬂ Dofdiffeomorphisms ofariemannian domain D.This group is theconﬁguration space foranideal ﬂuid ﬁlling thedomain D.Thekinetic energy deﬁnes aright-invariant metric onSDiﬂD. The number which we obtain byformally applying theformula forthecurvature ofaLiegroup to 97Cf.,forexample, thearticle ofL.D.Meshalkin andY.G.Sinai, “Investigation ofthestability ofastationary solution ofasystem ofequations fortheplane movement ofanincompressible viscous liquid.” J.Applied Math. Mech. 25(1962), 1700-1705. 337 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups thisinﬁnite-dimensional group isnaturally called thecurvature ofthegroup SDijfD. Calculation ofthecurvature ofagroup ofdiffeomorphisms hasbeen carried outcompletely only inthecase ofaﬂow onthetwo-dimensional torus with euclidean metric. Such atorus isobtained from theeuclidean plane R2byidentifying points whose difference liesinsome lattice (adiscrete subgroup oftheplane). Anexample ofsuch alattice isthesetofpoints with integral coordinates. Ingeneral, toobtain anarbitrary lattice Fwemay replace thesquare lying atthebasis ofthisspecial lattice byanyparallelogram. Now consider theLiealgebra ofvector ﬁelds with divergence zero onthe torus with asingle-valued stream function. The corresponding group S0DiﬂT2consists ofvolume-preserving diffeomorphisms which leave the center ofmass ofthetorus ﬁxed. Itisembedded inthegroup SDijj" T2ofall volume-preserving diffeomorphisms asatotally geodesic submanifold (i.e., asubmanifold such that each ofitsgeodesics isageodesic intheambient manifold). The proof consists ofthefactthat if,attheinitial moment, avelocity ﬁeld ofanideal ﬂuid hasasingle-valued stream function, then atallother moments oftime thestream function willalsobesingle-valued; thisfollows from thelawofconservation ofmomentum. Wewillnow investigate thecurvature ofthegroup S0Diﬂ T2inallpos- sible two-dimensional directions passing through theidentity ofthegroup (thecurvature ofthegroup SDiﬂ T2inevery such direction isthesame, since thesubmanifold S0DiﬂT2istotally geodesic). Choose anorientation onIR2.Then elements oftheLiealgebra ofthe group S0Dijf T2canbethought ofasrealfunctions onthetorus having average value zero (aﬁeld with divergence zero isobtained from such a function byconsidering ittobeastream function). Therefore, atwo-dimen- sional direction inthetangent space tothegroup S0Diﬂ T2isdetermined by apairoffunctions onthetorus with average value zero. Wewillgivesuch afunction bythesetofitsFourier coefﬁcients. Itiscon- venient tocarry outallcalculations with Fourier series inthecomplex do- main. Welete,,(where k,called awave vector, isapoint oftheeuclidean plane) denote thefunction whose value atapoint xofourplane isequal to e“"~"’. Such afunction determines afunction onthetorus ifitisF-periodic, i.e.,ifadding avector from thelattice Ftoxdoes notchange thevalue ofthe function. Inother words, thescalar product (k,x)must beamultiple of2nforall xeF.Allsuch vectors kbelong toalattice F*onR2.Thefunctions e,,,where keF*,form acomplete system inthespace ofcomplex functions onthetorus. Wenow complexify ourLiealgebra, scalar product <,),commutator [,]andoperation Binthealgebra, aswellastheriemannian connection andcurvature tensor Q,sothatallthese functions become (multi-) linear in thecomplex vector space ofthecomplexiﬁed Liealgebra. Thefunctions ek (where k6F*,kqéO)form abasis ofthisvector space. 338 Appendix 2:Geodesics ofleft-invaria'nt metrics onLiegroups Theorem 14.Theexplicit_formulas forthescalar product, commutator, opera- tionB,connection, andcurvature ofaright-invariant metric onthegroup S0Diff T2have thefollowing form; (e,,,e,) =Ofork +I950, (eh: 9-t> =kzs; [em er]=(k/\l)9k+i§ k2 B(@it, er)=bk,lek+l> Where but=(k/\Dig(k+l) (v/\u)(u-v) Vekel :dl,k+lek+l: where du,v : U2 i R;,!,,,,,_,,=Oifk+l+m+n7'$O;ifk+l+m+n=O,thenR,,,,,,,,,,,= (a,,,ak,,, —a,,,,ak,,)S, where am,=(u/\v)2/lu +v|'. Inthese formulas, Sisthearea ofthetorus, andu/\vthearea ofthe parallelogram spanned byuandv(with respect tothechosen orientation of R2).Theparentheses denote theeuclidean scalar product intheplane, and angled brackets denote thescalar product intheLiealgebra. Theproof ofthistheorem isintheﬁrstarticle listed intheintroduction to thisappendix. The formulas above allow ustocalculate thecurvature inany two- dimensional direction. These calculations show thatinmost directions the curvature isnegative, butinafewitispositive. Consider, forinstance, some ﬂuid ﬂow, i.e.ageodesic ofourgroup. ByJacobi’s equations, thestability of thisgeodesic isdetermined bythecurvatures inthedirections ofallpossible two-dimensional planes passing through thevelocity vector ofthegeodesic ateach ofitspoints. Assume nowthattheﬂowunder consideration isstationary. Then thegeo- desic isaone-parameter subgroup ofourgroup. From thisitfollows thatthe curvatures inthedirections ofallplanes passing through velocity vectors of thegeodesic atallofitspoints areequal tothecurvatures inthecorresponding planes going through thevelocity vector ofthisgeodesic attheinitial moment oftime (Proof: right translate totheidentity element ofthegroup). Thus the stability ofastationary ﬂowdepends only onthecurvatures inthedirections ofthose two-dimensional planes intheLiealgebra which contain thevector oftheLiealgebra which isthevelocity ﬁeld ofthestationary ﬂow. Consider, forexample, thesimplest parallel sinusoidal stationary ﬂow. Such aﬂow isgiven bythestream function _€k+€_k€——i2 - Consider anyother realvector ofthealgebra, 17=Zx,e,(sox_,=x,).We deduce easily from Theorem 14that 339 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Theorem 15.Thecurvature ofthegroup S0DiffT2inanytwo-dimensional plane containing thedirection 6isnon-positive. Namely, S <n(§, ll»="ZZaillxl +Xz+2i<l2-1 From thisformula itfollows, inparticular, that 1.Thecurvature isequal tozeroonlyforthose two-dimensional planes which consist ofparallel ﬂows inthesame direction as6,sothat[5,11]=0; 2.The curvature intheplane deﬁned bytheﬂow functions C=coskx, 17=cosIxIS k2+l’ .2.2K-Tsin asin /5’, where Sisthearea ofthetorus, ozistheangle between kandI,andﬁisthe angle between k+land k—l; 3.Inparticular, thecurvature ofthegroup ofdilfeomorphisms ofthetorus {(x,y)mod 21:}indirections determined bythevelocity ﬁelds (siny,0) (O,sinx)isequal to K_-1 _81t2' LDiscussion Itisnatural toexpect that thecurvature ofagroup ofdilfeomorphisms is related tothestability ofgeodesics inthisgroup (i.e.tothestability ofﬂows ofanideal ﬂuid) inthesame wayasthecurvature ofaﬁnite-dimensional Lie group isrelated tothestability ofgeodesics onit.Namely, negative curvature causes exponential instability ofgeodesics. The characteristic path length (the average path length inwhich errors intheinitial conditions grow e times) hasorder ofmagnitude l/,/—K. Thus, knowing thecurvatures ofa group ofdilfeomorphisms allows ustoestimate thetime forwhich wecan predict thedevelopment oftheﬂow ofanideal ﬂuid bymeans ofanapproxi- mate initial velocity ﬁeld before theerror grows toalarge order. Itshould beemphasized thatinstability ofaﬂow ofanideal ﬂuid ishere understood dif- ferently than insection K;itisaquestion ofexponential instability ofthemotion oftheﬂuid, notofitsvelocity ﬁeld. Itispossible forastationary ﬂow tobeaLiapunov stable solution of Euler‘s equation while thecorresponding motion oftheﬂuid isexponentially unstable. The reason isthatasmall change inthevelocity ﬁeldofaﬂuid caninduce anexponentially growing change inthemotion oftheﬂuid. Insuch acase(stability ofthesolution ofEuler’s equation andnegative curvature ofthegroup) wecanpredict thevelocity ﬁeld, butwecannot predict themotion oftheﬂuid mass without agreat lossofaccuracy. Theformulas mentioned above forcurvature canbeused even forrough estimates ofthetime over which along-term dynamical prediction ofthe 340 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups weather isimpossible, ifweagree toafewsimplifying assumptions. These simplifying assumptions consist ofthefollowing: 1.Theearth hastheshape ofatorus obtained byfactoring theplane bya square lattice. 2.The atmosphere isatwo-dimensional homogeneous non-compressible non-viscous ﬂuid. 3.Themotion oftheatmosphere isapproximately a“tradewind current,” parallel totheequator ofthetorus andhaving sinusoidal velocity proﬁle. Tocalculate thecharacteristic path length wemust then estimate the curvature ofthegroup S0DiﬂT2 indirections containing the“tradewind current” 6from Theorem 15.Todothiswewilllook atT2as{(x,y)mod 21:}, k=(O,1).Inother words, welook at21:-periodic ﬂows onthe(x,y)-plane close toastationary ﬂow, parallel tothex-axis andwith sinusoidal velocity proﬁle v=(siny,0). Itiseasy toseefrom theformula inTheorem 15thatthecurvature ofthe group S0Diff T2intheplanes containing ourtradewind current vvaries within thelimits 2—g<K<0,where S=4rc2isthearea ofthetorus. Here thelower limit isobtained byarather crude estimate. However, a direction with curvature K=—l/2S certainty exists, andthere aremany other directions with curvature ofapproximately thesame size. Inorder to make arough estimate ofthecharacteristic path length, wemake therough guess K0=—1/2Sasvalue ofthe“mean curvature.” Ifweagree tostart from thisvalue K0ofthecurvature, weobtain the characteristic path length S=(,/—K0)" =\/Es. Thevelocity ofmotion with respect tothegroup which corresponds to ourtradewind current isequal to\/S/2 (since theaverage square value of thesineis%).Therefore, thetimeittakes forourﬂow totravel thecharacteristic pathlength isequal to2.Thefastest particles oftheﬂuid goadistance of2after thistime, i.e.,l/rtoftheentire orbit around thetorus. Thus, ifwetake ourvalue ofthemean curvature, then theerror grows by e"z20after thetime ofoneorbit ofthefastest particle. Taking thevalue 100km/hr asthemaximal velocity ofthetradewind current, weget400hours forthetime oforbit, i.e.,lessthan three weeks. Thus, ifattheinitial moment thestate oftheweather wasknown with small error e,then theorder ofmagnitude oftheerror ofprediction after n months would be 30-24l0""s, where kzW TClOg10€'-=: 2.5. 341 Appendix 2:Geodesics ofleft-invariant metrics onLiegroups Forexample, topredict theweather twomonths inadvance wemust have initial data with ﬁvemore digits ofaccuracy than theprediction accuracy. Practically, this means that calculating theweather forsuch aperiod is impossible. Itisclear thattheestimates mentioned here arenotvery sharp, andthe model wetook isverysimpliﬁed. Thechoice ofthevalue of“mean curvature ” alsorequires justiﬁcation. 342 Appendix 3:Symplectic structures onalgebraic manifolds Thesymplectic manifolds ofclassical mechanics aremost often phase spaces oflagrangian mechanical systems, i.e.,cotangent bundles ofconﬁguration spaces. Anentirely different series ofsymplectic manifolds arises inalgebraic geometry. Forexample, anysmooth complex algebraic manifold (given byasystem ofpolynomial equations incomplex projective space) hasanatural symplectic structure. Theconstruction ofasymplectic structure onanalgebraic manifold is based onthefactthatcomplex projective space itself hasaparticular sym- plectic structure, namely theimaginary partofitshermitian structure. AThehermitian structure ofcomplex projective space Recall thatn-dimensional complex projective space CP"isthemanifold ofall complex lines passing through thepoint 0inan(n+1)-dimensional com- plexvector space C”1.Toconstruct asymplectic structure onCP"weuse thehermitian structure inthecorresponding vector space C”1. Recall thatahermitian scalar product (orhermitian structure) onacomplex vector space isacomplex linear function onpairs ofvectors, which (I)islinear intheﬁrstandanti-linear inthesecond variable, (2)changes itsvalue tothecomplex conjugate when thearguments are interchanged, and(3)becomes apositive-deﬁnite realquadratic form ifwetake thearguments equal: <l€.r1> =l<€.r1> <ri.€> =<€.r1> <€.€> >0 forC9*O. Anexample ofahermitian scalar product is (ll <§.'l> =Z§t'_Ii¢, where Qand17,,arethecoordinates ofthevectors Candryinsome basis. Abasis forwhich ahermitian scalar product hastheform (1)always exists, andiscalled a hermitian-orthonormal basis. Therealandimaginary parts ofahermitian scalar product arerealbilinear forms. The ﬁrstissymmetric, andthesecond skew-symmetric, andboth arenondegenerate; <6.r/>=(in)+i[€.rt] (€.r!)=(r1.<i) [é.rt]=—[r!,5]- Thequadratic form (Qtf)ispositive-deﬁnite. Thus ahermitian structure (,)onacomplex vector space gives itaeuclidean structure (,)and asymplectic structure[ ,].These twostructures arerelated tothecomplex structure bytherelation [5,rt]=(£12121) Wewillnow deﬁne ariemannian metric oncomplex projective space. Todothis,consider theunitsphere S2"+1 ={zEC"+1: (z,z> =1} inthecorresponding vector space C”1.This sphere inherits theriemannian metric from C”1.Every complex lineintersects oursphere inagreat circle. i 343 Appendix 3:Symplectic structures onalgebraic manifolds Deﬁnition. Thedistance between twopoints ofcomplex projective space is thedistance between thetwocorresponding circles ontheunitsphere. Wenote that these twocircles areparallel inthesense thatthedistance from anypoint ofone ofthecircles totheother isthesame (Proof: multiplica- tionofzbye1"’preserves themetric onthesphere). This circumstance allows usatonce towrite down anexplicit formula (2)fortheriemannian metric on thecomplex projective space given bytheconstruction deﬁned above. Infact,letpdenote themapping pit:"+1\0 ~»CP", taking apoint zac0ofthevector space CH1 tothecomplex linepassing through 0andz. Every vector Ztangent toCP"atthepoint pzcanberepresented (inmany ways) astheimage ofavector atthepoint z;under thismap §=p*§, §eTC§*1. Theorem. The square ofthelength ofavector Qintheriemannian metric deﬁned above isgiven bytheformula dS2(() =<69 €><Z9 Z> —<26» Z><2» ' (Z,Z) PROOF. Assume ﬁrstthatthepoint zliesontheunitsphere S2"*1. Decompose thevector 5intotwocomponents: oneinthecomplex linedetermined bythe vector zandtheother inthehermitian-orthogonal direction. Note thathermitian-orthogonal tothevector zmeans euclidean-orthogonal tothevectors zandiz.Thevector zisaeuclidean normal vector tothesphere S2"*1 atz.Thevector izisavector tangent tothecircle inwhich thesphere intersects thecomplex linepassing through z.Thus thecomponent 11ofthevector 5 which ishermitian-orthogonal tozistangent tothesphere S2"*1 andeuclidean-orthogonal tothecircle inwhich thesphere intersects thelinepz. Bythedeﬁnition ofthemetric onCP", theriemannian square ofthelength ofthevector Q isequal totheeuclidean square length ofthecomponent nofiwhich ishermitian-orthogonal to:. Wecalculate thecomponent I]ofi,hermitian-orthogonal toz.Wewrite ourdecomposition as E:c:+n,where (n,:) =0. Byhermitian multiplication withz,weﬁnd <6.I>=c<:.:>. SO <2,Z>é—<5.r>I H:iii (:.:§WW7’ Calculating thehermitian square ofthevector 1;.weﬁnd<l1,?1> =(r/,§)and <2.I><€. 5)-<5.r><:. €>(*1-'1> =—‘r""ij"2*4-- Thus, formula (2)isproved forpoints :oftheunitsphere. Thegeneral casefollows from looking atthehomothetic transformation z—>:/Izl. U 344 Appendix 3:Symplectic structures onalgebraic manifolds Note that ourconstruction allows ustodeﬁne notonly aeuclidean structure (2),butalso ahermitian structure onthetangent space toCP". Consider thehermitian-orthogonal complement Htothedirection ofthe vector zinthespace TCQ“, where zeS2"+1. The map p*IH—>T(CP"),,, maps Hisomorphically (asweshowed above) onto thetangent space toCP" andcarries over thehermitian structure from H. Itisclear thatthescalar square deﬁned bythishermitian structure isgiven byformula (2).Therefore, theformula forthehermitian scalar product in thetangent space toCP"canbewritten down without further calculations: <51, €2><Z, Z)-<61» Z><Z, 62> <1,Z>2 foranyvectors 5,,£2inT<D§+1 satisfying therelation p*5,,=Q,eT(CP"),,,,. Wenote thatinformula (3)thepoint zdoes notnecessarily lieontheunit sphere. The euclidean andhermitian structures (2)and(3)constructed onthe tangent spaces toCP"arenotinvariant under allprojective transformations ofthe manifold CP", butareinvariant under those which aregiven byunitary (preserving thehermitian structure) linear transformations ofthevector n+1space C.(3) (C1,C2)= BThesymplectic structure ofcomplex projective space Weconsider theimaginary part ofthehermitian form (3),taken with co- eﬂicient —l/it(thereason fortaking thiscoefficient isexplained inProblem 1, Section C): <4) Q<c1.t.>= -l1m<c,.c.>. Like theimaginary part ofanyhermitian form, therealbilinear form Qon thetangent space tocomplex projective space isskew-symmetric andnon- degenerate. Theorem. Thediﬂerential 2-form Qgives asymplectic structure oncomplex projective space. PROOF. Weneed only verify thattheform Qisclosed. Consider theexterior derivative (IQofthe form Q.Thisdifferential 3-form onCP"isinvariant withrespect tomappings induced byunitary transformations ofthe space C”1.Itfollows from thisthatitisequal tozero. Toseethis, welook atahermitian-orthonormal basis el,...,e,,ofthetangent space to ('P" atsome point z.Then thevectors el, e,,,iel, ,ie,,form aeuclidean-orthonormal R-basis. Wewillshow thatthevalue oftheform (IQonanytriple ofthese R-basis vectors is equal tozero. (Weassume thatn>l1forn=lthere isnothing toprove.) Note thatinanytriple ofR-basis vectors atleast oneishermitian-orthogonal tothetwo others. Denote thisvector bye.Itiseasytoconstruct aunitary transformation ofthespace C"*1 345 Appendix 3:Symplectic structures onalgebraic manifolds inducing amotion onCP"which ﬁxes thepoint zandthehermitian-orthogonal complement toe,andchanges thedirection ofe. Thevalue oftheform dQonourthree vectors, e,f,andgisequal toitsvalue onthetriple ~e,f,andgbytheinvariance oftheform Q,andishence equal tozero. El Remark. Another method ofconstructing thesame symplectic structure oncomplex projective space consists ofthefollowing. Consider small oscil- lations ofamathematical pendulum with an(n+1)-dimensional conﬁgura- tionspace. Wemake useoftheintegral ofenergy todecrease by1thedegree of freedom ofthesystem. Thephase space obtained after thisoperation isCP", andthesymplectic structure onitagrees with theform Qdescribed above up toafactor. One other method ofconstructing asymplectic structure onCP"uses thefactthatthis space may berepresented asoneoftheorbits oftheco-adjoint representation ofaLiegroup, andonevery such orbit there isalways astandard symplectic structure (cf.Appendix 2,Sec- tionA).FortheLiegroup wecantake thegroup ofunitary (preserving thehermitian metric) operators inan(n+l)-dimensional complex space. Theorbits ofthe co-adjoint representation inthiscasearethesame asofthe adjoint representation. Intheadjoint representation theoperator ofreﬂection through ahyperplane (which changes thesignoftheﬁrstcoordinate andleaves theothers ﬁxed) hasCP"asitsorbit, since thereﬂection operator isuniquely determined by thecomplex lineorthogonal tothehyperplane. CSymplectic structure onalgebraic manifolds Wewillnow obtain asymplectic structure onanycomplex submanifold M ofcomplex projective space. Letj:M—>CP"beanembedding ofthecomplex manifold Mintocomplex projective space. Theriemannian, hermitian, and symplectic structures onprojective space induce corresponding structures on M.Forexample, thesymplectic structure onMisgiven bytheformula QM Z j*Q. Theorem. Thediﬂerential form QMgives asymplectic structure onthemanifold M. PROOF. Thenondegeneracy ofthe2-form QMfollows from thefactthatM isacomplex submanifold. Infact, thequadratic form (Z5.<5)=Qiufé, ii) ispositive deﬁnite (itisinduced bytheriemannian metric onCP").Therefore, thebilinear form (5,n)=QM(§, in)isnondegenerate. This means thatthe form QMisalso nondegenerate. Theform QMisclosed since theform Qis closed. Cl Remark. Inthesame way asforcomplex projective space, wedeﬁne a hermitian structure onthetangent spaces ofitscomplex submanifolds; the symplectic structure istheimaginary part. 346 Appendix 3:Symplectic structures onalgebraic manifolds Acomplex manifold with ahermitian metric whose imaginary part isa closed form (i.e.asymplectic structure) iscalled aKdhler manifold andits hermitian metric aKdhler metric. Many important results have been obtained inthegeometry ofKéihler manifolds; inparticular, they have remarkable topological properties (cf.,forexample, A.Weil, “Variétés Kéihlériennes,” Hermann, 1958). Notallsymplectic manifolds admit aKéihler structure. PROBLEM l.Calculate thesymplectic structure Qintheafﬁne chart w=z,:zooftheprojective lineCF‘. ANSWER. Q=(1/rt)(dx /\dy)/(1 +x2+y2)2, where w=x+iy_The coefficient inthede- ﬁnition oftheform Qischosen toobtain theusual orientation ofthecomplex line(dxAdy) andsothattheintegral oftheform Qalong thewhole projective lineisequal tol. PROBLEM 2.Show thatthesymplectic structure Qintheaﬂine chart w,=z,25‘(k=1,...,n) ofthe projective space CP"={(z0:z,:...:z,,)}isgiven bytheformula Q:iZ05u<i5»(wu dwrTwIdwk)(Wk dwl“W1dwk) 2" (2,it=0(wkWr))2 Byconvention, we=1. Remark. Differential forms onacomplex space with complex values (such asdw,anddig) aredeﬁned ascomplex linear functions oftangent vectors; ifw,,=x,,+iy,,,then dwk =dxk + JW,‘ =dxk — Thespace ofsuchforms inC"hascomplex dimension 2n;the2nforms dw,,, dﬁk(k=1,.,.,n), forexample, form aC-basis, orthe2nforms dxk,dy,. Exterior multiplication isdeﬁned intheusual wayandobeys theusual rules. Forexample, dw/\dW=(dx+idy)/\(dx—idy) =—2idx Ady. Letfbeareal-smooth function onC"(with complex values, ingeneral). Anexample of such afunction islw|2=ZwkW,Thedifferential ofthefunction fisacomplex l-form. There- fore, itcanbedecomposed inthebasis dwi, dW,. Thecoefficients ofthisdecomposition are called thepartial derivatives “with respect tow,"and“with respect toW,"1 5f 5f_df=5;dW +55dW. Incalculating exterior derivatives itisalsoconvenient toseparate intodifferentiation d’ withrespect tothevariable wandd"with respect tothevariable W,sothatd=d’+d”. Forexample, forafunction f 6d'f=aldw d"f= dw.8w aw Forthedifferential l-form 0)=Za,(‘Wk +bkdvT',, 347 FAppend ix3:Symplectic structures onalgebraic manifolds theoperators d’andd"aredeﬁned analogously: I POBLEM 3.Show thatthesymplectic std'w=Zd'a,‘/\dwk+db,(/\ k d”w =Zd"ak /\dwk+d”bk Adﬁk. ructure Qonthealﬁne chart (wk=zkzg‘)ofthe projective R ' btheformula space CP" isgiven y 348- n Q=Law": |-|1.21: “go wk Appendix 4:Contact structures Anodd-dimensional manifold cannot admit asymplectic structure. The analogue ofasymplectic structure forodd-dimensional manifolds isalittle lesssymmetric, butalsoavery interesting structure—the contact structure. Thesource ofsymplectic structures inmechanics arephase spaces (i.e., cotangent bundles toconﬁguration manifolds), onwhich there isalways a canonical symplectic structure. Thesource ofcontact structures aremani- folds ofcontact elements ofconﬁguration spaces. Acontact element toann-dimensional smooth manifold atsome point is an(n-1)-dimensional plane tangent tothemanifold atthat point (i.e., an (n—1)-dimensional subspace ofthen-dimensional tangent space atthat point). Thesetofallcontact elements ofann-dimensional manifold hasanatural smooth manifold structure ofdimension 2n—1.Itturns outthatthere isan interesting additional “contact structure ”onthisodd-dimensional manifold (wedescribe thisbelow). Themanifold ofcontact elements ofariemannian n-dimensional manifold isclosely related tothe(2n—1)-dimensional manifold ofunittangent vectors ofthisriemannian n-dimensional manifold, ortothe(2n—1)-dimensional energy level manifold ofapoint mass moving ontheriemannian manifold under inertia. Thecontact structures onthese (2n—1)-dimensional mani- folds areclosely related tothesymplectic structure onthe2n-dimensional phase space ofthepoint (i.e., thecotangent bundle oftheoriginal n-dimen- sional riemannian manifold). ADeﬁnition ofcontact structure Deﬁnition. Acontact structure onamanifold isasmooth ﬁeld oftangent hyperplanes” satisfying anondegeneracy condition which willbeformu- lated later. Toformulate thiscondition weexamine what aﬁeldofhyperplanes looks likeingeneral inaneighborhood ofapoint inanN-dimensional manifold. EXAMPLE. LetN=2.Then themanifold isasurface andaﬁeld ofhyper- planes isaﬁeld ofstraight lines. Such aﬁeld inaneighborhood ofapoint is always constructed very simply, namely, asaﬁeld oftangents toafamily ofparallel lines inaplane. More precisely, oneofthebasic results ofthelocal theory ofordinary differential equations isthat itispossible tochange any smooth ﬁeldoftangent lines onamanifold intoaﬁeldoftangents toafamily ofstraight lines ineuclidean space byusing adilleomorphism inasuﬁiciently small neighborhood ofanypoint ofthemanifold. IfN>2,then ahyperplane isnotaline, and thequestion becomes signiﬁcantly more complicated. Forexample, most ﬁelds oftwo-dimensional 98Ahyperplane inavector space isasubspace ofdimension llessthan thedimension ofthe space (i.e.,thezero level setofalinear function which isnotidentically zero). Atangent hyper- plane isahyperplane inatangent space. 349 Appendix 4:Contact structures tangent planes inordinary three-dimensional space cannot bediffeo- morphically mapped onto aﬁeld ofparallel planes. Thereason isthatthere exist ﬁelds oftangent planes forwhich itisimpossible toﬁnd“integral sur- faces,” i.e.,surfaces which have theprescribed tangent plane ateach point. Thenondegeneracy condition foraﬁeld ofhyperplanes which enters into thedeﬁnition ofcontact structure consists ofthestipulation thattheﬁeldof hyperplanes must bemaximally farfrom aﬁeld oftangents toafamily of hyperplanes. Inorder tomeasure thisdistance, aswell astoconvince our- selves oftheexistence ofﬁelds without integral hypersurfaces, wemust make afewconstructions andcalculations.” BFrobenius’ integrability condition Wewillconsider some point onanN-dimensional manifold and tryto construct asurface passing through thispoint andtangent toagiven ﬁeld of(N—1)-dimensional planes ateach point (anintegral surface). Tothisendweintroduce acoordinate system onto aneighborhood of thispoint sothat atthepoint itself onecoordinate surface istangent toa plane oftheﬁeld. Wewillcallthisplane thehorizontal plane, andwillcall thecoordinate axisnotlying initthevertical axis. Construction ofanintegral surface. Anintegral surface, ifoneexists, isthe graph ofafunction ofN—1variables near theorigin. Toconstruct it,we cantake some smooth path onthehorizontal plane. Then thevertical lines over thispath form atwo-dimensional surface (cylinder); ourﬁeld ofplanes intersects itstangent planes inaﬁeld oftangent lines. Theintegral surface wearelooking for,ifitexists, intersects thiscylinder inanintegral curve ofthe ﬁeld oflines, starting attheorigin. Such anintegral curve always exists independent ofwhether anintegral surface exists. Thus wecanconstruct an integral surface over thehorizontal plane bymoving along smooth curves in thelatter. Inorder toobtain asmooth integral surface from alltheintegral curves weneed theresult ofourconstruction tobeindependent ofthepath, deter- mined only byitsendpoint. Inparticular, foracircuit ofaclosed path ina neighborhood oftheorigin inthehorizontal plane, theintegral curve onthe cylinder must close up. Itiseasytoconstruct examples ofﬁelds ofplanes forwhich such closure does nottake place and, therefore, forwhich anintegral surface does not exist. Such ﬁelds ofplanes arecalled nonintegrable. Example ofanonintegrable ﬁeld ofplanes. Inorder togiveaﬁeld ofplanes andmeasure numerically thedeviation from closure, weintroduce thefollow- ingnotation. Wenote ﬁrstofallthataﬁeldofhyperplanes canbegiven locally byadifferential l-form; aplane inthetangent space gives a1-form upto °°From now on,wewillomit thepreﬁx “hyper-". Ifwewish, wemay assume that wearein three-dimensional space andahypersurface isanordinary surface. Thehigher-dimensional caseisanalogous tothethree-dimensional case. 350 Appendix 4:Contact structures multiplication byanonzero constant. Wewillchoose thisconstant sothat thevalue oftheform onthevertical basic vector isequal to1. This condition canbesatisﬁed insome neighborhood oftheorigin since theplane oftheﬁeld atzero does notcontain thevertical direction. This condition determines theform uniquely (given theﬁeld ofplanes). Aﬁeld ofplanes inordinary threesspace which does nothave anintegral surface canbegiven, forexample, bythe1-form w=xdy-l-dz, where xandyarethehorizontal coordinates andzisthevertical. Theproof ofthefactthatthisﬁeld ofplanes isnonintegrable willbegiven below. Construction ofa2-form measuring nonintegrability. With thehelp ofthe form giving theﬁeld, wecanmeasure thedegree ofnonintegrability. This is done using thefollowing construction (Figure 236). Figure 236 Integral curves constructed foranon-integrable ﬁeldofplanes Consider apair ofvectors emanating from theorigin andlying inthe horizontal plane ofourcoordinate system. Construct aparallelogram on them. Weobtain twopaths from theorigin totheopposite vertex. Over each ofthese twopaths wecanconstruct anintegral curve (with twosections) as described above. Asaresult, ingeneral, there arise twodifferent points over thevertex oftheparallelogram opposite totheorigin. Thedifference inthe heights ofthese points isafunction ofourpairofvectors. This function is skew-symmetric andequal tozero ifoneofthevectors isequal tozero. Thus thelinear part oftheTaylor series ofthisfunction iszero atzero, andthe quadratic part ofitsTaylor series isabilinear skew-symmetric form onthe horizontal plane. Iftheﬁeld isintegrable, then this2-form isequal tozero. Therefore, this 2-form canbeconsidered asameasure ofthenonintegrability oftheﬁeld. The2-form iswelldeﬁned. Weconstructed the2-form above with thehelp ofcoordinates. However, thevalue ofour2-form onapairoftangent vectors does notdepend onthecoordinate system, butonly onthel-form used to givetheﬁeld. Toconvince ourselves ofthis,itisenough toprove thefollowing. Theorem. The2-form deﬁned above agrees with theexterior derivative ofthe l-form to,dco|,,,=O, onthenullspace ofa). 351 Appendix 4:Contact structures Pnoor. Wewillshow thatthedifference intheheights ofthetwopoints obtained asaresult ofourtwomotions along thesides ofthe parallelogram isthesame astheintegral ofthe l-form w over thefoursides oftheparallelogram, uptoaquantity small ofthird order with respect to thesides oftheparallelogram. Tothisendwenote thattheheight oftheriseofanintegral curve along anypath oflength s emanating from theorigin hasorder 62,since attheorigin theplane oftheﬁeld ishorizontal. Therefore, theintegrals ofthe2-form dwover allfourvertical areas over thesides oftheparal- lelogram bounded bytheintegral curves andthehorizontal plane, have order £3ifthesides areoforder s. Theintegrals oftheform walong integral curves areexactly equal tozero. Therefore, by Stokes" formula, theincrease inheight along theintegral curve lying overanyofthesides ofthe parallelogram isequal totheintegral ofthe1-form walong thissideuptoaquantity ofthird- order smallness. Now thetheorem follows directly from thedeﬁnition ofexterior differentiation. U Some arbitrariness remains intheChOiCe ofthe1-form towhich weused to construct our2-form. Namely, theform wisdeﬁned bytheﬁeld ofplanes only uptomultiplication byafunctionfwhich isnever zero. Inother words, wecould have started with theform fw.Then wewould have obtained the 2-form dfco=fdw+df/\ w, which, onourplane, dillers from the2-form dwbymultiplication bythe nonzero number f(0). Thus the2-form constructed ontheplane oftheﬁeld isdeﬁned invariantly uptomultiplication byanonzero constant. Condition forintegrability ofaﬁeld ofplanes Theorem. Ifaﬁeld ofhyperplanes isintegrable, then the2-form constructed above onaplane oftheﬁeld isequal tozero. Conversely, ifthe2-form con- structed onevery plane oftheﬁeld isequal tozero, thentheﬁeld isintegrable. PROOF. Theﬁrstassertion ofthetheorem isclear bytheconstruction ofthe2-form. Theproof ofthesecond assertion canbecarried outbyexactly theSame reasoning weused toprove the commutativity ofphase ﬂows forwhich thePoisson bracket ofthevelocity ﬁelds wasequal to zero. Wecansimply refer tothiscommutativity, applying ittotheintegral curves arising over thelines ofthecoordinate directions inthehorizontal plane. Cl Theorem. Theintegrability condition foraﬁeld ofplanes, dco=O for o)=O isequivalent tothefollowing condition ofFrobenius: to/\do)=O. Pnooi. Weconsider thevalue ofthe3-form above onanythrcc distinct coordinate vectors. Only oneofthese vectors canbethevertical. Therefore, ofalltheterms entering intothedeﬁni- tionofthe value oftheexterior product ofthe three vectors, only oneisnonzero: theproduct of 352 Appendix 4:Contact structures thevalue oftheform toonthevertical vector with thevalue oftheform do:onthepairof horizontal vectors. Iftheﬁeld given bytheform isintegrable, then thesecond factor iszero. soour3-form iszeroonarbitrary triples ofvectors. Conversely, ifthe3-form isequal tozero foranyvectors, then itisequal tozero forany triple ofcoordinate vectors, ofwhich oneisvertical and theother two horizontal. The value ofthe3-form onsuch atriple isequal totheproduct ofthevalue oftoonthevertical vector withthevalue ofdco onthepairofhorizontal vectors. Theﬁrstfactor isnotzero, sothesecond must bezero, andthus theform dwiszero onaplane oftheﬁeld. U CNondegenerate ﬁelds ofhyperplanes Deﬁnition. Aﬁeld ofhyperplanes issaidtobenondegenerate atapoint ifthe rank ofthe2-form dco|,,,:0 intheplane oftheﬁeld passing through this point isequal tothedimension oftheplane. This means thatforanynonzero vector inourplane, wecanﬁndanother vector intheplane such thatthevalue ofthe2-form onthispairofvectors isnotzero. Deﬁnition. Aﬁeld ofplanes iscalled nondegenerate onamanifold ifitisnon- degenerate atevery point ofthemanifold. Note thatonaneven-dimensional manifold there cannot beanondegen- erate ﬁeld ofhyperplanes; onsuch amanifold ahyperplane isodd-dimen- sional, and therank ofevery skew-symmetric bilinear form onan odd-dimensional space islessthan thedimension ofthespace (cf.Section 44). Nondegenerate ﬁelds ofhyperplanes doexist onodd-dimensional mani- folds. EXAMPLE. Consider aeuclidean space ofdimension 2m+1with coordinates x,y,andz(where xandyarevectors inanm-dimensional space andzisa number). Thel-form w=xdy+dz deﬁnes aﬁeldofhyperplanes. Theplane oftheﬁeldpassing through theorigin hasequation dz=O.Wetake xandyascoordinates inthishyperplane. Therefore, inthisplane oftheﬁeld our2-form canbewritten intheform dco|w=0 =dx/\dy=dxl/\dyl+ +dx,,, /\dy,,,. Therahk ofthisform is2m,soourﬁeld isnondegenerate attheorigin, and thus also inaneighborhood oftheorigin (infact, thisﬁeld ofplanes is nondegenerate atallpoints ofthespace). Now, ﬁnally, wecangivethedeﬁnition ofacontact structure onamani- fold: acontact structure onamanifold isanondegenerate ﬁeld oftangent hyperplanes. 353 Appendix 4:Contact structures DThemanifold ofcontact elements Theterm “contact structure” stems from thefactthatthere isalways such a structure onamanifold ofcontact elements ofasmooth n-manifold. Deﬁnition. Ahyperplane (dimension n—1)tangent toamanifold atsome point iscalled acontact element, andthispoint thepoint ofcontact. Thesetofallcontact elements ofann-dimensional manifold hasthestruc- tureofasmooth manifold ofdimension 2n—1. Infact,thesetofcontact elements withaﬁxed point ofcontact isthesetofall (n—1)-dimen- sional subspaces ofann-dimensional vector space, i.e.,aprojective space ofdimension n—l. Togive acontact element wemust therefore givethencoordinates ofthepoint ofcontact together with then—1coordinates deﬁning apoint ofan(n~1)-dimensional projective space “Zn—1coordinates inall. The manifold ofallcontact elements ofann-dimensional manifold isa ﬁber bundle whose base isourmanifold andwhose ﬁber is(n—1)-dimen- sional projective space. Theorem. Thebundle ofcontact elements istheprojectivization ofthecotangent bundle: itcanbeobtained from thecotangent bundle bychanging every cotangent n-dimensional vector space into an(n—1)-dimensional pro- jective space (apoint ofwhich isalinepassing through theorigin inthe cotangent space). PROOF. Acontact element isgiven bya1-form onthetangent space, forwhich thiselement is azero level set.This form isnotzero. anditisdetermined uptomultiplication byanonzero number. Butaform onthetangent space isavector ofthecotangent space. Therefore. a nonzero form onthetangent space, determined uptoamultiplication byanonzero number, isanonzero vector ofthecotangent space, determined uptoamultiplication byanonzero number, i.e.,apoint oftheprojectivized cotangent space. Cl Thecontact structure onthemanifold ofcontact elements. Inthetangent space tothemanifold ofcontact elements there isadistinguished hyperplane. Itiscalled thecontact hyperplane andisdeﬁned inthefollowing way. Weﬁxapoint ofthe(2n—1)-dimensional manifold ofcontact elements onann-dimensional manifold. Wecanthink ofthispoint asan(n—1)- dimensional plane tangent totheoriginal n-dimensional manifold. Deﬁnition. Atangent vector tothemanifold ofcontact elements ataﬁxed point belongs tothecontact hyperplane ifitsprojection onto then- dimensional manifold liesinthe(n~1)-dimensional plane which isthe given point ofthemanifold ofcontact elements. 354 Appendix 4:Contact structures Inother words, adisplacement ofacontact element istangent tothe contact hyperplane ifthevelocity ofthepoint ofcontact belongs tothis contact element, nomatter how theelement turns. EXAMPLE. Wetake some submanifold ofourn-dimensional manifold and consider all(n—l)-dimensional planes tangent toit(i.e.,contact elements). The setofallsuch contact elements forms asmooth submanifold ofthe (2n—1)-dimensional manifold ofallcontact elements. The dimension of thissubmanifold isequal ton—1,nomatter what thedimension ofthe original submanifold (which could be(n—l)-dimensional, orhave smaller dimension, down toacurve oreven apoint). This (n—1)-dimensional submanifold ofthe (2n-1)-dimensional manifold ofallcontact elements istangent ateach ofitspoints totheﬁeld of contact hyperplanes (bythedeﬁnition ofcontact hyperplane). Thus the ﬁeldof(2n—2)-dimensionalcontacthyperplaneshas an(n—l)-dimensional integral manifold. PROBLEM. Does thisﬁeldofplanes have integral manifolds ofhigher dimensions? ANSWER. No. PROBLEM. lsitpossible togivetheﬁeld ofcontact hyperplanes byadifferential 1-form onthe manifold ofall contact elements? ANSWER. No.even iftheunderlying n-dimensional manifold isaeuclidean space (forexample, theordinary two-plane). Wewillshow below thattheﬁeld ofcontact hyperplanes onthe(2n—1)- dimensional manifold ofallcontact elements ofann-dimensional manifold is nondegenerate. The proof uses thesymplectic structure ofthecotangent bundle. Themanifold ofcontact elements isrelated byasimple construction tothespace ofthecotangent bundle (the projectivization ofwhich isthe manifold ofcontact elements). Moreover, thenondegeneracy oftheﬁeld of contact planes oftheprojectivized bundle isclosely related tothenon- degeneracy ofthe2-form giving thesymplectic structure ofthecotangent bundle. Theconstruction weareconcerned with willbecarried outbelow ina somewhat more general situation. Namely, foranyodd-dimensional mani- fold with acontact structure wecanconstruct its“symplectiﬁcation”—a symplectic manifold whose dimension isonelarger. The inter-relation be- tween these twomanifolds—-the odd-dimensional contact manifold andthe even-dimensional symplectic manifold—is thesame asbetween themanifold ofcontact elements with itscontact structure andthecotangent bundle with itssymplectic structure. 355 Appendix 4:Contact structures ESymplectzﬁcation ofacontact manifold Consider anarbitrary contact manifold, i.e.,amanifold ofodddimension N with anondegenerate ﬁeldoftangent hyperplanes (ofeven dimension N—1). Wewillcallthese planes contact planes. Every contact plane istangent to thecontact manifold atonepoint. Wewillcallthispoint thepoint ofcontact. Deﬁnition. Acontact form isalinear form onthetangent space atthepoint of contact ofthemanifold such thatitszero setisthecontact plane. Itshould beemphasized that thecontact form isnotadifferential form butanalgebraic linear form ononetangent space. Deﬁnition. Thesymplectification ofacontact manifold isthesetofallcontact forms onthecontact manifold, provided with thestructure ofasym- plectic manifold asdeﬁned below. Wenote ﬁrstofallthatthesetofallcontact forms onacontact manifold hasanatural structure ofasmooth manifold ofeven dimension N+1. Namely, wecanconsider thesetofallcontact forms asthespace ofabundle over theoriginal contact manifold. Projection onto thebase isthemapping associating thecontact form tothepoint ofcontact. Theﬁber ofthis bundle isthesetofcontact forms with acommon point of contact. Allsuch forms areobtained from oneanother bymultiplication bya nonzero number (sothatthey determine thesame contact plane). Thus the ﬁber ofourbundle isone-dimensional: itisthelineminus apoint. Wealsonote thatthegroup ofnonzero realnumbers actsonthemanifold ofallcontact forms bytheoperation ofmultiplication, i.e.,theproduct ofa contact form andanonzero number isagain acontact form. Inthiswaythe group actsonourbundle, leaving every ﬁber ﬁxed (upon multiplication ofa form byanumber thepoint ofcontact isnotchanged). Remark. Sofarwehave notused thenondegeneracy oftheﬁeld ofplanes. Nondegeneracy isneeded only toinsure that themanifold obtained by symplectiﬁcation issymplectic. EXAMPLE. Consider themanifold (ofdimension 2n—1)ofallcontact elements ofann-dimensional smooth manifold. Onthemanifold ofelements there isa ﬁeld ofhyperplanes (which wedeﬁned above andcalled thecontact hyper- planes). Therefore, wecansymplectify themanifold ofcontact elements. Asaresult ofsymplectiﬁcation weobtain a2n-dimensional manifold. This manifold isthespace ofthecotangent bundle oftheoriginal n-dimen- sional manifold without zero vectors. Theaction bythemultiplicative group ofrealnumbers ontheﬁber reduces tomultiplication ofvectors oftheco- tangent space byanumber. 356 Appendix 4:Contact structures Onthecotangent bundle there isadistinguished l-form “pdq.”There is ananalogous 1-form onanymanifold obtained bysymplectiﬁcation from a contact manifold. Thecanonical 1-form onthesymplectiﬁed space Deﬁnition. The canonical l-form inthesymplectiﬁed space ofacontact manifold isthedifferential l-form Otwhose value onanyvector 5tangent tothesymplectiﬁed space atsome point p(Figure 237)isequal tothevalue ontheprojection ofthevector 5onto thetangent plane tothecontact manifold ofthe 1-form onthistangent plane which isthepoint p: a(t)=11(tt..6). where itistheprojection ofthesymplectiﬁed space onto thecontact manifold. i Figure 237 Symplectiﬁcation ofacontact manifold Theorem. Theexterior derivative ofthecanonical 1-form onthesymplectiﬁed space ofacontact manifold isanondegenerate 2-form. Corollary. The symplectiﬁed space ofacontact manifold hasasymplectic structure which iscanonically (i.e., uniquely, without arbitrariness) deter- mined bythecontact structure oftheunderlying odd-dimensional manifold. PROOF OFTHEOREM. Since theassertions ofthetheorem arelocal. itissufficient toprove itin asmall neighborhood ofapoint ofthe manifold. Inasmall neighborhood ofapoint onacontact manifold. aﬁeldofcontact planes canbegiven byadifferential form toonthecontact manifold. Weﬁxsuch al-form v). Bythesame token wecanrepresent thesymplectiﬁed space ofthecontact manifold over ourneighborhood asthedirect product ofthe neighborhood andthelineminus apoint. Namely, weassociate tothepair(x,A)-—where xisapoint ofthecontact manifold and/lisanonzero number —the contact form given bythedifferential l-form lrvonthetangent space atthepoint x. Thus inthepartofthesymplectiﬁed space weareconsidering. wehave deﬁned afunction A 357 Appendix 4:Contact structures whose values arenonzero numbers. ltshould beemphasized that/lisonlyalocal coordinate on thesymplectiﬁed manifold andthatthiscoordinate isnotdeﬁned canonically; itdepends on thechoice ofdifferential l-form to.Thecanonical 1-form atcanbewritten inournotation as at=/l1r*o) anddoes notdepend onthechoice ofa). Theexterior derivative ofthe l-form atthushastheform det=d/l/\rc*w +/li't*dto. Wewill show that the2-form dotisnondegenerate, i.e.,that forany vector iftangent to thesymplectiﬁcation, wecanﬁndavector nsuch thatdat(§,n) asO.Weselect from vectors tangent tothesymplectiﬁcation. those ofthefollowing type. Wecallavector 6vertical ifit istangent totheﬁber, i.e..ifrr,,§=O.Wecallthevector ihorizontal ifitistangent toalevel surface ofthefunction ,1.i.e.,ifd}.(§) =O.Wecallthevector 5acontact rector ifitsprojection onto thecontact manifold liesinthecontact plane, i.e.,ifw(rr*§) =0(inother words, ifat(i) =O). Wecalculate thevalue oftheform dotonapairofvectors (i,n): d0t(<f, I7)=(d/l/\1r*m)(§, ij)+(}.7t*doi)(§, ij). Assume that5isnotacontact vector. Forn,take anonzero vertical vector, sothatrc,,,nIO. Then thesecond term isequal tozero, andtheﬁrstterm isequal to -i1»1(»i>w(n..o which isnotzero since i1isanonzero vertical vector andZisnotacontact vector. Thus if5 isnotacontact vector, wehave found ani7forwhich dat(§, n)¢O. Now assume that cfisacontact vector andnotvertical. Then fori1wetake anycontact vector. Now theﬁrstterm isentirely Zero. andthesecond (and therefore thesum) isreduced toAdu)(rc* C.11*i1).Since Zisnotvertical, thevector i't,,,ﬁlying inthecontact plane isnotzero. Butthe2-form do.)isnondegenerate onthecontact plane (bythedeﬁnition ofcontact structure). Thus there isacontact vector i7such thatdv)(rr* Q",TC,,,'1) ¢O.Since A;¢0,wehave found a vector i1forwhich tl0t(C, I7)940. Finally, ifthevector 5isnonzero andvertical, then fornwecantake anyvector which is notacontact vector. 1:1 Remark. Theconstructions ofthe1-form ozandthe2-form dotarevalid foranarbitrary manifold with aﬁeld ofhyperplanes, anddonotdepend on thecondition ofnondegeneracy. However, the2-form dotwill deﬁne a symplectic structure only inthecasewhen theﬁeldofplanes isnondegenerate. PROOF. Assume thattheﬁeld isdegenerate, i.e.,thatthere exists anonzero vector Einaplane oftheﬁeld such thatdto(§’, rj’)=0forallvectors 11'inthisplane. Forsuch a5',thequantity dco(§’, i1’)asafunction ofn’isalinear form, identically equal tozero ontheplane oftheﬁeld. Therefore there isanumber ,unotdependent oni1’such that dw(€'. ii’)=uw(i1’) forallvectors i1’ofthe tangent space. Wenowtakefor6atangent vector tothesymplectiﬁed manifold forwhich 1r*§=cf’.Such avector Qisdetermined uptoaddition ofavertical summand, andwewillshow thatforasuitable choice ofthissummand wewillhave do<(ef, n):Oforallr1. 358 Appendix 4:Contact structures Theﬁrstterm oftheformula fordotisequal tod).(§)¢o(rt*n) (since w(rt* 5)=0).Thesecond term isequal toZdo)(1r* 5,11*:1)=lato(rc*r1). Wechoose thevertical component ofthevector § sothatdl(§) =—/la. Then fwillbeskew-orthogonal toallvectors I1. Thus ifdotisasymplectic structure, then theunderlying ﬁeld ofhyperplanes isacontact structure. l:| Corollary. Theﬁeld ofcontact hyperplanes deﬁnes acontact Structure onthe manifold ofallcontact elements ofanysmooth manifold. PROOF. The symplectiﬁcation ofthe(2n—1)-dimensional manifold ofall contact elements onann-dimensional smooth manifold, constructed with help oftheﬁeld of(2n—-2)-dimensional contact planes, isbyconstruction thespace ofthecotangent bundle oftheunderlying n-dimensional manifold without thezero cotangent vectors. The canonical l-form ozonthesym- plectiﬁcation is,byitsdeﬁnition, thesame 1-form onthecotangent bundle thatwecalled “pdq”andwhich isfundamental inhamilton mechanics (cf. Section 37).Itsderivative dotistherefore theform “dp/\dq”deﬁning the usual symplectic structure ofaphase space. Therefore theform docisnon- degenerate, and, bythepreceding remark, theﬁeld ofcontact hyperplanes is nondegenerate. U FContact dtﬂeomorphisms andvector ﬁelds Deﬁnition. Adiffeomorphism ofacontact manifold toitself iscalled a contact diﬂeomorphism ifitpreserves thecontact structure, i.e.,carries every plane ofagiven structure ofaﬁeld ofhyperplanes toaplane ofthe same ﬁeld. EXAMPLE. Consider the(2n—1)-dimensional manifold ofcontact elements ofann-dimensional smooth manifold with itsusual contact structure. To each contact element wecanascribe a“positive side” bychoosing oneofthe halves intowhich thiselement divides thetangent space tothen-dimensional manifold. Wewillcallacontact element with achosen sidea(transversally) oriented contact element. The oriented contact elements onourn-dimensional manifold form a (2n—l)-dimensional smooth manifold with anatural contact structure (it isadouble covering ofthemanifold ofordinary nonoriented contact elements). Now assume that wearegiven ariemannian metric ontheunderlying n-dimensional manifold. Then there isa“geodesic ﬂow”1°°onthemanifold oforiented contact elements. Thetransformation after time tbythisﬂow isdeﬁned asfollows. Wegooutfrom thepoint ofcontact ofacontact element along thegeodesic orthogonal toitanddirected totheside orienting the element. Inthecourse oftime twewillmove thepoint ofcontact along the ‘°°Strictly speaking, weneed torequire thattheriemannian manifold becomplete, i.e.,geodesics canbecontinued without limit. 359 Appendix 4:Contact structures geodesic, keeping theelement orthogonal tothegeodesic. After time twe obtain aneworiented element. Wehave deﬁned thegeodesic ﬂowoforiented contact elements. Theorem. Thegeodesic ﬂow oforiented contact elements consists ofcontact diﬂeomorphisms. Theproof ofthistheorem willnotbepresented since itisjustareformula- tioninnewterms ofHuygens’ principle (cf.Section 46). Deﬁnition. Avector ﬁeld onacontact manifold iscalled acontact vector ﬁeld ifitisthevelocity ﬁeld ofaone-parameter (local) group ofcontact diﬁeomorphisms. Theorem. ThePoisson bracket ofcontact vector ﬁelds isacontact vector ﬁeld. Thecontact vector ﬁelds form asubalgebra intheLie algebra ofallsmooth vector ﬁelds onacontact manifold. Theproof follows directly from thedeﬁnitions. GSymplectiﬁcation ofcontact diﬂeomorphisms andﬁelds Forevery contact diffeomorphism ofacontact manifold there isacanonically constructed symplectic dilleomorphism ofitssymplectiﬁcation. This sym- plectic dilfeomorphism commutes with theaction ofthemultiplicative group ofrealnumbers onthesymplectiﬁed manifold andisdeﬁned bythefollowing construction. Recall thatapoint ofthesymplectiﬁed manifold isacontact form onthe underlying contact manifold. Deﬁnition. Theimage ofacontact form pwith point ofcontact xunder the action ofacontact diffeomorphism fofthecontact manifold toitself is theform ﬁP=(ﬁ(x))— IP- Insimple terms, wecarry theform pfrom thetangent space atthepoint x tothetangent space atf(x)using thediﬁeomorphism f(whose derivative at xdetermines anisomorphism between these twotangent spaces). Theform f,pisacontact form since thediffeomorphism fisacontact diffeomorphism. Theorem. Themapping f.deﬁned above ofthesymplectiﬁcation ofacontact manifold toitself isasymplectic diﬁeomorphism which commutes with the action ofthemultiplicative group ofrealnumbers andpreserves thecanonical 1-form onthesymplectiﬁcation. 360 Appendix 4:Contact structures PROOF. Theassertion ofthetheorem follows from thefactthatthecanonical l-form, thesymp- lectic 2-form, andtheaction ofthegroup ofrealnumbers arealldetermined bythecontact structure itself (fortheir construction wedidnotusecoordinates oranyother noninvariant tools), andthedilleomorphism fpreserves thecontact structure. Itfollows from thisthatﬂ preserves allthatwhich wasinvariantly constructed using thecontact structure, inparticular the1-form oz,itsderivative dot,andtheaction ofthegroup. Theorem. Every symplectic diﬂeomorphism ofthesymplectiﬁcation ofacontact manifold which commutes with theaction ofthemultiplicative group (1) projects onto theunderlying contact manifold asacontact diﬂeomorphism and(2)preserves thecanonical l-form oz. Pnoor. Every diffeomorphism which commutes with theaction ofthemultiplicative group projects onto some diffeomorphism ofthecontact manifold. Toshow that thisisacontact diffeornorphism itissulﬁcient toprove thesecond assertion ofthetheorem (since only those vectors forwhich 01(5)=0project onto thecontact plane). Toprove thesecond assertion weexpress theintegral oftheform along anypath yinterms ofthesymplectic structure dot: J-at=lim da, 7 t—~0 a(t) where the2-chain 0(2)isobtained from 7bymultiplication byallnumbers intheinterval [5,1]. Theboundary of0contains, besides y,twovertical intervals andthepath cy.Theintegrals ofa overthevertical intervals areequal tozero, andtheintegral over 23'approaches 0asedoes. Now from theinvariance ofthe2-form dotandthecommutativity ofourdilleomorphism F withmultiplication bynumbers itfollows thatforanypath y w‘. 1 Z fa‘Fy F andthusthediffeomorphism Fpreserves thel-form oz. Cl Deﬁnition. Thesymplectiﬁcation ofacontact vector ﬁeld isdeﬁned bythe following construction. Consider theﬁeld asavelocity ﬁeld ofaone- parameter group ofcontact diﬁeomorphisms. Symplectify thediffeomor- phisms. Consider thevelocity ﬁeld ofthisgroup. Itiscalled thesym- plectiﬁcation oftheoriginal ﬁeld. Theorem. Thesymplectiﬁcation ofacontact vector ﬁeld isahamiltonian vector ﬁeld. Thehamiltonian canbechosen tobehomogeneous ofﬁrst order with respect totheaction ofmultiplication bythegroup ofrealnumbers: H(ix) =}.H(x). Conversely, every hamiltonian ﬁeld onasymplectiﬁed contact manifold, having ahamiltonian which ishomogeneous ofdegree 1,projects onto the underlying contact manifold asacontact vector ﬁeld. 361 Appendix 4:Contact structures PROOF. The fact that symplectiﬁcations ofcontact diffeomorphisms are symplectic implies that thesymplectiﬁcation ofacontact ﬁeld ishamil- tonian. Thehomogeneity ofthehamiltonian follows from thehomogeneity of symplectic diﬁeomorphisms (from commutativity with multiplication by2.). Thus theﬁrstassertion ofthetheorem follows from thetheorem onsym- plectiﬁcations ofcontact diffeomorphisms. The second part follows inthe same wayfrom thetheorem onhomogeneous symplectic diffeomorphisms.El Corollary. Symplectiﬁcation ofvector ﬁelds isanisomorphic map oftheLie algebra ofcontact vector ﬁelds onto theLiealgebra ofalllocally hamiltonian vector ﬁelds with hamiltonians which arehomogeneous ofdegree 1. Theproof isclear. HDarboux’s theorem forcontact structures Darboux’s theorem isatheorem onthelocal uniqueness ofacontact struc- ture. Itcanbeformulated inanyofthefollowing three ways. Theorem. Allcontact manifolds ofthesame dimension arelocally contact dijfeomorphic (i.e.,there isadiﬂeomorphism ofasujﬁcientl ysmall neighbor- hood ofanypoint ofonecontact manifold onto aneighborhood ofanypoint oftheother which carries thenoted point oftheﬁrst neighborhood tothe noted point ofthesecond andtheﬁeld ofplanes intheﬁrst neighborhood to theﬁeld ofplanes inthesecond). Theorem. Every contact manifold ofdimension 2m—1islocally contact diffeomorphic tothemanifold ofcontact elements ofm-dimensional space. Theorem. Every differential l-form deﬁning anondegenerate ﬁeld ofhyperplanes onamanifold ofdimension 2n+1,canbewritten insome local coordinate system inthe“normal form” to=xdy+dz, where x=(xl,...,x,,),y=(yl,...,y,,)andzarethelocal coordinates. Itisclear thattheﬁrsttwotheorems follow from thethird. Wewilldeduce thethird onefrom ananalogous theorem ofDarboux onthenormal form of the2-form giving asymplectic structure (cf.Section 43). PROOF orDARnoux’s THEOREM, Wesymplectify ourmanifold. Onthisnew(2n+2)-dimensional symplectic manifold there areacanonical l-form at,anondegenerate 2-form du_aprojection rt onto theunderlying contact manifold andavertical direction atevery point. Thegiven differential l-form toonthecontact manifold deﬁnes acontact form atevery point. These contact forms form a(Zn+1)-dimensional submanifold ofthesymplectic mani- fold. Theprojection rtmaps thissubmanifold diffeomorphically onto theunderlying contact manifold, andtheverticals intersect thissubmanifold atanonzero angle. 362 Appendix 4:Contact structures Consider apoint inthesurface justconstructed (inthesymplectic manifold) lying over the point ofthecontact manifold weareinterested in.lnthesymplectic manifold wecanchoose a local system ofcoordinates near thispoint such that dx=dpo /\dgo+---+dp,,/\dg,, andsuch thatthecoordinate surface po2Ocoincides with our(2n+l)-dimensional manifold (cf.Section 43,where intheproof ofthesymplectic Darboux‘s theorem theﬁrstcoordinate may bechosen arbitrarily). Wenote now that thel-form podgo+-~-p,,dg,,hasderivative dot.Thus, locally, <><=t>@dq@ +~-+P.dq..+d\~: where wisafunction which canbetaken tobezero attheorigin. Inparticular, onthesurface po=0theform attakes theform alpg=0 =P1dfIi+ +P»dqn +dW' Theprojection rtallows ustocarry thecoordinates p,,...,p,,;go;g1,...,g,,andthefunction wonto thecontact manifold. More precisely, wedeﬁne functions x,y,andzbytheformulas -\'.(ﬂ/1) =PM) .\'.'(rr/I) =q.-(A) :(nA) =w(/1). where Aisapoint onthesurface po=0. Then weobtain to=.\'dy+rl: anditremains only toverify thatthefunctions (xl,...,x,,;y1....,_t~,,: 2)form acoordinate system. Forthisitissuﬂicient toverify thatthepartial derivative ofwwith respect togoisnot zero, orinother words thatthel-form atisnotzero onavector ofthecoordinate direction go. Thelatter isequivalent tothe2-form dotbeing nonzero onthepairofvectors: thebasic vector inthedirection ofgoandthevertical vector. Butavector inthecoordinate direction goisskew-orthogonal toallvectors ofthe coordinate plane po=O.lfitwasalso skew-orthogonal tothevertical vector, then itwould beskew- orthogonal toallvectors_ which contradicts thenondegeneracy ofdoz. Thus thy,/dgo ¢0,andthe theorem isproved. El IContact hamiltonians Suppose thatthecontact structure ofacontact manifold isgiven byadif- ferential l-form co,andthatthisform isﬁxed. Deﬁnition. The0)-embedding ofthecontact manifold intoitssymplectiﬁcation isthemap associating toapoint ofthecontact manifold therestriction of theform toonthetangent plane atthispoint. Deﬁnition. The contact hamiltonian function ofacontact vector ﬁeld ona contact manifold with ﬁxed 1-form toisthefunction Konthecontact manifold whose value ateach point isthevalue ofthehomogeneous hamiltonian Hofthesymplectiﬁcation oftheﬁeld ontheimage ofthe given point under thew-embedding: K(/1)=H(w|A)- 363 Appendix 4:Contact structures Theorem. Thecontact hamiltonian function Kofacontact vector ﬁeld Xona contact manifold with agiven l-form toisequal tothevalue oftheform w onthiscontact ﬁeld : K=w(X). PROOF. Weusetheexpression fortheincrement oftheordinary hamiltonian function over a path interms ofthevector ﬁeldandthesymplectic structure (Section 48,C).Forthiswedraw a vertical interval {AB}. 0<Asl.through thepoint Bofthesymplectiﬁcation atwhich we want tocalculate thehamiltonian function. Thetranslations ofthisinterval over small time T under theaction ofthesymplectiﬁed ﬂow deﬁned byourﬁeld X.ﬁlloutatwo-dimensional region a(t). Thevalue ofthehamiltonian atthepoint Bisequal tothelimit H(B) =lim1"‘ dot, r—~O a(t) since H(}.B) —>0asA—>0.Buttheintegral oftheform dotover theregion istheintegral of thel-form atalong theedge formed bythetrajectory ofthe point B(the other parts ofthe boundary givezero integrals). Therefore. thedouble integral issimply theintegral ofthel-form atalong theinterval oftrajectories, andthelimit isthevalue ofozonthevelocity vector Yofthe symplec- tiﬁed ﬁeld. Thus K(rrB) =H(B) =at(Y)=tu(X)_ aswastobeshown. Cl JComputational formulas Suppose now thatwemake useofthecoordinates inDarboux’s theorem in which theform athasthenormal form w=xdy+dz, x=(x1,...,x,,),y=(y1,...,y,,). PROBLEM. Find thecomponents ofthecontact ﬁeld with agiven contact hamiltonian function K=K(x, y,z). ANSWER. Theequations ofthecontact ﬂow have theform X=—K,, +xK, J3=K. é=K—xK,,. Solution. Apoint ofthesymplectiﬁcation canbegiven bythe2n+2numbers xi,y,-,:. and/l,where (x,y.z)arethecoordinates ofapoint ofthecontact manifold andAisthenumber bywhich wemust multiply totoobtain thegiven point ofthe symplectiﬁed space. lnthese coordinates at=ixdy+Adz.Therefore, inthecoordinate system p.q,where P=(P-P0). P=1-who =1 it=(e.ea). 11=l‘~¢i0=:. theform attakes thestandard form: oc=pdq dot=dp/\ dq. Theaction Toofthemultiplicative group isnowreduced lomultiplication ofpbyanumber: 7;,(P- ‘ll=(up.Q). 364 Appendix 4:Contact structures The contact hamiltonian Kcan beexpressed interms ofthe ordinary hamiltonian HIH(/i. g.po.go)bytheformula K(x, _t',:):H(.\'. _\'.l.:). Thefunction Hishomogeneous ofdegree linp.Therefore, thepartial derivatives ofKatthe point (x._\'.:)arerelated tothederivatives ofHatthepoint (p=x,po=1,g=y,go=:)by therelations H4 =KY }{4o :KI’ H,,=K, H,,,_=K—.\"K_, Hamilton‘s equations with hamiltonian function Htherefore have thefollowing form atthe point under consideration: .»e+.\-2.: —K,. /.=~K:_ f'=K, .'§=K—.\'K,. from which weobtain theanswer above. PROBLEM. Find thecontact hamiltonian ofthePoisson bracket oftwocontact ﬁelds with contact hamiltonians KandK’. ANSWER. (K,K’)+KZEK’ ——KQEK, where thebrackets denote Poisson bracket inthevariables xandyandEistheEuler operator EF=F—xF,,. Solution. lnthenotation ofthesolution ofthepreceding problem wemust express the ordinary Poisson bracket ofthehomogeneous hamiltonians Hand H’atthepoint (p=.\'_po=I,g=y,:o=:)interms ofthe contact hamiltonians KandK’.Wehave W»H’)=H1”;—HIHﬁ=Hi”;—Hp”;+Hi..”}».. rHp.”-'1.» Substituting thevalues ofthe derivatives from thepreceding problem, weﬁndatthepoint under consideration (H,H'):K,.K_'\ —KXKQ. +K:(K' —XKQ) ~K':(K —.\'K_,). KLegendre manifolds Thelagrangian submanifolds ofasymplectic phase space correspond inthe contact casetoaninteresting class ofmanifolds which maybecalled Legendre manifolds since they areclosely related toLegendre transformations. Deﬁnition. ALegendre submanifold ofa(2n+1)-dimensional contact mani- foldisann-dimensional integral manifold oftheﬁeld ofcontact planes. Inother words, itisanintegral manifold ofthehighest possible dimension foranondegenerate ﬁeld ofplanes. EXAMPLE 1.Thesetofallcontact elements tangent toasubmanifold ofany dimension inanm-dimensional manifold isan(m—1)-dimensional Legendre submanifold ofthe(2m—1)-dimensional contact manifold ofallcontact elements. 365 Appendix 4:Contact structures EXAMPLE 2.Thesetofallplanes tangent tothegraph ofafunction f=(p(x) inan(n+1)-dimensional euclidean space with coordinates (x1,...,x,,;f) isaLegendre submanifold ofthe(2n+1)-dimensional space ofallnon- vertical hyperplane elements inthespace ofthegraph (thecontact structure isgiven bythe1-form o)=p1dx1+---+p,,dx,,-—df; theelement with coordinates (p,x,f)passes through thepoint with co- ordinates (x,f)parallel totheplane f=p1X1 + +p,,x,,). The Legendre transformation canbedescribed inthese terms inthe following way. Consider asecond (2n+1)-dimensional contact space with coordinates (P,X,F)andcontact structure given bytheform Q-=PdX—dF. TheLegendre involution isthemap taking apoint oftheﬁrstspace with coordinates (p,x,f)tothepoint ofthesecond space with coordinates P=>< X=P F=P><—f The Legendre involution, ascanbeeasily calculated, carries theﬁrst contact structure tothesecond. Clearly, wehave Theorem. Adiﬂeomorphism ofonecontact manifold ontoanother which carries contact planes tocontact planes, carries every Legendre manifold toa Legendre manifold. Inparticular, under theaction oftheLegendre involution theLegendre manifold ofplane elements tangent tothegraph ofafunction iscarried intoa newLegendre manifold. This newmanifold iscalled theLegendre transform oftheoriginal manifold. Theprojection ofthenewmanifold onto thespace with coordinates (X,F) (parallel totheP-direction) isingeneral notasmooth manifold, buthas singularities. This projection iscalled theLegendre transform ofthegraph of thefunction <p. Ifthefunction rpisconvex, then theprojection isitself thegraph ofa function F=(D(X). Inthiscase (Discalled theLegendre transform ofthe function (p. Asanother example weconsider themotion oforiented contact elements under theaction ofthegeodesic ﬂow onariemannian manifold. Asthe “initial wave front” wetake some smooth submanifold ofourriemannian manifold (thedimension ofthesubmanifold isarbitrary). Theoriented con- tactelements tangent tothissubmanifold form aLegendre manifold inthe space ofallcontact elements. From thepreceding theorem weobtain 366 Appendix 4:Contact structures Corollary. Thefamily ofallelements tangent toawave front istransformed under theaction ofthegeodesic ﬂow after time ttoaLegendre manifold of thespace ofallcontact elements. Itshould benoted thatthis_newLegendre manifold may notbethefamily ofallelements tangent tosome smooth manifold, since awave front may develop singularities. TheLegendre singularities which arise inthiswaycanbedescribed ina manner similar tolagrangian singularities (cf.Appendix 12).ALegendre ﬁbration ofa(2n+1)-dimensional contact manifold isaﬁbration allof whose ﬁbers aren-dimensional Legendre manifolds. ALegendre singularity isasingularity oftheprojection ofann-dimensional Legendre submanifold ofa(2n+l)-dimensional contact manifold Onto the(n+l)-dimensional base oftheLegendre ﬁbration. Consider thespace lR2"*‘ with contact structure given bytheform oz=xdy+dz,where x=(x1,...,x,,)andy=(yl,...,y,,).Theprojection (x,y,z)—>(y,z)gives aLegendre ﬁbration. Anequivalence ofLegendre ﬁbrations isadilfeomorphism ofthetotal spaces oftheﬁbrations carrying thecontact structure andﬁbers oftheﬁrst bundle tothecontact structure andﬁbers ofthesecond bundle. Itcanbe shown that every Legendre bundle isequivalent tothespecial bundle just described inaneighborhood ofevery point ofthespace ofthebundle. Thecontact structure ofthetotal space ofﬁbration gives theﬁbers alocal structure ofaprojective space. Legendre equivalence preserves thisstructure, i.e.,deﬁnes locally projective ﬁber transformations. The following theorem allows ustolocally describe Legendre sub- manifolds andmaps byusing generating functions. Theorem. Foranypartition I+Jofthesetofindices (1,...,n)intotwodis- joint subsets andfor anyfunction S(x,, yj)ofnvariables xi,ieI,je J,the formulas _@_5 X__@i -_S_ ‘ii y'_6x_, J- 6y, 4- "Hex, deﬁne aLegendre submanifold ofR2” 1.Conversely, every Legendre sub- manifold ofR2"*1 isdeﬁned inaneighborhood ofeverypoint bythese formulas foratleast oneofthe2"possible choices ofthesubset I. Theproof isbased onthefactthat, onaLegendre manifold, dz+xdy=O, 50d(-'7+XIV!) =Y1dxl—X1dyJ- D Intheformulas ofthepreceding theorem, wereplace Sbyafunction from thelistofthesimple lagrangian singularities given inAppendix 12.We obtain Legendre singularities which arepreserved under small deformations oftheLegendre mapping (x,y,z)—>(y,z)(i.e., arecarried toequivalent 367 jzzn-""-*7?r+44Appendix 4:Contact structures singularities forsmall deformations ofthefunction S).Every Legendre mapping forn<6canbeapproximated byamap, allofwhose singularities arelocally equivalent tosingularities from thelistAk(1£ks6),D, (43k36),E6. Inparticular, weobtain alistofthesingularities ofawave front ingeneral position inspaces ofdimension lessthan 7. Inordinary three-space thislistisasfollows: A1:S=ixf A2:S=ix? A3:S=ix? +xfyz where I={1},J={2},andn=2. Theprojections oftheLegendre manifolds indicated here onto thebase oftheLegendre bundle (i.e.,onto thespace with coordinates y,,yz,andz) are:asimple point intheeaseofA1,acuspidal edge inthecaseofA2,anda swallowtail (cf.Figure 246) inthecase ofA3. Thus awave front ingeneral position inthree-space hasonly cusps and “swallowtail” points assingularities. Atisolated moments oftime during the motion ofthefront wecanobserve transitions ofthethree types A4,D;and DI(cf.Appendix 12,where thecorresponding caustics ﬁlled outbythe singularities ofthefront during itsmotion aredrawn). PROBLEM I.Layoutaninterval oflength tonevery interior normal toanellipse intheplane. Draw thecurve obtained and investigate itssingularities anditstransitions astchanges. PROBLEM 2.Dothesame thing foratriaxial ellipsoid inthree-dimensional space. LContactiﬁcation Along with symplectiﬁcation ofcontact manifolds, there isacontactiﬁcation ofsymplectic manifolds with symplectic structure cohomologous tozero. Thecontactiﬁcation E2"*‘ ofthesymplectic manifold (M2",012)iscon- structed asthespace ofabundle with ﬁber Rover M2".LetUbeasuﬁiciently small neighborhood ofapoint xinM,sothatthere isacanonical coordinate system p,qonUwith 0)=dp/\dq.Consider thedirect product U><R with coordinates p,q,z.LetV><Rbethesame kind ofproduct constructed onanother (orthesame) neighborhood V,with coordinates P,Q,Z;dP/\dQ =co.Iftheneighborhoods Uand VonMintersect, then weidentify the ﬁbers above thepoints ofintersection inboth representations sothat the form dz+pdq=dZ+PdQ=atisdeﬁned onthewhole (this ispossible since PdQ—pdqisatotal differential onUmV). Itiseasy toverify thatafter thispasting together wehave abundle E2'”1 onM2"andthattheform atdeﬁnes acontact structure onE.Themanifold E iscalled thecontactiﬁcation ofthesymplectic manifold M.Ifthecohomology class oftheform (ozisintegral, then wecandeﬁne acontactiﬁcation with ﬁber S1. 368 Appendix 4:Contact structures MIntegration ofﬁrst-order partial dtﬁerential equations LetM2"*1 beacontact manifold, andE2"ahypersurface inM2"“. The contact structure onMdeﬁnes some geometric structure onEMinparticular, theﬁeld ofso-called characteristic directions. Ananalysis ofthisgeometric structure canreduce theintegration ofgeneral ﬁrst-order nonlinear partial differential equations totheintegration ofasystem ofordinary differential equations. Weassume thatthemanifold E2"istransverse tothecontact planes atall itspoints. Inthiscase, theintersection ofthetangent plane toE2"ateach ofits points with thecontact plane hasdimension 2n—1,sothatwehave aﬁeld ofhyperplanes onE2".Furthermore, thecontact structure onM2”1deﬁnes onE2"aﬁeld oflines lying inthese (2n—1)-dimensional planes. Infact, letatbea1-form onM2"*1 locally giving thecontact structure; letto=dotandletR2"beacontact plane atthepoint xinE2".Let<1)=O bethelocal equation ofE2"(sodd)isnotzero atx).Therestriction ofdd)to R2"deﬁnes anonzero linear form onR2".The2-form wgives R2”thestructure ofasymplectic vector space andthus anisomorphism ofthisspace with its dual. Thenonzero 1-form d<D|R1.. corresponds toanonzero vector CofR2", sothatd<D(-) =oo(§, -).Thevector 5iscalled thecharacteristic vector ofthe manifold E2"atthepoint x.The characteristic vector 5liesintheinter- section ofR2"with thetangent plane toE2",sothata'(§) =0. Thevector 5isnotuniquely deﬁned bythemanifold E2"andthecontact structure onM,butonly uptomultiplication byanonzero number. Infact, likethe2-form 0)onR2",the1-form donR2"isdeﬁned only uptomulti- plication byanonzero number. Thedirection ofthecharacteristic vector (i.e., thelinccontaining it)is determined uniquely bythecontact structure atevery point ofthemanifold E.Thus wehave aﬁeld ofcharacteristic directions onthehypersurface Eof thecontact manifold M.Theintegral curves ofthisﬁeld ofdirections are called thecharacteristics. Now suppose wearegiven an(n—1)-dimensional submanifold Iofour hypersurface E2",which isintegral forthecontact ﬁeld (sothatthetangent plane toIateach point iscontained inthecontact plane). Theorem. [fat apoint xof!thecharacteristic onE2"isnottangent toI,then inaneighborhood ofthepoint xthecharacteristics onE2”passing through points ofIform aLegendre submanifold L"inM2” ‘. PR()()l-. LetEbeavector ﬁeld onE2"made upofcharacteristic vectors. By thehomotopy formula (cf.Section 36G) wehave onE2" L50! =dléa 'l' da. Butigot=0since thecharacteristic vector belongs tothecontact plane. Therefore, onE2"wehave Lgot=igw. Butthel-form igwiszero onthe 369 -Iwtvi-1-"-'"-~Appendix 4:Contact structures intersection ofthetangent plane toE2"with thecontact plane (since onthe contact plane igw=d<D,andonthetangent plane d=0).Therefore, on thetangent plane toE2"wehave iéoa=ca.Thus onthehypersurface E, Lgot=ca (where cisafunction smooth inaneighborhood ofx). Now let{g'}bethe(local) phase ﬂow oftheﬁeld 5andr;avector tangent toE2". Seti1(t)=gjknandy(t)=ot(n(t)). Then thefunction ysatisﬁes the linear differential equation g=c(t)y(t)- Ifn(0)istangent toI,then y(O) =ot(r7(0)) =0.This means y(t)=tX(i1(l)) =0,i.e.,forallt,q(t)liesinthecontact plane. Therefore, g'Iisanintegral manifold ofthecontact ﬁeld. Therefore themanifold formed byall{g'I} for small zisaLegendre manifold. Cl EXAMPLE. Consider [Fl2"*‘ with coordinates x,, x,,;p,, p,,;uwith contact structure deﬁned bythel-form Ol=du—pdx.Afunction (x,p,u) deﬁnes adifferential equation <l>(x, ou/ﬁx, u)=0andasubmanifold E= CD‘1(0)inthespace Rh“ (called thespace of1-jets offunctions onIR”). Aninitial condition fortheequation (D=0isanassignment ofavalue f tothefunction uonan(n—1)-dimensional hypersurface Finthen-dimen- sional space with coordinates x1,...,x,,. Aninitial condition determines thederivatives ofuinthen—1indepen- dent directions ateach point ofF.Thederivative inadirection transverse to Fcangenerally befound from theequation; iftheconditions oftheimplicit function theorem arefulﬁlled, then theinitial condition iscalled noncharacter- istic. Anoncharacteristic initial condition deﬁnes an(n—l)-dimensional inte- gralsubmanifold Ioftheformat (thegraph ofthemappingu =f(x),p=p(x), x6F).Thecharacteristics onEintersecting Iform aLegendre submanifold ofIR2"*‘, thegraph ofthemapping u=u(x), p=éiu/o‘x. Thefunction u(x) isasolution oftheequation <D(x, éiu/(ix, u)=0with initial condition ulr=f Note thattoﬁndthefunction uweneed only solve thesystem of2nﬁrst- order ordinary differential equations forthecharacteristics onE,andperform aseries of“algebraic” operations. 370 Appendix 5:Dynamical systems with symmetries Bythetheorem ofE.Noether, one-parameter groups ofsymmetries ofa dynamical system determine ﬁrstintegrals. Ifasystem admits alarger group ofsymmetries, then there areseveral integrals. Simultaneous level manifolds ofthese ﬁrstintegrals inthephase space areinvariant manifolds ofthephase ﬂow. Thesubgroup ofthegroup ofsymmetries mapping such aninvariant manifold intoitself actsonthemanifold. Inmany cases, wecanlook atthe quotient manifold ofaninvariant manifold bythissubgroup. This quotient manifold, called thereduced phase space, hasanatural symplectic structure. Theoriginal hamiltonian dynamical system induces ahamiltonian system onthereduced phase space. The partition ofthephase space into simultaneous level manifolds generally hassingularities. Anexample isthepartition ofaphase plane into energy level curves. Inthisappendix Wewillbrieﬁy discuss dynamical systems inreduced phase space andtheir relationship with invariant manifolds intheoriginal space. Allthese questions were investigated byJacobi andPoincare (“elimin- ation ofthenodes” inthemany-body problem, “reduction oforder” in systems with symmetries, “stationary rotations” ofrigid bodies, etc.). A detailed presentation incurrent terminology canbefound inthefollowing articles: S.Smale, “Topology andmechanics,” Inventiones Mathematicae 10:4 (1970) 305-331, ll:1(1970), 45-64; andJ.Marsden andA.Weinstein, “Reduction ofsymplectic manifolds with symmetries,” Reports onMathe- matical Physics 5(1974) l2l—l30. APoisson action ofLiegroups Consider asymplectic manifold (M2",012)andsuppose aLiegroup Gacts onitasagroup ofsymplectic diffeomorphisms. Every one-parameter sub- group ofGthen actsasalocally hamiltonian phase ﬂow onM.Inmany important cases, these ﬂows have single-valued hamiltonian functions. EXAMPLE. LetVbeasmooth manifold andGsome Liegroup ofdiffeomorphisms ofV.Since every dilleomorphism takes l-forms onVto1-forms, thegroup Gactsonthecotangent bundle M2T*V. Recall that onthecotangent bundle there isalways acanonical l-form 1(“pdq”) and a natural symplectic structure to=dz.Theaction ofthegroup GonMissymplectic since it preserves thel-form ozandhence alsothe2-form det. Aone-parameter subgroup {g’}ofGdeﬁnes aphase ﬂowonM.itiseasy toverify thatthis phase ﬁow hasasingle-valued hamiltonian function. Infact,thehamiltonian function isgiven bytheformula from Noether‘s theorem: iH(x) =oz((— g'.\‘), where xeM. dt i=0 Wenow assume thatwearegiven asymplectic action ofaLiegroup G onaconnected symplectic manifold Msuch that, toevery element aofthe Liealgebra ofG,there corresponds aone-parameter group ofsymplectic diffeomorphisms with asingle-valued hamiltonian Ha.These hamiltonians 371 Appendix 5:Dynamical systems with symmetries aredetermined uptotheaddition ofconstants which canbechosen sothat thedependence ofHaupon aislinear. Todothis, itissufficient tochoose arbitrarily theconstants inthehamiltonians forasetofbasis vectors ofthe Liealgebra ofG,andtothendeﬁne thehamiltonian function foreach element ofthealgebra asalinear combination ofthebasis functions. Thus, given asymplectic action ofaLiegroup Gandasingle-valued hamiltonian onM,wecanconstruct alinear mapping oftheLiealgebra of Ginto theLiealgebra ofhamiltonian functions onM.Thefunction H[,,,,,] associated tothecommutator oftwoelements oftheLiealgebra isequal to thePoisson bracket (Ha, H,,),orelseitdiffers from thisPoisson bracket bya constant: H[a,b] I(Has Hb) +C(as Remark. Theappearance ofthe constant Cinthisformula isaconsequence ofaninteresting phenomenon: theexistence ofatwo-dimensional cohomology class oftheLiealgebra of (globally) hamiltonian ﬁelds. Thequantity C(a,b)isabilinear skew-symmetric function ontheLiealgebra. TheJacobi identity gives us C([a_ b],c)+C'([b, c],a)+C([t‘, ti].h):0. Abilinear skew-symmetric function onaLiealgebra withthisproperty iscalled atwo-dimensional cocycle oftheLiealgebra. Ifwechoose theConstants inthehamiltonian functions differently, then thecocycle Cis replaced byC’,where C’(a, b)=C'(a,b)+p([u. b]) where pisalinear function ontheLiealgebra. Such acocycle C’issaidtobecohomologous to thecocycle C.Aclass ofcocycles which arecohomologous tooneanother iscalled acohomology class oftheLiealgebra. Thus, asymplectic action ofagroup Gforwhich single-valued hamiltonians exist deﬁnes a two-dimensional cohomology class oftheLiealgebra ofG.This cohomology class measures thedeviation oftheaction from oneinwhich thehamiltonian function ofacommutator canbe chosen equal tothePoisson bracket ofthehamiltonian functions. Deﬁnition. Anaction ofaconnected Liegroup onasymplectic manifold is called aPoisson action ifthehamiltonian functions forone-parameter groups aresingle-valued, andchosen sothat thehamiltonian function depends linearly onelements oftheLiealgebra andsothatthehamiltonian function ofacommutator isequal tothePoisson bracket ofthehamil- tonian functions: H[a,b] :(Ha: Inother words, aPoisson action ofagroup deﬁnes ahomomorphism from theLiealgebra ofthisgroup totheLiealgebra ofhamiltonian functions. 372 Appendix 5:Dynamical systems with symmetries EXAMPLE. LetVbeasmooth manifold andGaLiegroup acting onVasagroup ofdiffeo- morphisms. LetM=T*V bethecotangent bundle ofthe manifold Vwith theusual symplectic structure to:dz.The hamiltonian functions ofone-parameter groups aredeﬁned asabove: (1) H,(x) :4%‘, 0g1X)_ XeT*V. Theorem. This action isPoisson. PROOF. Bydeﬁnition ofthel-form at.thehamiltonian functions H,arelinear “inp“(i.e.,on every cotangent SptiCe). Therefore. their Poisson brackets arealso linear. Thus thefunction H[,,_,,] ~(Ha. H,,)islinear inp.Since itisconstant. itisequal tozero. l:l Inthesame way. wecanshow thatthesymplectiﬁcation ofanycontact action isaPoisson action. EXAMPLE. LetVbethree-dimensional euclidean space andGthesix-dimensional group ofits motions. Thefollowing sixone-parameter groups form abasis oftheLiealgebra: thetrans- lations with velocity lalong thecoordinate axes ql,qZ_andq3andtherotations with angular velocity Iaround these axes. Byformula (1),thecorresponding hamiltonian functions are(in theusual notation) pl.P2.p3:M,,M2,M3,where M,=q2p3 -q3p2_ etc.Thetheorem im- plies thatthepairwise Poisson brackets ofthese sixfunctions areequal tothehamiltonian functions ofthecommutators ofthecorresponding one-parameter groups. APoisson action ofagroup Gonasymplectic manifold Mdeﬁnes a mapping ofMintothedual space oftheLiealgebra ofthegroup P:M—>9*. That is,weﬁxapoint xinMandconsider thefunction ontheLiealgebra which associates toanelement aoftheLiealgebra thevalue oftheHamil- tonian Haattheﬁxed point x: .v,.(a)=H..(x)~ This pxisalinear function ontheLiealgebra andistheelement ofthedual space tothealgebra associated tox: P(><)=P..- Following Souriau (Structure dessystemes Dynamiques, Dunod, 1970), we willcallthemapping Pthemomentum. Note thatthevalue ofthemomentum isalways avector inthespace g*. EXAMPLE. LetVbeasmooth manifold, GaLiegroup acting onVasagroup ofdiffeomorphisms .\/I=T*V thecotangent bundle andH“thehamiltonian functions constructed above ofthe action ofGonM(cf.(ll). Then the"momentum" mapping P:.\/1'—»g*canbedescribed inthefollowing way. Con- sider themap CD:G—>Mgiven bytheaction ofalltheelements ofGonaﬁxed point xinM (so<D(g) =gx).Thecanonical l-form atonMinduces al-form <D*otonG.Itsrestriction tothe tangent space attheidentity ofGisalinear form ontheLiealgebra. Thus toevery point xinMwehave associated alinear form ontheLiealgebra. Itiseasy toverify thatthismapping isthemomentum ofourPoisson action. 373 Appendix 5:Dynamical systems withsymmetries Inparticular, ifViseuclidean three-space andGISthegroup ofrotations around thepoint 0. then thevalues ofthe momentum aretheusual vectors ofangular momentum; ifGisthegroup ofrotations around anaxis, thenthevalues ofthemomentum aretheangular momenta relative tothisaxis: ifGisthegroup ofparallel translations, then thevalues ofthe momentum arethe vectors oflinear momentum. Theorem. Under themomentum mapping P,aPoisson action ofaconnected Liegroup Gistaken totheco-adjoint action ofGonthedualspace g*ofits Liealgebra (cf.Appendix 2),i.e.,thefollowing diagram commutes: (.?___“U “i"_wT:Ad}.9*————> Corollary. Suppose thatahamiltonian function H:M->[Risinvariant under thePoisson action ofagroup GonM.Then themomentum isaﬁrst integral ofthesystem with hamiltonian function H. PROOF orTHETHEOREM Thetheorem asserts thatthehamiltonian function H,oftheone- parameter group h’iscarried over bythedilleomorphism gtothehamiltonian ll.I1'lCll0l1 H,,,,°,, oftheone-parameter group gh'g' '. Letgsbeaone-parameter group with hamiltonian function H,,.Itissulticient toshow that thederivatives with respect tos(fors=0)ofthefunctions H,,(g’x) andH,,,v,,_(x) arethesame. Theﬁrstofthese derivatives isthevalue atxofthePoisson bracket (Ha, H,,).Thesecond is H,“,,,(x). Since theaction isPoisson. thetheorem isproved. Cl Pnoor orrutCOROLLARY. Thederivative, inthedirection ofthephase ﬂowwithhamiltonian function H,ofeach component ofthemomentum iszero. since itisequal tothederivative of function Hinthedirection ofthephase ﬂowcorresponding toaone-parameter subgroup ofG. Cl BThereduced phase space Suppose thatwearegiven aPoisson action ofagroup Gonasymplectic manifold M.Consider alevel setofthemomentum, i.e.,theinverse image of some point pe9"‘under themap P.Wedenote thissetbyMP,sothat (Figure 238) M,,=P“lo). Inmany important cases thesetM,,isamanifold. Forexample, thiswill besoifpisaregular value ofthemomentum, i.e.,ifthedifferential ofthemapP ateach point ofthesetMpmaps thetangent space toMonto thewhole tangent space tog*. Ingeneral, aLiegroup Gacting onMtakes thesetsMPintooneanother. However, thestationary subgroup ofapoint pintheco-adjoint representa- tion (i.e., thesubgroup consisting ofthose elements gofthegroup Gfor which Ad;‘p =p)leaves MPﬁxed. Wedenote thisstationary subgroup by 374 Appendix 5:Dynamical systems with symmetries <0 Ui P Figure 238 Reduced phase space GP.Thegroup GPisaLiegroup, anditactsonthelevel setMPofthemo- mentum. Thereduced phase space isobtained from MPbyfactoring bytheaction ofthegroup GP.Inorder forsuch afactorization tomake sense, itisnecessary tomake several assumptions. Forexample, itissuﬁicient toassume that 1.pisaregular value, sothatMPisamanifold, 2.Thestationary subgroup GPiscompact, and 3.Theelements ofthegroup GPactonMPwithout ﬁxed points. Remark. These conditions canbeweakened. Forexample, instead ofcompactness ofthe group GPwecanrequire thattheaction beproper (i.e.,thattheinverse images ofcompact sets under themapping (g.x)—>(g(x). x)arecompact). Forexample, theactions ofagroup on itself byleftandright translation arealways proper. Ifconditions (1),(2),and(3)aresatisﬁed, then itiseasy togivethesetof orbits oftheaction ofGPonMPthestructure ofasmooth manifold. Namely, achart onaneighborhood ofapoint xeMPisfurnished byanylocal trans- versal totheorbit GPx,whose dimension isequal tothecodimension ofthe orbit. Theresulting manifold oforbits iscalled thereduced phase space ofa system withsymmetry. Wewilldenote thereduced phase space corresponding toavalue ofthe momentum byFP.Themanifold FPisthebasespace ofthebundle rt:MP—>FP with ﬁber diffeomorphic tothegroup GP. There isanatural symplectic structure onthereduced phase space FP. Namely, consider anytwovectors Cand17tangent toFPatthepoint fThe point fisoneoftheorbits ofthegroup GPonthemanifold MP. Letxbe oneofthepoints ofthisorbit. Thevectors 6and17tangent toFPareobtained from some vectors 5’andn’tangent toMPatsome point xbytheprojection rt:MP—+FP. Deﬁnition. Theskew-scalar product oftwovectors ifand27which aretangent toareduced phase space atthesame point, istheskew-scalar product of 375 Appendix 5:Dynamical systems with symmetries thecorresponding vectors 6’and11',tangent totheoriginal symplectic manifold M: [6,'1],=[5,'1']- Theorem.1°‘ Theskew-scalar product ofthevectors 5and11does notdepend onthechoices ofthepoint xandrepresentatives C’andrt’,andgives a symplectic structure onthereduced phase space. Corollary. Thereduced phase space iseven-dimensional. PROOF orTHETHEOREM. Welook atthefollowing twospaces inthetangent space toMatx: T(MP), thetangent space tothelevel manifold MP,and T(G,,), thetangent space totheorbit ofthegroup G. Lemma. These twospaces areskew-orthogonal complements tooneanother inTM. PROOF. Avector Qliesintheskew-orthogonal complement tothetangent plane ofanorbit of thegroup Gifandonly iftheskew-scalar product ofthevector Qwith velocity vectors ofthe hamiltonian ﬂowofthegroup Gisequal tozero(bydeﬁnition). Butthese skew-scalar products areequal tothederivatives ofthecorresponding hamiltonian functions inthedirection Q. Therefore, thevector Qliesintheskew-orthogonal complement totheorbit ofGifandonly if thederivative ofthe momentum inthedirection Qisequal tozero, i.e.,if;liesinT(MP). II] Therepresentatives 5'andn’aredeﬁned uptoaddition ofavector from thetangent plane totheorbit ofthegroup GP.Butthistangent plane istheintersection ofthetangent planes to theorbit (ixandtothemanifold MP(bythelasttheorem ofpartA).Consequently. theaddition toZ’ofavector from T(GPx) does notchange theskew-scalar product with anyvector ii’from T(MP) (since bythelemma T(GPx) isskew-orthogonal toT(MP)). Thus, wehave shown the independence from therepresentatives 5'and11'. Theindependence ofthequantity [5,n]Pfrom thechoice ofthe point xoftheorbit /'follows from thesymplectic nature oftheaction ofthegroup GonMandtheinvariance ofMP.Thus wehave deﬁned adifferential 2-form onI-"P: Qpti. '1)=[iPt],- Itisnondegenerate. since if[5,n]P=Oforevery :1,then thecorresponding representative Cisskew-orthogonal toallvectors inT(MP). Therefore, 5'must betheskew-orthogonal com- plement toT(MP) inTM. Then bythelemma 5’eT(Gx). i.e..§ I0. Theform QPisclosed. Inorder toverify thisweconsider achart. i.e.apiece ofsubmanifold inM,,.transversally intersecting theorbit ofthe group GPinonepoint. Theform Q,isrepresented inthischart bya2-form induced from the2-form mwhich deﬁnes thesymplectic structure inthewhole space M,bymeans oftheembedding ofthesubmanifold piece. Since theform toisclosed, theinduced form ISalsoclosed. Thetheorem isproved. Cl '°‘Thetheorem wasﬁrstformulated inthisform byMarsden andWeinstein. Many special cases have been considered since thetime ofJacobi andused byPoincare andhissuccessors in mechanics. byKirillov andKostant ingroup theory. andbyFaddeev inthegeneral theory of relativity. 376 Appendix 5:Dynamical systems with symmetries EXAMPLE 1.LetM=R2"beeuclidean space ofdimension 2nwith coordin- atespk,qkand2-form 2dp,,/\dqk. LetG=S‘bethecircle, andletthe action ofGonMbegiven bythehamiltonian ofaharmonic oscillator H=%X(Pf+qi)- Then themomentum mapping issimply H:R2"—>R,anonzero momen- tumlevel manifold isasphere 52"“ 1,andthequotient space isthecomplex projective space CP"‘ 1. The preceding theorem deﬁnes asymplectic structure onthiscomplex projective space. Itiseasy toverify that thisstructure coincides (uptoa multiple) with theoneweconstructed inAppendix 3. EXAMPLE 2.LetVbethecotangent bundle ofaLiegroup, Gthesame group andtheaction deﬁned bylefttranslation. Then MPisasubmanifold ofthe cotangent bundle ofG,formed bythose vectors which, after right translation totheidentity ofthegroup, deﬁne thesame element inthedual space tothe Liealgebra. The manifolds MParediffeomorphic tothegroup itself andareright- invariant cross-sections ofthecotangent bundle. Allthevalues pareregular. Thestationary subgroup G,ofthepoint pconsists ofthose elements of thegroup forwhich leftandright translation ofpgivethesame result. The actions ofelements different from theidentity ofGPonMphave noﬁxed points (since there arenone byright translation ofthegroup onto itself). Thegroup GPactsproperly (cf.remark above). Consequently, thespace oforbits ofthegroup GPonMPisasymplectic manifold. Butthisspace oforbits iseasily identiﬁed with theorbit ofthepoint p intheco-adjoint representation. Actually, wemap theright-invariant section MPofthecotangent bundle intothecotangent space tothegroup at theidentity with lefttranslations. Wegetamapping 1t:Mp-+g*. Theimage ofthismapping istheorbit ofthepoint pintheco-adjoint representation, andtheﬁbers aretheorbits oftheaction ofthegroup GP. Thesymplectic structure ofthereduced phase space thusdeﬁnes asymplectic structure intheorbits oftheco-adjoint representation. Itisnothard toverify bydirect calculation thatthisisthesame structure which wediscussed inAppendix 2. EXAMPLF 3.Letthegroup G=S‘,thecircle, andletitactwithout ﬁxed points onamanifold V.Then there isanaction ofthecircle onthecotangent bundle M=T*V. Wecandeﬁne momentum level manifolds MP(ofco- dimension 1inM)andquotient manifolds FP(thedimension ofwhich is2 lessthan thedimension ofM). 377 Appendix 5:Dynamical systems with symmetries Inaddition, wecanconstruct aquotient manifold oftheconﬁguration space Vbyidentifying thepoints ofeach orbit ofthegroup onV.Wedenote thisquotient manifold byW. Theorem. Thereduced phase space FPissymplectic anddiﬂeomorphic tothe cotangent bundle ofthequotient conﬁguration manifold W. PROOF. Let1::V~Wbethefactorization map, andw6T*W a1-form onWatthepoint w=1tv. Theform n"wonVatthepoint vbelongs toM0andprojects toapoint inthequotient F0. Conversely, theelements ofF0aretheinvariant 1-forms onVwhich arecqual tozero onthe orbits; they deﬁne l-forms inWWehave constructed amapping T"W —+F0;itiseasytosee thatthisisasymplectic dilleomorphism. Thecasepaé0isreduced tothecase p=0asfollows. Consider ariemannian metric on V,invariant withrespect toG.Theintersection ofM,,withthecotangent plane toVatthepoint v isahyperplane. Thequadratic form deﬁned bythemetric hasaunique minimum point S(v)in this hyperplane. Subtraction ofthevector S(v) carries thehyperplane M,nT"V,into MorwT"‘V,,, andweobtain apossibly nonsymplectic diffeomorphism Fl,—»F0. Thedifference between thesymplectic structures onT*W induced bythatofFpandF0isa 2-form, induced bya2-form onW. El CApplications tothestudy ofstationary rotations andbifurcations ofinvariant manifolds Suppose that wearegiven aPoisson action ofagroup Gonasymplectic manifold M;letHbeafunction onMinvariant under G.LetFPbeareduced phase space (weassume thattheconditions under which thiscanbedeﬁned aresatisﬁed). The hamiltonian ﬁeld with hamiltonian function Histangent toevery momentum level manifold MP(since momentum isaﬁrst integral). The induced ﬁeld onMPisinvariant with respect toGPanddeﬁnes aﬁeld onthe reduced phase space FP.This vector ﬁeld onFPwillbecalled thereduced ﬁeld. Theorem. Thereduced ﬁeld onthereduced phase space ishamiltonian. The value ofthehamiltonian function ofthereduced ﬁeld atanypoint ofthe reduced phase space isequal tothevalue oftheoriginal hamiltonian function atthecorresponding point oftheoriginal phase space. PROOF. Therelation deﬁning ahamiltonian ﬁeld X”with hamiltonian Honamanifold M with form w dH(§) =w(§, X”) forevery § implies ananalogous relation forthereduced ﬁeld inview ofthedeﬁnition ofthesymplectic Structure onFP. El EXAMPLE. Consider anasymmetric rigid body, ﬁxed atastationary point, under theaction oftheforce ofgravity (oranypotential force symmetric with respect tothevertical axis). 378 Appendix 5:Dynamical systems with symmetries Thegroup S‘ofrotations with respect toavertical lineactsonthecon- ﬁguration space SO(3). The hamiltonian function isinvariant under rota- tions, andtherefore weobtain areduced system onthereduced phase space. The reduced phase space is,inthiscase, thecotangent bundle ofthe quotient conﬁguration space (cf.Example 3above). Factorization ofthe conﬁguration space bytheaction ofrotations around thevertical axiswas done byPoisson inthefollowing way. Wewillspecify theposition ofthebody bygiving theposition ofanortho- normal frame (el,e2,ea).Thethree vertical components ofthebasic vectors giveavector inthree-dimensional euclidean space. Thelength ofthisvector is1(why?). This Poisson vector‘°2 7determines theoriginal frame upto rotations around avertical line(why ?). Thus thequotient conﬁguration space isrepresented byatwo-dimensional sphere S2,andthereduced phase space isthecotangent bundle T*S2 with a nonstandard symplectic structure. Thereduced hamiltonian function onthe cotangent bundle isrepresented asthesum ofthe“kinetic energy ofthe reduced motion,” which isquadratic inthecotangent vectors, and the “effective potential” (thesumofthepotential energy andthekinetic energy of rotation around avertical line). Thetransition tothereduced phase space inthiscaseisalmost by“elimination ofthecyclic coordinate tp.“Thedifference isthattheusual procedure ofelimination requires thatthecon- ﬁguration orphase space beadirect product bythecircle, whereas inourcasewehave only a bundle. This bundle canbemade adirect product bydecreasing thesizeoftheconﬁguration space (i.e., byintroducing coordinates with singularities atthepoles); theadvantage ofthe approach above isthatitmakes itclear thatthere arenorealsingularities (except singularities ofthe coordinate system) near thepoles. Deﬁnition. Thephase curves inMwhich project toequilibrium positions in thereduced system onthereduced phase space F,arecalled therelative equilibria oftheoriginal system. EXAMPLE. Stationary rotations ofarigid body which isﬁxed atitscenter of mass arerelative equilibria. Inthesame way, rotations ofaheavy rigid body with constant speed around thevertical axisarerelative equilibria. Theorem. Aphase curve ofasystem withaG-invariant hamiltonian function isa relative equilibrium ifandonlyifitistheorbit ofaone~parameter subgroup ofGintheoriginal phase space. PROOF. Itisclear thataphase curve which isanorbit projects toapoint. Ifaphase curve x(t) projects toapoint, then itcanbeexpressed uniquely intheform x(t)=g(t)x(O), anditisthen easytoseethat{g(t)} isasubgroup. El “)2Poisson showed thattheequations ofmotion ofaheavy rigid body canbewritten interms ofyinaremarkably simple form, the“Euler- Poisson equations“: ‘M-[M — ldy- dt iwl—/1a[r, lI—[L">1 379 Appendix 5:Dynamical systems with symmetries Corollary 1.Anasymmetrical rigid body inanaxially symmetric potential ﬁeld, ﬁxed atapoint ontheaxis oftheﬁeld, hasatleast twostationary rotations (forevery value oftheangular momentum withrespect totheaxis ofsymmetry). Corollary 2.Anaxially symmetric rigid body ﬁxed atapoint ontheaxisof symmetry, hasatleast twostationary rotations (forevery value oftheangular momentum withrespect totheaxisofsymmetry). Both corollaries follow from thefactthatafunction onthesphere hasat least twocritical points. Another application ofrelative equilibria isthat they canbeused to investigate modiﬁcations ofthetopology ofinvariant manifolds under changes oftheenergy andmomentum values. Theorem. Thecritical points ofthemomentum andenergy mapping PxH:M—>g*><R onaregular momentum level setareexactly therelative equilibria. PROOF. Thecritical points ofthemapping P><Haretheconditional extrema ofHonthe momentum level manifold M,(since thislevel manifold isregular, i.e.,forevery xinM,,we have P*TM,, =Tg;). After factorization by6,,theconditional extrema ofHonM,deﬁne thecritical points of thereduced hamiltonian function (since Hisinvariant under G,). Cl Thedetailed study ofrelative equilibria andsingularities oftheenergy- momentum mapping isnotsimple andhasnotbeen completely carried out, even intheclassical problem ofthemotions ofanasymmetrical rigid body inagravitational ﬁeld. Thecasewhen thecenter ofgravity liesononeofthe principal axes ofinertia istreated inthesupplement written byS.B.Katok totheRussian translation‘°3 ofthearticle byS.Smale cited inthebeginning ofthisappendix. Inthisproblem thedimension ofthephase space issix,and thegroup isthecircle; thereduced phase space T"‘S2 isfour-dimensional. Thenonsingular energy level manifolds inthereduced phase space are (depending onthevalues ofmomentum andenergy) ofthefollowing four forms: S3,S2><S1,RP3, anda“pretzel” obtained from thethree-sphere S3 byattaching two“handles” oftheform S‘xD2 (D2=thedisc{(x,y)|x2 +y231}). '03Uspekhi Mutematicheskikh Nuuk 27,no.2(I972) 78433. 380semi:'33.4 Appendix 6:Normal forms ofquadratic hamiltonians Inthisappendix wegive alistofnormal forms towhich wecanreduce a quadratic hamiltonian function bymeans ofarealsymplectic transformation. This listwascomposed byD.M.Galin based onthework ofJ.Williamson in“On analgebraic problem concerning thenormal forms oflinear dynamical systems,” Amer. J.ofMath. 58,(1936), 141-163. Williamson’s paper gives thenormal forms towhich aquadratic form inasymplectic space over any ﬁeld canbereduced. ANotation Wewillwrite thehamiltonian as H=%(/Ix, X), where x=(p1,..., p,,;q1,...,q,,) isavector written inasymplectic basis andAisasymmetric linear operator. Thecanonical equations then have the form x=IAx, whereI= 0 E.E 0 Bytheeigenvalues ofthehamiltonian wewillmean theeigenvalues ofthe linear inﬁnitesimally-symplectic operator IA.Inthesame way, byaJordan block wewillmean aJordan block oftheoperator IA. Theeigenvalues ofthehamiltonian areoffour types: realpairs (a,—a), purely imaginary pairs (ib,—ib),quadruples (iaiib),andzeroeigenvalues. TheJordan blocks corresponding tothetwomembers ofapairorfour members ofaquadruple always have thesame structure. Inthecase when therealpart ofaneigenvalue iszero, wehave todis- tinguish theJordan blocks ofeven andoddorder. There areaneven number of blocks ofoddorder with zero eigenvalue andthey canbenaturally divided intopairs. Acomplete listofnormal forms follows. BHamiltonians ForapairofJordan blocks oforder kwith eigenvalues ia,thehamiltonian is k km1 H=“dz pjqj+ P1‘11+1- _i=1 j=1 Foraquadruple ofJordan blocks oforder kwith eigenvalues iaibi thehamiltonian is 2k k 2k-2 H=-9 Pjqj +bz(P21-1112; _P214121-1) "l‘ Pjqj+2- ]=1 _]=1 _]=1 381 Appendix 6:Normal forms ofquadratic hamiltonians ForapairofJordan blocks oforder kwitheigenvalue zerothehamiltonian is t-1 j=1 ForaJordan block oforder 2kwith eigenvalue zero, thehamiltonian is ofoneofthefollowing twoinequivalent types: 1k—1 It k—1 H: i‘ Zpjpk—j_ 2qjqk—j+1)_Zpjqj+1 1 J1‘=1 1': ': 2l. (fork=1thisisH=1-%qf). ForapairofJordan blocks ofoddorder 2k+1with purely imaginary eigenvalues ibi, thehamiltonian isofoneofthefollowing twoinequivalent types: 1"2 H=i5Z1“? p2jp2k—2j+2 +q2jq2k-2j+2) k+1 2k "Z(b2P2j~ 1P2k-2j+3 +q2j— 1q2k—2j+3)] _2Pj¢Ij+1- j=I j=1 Fork=0,H=i%(b’r>i +(Ii)- ForapairofJordan blocks ofeven-order 2kwith purely imaginary eigen- values 1*bi,thehamiltonian isofoneofthefollowing twoinequivalent types: 1" 1 H=i‘ q2j—1q2k—2j+1 +q2jq2k-2j+2) k—l —Zb2p2j+1p2k—2j+1 +P2j+2P2|<—2j+2)]1 j: lt ll “b2ZP2;-iqzj +2P2i"‘121—1j=1 j=1 1 1 2 Z 2f°Yk=1»H=i§ F4l1+q2 _bP1412+P2q1~ Williamson’s theorem. Arealsymplectic vector space withagiven quadratic form Hcanbedecomposed intoadirect sumofpairwise skew orthogonal real symplectic subspaces sothattheform Hisrepresented asasumofforms of thetypes indicated above onthese subspaces. CNonremovable Jordan blocks Anindividual hamiltonian in“general position” does nothave multiple eigenvalues andreduces toasimple form (alltheJordan blocks areofﬁrst order). However, ifweconsider notanindividual hamiltonian butawhole 382ii Appendix 6:Normal forms ofquadratic hamiltonians family ofsystems depending onparameters, then forsome exceptional values oftheparameters more complicated Jordan structures canarise. We cangetridofsome ofthese byasmall change ofthefamily; others arenon- removable andonly slightly deformed after asmall change ofthefamily. If thenumber lofparameters ofthefamily isﬁnite, then thenumber ofnon- removable types inl-parameter families isﬁnite. The theorem ofGalin formulated below allows ustocount allthese types foranyﬁxed l. Wedenote byn1(z) 2n2(z) 2 2n,(z) thedimensions oftheJordan blocks with eigenvalues zre0,and byml2m23 2m,and rﬁl2 iii,3 Zrh,thedimensions oftheJordan blocks with eigenvalues zero, where themiareeven andtherh,areodd (ofevery pair ofblocks ofodd dimension, only oneisconsidered). Theorem. Inthespace ofallhamiltonians, themanifold ofhamiltonians with Jordan blocks oftheindicated dimensions hascodimension N)»-c=—2[/s(i)(2j— 1)n,(z)— 1]+%i(2j—1)m,- :#0 '=1 i=1 +i[2(2j —-1)n'i,- +1]+2i imin{m,-, nik}. i=1 ;=11<=1 (Note that, ifzero isnotaneigenvalue, then only theﬁrst term inthesum isnotzero.) Corollary. Inl-parameter families ingeneral position oflinear hamiltonian systems, theonlysystems which occur arethose withJordan blocks such that thenumber ccalculated bytheformula above isnotgreater than l:all cases with larger ccanbeeliminated byasmall change ofthefamily. Corollary. In0ne-andtwo-parameter families, nonremovable Jordan blocks of only thefollowing 12types occur: 1=1I(ir1)2. (iia)2. 0’ (here theJordan blocks aredenoted bytheir determinants; forexample, (1-a)2 denotes apairofJordan blocks oforder 2with eigenvalues aand a,respectively, I=2:(:a)3. tiai)’. (eaibi)’.0‘.(ie)2(ib)2. (iai)2(ibi)2, (ia)2(1bi)’,(1(1)102, (iai)2O2 (theremaining eigenvalues aresimple). 383 Appendix 6:Normal forms ofquadratic hamiltonians Galin hasalsocomputed thenormal forms towhich onecanreduce any family oflinear hamiltonian systems which depend smoothly onparameters, byusing asymplectic linear change ofcoordinates which depends smoothly ontheparameters. Forexample, forthesimplest Jordan square (ia)2, the normal form ofthehamiltonian willbe H(/1) I—a(P1¢I1 +P2q2) +P142 +/lipiqi +}~2P2q1 (/lland/12aretheparameters). 384 Appendix 7:Normal forms ofhamiltonian systems near stationary points andclosed trajectories Instudying thebehavior ofsolutions toHamilton’s equations near an equilibrium position, itisoften insuﬁicient tolook only atthelinearized equation. Infact, byLiouville’s theorem ontheconservation ofvolume, itisimpossible tohave asymptotically stable equilibrium positions forhamil- tonian systems. Therefore, thestability ofthelinearized system isalways neutral: theeigenvalues ofthelinear part ofahamiltonian vector ﬁeld ata stable equilibrium position alllieontheimaginary axis. For systems ofdifferential equations ingeneral form, such neutral stability canbedestroyed bytheaddition ofarbitrarily small nonlinear terms. Forhamiltonian systems thesituation ismore complicated. Suppose, forexample, that thequadratic part ofthehamiltonian function atan equilibrium position (which determines thelinear part ofthevector ﬁeld) is (positive ornegative) deﬁnite. Then thehamiltonian function hasamaximum orminimum attheequilibrium position. Therefore, thisequilibrium position isstable (inthesense ofLiapunov, butnotasymptotically), notonly forthe linearized system butalsofortheentire nonlinear system. Ontheother hand, thequadratic part ofthehamiltonian function ata stable equilibrium position may notbedeﬁnite. Asimple example issupplied bythefunction H=pf+qf—p§—q§.Toinvestigate thestability of systems with thiskind ofquadratic part, wemust take intoaccount terms of degree 23intheTaylor series ofthehamiltonian function (i.e.,theterms of degree 22forthephase velocity vector ﬁeld). Itisuseful tocarry outthis investigation byreducing thehamiltonian function (and, therefore, the hamiltonian vector ﬁeld) tothesimplest possible form byasuitable canonical change ofvariables. Inother words, itisuseful tochoose acanonical co- ordinate system, near theequilibrium position, inwhich thehamiltonian function andequations ofmotion areassimple aspossible. Theanalogous question forgeneral (non-hamiltonian) vector ﬁelds can besolved easily: there thegeneral caseisthatavector ﬁeldinaneighborhood ofanequilibrium position islinear inasuitable coordinate system (the relevant theorems ofPoincare andSiegel canbefound, forinstance, inthe book, Lectures onCelestial Mechanics, byC.L.Siegel and J.Moser, Springer-Verlag, 1971.) Inthehamiltonian casethepicture ismore complicated. Theﬁrstdifﬁculty isthat reduction ofthehamiltonian ﬁeld toalinear normal form bya canonical change ofvariables isgenerally notpossible. Wecanusually kill thecubic partofthehamiltonian function, butwecannot killalltheterms of degree four(this isrelated tothefactthat, inalinear system, thefrequency of oscillation does notdepend ontheamplitude, while inanonlinear system it generally does). Thisdifﬁculty canbesurmounted bythechoice ofanonlinear normal form which takes thefrequency variations into account. Asaresult, wecan(inthe“non-resonance” case) introduce action-angle variables near anequilibrium position sothatthesystem becomes integrable uptoterms of arbitrary high degree intheTaylor series. 385 Appendix 7:Normal forms ofhamiltonian systems near stationary points This method allows ustostudy thebehavior ofsystems over thecourse of large intervals oftime forinitial conditions close toequilibrium. However, itisnotsuﬂicient todetermine whether anequilibrium position willbe Liapunov stable (since onaninﬁnite time interval theinﬂuence ofthedis- carded remainder term oftheTaylor series candestroy thestability). Such stability would follow from anexact reduction toananalogous normal form which didnotdisregard remainder terms. However, wecanshow that thisexact reduction isgenerally notpossible, andformal series forcanonical transformations reducing asystem tonormal form generally diverge. Thedivergence ofthese series isconnected with thefactthat reduction tonormal form would imply simpler behavior ofthephase curves (they would have tobeconditionally-periodic windings oftori) than that which infactoccurs. Thebehavior ofphase curves near anequilibrium position is discussed inAppendix 8.Inthisappendix wegivetheformal results onnor- malization uptoterms ofhigh degree. Theidea ofreducing hamiltonian systems tonormal forms goes back to Lindstedt andPoincaré;‘°4 normal forms inaneighborhood ofanequi- librium position were extensively studied byG.D.Birkhoff (G.D.Birkhoff, Dynamical Systems, American Math. Society, 1927). Normal forms fordegenerate cases canbefound inthework ofA.D. Bruno, “Analytic forms ofdifferential equations,” (Trudy Moskovskovo matematischeskovo obschchestva, v.25andv.26). ANormal form ofaconservative system near an equilibrium position Suppose that inthelinear approximation anequilibrium position ofa hamiltonian system with ndegrees offreedom isstable, andthatallncharac- teristic frequencies col,.(1),,aredifferent. Then thequadratic part ofthe hamiltonian canbereduced byacanonical linear transformation tothe form H=%(w1(Pl +qi)++%w.(1>§ +q.’f))- (Some ofthenumbers wkmay benegative). Deﬁnition. The characteristic frequencies ml,...,0),,satisfy aresonance relation oforder Kifthere exist integers k,notallequal tozero such that k1w1+"'+knC0n:0, lk1l+"'+|k"l:K. Deﬁnition. ABirkhoff normal form ofdegree sforahamiltonian isapoly- nomial ofdegree sinthecanonical coordinates (P,,Q,)which isactually apolynomial (ofdegree [s/2]) inthevariables r,=(P12+Qf)/2. 1°‘Cf.H.Poincare, LesMéthodes Nouvelles delaMécanique Celeste, Vol.1,Dover, 1957. 386 Appendix 7:Normal forms ofhamiltonian systems near stationary points Forexample, forasystem withonedegree offreedom thenormal form ofdegree 2m(or2m+1) looks like H2m:H2m+1:a1T+a2TZ+""l'amTmv r:(P2+Q2)./2* andforasystem with twodegrees offreedom theBirkhoff normal form ofdegree 4willbe H4="111 'l'azlz 'l'“riff ‘l“allrlrl +auti- Thecoelﬁcients u,anda2arecharacteristic frequencies, andthecoeﬁicients a,-jdescribe the dependence ofthe frequencies ontheamplitude. Theorem. Assume that thecharacteristic frequencies co,donotsatisfy any resonance relation oforder sorsmaller. Then there isacanonical co- ordinate system inaneighborhood oftheequilibrium position such that thehamiltonian isreduced toaBirkhoff normal form ofdegree suptoterms oforder s+1: H(P,q) =H.(P.Q) +R R=0(|P| +|Q|)‘“- Pnoor. Theproof ofthistheorem iseasytocarry outinacomplex coordinate system -7!=Pi'l'lqi “'1=P1_lqi (upon passing tothiscoordinate system wemust multiply thehamiltonian by—2i). Ifthe terms ofdegree lessthan Nentering intothenormal form arenotalready killed. thenthetransformation with generating function Pq+S,-(P, q)(where SNisahomogeneous polynomial ofdegree N) changes only terms ofdegree Nandhigher intheTaylor expansion ofthehamiltonian function. Under thistransformation thecoellicient foramonomial ofdegree Ninthehamiltonian function having theform Ii‘--'ZZ"Wf‘-:-Wf" (@<i+--'+<1i+l3i+~--+l3,.=N) ischanged intothequantity -tail/lillli —<1.)++»1..(/3..~1.)]- where it,=ico,andwhere s,,,isthecoefﬁcient forz’w“ intheexpansion ofthe function S_,i(P, q) inthevariables zandw. Under theassumptions about theabsence ofresonance. thecoefﬁcient ofs,,,,»inthesquare brackets isnotzero, except inthecase when ourmonomial canbeexpressed interms ofthc product 2,w,=Zr,(i.e.,when alltheoz,areequal tothe/i’,).Thus wecankillallterms ofdegree N except those expressed interms ofthe variables r,.Setting N='3,4,...,s,weobtain thetheorem. El TouseBirkholf’s theorem, itishelpful tonote thatahamiltonian innormal form isintegrable. Consider the“canonical polar coordinates” 1,,<p,,in which P;andQ,canbeexpressed bytheformulas P,=./2r,cosgo, Q,=./2r,sin<p,. Since thehamiltonian isexpressed interms ofonly theaction variables r,, thesystem isintegrable anddescribes conditionally periodic notions onthe tori‘I.’=const with frequencies co=0H/dt. Inparticular, theequilibrium position P=Q=Oisstable forthenormal form. 387 Appendix 7:Normal forms ofhamiltonian systems near stationary points BNormal form ofacanonical transformation near astationary point Consider acanonical (i.e.area-preserving) mapping ofthetwo-dimensional plane toitself. Assume thatthistransformation leaves theorigin ﬁxed, and thatitslinear part haseigenvalue A=eii“(i.e.,isarotation byangle atina suitable symplectic basis with coordinates p,q).Wewillcallsuch atrans- formation elliptic. Deﬁnition. ABirkhoff normal form ofdegree sforatransformation isacanon- icaltransformation oftheplane toitself which isarotation byavariable angle which isapolynomial ofdegree notmore than m=[s/2] —1 intheaction variable Tofthecanonical polar coordinate system: (T>(P)_*(Ti(P'l'O‘0'l' a1T+"' +amTm)s p=,/Zrcosrp q=\/2Tsin ¢_where Theorem 2.Iftheeigenvalue /lofanelliptic canonical transformation isnota rootofunityofdegree sorless,thenthistransformation canbereduced bya canonical change ofvariables toaBirkhoff normal form ofdegree swith error terms ofdegree s+1andhigher. Themulti-dimensional generalization ofanelliptic transformation isthe direct product ofnelliptic rotations oftheplanes (p,,q,)with eigenvalues /l,=e*'“'. ABirkhoff normal form ofdegree sisgiven bytheformula +55(M/>)—> no 5,. where Sisapolynomial ofdegree notmore than [s/2] intheaction variables r1,...,r,,. Theorem 3.Iftheeigenvalues /l,ofamulti-dimensional elliptic canonical transformation donotadmit resonances Ail.-.A:II:1, |k1|+..._i_|kn|sS’ thenthistransformation canbereduced toaBirkhoff normal form ofdegree s (with error interms ofdegree sintheexpansion ofthemapping inaTaylor series atthepoint p:q=O). CNormal form ofanequation withperiodic coeﬂicients near anequilibrium position Letp=q=0beanequilibrium position ofasystem whose hamiltonian function depends 21:-periodically ontime. Assume thatthelinearized equa- tioncanbereduced byalinear symplectic time-periodic transformation toan autonomous normal form with characteristic frequencies col,...,w,,. 388 Appendix 7:Normal forms ofhamiltonian systems near stationary points Wesaythatasystem isresonant oforder K>Oifthere isarelation klcol + +k,,co,+ k0=0 with integers k0,k,,...,k,, forwhich |k,|+ +lk,,|=K. Theorem. Ifasystem isnotresonant oforder sorless, then there isa21:- periodic time-dependent canonical transformation reducing thesystem ina neighborhood ofanequilibrium position tothesame Birkhoff normal form ofdegree sasifthesystem were autonomous, with only thediﬂerence that theremainder terms Rofdegree s+landhigher willdepend periodically ontime. Finally, suppose thatwearegiven aclosed trajectory ofanautonomous hamiltonian system. Then, inaneighborhood ofthistrajectory, wecan reduce thesystem tonormal form byusing either ofthefollowing two methods: 1.Isoenergetic reduction: Fixanenergy constant andconsider aneighbor- hood oftheclosed trajectory onthe(2n—1)-dimensional energy level manifold astheextended phase space ofasystem with n~1degrees of freedom, periodically depending ontime. 2.Surface ofsection: Fixanenergy constant andvalue ofoneoftheco- ordinates (sothattheclosed trajectory intersects theresulting (2n 2)- dimensional manifold transversally). Then phase curves near thegiven onedeﬁne amapping ofthis(2n—2)-dimensional manifold toitself, with aﬁxed point ontheclosed trajectory. This mapping preserves the natural structure onour(2n—2)-dimensional manifold, and wecan study itbyusing thenormal form inSection B. Ininvestigating closed trajectories ofautonomous hamiltonian systems, aphenomenon arises which contrasts with thegeneral theory ofequilibrium positions ofsystems with periodic coefficients. The factisthat theclosed trajectories ofanautonomous system arenotisolated, butform (asarule) one-parameter families. Theparameter ofthefamily isthevalue oftheenergy constant. Infact, assume that forsome choice oftheenergy constant the closed trajectory intersects transversally the(2n—2)-dimensional manifold described above inthe(2n—l)-dimensional energy level manifold. Then fornearby values oftheenergy, there willexist asimilar closed trajectory. Bytheimplicit function theorem wecaneven saythatthisclosed trajectory depends smoothly ontheenergy constant. Ifwenow wish tousetheBirkhoff normal form toinvestigate aone- parameter family ofclosed trajectories, weencounter thefollowing dilﬁculty. Astheparameter describing thefamily varies, theeigenvalues ofthelinearized problem willgenerally change. Therefore, forsome values oftheparameter wewillinevitably encounter resonances, obstructing reduction tothenormal form. 389 Appendix 7:Normal forms ofhamiltonian systems near stationary points Especially dangerous areresonances ofloworder, since they inﬂuence theﬁrstfewterms oftheTaylor series. Ifweareinterested inaclosed trajectory forwhich theeigenvalues nearly satisfy aresonance relation ofloworder, then theBirkhoff form must besomewhat modiﬁed. Namely, forresonance oforder Nsome oftheexpressions kO_Ew1(ﬁ1_al)+"'+u)n(ﬂn_an):lv bywhich wemust divide tokilltheterms oforder Ninthehamiltonian function, may become zero. Fornon-resonant values oftheparameter which areclose toresonance, thiscombination ofcharacteristic frequencies is generally notzero, butvery small (this combination istherefore called a “small denominator ”). Division byasmall denominator leads tothefollowing difﬁculties: 1.The transformation which reduces tonormal form depends discon- tinuously ontheparameter (ithaspoles forresonant values oftheparam- eter); 2.Theregion inwhich theBirkhoff normal form accurately describes the system contracts tozero atresonance. Inorder togetridofthese deﬁciencies, wemust giveuptrying toannihilate some oftheterms ofthehamiltonian (namely, those which become resonant forresonance values oftheparameter). Moreover, these terms must be preserved notonly forresonance, butalso fornearby values oftheparam- eter.‘°5 Thenormal form thus obtained issomewhat more complicated than theusual normal form, butinmany cases itgives ususeful information on thebehavior ofsolutions near resonance. DExample :Resonance oforder 3 Asasimple example, wewillstudy what happens toaclosed trajectory ofan autonomous hamiltonian system with twodegrees offreedom, forwhich theperiod ofoscillation (about theclosed trajectory) ofneighboring trajec- tories isthree times theperiod oftheclosed trajectory itself. Bywhat wesaid above, thisproblem may bereduced toaninvestigation ofaone-parameter system ofnon-autonomous hamiltonian systems with onedegree offreedom, 21:-periodically depending ontime, inaneighborhood ofanequilibrium position. This equilibrium position canbetaken astheorigin forallvalues of theparameter (toachieve thiswemust make achange ofvariables depending ontheparameter). Furthermore, thelinearized system attheequilibrium position canbe converted intoalinear system with constant coeﬂicients bya21:-periodically time-dependent linear canonical change ofvariables. Inthenewcoordinates thephase ﬂow ofthelinearized system isrepresented asauniform rotation ‘°5Themethod indicated hereisuseful notonly ininvestigating hamiltonian systems, butalso inthegeneral theory ofdifferential equations. Cf.,forexample, V.I.Arnold, “Lectures on bifurcations andversal families,” Russian Math. Surveys 27,No.5,I972, 54-123. 390 Appendix 7:Normal forms ofhamiltonian systems near stationary points around theequilibrium position. The angular velocity toofthisrotation depends ontheparameter. Attheresonance value oftheparameter, to=%(i.e., after time 21:,wehave gone one-third ofthewayaround theorigin). Thederivative oftheangular velocity towith respect totheparameter isgenerally notzero. Therefore, we cantake asaparameter thisangular velocity or,even better, itsdifference from Wewilldenote thisdifference bye.The quantity siscalled the frequency deviation ordetuning. The resonance value oftheparameter is c=0,andweareinterested inthebehavior ofthesystem forsmall s. Ifwedisregard thenonlinear terms inHamilton’s equations and dis- regard thefrequency deviation e,then alltrajectories ofoursystem become closed after making three revolutions (i.e., they have period 61:).Wenow want tostudy theinﬂuence ofthenonlinear terms andfrequency deviation onthebehavior ofthe trajectories. Itisclear thatinthegeneral casenotallthe trajectories willbeclosed. Tostudy their behavior, itisuseful tolook at thenormal form. Inthechosen coordinate system, z=p+iq,Z=p—iq,thehamiltonian function hastheform +00—2iH=—iwzZ +ZZh,,,,,z“2”e'*' + 1+ﬂ= 3k=—oo where thedots indicate terms oforder higher than three, andwhere to= (%)+@- Inthereduction tonormal form wecankillallterms ofdegree three except those forwhich thesmall denominator o)(ot—[i)+k becomes zero atresonance. These terms canbedescribed also asthose which areconstant along trajectories oftheperiodic motion obtained by disregarding thefrequency deviation andnonlinearity. They arecalled the resonant terms. Thus, forresonance to=%,theresonant terms arethose for which or—ll+3k=0. Oftheterms ofthird order, only z3e'“ andZ3e" turn outtoberesonant. Thus wecanreduce thehamiltonian function totheform —2iH =—it0zZ +hz3e_“ —hZ3e“ + (theconjugacy ofhandhcorresponds tothefactthatHisreal). Note that, inorder toreduce thehamiltonian function tothisnormal form, wemade a21:-periodic time-dependent smooth canonical transforma- tionwhich depends smoothly ontheparameter, even inthecaseofresonance. This transformation differs from theidentity only byterms thataresmall of second order relative tothedeviation from theclosed trajectory (and its generating function differs from thegenerating function oftheidentity only bycubic terms). 39] Appendix 7:Normal forms ofhamiltonian systems near stationary points Further investigation ofthebehavior ofsolutions ofHamilton’s equations proceeds inthefollowing way. First, wethrow outofthehamiltonian function allterms oforder higher than three andstudy thesolutions oftheresulting truncated system. Then wemust seehow thediscarded terms canaffect the behavior ofthetrajectories. The study ofthetruncated system canbesimpliﬁed byintroducing a coordinate system inthecomplex z-plane which rotates uniformly with angular velocity %,i.e.,bythesubstitution z=Ce"/3. Then forthevariable Q weobtain anautonomous hamiltonian system withhamiltonian function —2iHO =—is§§ +M3—E53, where 1;=w—(§). Thefactthat, inarotating coordinate system, thetruncated system isautonomous isvery good luck. Thetotal system ofHamilton’s equations (including terms ofdegree higher thanthree inthehamiltonian) isnotonly notautonomous inarotating coordinate system. butisnot even 21:-periodic (butonly 6n-periodic) intime. Theautonomous system with hamiltonian HO isessentially theresult ofaveraging theoriginal system over closed trajectories ofthelinear system with 2=0(where wedisregard terms ofdegree higher than three)r Thecoefﬁcient hcanbemade real(byarotation ofthecoordinate system). Thus thehamiltonian function intherealcoordinates (x,y)isreduced to theform H0=g(xz+yz)+a(x3 —3xy2). Thecoefﬁcient adepends onthefrequency deviation sasonaparameter. Fors=0thiscoefﬁcient isgenerally notzero. Therefore, wecanmake this coeﬂicient equal tolbyasmooth change ofcoordinates depending ona parameter. Thus wemust investigate thedependence onthesmall parameter eofthephase portrait ofthesystem with hamilton function H0=3(x2+yz)+(x3—3xy2) inthe(x,y)-plane. Itiseasy toseethatthisdependence consists ofthefollowing (Fig. 239). \/ & < 3NOT’M0/6 7 C7 —w .( /\ 6<0 6=0 G> 0 Figure 239 Passage through resonance 3:1 392 Appendix 7:Normal forms ofhamiltonian systems near stationary points Fors=Othezero level setofthefunction H0consists ofthree straight lines through 0,intersecting atangles of60°.Under achange ofsthelevel line always consists ofthree straight lines, where these three lines aremoved forward asschanges, always forming anequilateral triangle with center at theorigin. Thevertices ofthistriangle aresaddle points ofthehamiltonian function. Asspasses through zero (i.e., upon passage through resonance), thecritical point attheorigin changes from aminimum toamaximum. Thus, forasystem with hamiltonian function H0,theorigin isastable equilibrium position forallvalues oftheparameter except atresonance, andatresonance theorigin isunstable. Forvalues oftheparameter close to resonance, thetriangle close totheorigin ﬁlled byclosed phase curves is small (oforder e),sothe“radius ofstability” oftheorigin approaches zero as s->0:asmall (oforder s)perturbation oftheinitial condition issuﬂicient tomake aphase point move outside thetriangle andbegin togoaway from theequilibrium position. Returning totheoriginal problem oftheperiodic trajectory, wecome to thefollowing conclusions (which, ofcourse, arenotproven, since wethrew outterms ofdegree higher than three, butwhich canbejustiﬁed): 1.Atthemoment ofpassage through theresonance 3:1aperiodic trajectory generally loses itsstability. 2.Forvalues oftheparameter close toresonance there isanunstable periodic trajectory near theperiodic trajectory under consideration onthesame energy level manifold. Itisclosed after making three circulations along theoriginal trajectory andonerevolution around it.Fortheresonance value oftheparameter, thisunstable trajectory merges with theoriginal one. 3.The distance ofthis unstable periodic trajectory from theoriginal decreases, asweapproach resonance, toﬁrst order inthefrequency deviation (i.e.,astheﬁrstorder ofthedifference oftheparameter from the resonance value). 4.Through thisunstable trajectory onthesame three-dimensional energy level manifold there pass two two-dimensional invariant surfaces, ﬁlled with trajectories approximating thisunstable periodic trajectory ast—+ooononesurface andast-—>—ooontheother. 5.Thelocation oftheseparatrices issuch that, byintersecting with amani- foldtransversal totheoriginal trajectory, weobtain aﬁgure close tothe three sides ofanequilateral triangle andtheir continuations. Thevertices ofthetriangle arethepoints ofintersection oftheunstable periodic trajectory with thetransversal manifold. 6.Forinitial conditions inside thetriangle formed bytheseparatrices, a phase point stays near theoriginal periodic trajectory (atadistance of order s)foralongtime(oforder notlessthan 1/s),andforinitial conditions outside thetriangle itgoes offquite rapidly toadistance which islarge in comparison with a. 393 Appendix 7:Normal forms ofhamiltonian systems near stationary points ESplitting ofseparatrices Inreality, theseparatrices wetalked about instatements 4,5,and6above have avery complicated structure (because oftheinﬂuence oftheterms oforder higher than three which wedisregarded inourapproximation). In order tounderstand thesituation, itisconvenient tolook atatwo-dimen- sional surface transversally intersecting theoriginal closed trajectory at some point onit(and lying entirely inoneenergy level manifold).‘°° Trajec- tories beginning onthissurface intersect itagain after atime close tothe time ofcirculation around theoriginal closed trajectory. Thus wehave a mapping ofaneighborhood ofthepoint ofintersection oftheclosed trajec- tory with thesurface onto apart ofthesurface. This mapping hasaﬁxed point (atthepoint where theclosed trajectory intersects thesurface) andis approximately arotation by120° around thispoint, which wetake forthe origin inoursurface. Wenow consider thethird power ofthemapping indicated above. This isagain amapping ofsome neighborhood oftheorigin toapartofthesur- face, leaving theorigin ﬁxed. Butnowthismapping isapproximately rotation by360°, i.e.,theidentity: itisrealized bythetrajectories ofoursystem after approximately three periods ofourclosed trajectory. Thecalculations above give nontrivial information about thestructure ofthis“mapping after three periods.” Infact, bythrowing outtheterms of degree four andhigher inthehamiltonian function, wechange theterms of degree three andhigher ofthemapping. Therefore, themapping after three periods which corresponds tothetruncated hamiltonian function approxi- mates (with cubic error) theactual mapping after three periods. Butweknow theproperties ofthemapping after three periods correspond- ingtothetruncated hamiltonian function, since itisthemapping ofthe phase ﬂow ofthesystem with hamiltonian function H0(x, y)after time 6n(theproof isbased onthefactthatafter time 61:ourrotating coordinate system returns totheoriginal position). Wenow look atwhich ofthese properties arepreserved forperturbations ofthird-order smallness relative tothedistance from theﬁxed point, andwhich arenot. WeletA0denote themapping after three periods forthetruncated system, andAtheactual mapping after three periods. 1.Themapping A0isincluded inaﬂow: itisthetransformation after time 61:inthephase ﬂow with hamiltonian H0. There isnoreason tothink thatthemapping Aisincluded inaﬂow. 2.Themapping A0issymmetric under arotation by120°: there isanon- trivial diffeomorphism gforwhich g3=Eandwhich commutes with A0. There isnoreason tothink that themapping Acommutes with any nontrivial diffeomorphism gsatisfying g3=E. '°°Here wehave thefollowing general phenomenon: itiseasier tothink about mappings after aeriod, andeasier tocalculate with ﬂows. P 394“.1-1i:<.‘.u.ttt'.1. .4ﬁn ! Appendix 7:Normal forms ofhamiltonian systems near stationary points 3.Themapping A0hasthree unstable ﬁxed points atadistance sfrom the origin, approximately thevertices ofanequilateral triangle. Forsuﬂiciently small deviations from resonance (i.e.,forsuiﬁciently small s)themapping Aalso hasthree unstable ﬁxed points near thevertices ofanequilateral triangle. This follows from theimplicit function theorem. 4.Theseparatrices ofﬁxed points ofthemapping A0form, forvalues ofthe parameter close to(but notat)resonance, aﬁgure approximating the sides andextended sides ofanequilateral triangle. Ifwebegin with a point ononeofthesides ofthetriangle, then after repeated applications ofA0weobtain asequence ofpoints onthesame side ofthetriangle approaching oneofthevertices bounding theside, sayM0. Applying A01,weobtain asequence approaching theother vertex, which wewill denote byN0. Each ofthethree unstable ﬁxed points ofthemapping Aalsohassepara- trices approximating thesides ofatriangle (Figure 240). Namely, those points oftheplane which approach theﬁxed point Mafter applying themappings A",n—>+00,form asmooth curve I“invariant under A,passing through Mand, near M,close tothesideM0N0oftheseparatrices ofA0.Thepoints which approach Nafter applications ofA",where n—>—-oo,form another smooth invariant curve F‘,passing through Nandalso near M0N0near N0. 1"” FT ,\‘~ "J .\'\ IM‘ I’ NI Figure 240 Splitting ofseparatrices However thetwocurves I“andF‘,both near thelineM0N0,arenotat allobliged tocoincide. This isthephenomenon ofsplitting ofseparatrices, which accounts forthediffering behavior ofthetrajectories ofthetruncated andtotal systems. Themagnitude ofthesplitting ofseparatrices isexponentially small forsmall 2:therefore itiseasytooverlook thephenomenon ofsplitting incalculations inoneoranother scheme of “perturbation theory.” However, thisphenomenon isveryimportant infundamental questions. Forexample, itsexistence immediately implies thedivergence ofthe series innumerous versions ofperturbation theory (since iftheseries converged, there would benosplitting). lngeneral. thedivergence ofseries inperturbation theory (while agood approximation is given byafewinitial terms) isusually related tothefactthatwearelooking foranobject which 395 Appendix 7:Normal forms ofhamiltonian systems nearstationary points does notexist. Ifwe trytoﬁtaphenomenon toascheme which actually contradicts theessential features ofthephenomenon, then itisnotsurprising thatourseries diverge. The Birkhoff series (which areobtained ifonecontinues inﬁnitely thenormalizations of theinitial terms oftheTaylor series ofthehamiltonian function) areoneexample ofaformally convergent, butactually divergent, scheme ofperturbation theory. Ifthese series converged, then ageneral oscillating system with onedegree offreedom with periodic coeﬁicients would be reduced near anequilibrium position toanautonomous normal form andthere would beno splitting ofseparatrices init(whereas infactthere is). Returning totheoriginal closed trajectory, weseethatthethree unstable ﬁxed points ofthemapping Acorrespond toanunstable closed trajectory near theoriginal triple. There isafamily oftrajectories approaching this unstable trajectory ast—>+00, and another family oftrajectories ap- proaching theunstable oneas1-»—oo.The points ofthetrajectories of each ofthese families form asmooth surface containing ourunstable trajec- tory. These twosurfaces arealso theseparatrices wetalked about instate- ments 4,5,and6ofSection D.Byintersecting them with ourtransversal surface weobtain theinvariant curves F*andF‘ofthemapping A.The intersections ofthese twocurves form acomplicated network about which H.Poincare, whoﬁrstdiscovered thephenomenon ofsplitting ofseparatrices, wrote, “The intersections form atype oflattice, tissue, orgridwith inﬁnitely ﬁnemesh. Neither ofthetwocurves must ever cutacross itself again, but itmust bend back upon itself inavery complex manner inorder tocut across allofthesquares inthegridaninﬁnite number oftimes. “One willbestruck bythecomplexity ofthisﬁgure, which Ishall noteven attempt todraw. Nothing ismore suitable forproviding uswith anideaof thecomplex nature ofthethree-body problem, andofalltheproblems of dynamics ingeneral,where there isnouniform integral andwhere theBohlin series aredivergent.” (H.Poincare, “Les Méthodes Nouvelles delaMechan- ique Celeste,” Vol. III,Dover, 1957, 389.) Weshould note thatmuch isstillunclear about thepicture ofintersecting separatrices. FResonances ofhigher order Resonances ofhigher order canalso bestudied using anormal form. In thisconnection, wenote that resonances oforder higher than 4donot usually induce instability, since inthenormal form terms ofdegree 4appear, guaranteeing aminimum ormaximum ofthefunction H0even atresonance. Inthecase ofresonance oforder n>4,thetypical development ofthe phase portrait ofthesystem with hamiltonian function H0isgiven bythe formula H0=er+r2ot(r) +ar"/2 sinngo, Zr=P2+q’, 1(0)=il, andconsists ofthefollowing (Figure 241). 396 Appendix 7:Normal forms ofhamiltonian systems near stationary points Figure 241 Averaged hamiltonian ofphase oscillations near resonance 5:1 Forsmall (oforder s)deviations ofthefrequency from resonance, andat asmall (oforder \/m) distance from theequilibrium position attheorigin, thefunction H0has2ncritical points near thevertices ofaregular n-gon with center attheorigin. Half ofthese critical points aresaddle points, andtheother halfaremaxima iftheorigin isaminimum orminima ifthe origin isamaximum. Thesaddle points andstable points alternate. Alln saddle points lieononelevel ofthefunction H0;their separatrices, con- necting successive saddle points, form n“islands,” each ofwhich isﬁlled with closed phase curves encircling astable point. Thewidth oftheislands isoforder s‘""“"1/2’. Theclosed phase curves inside each island arecalled “phase oscillations” (since what varies essentially isthephase oftheoscilla- tions around theorigin). The period ofthephase oscillations grows with decreasing frequency deviation slikes'"/4. Inside thenarrow ringformed bytheislands, closer totheorigin, there are closed phase curves encircling theorigin; outside thering thephase curves areclosed, butmotion along them proceeds inthedirection opposite tothatinside thering. Wenote thattheradius ofthering hasorder \/ls] independently oftheorder ofresonance, ifthisorder isgreater than 4.Also, theringofislands exists foronly oneofthetwosigns ofs. Ifwepass from thetruncated system with hamiltonian H0tothetotal system, theseparatrices split inaway similar tothat described above for resonance oforder 3.Thesizeofthesplitting oftheseparatrices isexpo- nentially small (ororder e"‘/‘"/4), butthesplitting isoffundamental im- portance forinvestigating stability, especially inthemulti-dimensional case. Returning toouroriginal closed trajectory, wehave thefollowing picture. Asweapproach resonance along thesaxisfrom oneside,‘°7 twoperiodic trajectories split offfrom ourperiodic trajectory: astable oneandanun- stable one. These new trajectories close upafter ncirculations along the original trajectory andlieatadistance oforder ,/Islfrom theoriginal trajectory. Near thestable trajectory there isazone ofslow phase oscillations ‘O7Unlike resonance oforder 3.forwhich there isanunstable periodic trajectory branching offfrom both sides oftheresonance. 397 Appendix 7:Normal forms ofhamiltonian systems near stationary points with period oforder s'"/4 and amplitude oforder rt/nintheazimuthal direction andoforder al"/“H1/2’ intheradial direction. Loss ofstability ofthe original periodic trajectory atthemoment ofpassage through resonance does notoccur, atleast intheapproximation which wehave considered. Thecase ofresonance offourth order issomewhat exceptional. Inthiscase, inthenormal form there areboth resonant andnon-resonant terms oforder 4.Theshape ofthephase curves ofthetruncated system depends onwhich ofthese terms ofthenormal form dominates. a resonant oneoranon-resonant one.Intheﬁrstcasethedevelopment isthesame asforthird- order resonance, except thatinplace ofatriangle there isasquare. Inthesecond casethedevelop- ment isthesame asforn>4. Inconclusion, weremark that thegiven normal form becomes abetter approximation aswegetcloser toresonance (a<1)andasthedeviation oftheinitial point from theperiodic trajectory getssmaller. That is,asthe period oftheclosed trajectory andtheperiod ofoscillation ofneighboring trajectories near itbecome more exactly commensurable, andastheinitial condition approaches theclosed trajectory, theinterval oftime grows on which ourapproximation accurately describes thebehavior ofthephase curves. Noconclusion about thebehavior ofnon-closed phase curves oninﬁnite intervals oftime (forexample, about theLiapunov stability oftheoriginal periodic trajectory) follows from ourarguments, since theterms ofhigher order which were thrown outinreducing tonormal form can,over aninﬁnite period oftime, completely change thecharacter ofthemotion. Actually, under theconditions considered, theoriginal periodic trajectory isLiapunov stable, buttheproof requires substantially new techniques beyond the Birkhoff normal form (cf.Appendix 8). 398 Appendix 8:Theory ofperturbations ofconditionally periodic motion, andKolmogorov’s theorem The collection ofsolvable “integrable” problems which wehave atour disposal isnotlarge (one-dimensional problems, motion ofapoint ina central ﬁeld, eulerian andlagrangian motions ofarigid body, theproblem of twoﬁxed centers, andmotion along geodesics ontheellipsoid). However, with thehelp ofthese “integrable cases,” wecanobtain meaningful informa- tionabout motions ofmany important systems byconsidering anintegrable problem asaﬁrstapproximation. Anexample ofsuch asituation istheproblem ofmotion oftheplanets around thesununder thelawofuniversal gravitation. Themass oftheplanets isapproximately 0.001 ofthemass ofthesun,soinaﬁrstapproximation we candisregard theinteraction oftheplanets ononeanother andconsider only theattraction bythesun.Asaresult, weobtain theexactly integrable problem ofthemotion ofnon-interacting planets around thesun;each planet willdescribe itskeplerian ellipse independently oftheothers, andthemotion ofthesystem asawhole willbeconditionally periodic. Ifwenowconsider the interactions oftheplanets ononeanother, thekeplerian motion ofeach planet willbeslightly changed. Wecallupon thetheory ofperturbations from celestial mechanics to study thisinteraction. Itisclear that calculations fortime oftheorder of 1,000 years donotpresent anyfundamental difﬁculties. However, ifwewant tostudy longer intervals oftime, andespecially ifweareinterested inqualita- tivequestions about thebehavior ofexact solutions oftheequations of motion onaninﬁnite time interval, then such diﬁiculties arise. The ac- cumulation ofperturbations after aninterval oftime which islarge in comparison to1,000 years could cause acomplete change inthecharacter of themotion: forexample, theplanets could fallintothesun,escape from it,or collide with oneanother. Note thatthequestion ofthe behavior ofsolutions ofthe equations ofmotion onaninﬁnite time interval hasonly anindirect relation totheproblem ofthemotion ofrealplanets. The reason isthat, after intervals ofbillions ofyears, small non-conservative effects notconsidered inNewton‘s equations become important. Thus, theeffects ofthegravitational interaction of theplanets areofrealimportance onlywhen theyseriously change thepicture ofmotion within a ﬁnite time which issmall incomparison with thetime ofdevelopment ofnon-conservative effects. lncalculating motion over such ﬁnite times, computers prove tobevery useful, quickly determining themotion oftheplanets formany thousands ofyears inthefuture orpast. How- ever, weshould notethateven theapplication ofmodern calculating methods maybeinsuﬁicient topredict theinﬂuence ofperturbations ifaphase point fallsinthezone ofexponential in- stability. Asymptotic andqualitative methods have even greater value forthestudy ofcharged particles inmagnetic ﬁelds, since inthissituation aparticle outstrips thecomputer andmakes somany orbits thatmechanical calculation ofitstrajectory isimpossible even intheabsence of exponential instability. Awhole series ofmethods hasbeen devised forcalculating perturbations incelestial mechanics. (Adetailed analysis ofthem canbefound inthebook, 399 Appendix 8:Theory ofpertubrations ofconditionally periodic motion “Les Methodes Nouvelles delaMecanique Celeste,” byH.Poincare, Dover, 1957.) Adiﬂiculty with allofthese methods isthatthey leadtodivergent series andtherefore givenoinformation about thebehavior ofmotion asawhole over inﬁnite intervals oftime. Thereason forthedivergence ofseries inthe theory ofperturbations is“small denominators”: integral linear combina- tions offrequencies ofunperturbed motions bywhich itisnecessary todivide incalculating theinﬂuence ofperturbations. Forexact resonance (i.e.,for commensurable frequencies) these denominators vanish, and thecor- responding term oftheseries inthetheory ofperturbations becomes in- ﬁnitely large. Close toresonance, thisterm oftheseries isvery large. Thus, forexample, intheir motion around thesun,Jupiter andSaturn, inoneday,gothrough approximately 299and120.5 seconds ofarcrespectively. Therefore, thedenominator 2w,~Sws isverysmall incomparison witheach oftheir frequencies. This amounts toalarge long-period perturbation oftheplanets ononeanother (itsperiod isabout 800years); thestudy byLaplace ofthis effect wasoneofthe ﬁrstsuccesses ofthetheory ofperturbations. Wenote thatthediﬂiculty caused bysmall denominators isessential. The rational numbers form adense set;thusinthephase space ofanunperturbed problem, initial conditions forwhich wehave resonance and thesmall denominators vanish form adense set.Hence, thefunctions given bythe series ofperturbation theory have adense setofsingular points. Thedifficulty mentioned here ischaracteristic notonly forproblems of celestial mechanics, butforallproblems which areclose tointegrable (for instance, fortheproblem ofanasymmetrical rigid topunder very fastrota- tion). Poincare himself called theproblem ofstudying perturbations of conditionally-periodic motions inasystem given bythehamiltonian H=110(1) +8H1(I, (P), 8<1, inaction-angle variables Iandgo,thefundamental problem ofdynamics. Here H0isthehamiltonian oftheunperturbed problem, andeH1aperturbation which isa21:-periodic function oftheangle variables (pl,...,<p,,.Intheunper- turbed problem (e=0)theangles (pchange uniformly with constant frequencies 6H (Uk = J,dlj andalltheaction variables areﬁrstintegrals. Wemust investigate thephase curves ofHamilton’s equations i_an __611 _ dtp (P_61 inaphase space which isadirect product ofaregion inn-dimensional space with coordinates Iandthen-dimensional torus with angular coordinates (p. Asubstantial advance inthestudy ofphase curves ofthisperturbed problem wasbegun in1954 with thework ofA.N.Kolmogorov in“On con- 400 Appendix 8:Theory ofpertubrations ofconditionally periodic motion servation ofconditionally-periodic motions forasmall change inHamilton’s function,” Dokl. Akad. Nauk SSSR 98:4 (1954) 525-530 (Russian). Inthis appendix wepresent thebasic results obtained since then inthisarea. The proofs canbefound inthefollowing works: V.I.Arnold, “Small denominators I,Mapping thecircle onto itself,” Izv.Akad. Nauk SSSR Ser.Mat. 25(I961), 21-86. V.I.Arnold, “Small denominators ll,Proof ofatheorem ofA.N.Kolmogorov onthepreserva- tionofconditionally-periodic motions under asmall perturbation oftheHamiltonian," Russian Math. Surveys 18:5(I963). V.I.Arnold, “Small denominators III.Small denominators andproblems ofstability ofmotion inclassical andcelestial mechanics.” Russian Math. Surveys I826(I963). V.I.Arnold, A.Avez, Ergodic problems ofclassical mechanics, New York, Benjamin, 1968. J.Moser, Oninvariant curves ofarea-preserving mappings ofanannulus (Nachr. Akad. Wiss. Gottingen, Math. Phys. KlIla,(I962) I-20). J.Moser, Arapidly converging iteration method andnonlinear diﬂerential equations_ (Annali della Scuola Norm. Sup. dePisa, (3),20(1966), 265-315; (1966), 499-535. I.Moser, Convergent series expansions forquasi-periodic motions, Math. Ann. I69(1967), 136-176. C.L.Siegel, J.K.Moser, Lectures onCelestial Mechanics, Springer-Verlag, I971. S.Sternberg, Celestial Mechanics, I,II,New York, Benjamin, I969. Before formulating ourresults, wewillbrieﬂy discuss thebehavior of phase curves intheunperturbed problem already studied inChapter 10. AUnperturbed motion Thesystem with hamiltonian H0(1)hasnﬁrstintegrals ininvolution (then action variables). Every level setofallthese integrals isann-dimensional torus in2n-dimensional phase space. This torus isinvariant with respect to thephase ﬂow oftheunperturbed system: every phase curve starting ata point ofourtorus remains onit. Themotion ofaphase point ontheinvariant torus I=const iscondi- tionally-periodic. Thefrequencies ofthismotion arethederivatives ofthe unperturbed hamiltonian with respect totheaction variables: c/3,,=w,,(I), where wk=%. It Therefore, thephase curve densely ﬁllsatorus whose dimension isequal tothenumber offrequencies wkwhich arearithmetically independent. Wenote thatthefrequencies depend onwhich torus wearelooking at; i.e.,which values oftheﬁrstintegrals wehave ﬁxed. Asystem ofnfunctions wofnvariables Iisgenerally functionally independent; insuch acase we cansimply number thetoribytheir frequencies, choosing thevariables w forcoordinates inaneighborhood ofthepoint under consideration inthe space ofaction variables I. 401 Appendix 8:Theory ofperturbations ofconditionally periodic motion Thecasewhen thefrequencies arefunctionally independent willbecalled thenondegenerate case. Theconditions fornondegeneracy have theform tar ‘ll612F605' H0e—=e-i Thus, inthenondegenerate case, theunperturbed problem determines onthe different invariant toriinphase space conditionally-periodic motions with different frequencies. Inparticular, theinvariant torionwhich thenumber of frequencies ismaximal (i.e.,n)form adense setinphase space; such toriare called non-resonant tori. Itcanbeshown that thenon-resonant toriform asetoffullmeasure, i.e.,theLebesgue measure oftheunion ofallinvariant resonant toriofthe unperturbed non-degenerate system isequal tozero. Nevertheless, invariant resonant toriexist andaremixed inwith thenon-resonant toriinsuch away thatthey tooform adense set.Furthermore, thesetofresonant toriwith any nun‘ber ofindependent frequencies from 1ton—1isdense. Inparticular, theinvariant torionwhich allphase curves areclosed (thenumber ofin- dependent frequencies is1)form adense set.Nevertheless, wenote thatthe probability oflanding onaresonant torus byarandom choice ofinitial point inthephase space oftheunperturbed system, isequal tozero(since the probability oflanding onarational number byarandom choice ofareal number iszero). Thus, bydisregarding setsofmeasure zero, wecansaythat almost allinvariant toriinanondegenerate unperturbed system arenon- resonant andhave atotal setofnarithmetically independent frequencies. Onanon-resonant torus, thetrajectory ofaconditionally-periodic motion isdense. Thus, foralmost allinitial conditions, aphase curve ofanon-de- generate unperturbed system densely ﬁllsaninvariant torus whose dimension isequal tothenumber ofdegrees offreedom (i.e.,halfthedimension ofthe phase space). Tobetter understand thewhole picture, weconsider thecase oftwo degrees offreedom (n=2).Inthiscase, thephase space isfour-dimensional soeach energy level setisthree-dimensional. Weﬁxonesuch level set.This three-dimensional manifold, ﬁbered bytwo-dimensional tori, canberepre- sented inordinary three-dimensional space asafamily ofconcentric tori lying inside oneanother (Figure 242). Figure 242 Invariant toriinathree-dimensional energy level manifold 402 Appendix 8;Theory ofperturbations ofconditionally periodic motion Thephase curves arewindings ofthese tori; both frequencies ofcirculation change from torus totorus. Ingeneral, notonly both frequencies butalso their ratio willchange from torus totorus. Ifthederivative oftheratio of frequencies with respect totheaction variable numbering thetorionthe given level setofthefunction H0isnotzero, then wesaythatoursystem is isoenergeticall ynondegenerate. Thecondition forisoenergetic nondegeneracy has(asiseasy tocalculate) theform 02H0 % 012 61 det 8 7&0. H0—— 051 Theconditions fornondegeneracy andisoenergetic nondegeneracy areindependent from oneanother: i.e.,anondegenerate system could beisoenergetically degenerate. andaniso- energetically nondegenerate system could bedegenerate. Inthemany-dimensional case(11>2) isoenergetic nondegeneracy means nondegeneracy ofthefollowing mapping ofthe(n-l)- dimensional level manifold ofthefunction H0ofnaction variables totheprojective space of dimension n—I: I—>(r/11(1): 012(1): ---Iw,,(I)). Now consider anisoenergetically nondegenerate system with twodegrees offreedom. Itiseasy toconstruct atwo-dimensional plane inthethree- dimentional energy level settransversally intersecting thetwo-dimensional toriofourfamily (inafamily ofconcentric circles inthemodel inthree- dimensional euclidean space). Aphase curve beginning insuch aplane returns toitafter making a circuit around thetorus. Asaresult weobtain anewpoint onthesame circle inwhich thetorus intersects theplane. Inthiswaythere arises amapping of theplane toitself. This mapping oftheplane toitself ﬁxes theconcentric meridian circles in which theplane intersects theinvariant tori. Every circle isrotated through some angle, namely through that fraction ofanentire revolution that the frequency along themeridian constitutes ofthefrequency along theequator. Ifthesystem isisoenergetically nondegenerate, theangle ofrevolution of invariant circles intheplane ofintersection changes from onecircle to another. Therefore, onsome circles thisangle willbecommensurable with a whole revolution, andonothers itwillbeincommensurable. Each ofthese classes ofcircles willform adense set,butonalmost allcircles (inthesense of Lebesgue measure) theangle ofrotation willbeincommensurable with a whole revolution. Thecommensurability orincommensurability ismanifested inthefollow- ingwayonthebehavior ofpoints ofacircle under themapping oftheregion toitself. Iftheangle ofrotation iscommensurable with awhole rotation, then 403 Appendix 8:Theory ofperturbations ofconditionally periodic motion after several iterations ofthemapping thepoint willreturn toitsinitial position (thenumber ofiterations willbelarger asthedenominator ofthe fraction expressing theangle ofrotation islarger). Iftheangle ofrotation is incommensurable with awhole rotation, thesuccessive images ofthepoint under repetitions ofthemapping willdensely ﬁllupthemeridian circle. Wenote further thatcommensurability corresponds toresonant toriand incommensurability tonon-resonant tori. Also, theexistence ofresonant toriimplies thefollowing property. Consider some power ofthemapping of ourregion toitself induced bythephase curves. Lettheexponent bethe denominator ofthefraction expressing theratio ofthefrequencies ononeof theresonant tori. Then themapping raised totheindicated power hasa whole circle consisting entirely ofﬁxed points (namely, themeridian ofthe resonant torus under consideration). Such behavior ofﬁxed points isunnatural formappings inanysortof general form, even canonical mappings (ﬁxed points areusually isolated). Inthegiven case, awhole circle ofﬁxed points arises because wehave con- sidered anunperturbed integrable system. Forarbitrarily small perturbations ofgeneral form, thisproperty ofthemapping (having awhole circle ofﬁxed points) must fail.Thecircle ofﬁxed points must bedispersed sothatonly a ﬁnite number remain. Inother words, under small perturbations ofourintegrable system, we expect achange inthequalitative picture ofthephase curves, ifonly inthe respect thatentire invariant toriﬁlled outbyclosed phase curves willdis- integrate sothat there remain only aﬁnite number ofclosed curves, near those fortheunperturbed system, andtheremaining phase curves willbe more complicated. Wehave already encountered such acaseinAppendix 7 ininvestigating phase oscillations near resonance. Wenow consider what happens tonon-resonant invariant toriunder a small perturbation ofahamiltonian function. Formal application ofthe principle ofaveraging (i.e.,theﬁrstapproximation oftheclassical theory of perturbations, cf.Section 52)leads ustotheconclusion thatanon-resonant torus does notundergo anyevolution. Wenote that thefactthat theperturbations arehamiltonian isessential, since fornon- conservative perturbations itisclear thattheaction variables mayevolve. Incelestial mechanics. their evolution means asecular change inthemajor semi-axes ofthekeplerian ellipses, i.e..the planets falling intothesun,colliding, orescaping toalarge distance inatime which isinversely proportional tothesizeoftheperturbation. Ifconservative perturbations ledtoevolutions in aﬁrstapproximation, thiswould manifest itself inthefateoftheplanets after atime onthe order of1,000 years. Fortunately, theorder ofmagnitude ofthe non-conservative perturbations ismuch less. Thetheorem ofKolmogorov, formulated below, furnishes onejustiﬁcation fortheconclusion, drawn from thenon-rigorous theory ofperturbations, about theabsence ofevolution ofaction variables. 404 Appendix 8:Theory ofperturbations ofconditionally periodic motion BInvariant toriinaperturbed system Theorem. Ifanunperturbed system isnondegenerate, then forsuﬂlciently small conservative hamiltonian perturbations, most non-resonant invariant toridonotvanish, butareonlyslightly deformed, sothatinthephase space oftheperturbed system, too,there areinvariant toridensely ﬁlled withphase curves winding around them conditionally-periodically, with anumber of independent frequencies equal tothenumber ofdegrees offreedom. These invariant toriform amajority inthesense thatthemeasure ofthe complement oftheir union issmall when theperturbation issmall. A.N.Kolmogorov’s proof ofthistheorem isbased onthefollowing two observations. 1.Weﬁxanon-resonance setoffrequencies oftheunperturbed system so thatthefrequencies arenotonly independent, butdonoteven approximately satisfy anyresonance conditions ofloworder. More precisely, weﬁxaset offrequencies toforwhich there exist Candvsuch that |(ro,k)|>C|k|'“ forallintegral vectors kaé0. Itcanbeshown that, ifvissuﬂiciently large (say v=n+1),then the measure ofthesetofsuch vectors w(lying inaﬁxed bounded region) for which theindicated condition ofnon-resonance isviolated, issmall when C issmall. Next, near anon-resonant torus oftheunperturbed system corresponding toaﬁxed value ofthefrequencies, wewilllook foraninvariant torus ofthe perturbed system onwhich there isconditionally-periodic motion with exactly thesame frequencies astheones weﬁxed, and which necessarily satisfy thecondition ofbeing non-resonant described above. Inthisway, instead ofthevariations offrequency customary inperturba- tionschemes (consisting oftheintroduction offrequencies depending onthe perturbation), wemust hold constant thenon-resonant frequencies, while selecting initial conditions depending ontheperturbation inorder to guarantee motion with thegiven frequencies. This canbedone byasmall (when theperturbation issmall) change ofinitial conditions, because the frequencies change with theaction variables according tothenon-degen- eracy condition. 2.Thesecond observation isthat, toﬁndaninvariant torus, instead of using theusual series expansion inpowers oftheperturbation parameter, we canusearapidly convergent method similar toNewton’s method oftangents. Newton’s method oftangents forﬁnding roots ofalgebraic equations with initial error egives, after napproximations, anerror oforder 62”.Such super-convergence allows ustoparalyze theinﬂuence ofthesmall denomin- ators appearing inevery approximation, andintheendsucceeds notonly in carrying outaninﬁnite number ofapproximations, butalsoinshowing the convergence oftheentire procedure. 405 Appendix 8:Theory ofperturbations ofconditionally periodic motion Theassumption under which allthiscanbedone isthattheunperturbed hamiltonian function H0(I)isanalytic andnondegenerate, andtheperturbing hamiltonian function aH,(I, (p)isanalytic and21:-periodic intheangle vari- ables (p.Thepresence ofthesmall parameter aisimmaterial: itisimportant only thattheperturbation besufﬁciently small insome complex neighbor- hood ofradius poftherealplane ofthevariables (p(less than some positive function M(p, H0)). AsJ.Moser showed, therequirement ofanalyticity canbechanged to dilferentiability ofsufﬁciently high order ifwecombine Newton’s method with anidea ofJ.Nash, theapplication ofasmoothing operator ateach approximation. Theresulting conditionally-periodic motions oftheperturbed system with ﬁxed frequencies coturn outtobesmooth functions oftheparameter eof perturbation. Therefore, they could have been sought, without Newton’s method, intheform ofaseries inpowers ofs.Thecoefﬁcients ofthisseries, called theLindstedt series, canactually befound; however, wecanprove its convergence only indirectly, with thehelp ofnewtonian approximations. CZones ofinstability Thepresence ofinvariant toriinthephase space oftheperturbed problem means that, formost initial conditions inasystem which isnearly integrable, motion remains conditionally periodic with amaximal setoffrequencies. The question naturally arises ofwhat happens totheremaining phase curves, with initial conditions falling intothegaps between theinvariant tori which replace theresonant invariant toriofthenon-perturbed problem. Thedisintegration ofaresonant torus onwhich thenumber offrequencies isonelessthan themaximum iseasy toinvestigate inaﬁrst-order perturba- tiontheory. Todothis, wemust average theperturbation over the(n—I)- dimensional invariant tori into which theresonant invariant torus is decomposed andwhich aredensely ﬁlled outbyphase curves oftheun- perturbed system. After averaging, weobtain aconservative system with one degree offreedom (cf.theinvestigation ofphase oscillations near resonance inAppendix 7),which iseasy tostudy. Intheapproximation under consideration wehave, nearthen-dimensional reducible torus, stable andunstable (n—1)-dimensional tori, with phase oscillations around thestable ones. The corresponding conditionally- periodic motions have afullsetofnfrequencies, ofwhich n—1arethefast frequencies oftheoriginal oscillations andoneistheslow (oforder ,/5) frequency ofthephase oscillations. However, onemust notconclude thattheonly difference between motions intheunperturbed andperturbed systems istheappearance of“islands” ofphase oscillations. Infact, theactual phenomena aremuch more compli- cated than theﬁrst approximation described above. One manifestation of thiscomplicated behavior ofthephase curves oftheperturbed problem is thesplitting ofseparatrices discussed inAppendix 7. 406 Appendix 8:Theory ofperturbations ofconditionally periodic motion Tostudy motions ofaperturbed system outside oftheinvariant toriwe must distinguish thecases oftwoandhigher degrees offreedom. Fortwo degrees offreedom, thedimension ofthephase space isfour, andanenergy level manifold isthree-dimensional. Therefore, theinvariant two-dimensional toridivide each energy level set.Thus, aphase curve beginning inthegap between twoinvariant torioftheperturbed system remains forever conﬁned between those tori. Nomatter how complicated thiscurve appears, itdoes notleave itsgap,andthecorresponding action variables remain forever near their initial conditions. Ifthenumber nofdegrees offreedom isgreater than two, then-dimen- sional invariant toridonotdivide the(2n—1)-dimensional energy level manifold butarearranged initlikepoints onaplane orlines inspace. Inthis casethe“gaps” corresponding todifferent resonances areconnected toone another, sotheinvariant toridonotprevent phase curves starting near resonance from going faraway. Hence, there isnoreason toexpect thatthe action variables along such aphase curve willremain close totheir initial values foralltime. Inother words, under suﬂiciently small perturbations ofsystems with twodegrees offreedom (satisfying thegenerally fulﬁlled condition ofiso- energetic nondegeneracy), notonly dotheaction variables along aphase trajectory have nosecular perturbations inanyapproximation ofperturba- tiontheory (i.e.,theychange little inatime interval ontheorder of(1/a)” for anyN,where aisthemagnitude oftheperturbation), butthese variables remain forever near their initial values. This istrue, both fornon-resonant phase curves conditionally-periodically ﬁlling outtwo-dimensional tori(and comprising most ofthephase space), andfortheremaining initial conditions. Atthesame time, there exist systems with more than two degrees of freedom satisfying allthenondegeneracy conditions, inwhich, although for most initial conditions motion isconditionally periodic, forsome initial conditions aslow drift oftheaction variables away from their initial values occurs. The average velocity ofthisdrift inknown examples‘°8 isonthe order ofe“/“S5, i.e.,thisvelocity decreases faster than anypower ofthe perturbation parameter. Thus itisnotsurprising thatthisdrifting away does notappear inanyapproximation ofperturbation theory. (Byaverage vel- ocity, wemean theratio oftheincrease ofaction variables totime, sothat weareactually dealing with anincrease oforder 1after atime oforder e1/*/E). Anupper bound ontheaverage velocity ofthedrift oftheaction variables ingeneral nearly integrable systems ofhamiltonian equations with ndegrees offreedom isincluded intherecent work ofN.N.Nehoroshev.1°9 '08Cf.V.I.Arnold, Instability ofdynamical systems with many degrees offreedom. Soviet Mathematics 5:3(I964) 581-585. 10°N.N.Nehoroshev, Thebehavior ofhamiltonian systems thatareclose tointegrable ones, Functional Analysis andItsApplications, 5:4(I971); Uspekhi Mat. Nauk 32:6 (1977). 407 Appendix 8:Theory ofperturbations ofconditionally periodic motion This bound, likethelower bound mentioned above, hastheform e"/"1; thus theincrease oftheaction variables issmall while thetime issmall in comparison with e‘/‘d, ife<e0.Here sisthemagnitude oftheperturbation, anddisanumber between 0and1deﬁned, likee0,bytheproperties ofthe unperturbed hamiltonian H0. Inaddition, anondegeneracy condition is imposed ontheunperturbed hamiltonian (this condition hasalong formula- tion, butisgenerally satisﬁed; inparticular, strong convexity oftheun- perturbed hamiltonian issuﬂicient, i.e.,positive ornegative deﬁniteness of thesecond differential ofH0). From thisupper bound itisclear thatsecular changes oftheaction vari- ables arenotdetected byanyapproximation ofperturbation theory, since theaverage velocity ofthese changes isexponentially small. Wenote also that secular changes oftheaction variables obviously have nodirectional character, butarerepresented bymore orlessrandom wandering inthe resonant regions between theinvariant tori. Amore detailed discussion of thequestions arising here canbefound inthearticle, “Stochastic instability ofnonlinear oscillations,” byG.M.Zaslavski andB.V.Chirikov, Soviet Physics Uspekhi, v.105,no.1(1971), 3-39. DVariants ofthetheorem oninvariant tori Statements analogous tothetheorem onconservation ofinvariant toriinan autonomous system have been proved fornon-autonomous equations with periodic coeﬂicients andforsymplectic mappings. Analogous statements are valid inthetheory ofsmall oscillations inaneighborhood ofanequilibrium position ofanautonomous system orasystem with periodic coefficients, as well asinaneighborhood ofaclosed phase curve ofaphase ﬂow orina neighborhood ofaﬁxed point ofasymplectic mapping. Thenondegeneracy conditions necessary inthevarious cases aredifferent. Forreference, wewillnow give these nondegeneracy conditions. Wewill limit ourselves tothesimplest requirements ofnondegeneracy, which areall fulﬁlled bysystems in“general position.” Inmany cases, therequirements ofnondegeneracy canbeweakened, buttheadvantage gained bythisisoffset bythecomplication oftheformulas. 1.Autonomous systems. Thehamiltonian function is H=H0(I) +eH,(I, cp), IEGCR",rpmod 21r€ T“. Thenondegeneracy condition 62 d€t * 75O guarantees preservation“° ofmost invariant toriunder small perturbations (e<1). 11°Itisunderstood thatthetoriareslightly deformed under perturbations. 408 Appendix 8:Theory ofperturbations ofconditionally periodic motion Thecondition forisoenergetic nondegeneracy 62H06H0 012 01 det 9*0 6H0 guarantees theexistence onevery energy level manifold ofasetofinvariant toriwhose complement hassmall measure. The frequencies onthese tori generally depend onthesizeoftheperturbation, buttheratios offrequencies arepreserved under changes ine. Ifn=2,thenthecondition forisoenergetic nondegeneracy alsoguarantees stability oftheaction variables, inthesense thattheyremain forever close to their initial values forsuﬁiciently small perturbations. 2.Periodic systems. Thehamiltonian function is H= H0(I) +eH,(I,cp,t), IeG clR",rpmod 21:6 T"; theperturbation is21:-periodic notonly inrp,butalsoint.Itisnatural tolook attheunperturbed system inthe(2n+l)-dimensional space {(1,(,0,t)}= IR"xT"+1. Theinvariant torihave dimension n+1.The nondegeneracy condition 82 dCt Fgg ¢0 guarantees thepreservation ofmost (n+1)-dimensional invariant toriunder asmall perturbation (e<1). Ifn=I,thisnondegeneracy condition also guarantees stability ofthe action variable, inthesense thatitremains forever near itsinitial value for sufficiently small perturbations. 3.Mappings (I,(,0)—>(I’,cp’)ofthe“2n-dimensional annulus.” Thegener- ating function is S(I,1 = SO([') + 8S1(I’s (P): FEG C R": Thenondegeneracy condition 52 guarantees thepreservation ofmost invariant torioftheunperturbed map- ping (I,go)—->(I,(,0+(080/51) under small perturbations (e<1). Ifn=1,weobtain anarea-preserving mapping oftheordinary annulus to itself. Theunperturbed mapping isrepresented oneach circle I=const asa rotation. Inthiscase thenondegeneracy condition means that theangle of rotation changes from onecircle toanother. Theinvariant toriinthecase n=1areordinary circles. Inthiscase, the theorem guarantees thatunder iterations ofthemapping alltheimages ofa 409 Appendix 8:Theory ofpertubrations ofconditionally periodic motion point willremain near thecircle onwhich theoriginal point lay,ifthe perturbation issufficiently small. 4.Neighborhoods ofequilibrium positions (autonomous case). Anequili- brium position isassumed tobestable inalinear approximation sothatn characteristic frequencies to1,...,to,aredeﬁned. Weassume thatthere areno resonance relations among thecharacteristic frequencies, i.e.,norelations klcol + +k,,o),, =0with integers k,such that0<2lk,-I34. Then thehamiltonian function canbereduced totheBirkhoﬂ normal form (cf.Appendix 7) H: Ho(T)+ "', where H0(-r) =Zwkrk+ w,,,r,, r,andthedots denote terms ofdegree higher than four with respect tothedistance from theequilibrium position. Thenondegeneracy condition detlcok,l aé0 guarantees theexistence ofasetofinvariant toriofalmost fullmeasure ina sufficiently small neighborhood oftheequilibrium position. Thecondition forisoenergetic nondegeneracy, detwill wkab0, CU] O guarantees theexistence ofsuch asetofinvariant torionevery energy level set(suﬂiciently close tothecritical point). Inthecasen=2,thecondition forisoenergetic nondegeneracy issatisﬁed ifthequadratic part ofthefunction H0isnotdivisible bythelinear part. In thiscase, isoenergetic nondegeneracy guarantees Liapunov stability ofthe equilibrium position. 5.Neighborhoods ofequilibrium positions (periodic case). Here again we assume stability inalinear approximation, sothat ncharacteristic fre- quencies wl,...,canaredeﬁned. Weassume that there arenoresonance relations I1 k,w,+---+k,,co,,+k0=0 with0< Z|k,|g4 i=1 among thecharacteristic frequencies andthefrequency ofthetime-depen- dence ofthecoeﬂicients (which wewillassume equal to1). Then thehamiltonian function canbereduced toaBirkhoff normal form inthesame way asintheautonomous case, butwith 21:-periodicity with respect totime intheremainder term. Thenondegeneracy condition det|co,,,l aé0 410 Appendix 8:Theory ofperturbations ofconditionally periodic motion guarantees theexistence of(n+1)-dimensional invariant toriinthe (2n+1)- dimensional extended phase space, near thecircle t=0representing the equilibrium position. Inthecasen=Ithenondegeneracy condition reduces tothenon-vanish- ingofthederivative oftheperiod ofsmall oscillations with respect tothe square oftheamplitude ofsmall oscillations. Inthiscase, nondegeneracy guarantees thattheequilibrium position isLiapunov stable. 6.Fixed points ofmappings. Here weassume thatall2neigenvalues ofthe linearization ofacanonical mapping ataﬁxed point have modulus 1anddo notsatisfy anylow-order resonance relations oftheform: /l'{'---»l§"=1, |k,|+-~-+|k,,l$4 (where the2neigenvalues are1,,...,/l,,,/ll,...,/in). Then ifwedisregard terms ofhigher than third order intheTaylor series attheﬁxed point, themapping canbewritten inBirkhoff normal form (1,(p)—>(r,(p+01(1)), where a(t)=g, S=Z0),,rk+ixw,,,r,,r, (theusual coordinates inaneighborhood ofthe equilibrium position arepk=./2r,‘cos(pk,qk=./21,,sin(pk). Thenondegeneracy condition det|o),,,| 75O guarantees theexistence ofn-dimensional invariant tori(close tothetori r=const), forming asetofalmost fullmeasure inasuﬂiciently small neighborhood oftheequilibrium position. Ifn=1,wehave amapping oftheordinary plane toitself, and the invariant toribecome circles. Thenondegeneracy condition means that, for thenormal form, thederivative oftheangle ofrotation ofacircle with respect totheareabounded bythecircle isnotzero (attheﬁxed point and,therefore, insome neighborhood ofit). Inthecase n=1thenondegeneracy condition guarantees Liapunov stability oftheﬁxed point ofthemapping. Wenote thatinthiscasethecon- dition ofabsence oflower resonance hastheform P;-e1 ,t“¢1. Thus aﬁxed point ofanarea-preserving mapping oftheplane toitself is Liapunov stable ifthelinear partofthemapping isrotation through anangle which isnotamultiple of90°or120°andifthecoefficient to,1inthenormal Birkhoff form isnotzero (guaranteeing nontrivial dependence oftheangle ofrotation ontheradius). Wehave notgone into thesmoothness conditions assumed inthese theorems. Theminimal smoothness needed isnotknown ineven onecase. 4ll Appendix 8:Theory ofperturbations ofconditionally periodic motion Forexample, wepoint outthat thelastassertion about stability ofﬁxed points ofamapping oftheplane toitself wasﬁrstproved byJ.Moser under theassumption of333-times differentiability, andonly later (byMoser and Russman) wasthenumber ofderivatives reduced to6. EApplications ofthetheorem oninvariant tori anditsgeneralizations There aremany mechanical problems towhich wecanapply thetheorem formulated above. One ofthesimplest ofthese problems isthemotion ofa pendulum under theaction ofaperiodically changing exterior ﬁeld orunder theaction ofvertical oscillations ofthepoint ofsuspension. Itiswellknown that, intheabsence ofparametric resonance, thelower equilibrium position ofapendulum isstable inthelinear approximation. The stability ofthisposition with regard tononlinear effects (under thefurther assumption oftheabsence ofresonances oforder 3and4)canbeproved only with thehelp ofthetheorem oninvariant tori. Inananalogous waywecanusethetheorem oninvariant toritoinvestigate conditionally-periodic motions ofasystem ofinteracting nonlinear os- cillators. Another example isthegeodesic ﬂow onaconvex surface close toan ellipsoid. There aretwodegrees offreedom inthissystem, andwecanshow thatmost geodesics onathree-dimensional near-ellipsoidal surface oscillate between two“caustics” close tothelines ofcurvature ofthesurface, densely ﬁlling outtheringbetween them. Atthesame time, wecanarrive attheorems onthestability ofthetwoclosed geodesics obtained, after deforming the surface, from thetwoellipses containing themiddle axisoftheellipsoid (in theabsence ofresonances oforders 3and4). Asonemore example, wecanlook atclosed trajectories onabilliard table ofanyconvex shape. Among theclosed billiard trajectories arethose which arestable inthelinear approximation, andwecanconclude that inthe general case they areactually stable. Anexample ofsuch astable billiard trajectory istheminor axisofanellipse; therefore, aclosed billiard trajec- tory, close totheminor axisofanellipse onabilliard table which isalmost theellipse, isstable. Application ofthetheorem oninvariant toritotheproblem ofrotations ofanasymmetric heavy rigid body allows ustoconsider thenonintegrable case ofarapidly rotating body. The problem ofrapid rotation ismathe- matically equivalent totheproblem ofmotion with moderate velocity ina weak gravitational ﬁeld: theessential parameter istheratio ofpotential to kinetic energy. Ifthisparameter issmall, then wecanuseeulerian motion of arigid body asaﬁrstapproximation. Byapplying thetheorem oninvariant toritotheproblem with twodegrees offreedom obtained after eliminating cyclic coordinates (rotations around thevertical) wecome tothefollowing conclusion about themotion ofa rapidly rotating body: ifthekinetic energy ofrotation ofabody issuﬂiciently 412 Appendix 8:Theory ofperturbations ofconditionally periodic motion large incomparison with thepotential energy, then thelength ofthevector of angular momentum anditsangle with thehorizontal remain forever close totheir initial values. Itfollows from thisthatthemotion ofthebody willforever beclose toa combination ofEuler-Poinsot motion andazimuthal procession, except in thecase when theinitial values ofkinetic energy andtotal momentum are close tothose forwhich thebody canrotate around themiddle principal axis. Inthislastcase, realized only forspecial initial conditions, thesplitting of separatrices near themiddle axis implies amore complicated undulation about themiddle axisthan inEuler-Poinsot motion. One generalization ofthetheorem oninvariant torileads tothetheorem ontheadiabatic invariance foralltime oftheaction variable inaone- dimensional oscillating system with periodically changing parameters. Here wemust assume that theruleforchanging parameters isgiven byaﬁxed smooth periodic function of“slow time,” andthesmall parameter ofthe problem istheratio oftheperiod ofcharacteristic oscillations andtheperiod ofchange ofparameters. Then, iftheperiod ofchange ofparameters issulﬁ- ciently large, thechange intheadiabatic invariant ofaphase point remains small inthecourse ofaninﬁnite interval oftime. Inananalogous way wecanprove theadiabatic invariance foralltime oftheaction variable intheproblem ofacharged particle inanaxially- symmetric magnetic ﬁeld. Violation ofaxial symmetry inthisproblem in- creases thenumber ofdegrees offreedom from twotothree, sothat the invariant toricease todivide theenergy level manifolds, andthephase curve wanders about theresonance zones. Finally, applying thetheory tothethree- (ormany-) body problem, we succeed inﬁnding conditionally-periodic motions of“planetary type.” To describe these motions, wemust sayafewwords about thenext approxima- tionafter thekeplerian oneintheproblem ofthemotion oftheplanets. For simplicity wewilllimit ourselves totheplanar problem. Foreach keplerian ellipse, consider thevector connecting thefocus ofthe ellipse (i.e.,thesun)tothecenter oftheellipse. This vector, called theLaplace vector, characterizes both themagnitude oftheeccentricity oftheorbit andthe direction totheperihelion. The interaction oftheplanets onone another causes thekeplerian ellipse (and therefore theLaplace vector) tochange slowly. Inaddition, there isanimportant difference between changes inthemajor semi-axis and changes intheLaplace vector. Namely, themajor semi-axis hasnosecular perturbations, i.e.,intheﬁrst approximation itmerely oscillates slightly around itsaverage value (“Laplace’s theorem”). TheLaplace vector, onthe other hand, performs both periodic oscillations and secular motion. The secular motion may beobtained ifwespread each planet over itsorbit proportionally tothetime spent intravelling each piece oftheorbit, and replace theattraction oftheplanets bytheattraction oftherings obtained, that is,ifweaverage theperturbation over therapid motions. The true 413 lAppendix 8:Theory ofperturbations ofconditionally periodic motion motion oftheLaplace vector isobtained from thesecular onebytheaddi- tionofsmall oscillations; these oscillations areessential ifweareinterested insmall intervals oftime (years), buttheir effect remains small incomparison totheeffect ofthesecular motion ifweconsider alarge interval oftime (thousands ofyears). Calculations (carried outbyLagrange) show that thesecular motion of theLaplace vector ofeach ofnplanets moving inoneplane consists ofthe following (ifweignore thesquares oftheeccentricities oftheorbits which aresmall incomparison with theeccentricities themselves). Intheorbital plane ofaplanet wemust arrange nvectors ofﬁxed lengths, each rotating uniformly with itsangular velocity. TheLaplace vector istheir sum. This description ofthemotion oftheLaplace vector isobtained because thehamiltonian system averaged with respect torapid motions, which describes thesecular motion oftheLaplace vector, hasanequilibrium posi- tioncorresponding tozero eccentricities. Thedescribed motion oftheLap- lacevector isthedecomposition ofsmall oscillations near thisequilibrium position into characteristic oscillations. Theangular velocities oftheuni- formly rotating components oftheLaplace vector arethecharacteristic frequencies, andthelengths ofthese components determine theamplitudes ofthecharacteristic oscillations. Wenote thatthemotion oftheLaplace vector oftheearth is,apparently, oneofthe factors involved intheoccurrence oficeages. The reason isthat, when theeccentricity oftheearth’s orbit increases, thetime itspends near thesundecreases, while thetime itspends farfrom the sunincreases (bythelawofareas); thus theclimate becomes more severe astheeccentricity increases. Themagnitude ofthiseffect issuch that, forexample, theamount ofsolar energy received inayear atthelatitude ofLeningrad (60°N) may attain thevalue which now corresponds tothelatitudes ofKiev (50°N) (fordecreased eccentricity) andTaimir (80°N) (forincreased eccentricity). Thecharacteristic time ofvariation oftheeccentricity (tens ofthousands ofyears) agrees well with theinterval between iceages. Thetheorems oninvariant torilead totheconclusion thatforplanets of suﬂiciently small mass, there is,inthephase space oftheproblem, asetof positive measure ﬁlled with conditionally-periodic phase curves such that thecorresponding motion oftheplanets isnearly motion over slowly changing ellipses ofsmall eccentricities, and themotion oftheLaplace vectors isalmost thatgiven bytheapproximation described above. Further- more, ifthemasses oftheplanets aresufﬁciently small, then motions ofthis type ﬁllupmost oftheregion ofphase space corresponding inthekeplerian approximation tomotions oftheplanets inthesame direction over non- intersecting ellipses ofsmall eccentricities. Thenumber ofdegrees offreedom intheplanar problem with nplanets isequal to2nifwetakethesuntobeﬁxed. Theintegral ofangular momentum allows ustoeliminate onecyclic coordinate; however, there arestilltoo many variables fortheinvariant toritodivide anenergy level manifold (even ifthere areonly twoplanets thismanifold isﬁve-dimensional, andthetori arethree-dimensional). Therefore, inthisproblem wecannot draw anycon- 414 Appendix 8:Theory ofperturbations ofconditionally periodic motion clusions about thepreservation ofthelarge semi-axes over aninﬁnite interval oftime forallinitial conditions, butonly formost initial conditions. Aproblem with twodegrees offreedom isobtained byfurther idealization. Wereplace oneofthetwoplanets byan“asteroid” which moves intheﬁeld ofthesecond planet (“Jupiter”), notperturbing itsmotion. Theproblem ofthemotion ofsuch anasteroid iscalled therestricted three-body problem. Theplanar restricted three-body problem reduces toa system with twodegrees offreedom, periodically depending ontime, forthe motion oftheasteroid. If,inaddition, theorbit ofJupiter iscircular, then ina coordinate system rotating together with itweobtain, forthemotion ofthe asteroid, anautonomous hamiltonian system with twodegrees offreedom—- called theplanar restricted circular three-body problem. Inthisproblem, there isasmall parameter—the ratio ofthemasses of Jupiter andthesun. The zero value oftheparameter corresponds toun- perturbed keplerian motion oftheasteroid, represented inourfour-dimen- sional phase space asaconditionally-periodic motion onatwo-dimensional torus (since thecoordinate system isrotating). One ofthefrequencies ofthis conditionally-periodic motion isequal toIforallinitial conditions; thisis theangular velocity oftherotating coordinate system, i.e.,thefrequency of therevolution ofJupiter around thesun.Thesecond frequency depends on theinitial conditions (this isthefrequency oftherevolution oftheasteroid around thesun) andisﬁxed onanyﬁxed three-dimensional level manifold ofthehamiltonian function. Therefore, thenondegeneracy condition isnotfulﬁlled inourproblem, but thecondition forisoenergetic nondegeneracy isfulﬁlled. Kolmogorov’s theorem applies, andweconclude that most invariant toriwith irrational ratios offrequencies arepreserved inthecasewhen themass oftheperturbing planet (Jupiter) isnotzero, butsufficiently small. Furthermore, thetwo-dimensional invariant tori divide the three- dimensional level manifolds ofthehamiltonian function. Therefore, the magnitude ofthemajor semi-axis and theeccentricity ofthekeplerian ellipse oftheasteroid willremain forever near their initial values if,atthe initial moment, thekeplerian ellipse does notintersect theorbit ofthe perturbing planet, andifthemass ofthisplanet issuﬂiciently small. Inaddition, inastationary coordinate system, thekeplerian ellipse ofthe asteroid could slowly rotate, since oursystem isonly isoenergetically non- degenerate. Therefore under perturbations ofaninvariant torus frequencies arenotpreserved, butonly their ratios. Asaresult ofaperturbation, the frequency ofazimuthal motion oftheperihelion oftheasteroid inastationary coordinate system could beslightly different from Jupiter’s frequency, and then inthestationary system theperihelion would slowly rotate. 415 Appendix 9:Poincaré’s geometric theorem, its generalizations andapplications Inhisstudy ofperiodic solutions ofproblems incelestial mechanics, H. Poincaré constructed averysimple model which contains thebasic difficulties oftheproblem. This model isanarea-preserving mapping oftheplanar circular annulus toitself. Mappings ofthisform arise inthestudy ofdynam- icalsystems with two degrees offreedom. Infact, amapping ofatwo- dimensional surface ofsection toitself isdeﬁned asfollows: each point pof thesurface ofsection istaken tothenext point atwhich thephase curve originating atpintersects thesurface (cf.Appendix 7).Thus, aclosed phase curve corresponds toaﬁxed point ofthemapping orofapower ofthe mapping. Conversely, every ﬁxed point ofthemapping orofapower of themapping determines aclosed phase curve. Inthisway, aquestion about theexistence ofperiodic solutions ofprob- lems indynamics isreduced toaquestion about ﬁxed points ofarea-pre- serving mappings oftheannulus toitself. Instudying such mappings, Poincaré arrived atthefollowing theorem. AFixed points ofmappings oftheannulus toitself Theorem. Suppose thatwearegiven anarea-preserving homeomorphic mapping oftheplanar circular annulus toitself. Assume thattheboundary circles of theannulus areturned indiﬂerent directions under themapping. Then this mapping hasatleast twoﬁxed points. Thecondition thattheboundary circles areturned indifferent directions means that, ifwechoose coordinates (x,ymod 21:)ontheannulus sothatthe boundary circles arex=aandx=b,then themapping isdeﬁned bythe formula (X,y)—>(f(X,y),y+90¢,y)), where thefunctions fandgarecontinuous and 21:-periodic iny,with f(a,y)2a,f(b,y)Eb,andg(a,y)<O,g(b,y)>Oforally. The proof ofthistheorem, announced byPoincare notlong before his death, was given only later byG.D.Birkhoff (cf.hisbook, Dynamical Systems, Amer. Math. Soc., 1927). There remain many open questions related tothistheorem; inparticular, attempts togeneralize ittohigher dimensions areimportant forthestudy ofperiodic solutions ofproblems with many degrees offreedom. Theargu- ment Poincare used toarrive athistheorem applies toawhole series ofother problems. However, theintricate proof given byBirkhoff does notlend itself togeneralization. Therefore, itisnotknown whether theconclusions sug- gested byPoincaré’s argument aretruebeyond thelimits ofthetheorem on thetwo-dimensional annulus. Theargument inquestion isthefollowing. 416 Appendix 9:Poincaré’s geometric theorem, itsgeneralizations andapplications BTheconnection between ﬁxed points ofa mapping andcritical points ofthegenerating function Wewilldeﬁne asymplectic diffeomorphism oftheannulus (X,y)—>(X,Y) with thehelp ofthegenerating function Xy+S(X, y),where thefunction S is21:-periodic iny.Forthistobeadilfeomorphism weneed that6X/6x 750. Then dS=(x—X)dy +(Y—y)dX, and, therefore, theﬁxed points ofthediffeomorphism arecritical points of thefunction F(x,y)=S(X(x, y),y).Thisfunction Fcanalways beconstructed bydeﬁning itastheintegral oftheform (x—X)dy +(Y—y)dX. The gradient ofthisfunction isdirected either inside theannulus oroutside on both boundary circles atonce (bythecondition onrotation indifferent directions). Butevery smooth function ontheannulus whose gradient onboth bound- arycircles isdirected inside theannulus (oroutfrom it)hasacritical point (maximum orminimum) inside theannulus. Furthermore, itcanbeshown thatthenumber ofcritical points ofsuch afunction ontheannulus isatleast two. Therefore, wecould assert that ourdiffeomorphism hasatleast two critical points ifwewere surethatevery critical point ofFisaﬁxed point of themapping. Unfortunately, thisistrue only under thecondition that 8X/6x aé0,so that wecanexpress Finterms ofXandy.Thus ourargument isvalid formappings which arenottoodifferent from theidentity. Forexample, itis sufficient thatthederivatives ofthegenerating function Sbelessthan 1. Areﬁnement ofthis argument (with adifferent choice ofgenerating function‘ 1‘)shows thatitiseven sufficient thattheeigenvalues oftheJacobi matrix D(X, Y)/D(x, y)never beequal to--1atanypoint, i.e.,that our mapping never ﬂips thetangent space atanypoint. Unfortunately, allsuch conditions areviolated atsome points formappings farfrom theidentity. Theproof ofPoincaré’s theorem inthegeneral case uses entirely different arguments. Theconnection between ﬁxed points ofmappings andcritical points of generating functions seems tobeadeeper factthan thetheorem onmappings ofatwo-dimensional annulus intoitself. Below, wegiveseveral examples in which thisconnection leads tomeaningful conclusions which aretrueunder some restrictions whose necessity isnotobvious. lll Y'__, d<D=% " ’I dX+dx dl’+dy 417 Appendix 9:Poincaré’s geometric theorem, itsgeneralizations andapplications CSymplectic diﬂeomorphisms ofthetorus Consider asymplectic diffeomorphism ofthetorus which ﬁxes thecenter of gravity (x,y)e(X+f(><, y),y+a(x,y))=(X,Y), where xandymod 21:areangular coordinates onthetorus, “symplectic” means theJacobian D(X, Y)/D(x, y)isequal to1,andthecondition on preserving thecenter ofgravity means thattheaverage values ofthefunctions fandgareequal tozero. Theorem. Such adiﬂeomorphism hasatleast four ﬁxed points, counting multiplicity, andatleast three geometrically diﬂerent ones, atleast under the assumption thattheeigenvalues oftheJacobi matrix arenotequal to—lat anypoint. Theproof isbased onconsideration ofthefunction onthetorus given by theformula <1><><.y)=ifor-xxdr+dw-(Y-nwr+dx). andonthefactthatasmooth function onthetorus hasatleast fourcritical points (counting multiplicity) ofwhich atleast three aregeometrically different. Attempts atproving thistheorem without restrictions ontheeigenvalues meet with difficulties very similar tothose encountered byPoincare inthe theorem about theannulus. Wenote thatthetheorem about theannulus would follow from thetheorem about thetorus ifinthelatter wecould throw outthecondition ontheeigenvalues. Infact.wecanputtogether atorus from twocopies ofourannulus, inserting anarrow connecting annulus along each of thetwoboundary circles. Then wecanextend ourmapping oftheannulus toasymplectic diffeomorphism ofthe torus such that: (l)oneach ofthetwolarge annuli thediffeomorphism coincides with the original, (2)oneach oftheconnecting annuli thediffeomorphism hasnoﬁxed points. and(3) thecenter ofgravity remains ﬁxed. Theconstruction ofsuch adiffeomorphism ofthe torus usestheproperty thattheboundary circles rotate indifferent directions. Oneach connecting annulus allpoints aretranslated inthe same direction asonboth circles bounding theconnecting annulus. Since thetranslations on theconnecting annuli areinopposite directions, thesizeofthetranslations canbechosen to ensure preservation ofthecenter ofgravity. Now outoffourﬁxed points onthetorus, twomust lieintheoriginal annulus. andweobtain thetheorem onannuli from thetheorem ontori. The theorem ontori formulated above can begeneralized toother symplectic manifolds, both two-dimensional and many-dimensional. To formulate these generalizations, wemust ﬁrstreformulate thecondition of preservation ofthecenter ofgravity. 418 Appendix 9:Poincare’s geometric theorem, itsgeneralizations andapplications Letg:M—>Mbeasymplectic diffeomorphism. Wesaythatgishomolo- gous totheidentity ifitcanbeconnected totheidentity diffeomorphism byasmooth curve g,consisting ofsymplectic diffeomorphisms such that theﬁeld ofvelocities g,ateach moment oftime thasasingle-valued hamil- tonian function. Itcanbeshown thatthesymplectic diffeomorphisms homo- logous totheidentity form thecommutator subgroup oftheconnected component oftheidentity inthegroup ofallsymplectic diffeomorphisms of themanifold. Inthecase when ourmanifold isthetwo-dimensional torus, thesym- plectic diffeomorphisms homologous totheidentity areexactly those which preserve thecenter ofgravity. Thus wecome tothefollowing generalization ofPoincaré’s theorem. Theorem. Every symplectic dijfeomorphism ofacompact symplectic manifold, homologous totheidentity, hasatleast asmany ﬁxed points asasmooth function onthismanifold hascritical points (atleast ifthisdiffeomorphism isnottoofarfrom theidentit y).112 Wenote that thecondition ofthemapping being homologous tothe identity isessential, asweseealready from theexample ofatranslation on thetorus, which hasnoﬁxed points atall. Astothelastrestriction (that thediffeomorphism benottoofarfrom the identity), itisnotclear whether itisessential.‘ 12”Inthecase that ourmanifold isthetwo-dimensional torus, itissufficient thatnone oftheeigenvalues ofthe Jacobi matrix ofthediffeomorphism (inanyglobal symplectic coordinate system onR2") beequal tominus one. Arestriction ofthissortmay benecessary inhigher-dimensional problems. Itisnotim- possible that Poincaré's theorem ISduetoanessentially two-dimensional effect, asisthe following theorem ofA. I.Snirel’man andN.A.Nikishin: Every area-preserving diffeomorphism ofthetwo-dimensional sphere toitself hasatleast twogeometrically different ﬁxed points. Theproof ofthistheorem isbased onthefactthattheindex ofthegradient vector ﬁeld ofasmooth function oftwovariables atanisolated critical point cannot begreater than l (although itcanbeequal to1.0,—l,-2,~3,...),andthesumoftheindices ofalltheﬁxed points ofanorientation-preserving diffeomorphism ofthetwo-dimensional sphere toitself isequal to2.Ontheother hand, theindex ofthe gradient ofasmooth function ofalarge number ofvariables atacritical point cantakeanyinteger value. DIntersections oflagrangian manifolds Poincaré’s argument canbegiven aslightly different form ifonevery radius oftheannulus weconsider thepoints shifted only radially. There are such points onevery radius, since theboundary circles oftheannulus turn “Z[For aproof, seeV.Arnold, Surlesproprietes topologiques desapplications globalement canoniques delamécanique classique, C.R.Acad. Sci.Paris, 1965 andA.Weinstein, Symplectic manifolds andtheir lagrangian submanifolds, Advances inMath. 6(1971) 329-346.] “Z”[Recently, Conley andZehnder, followed byothers, have proved thetheorem fortori, surfaces, andother manifolds, without therestriction ofcloseness totheidentity.] 419 Appendix 9:Poincaré’s geometric theorem, itsgeneralizations andapplications indifferent directions. Assume thatwecanmake asmooth curve ofradially shifting points, separating theinterior andexterior circles oftheannulus. Then theimage ofthiscurve under ourmapping must intersect thecurve (since theregions into which thecurve divides theannulus arecarried to regions ofequal area). Ifthiscurve anditsimage each intersect each radius once, then thepoints ofintersection ofthecurve with itsimage areobviously ﬁxed points ofthe mapping. Part ofthisargument canbecarried outinhigher dimensions, andthis gives useful results about periodic solutions ofproblems indynamics. The role oftheannulus inthemany-dimensional case isplayed bythephase space: thedirect product ofaregion ineuclidean space with atorus ofthe same dimension (theannulus istheproduct ofaninterval with thecircle). Asymplectic structure onthephase space isdeﬁned intheusual way, i.e.,ithas theform Q=Zdx,/\dy,,,where thex,,areaction variables andykareangle variables. Itisnotdifficult toexplain which symplectic diffeomorphisms ofour phase space arehomologous totheidentity. Namely, asymplectic diffeo- morphism Aishomologous totheidentity ifitcanbeobtained from the identity byacontinuous deformation andif 4ixdy= 4;xdy Y A‘! foranyclosed contour y(notnecessarily homologous tozero). Thecondition thatthetransformation behomologous totheidentity prohibits systematic shifts along thex-direction (“evolution oftheaction variables”), butpermits shifts along thetori. Weconsider oneofthen-dimensional torix=c=const andapply to itoursymplectic diffeomorphism homologous totheidentity. Itturns out that theoriginal torus intersects itsimage inatleast 2"points (counting multiplicities), ofwhich atleast n+1aregeometrically different, atleast under theassumption that theimage torus hasanequation oftheform x=f(y),where fissmooth. Forn=1,thisassertion means that each oftheconcentric circles con- stituting theannulus intersects itsimage inatleast twopoints. This also follows from thepreservation ofarea, sothattheassumption thattheimage hasequation x=f(y) isnotnecessary. Whether ornotthisassumption isnecessary inhigher dimensions isnot known. Ifwemake thisassumption, theproof proceeds inthefollowing way. Wenote that theoriginal torus, isalagrangian submanifold ofphase space. Ourdiffeomorphism issymplectic, sotheimage torus isalsolagrang- ian.Therefore, the1-form (x—c)dy onitisclosed. Furthermore, thisform onthetorus isthetotal differential ofsome single-valued smooth function F, since ourdiffeomorphism ishomologous totheidentity, andtherefore for 420 Appendix 9:Poincare’s geometric theorem, itsgeneralizations andapplications anyclosed contour ywehave ff(x—c)dy= §xdy—tl;cdy= §xdy—— fl;cdy Av Ar Av r Ar =c§idy——c§; dy=0. ‘H Av Wenote thatpoints ofintersection ofthetorus with itsimage arecritical points ofthefunction F(since atthem dF=(x—c)dy =O). From thecondition ofsingle-valued projection oftheimage torus (i.e., from thefactthat theimage torus hasequation x=f(y)) itfollows that, conversely, allcritical points ofthefunction Farepoints ofintersection of ourtori.Infact,under these conditions ycanbetaken forlocal coordinates onthetorus, andtherefore thefactthatdFiszero forallvectors tangent to theimage torus implies x=c. Asmooth function onann-dimensional torus hasatleast 2"critical points, counting multiplicities, ofwhich atleast n+1aregeometrically different (cf.,forexample, Milnor, Morse Theory, Princeton University Press, 1967). Therefore, ourtoriintersect inatleast 2"points (counting multiplicities), andthere areatleast n+lgeometrically different points ofintersection. Exactly thesame argument shows that anylagrangian torus intersects itsimage inatleast 2"points (ofwhich atleast n+1aregeometrically different), under theassumption that both theoriginal torus anditsimage project single-valued onto they-space, i.e.,aregiven byequations y=f(x) andx=g(y), respectively. Besides, thisstatement reduces totheprevious onebythecanonical transformation (x,y)—>(x-f(y),y). EApplications todetermining ﬁxed points andperiodic solutions Wenow consider asymplectic transformation, homologous totheidentity, ofthespecial form which arises inintegrable problems indynamics, i.e.,of theform A0(x, y)=(x,y+w(x)), where to= Here xeR"istheaction variable andymod 21:eT"istheangular coordin- ate. Weassume thatonthetorus x=x0allthefrequencies arecommensur- able: l<- ..vs,-(x0) =ﬁ'21:with integers k,-,N;o)(X0) 9'50, andthatthenondegeneracy condition det62 96OﬁxX0 issatisﬁed. 421 Appendix 9:Poincaré’s geometric theorem, itsgeneralizations andapplications Theorem. Every symplectic dijfeomorphism Ahomologous totheidentity and sufﬁciently close toA0has, near thetorus x=xo,atleast 2"periodic points 5ofperiod N(such thatANC =6),counting multiplicity. Theproof could bereduced toinvestigating theintersection oftwo lagrangian submanifolds ofa4n-dimensional space (iR"><T"xill"xT")with Q=dx/\dy—a'X/\dY,oneofwhich isthediagonal (X=x,Y=_i')andtheother thegraph ofthemapping A”. However, itiseasier todirectly construct asuitable function onthetorus. Infact.themap- ping Aghastheform Fat(X,j»)—)(X,y+1(x)), where 1(x0) :0,det ¢0. Bytheimplicit function theorem, themapping A"has,near thetorus x=x0.atorus which is displaced only radially ((x,y)—>(X.Y))andisgiven byanequation oftheform x=f(y): itsimage isalsogiven byanequation x=g(y)ofthe same form. Inthisnotation, X(f(_\')_ y)= eh"),Y(.f(_v). y)=)1 Since Aishomologous totheidentity, itfollows thatA"hasasingle-valued global generating function ofthe form Xy+S(X, y),where Shasperiod 21:inthevariable y. Thefunction F(y) =S(X(f(_v). y),3')hasatleast 2"critical points y,onthetorus. Allthe points it=(_l_U'tl- .11.)areﬁxed points forA“.Infact. (IF=(.\'~X)tl_t'+(Y—_\')dX=(x—X)dy=(_/'(_\‘l —g(y)) dy. Therefore. since rlF|_,.k =0,itfollows thatf'(y,) =g(t,), i.e.,ANQ, =tfk,aswastobeshown. Weturn now toclosed orbits ofconservative systems. Using theterm- inology ofAppendix 8,wecanformulate theresult asfollows. Corollary. Upon disintegration ofann-dimensional torus, entirely ﬁlled upby closed trajectories ofanisoenergetically nondegenerate system, atleast 2"" closed trajectories oftheperturbed problem areformed (counting multiplicities), among which atleast naregeometrically distinct, atleast ifthe perturbation issufficiently small. Theproof isreduced tothepreceding theorem with thehelpofa(2n—2)- dimensional surface ofsection. Wemust ﬁrstchoose angular coordinates y such that theclosed trajectories oftheunperturbed problem onthetorus aregiven bytheequations y,= =y,,=0,andthen deﬁne asurface of section byy,=0. Inthecaseoftwodegrees offreedom wecanapply Poincaré’s theorem to theannuli formed byintersecting invariant tori with atwo-dimensional intersecting surface. Weobtain thefollowing result: Inthegapbetween twotwo-dimensional invariant toriofasystem with twodegrees offreedom there arealways atleast twoclosed phase trajectories, iftheratio ofthefrequencies ofconditionally-periodic motions onthese tori aredifferent. Inthiswayweobtain many periodic solutions inallproblems with two 422 Appendix 9;Poincaré‘s geometric theorem, itsgeneralizations andapplications degrees offreedom, where invariant toriarefound (forexample, inthebound- edcircular three-body problem, intheproblem ofclosed geodesics, etc.). There iseven aconjecture thatinhamiltonian systems of“general form ”with compact phase spaces, theclosed phase curves form adense set.“3 How- ever, ifthisistrue, theclosedness ofmost ofthese curves haslittle importance since their periods areextremely large. Asanexample ofapplying Poincaré’s methods tosystems with more than twodegrees offreedom, wehave atheorem ofBirkhoff about theexistence of inﬁnitely many periodic solutions close toagiven linearly stable periodic solution ofgeneral form (orabout theexistence ofinﬁnitely many periodic points inaneighborhood ofaﬁxed point ofalinearly stable nondegenerate symplectic mapping ofaspace toitself). Intheproof, themapping isﬁrst approximated byitsnormal form, andthen theconnection between ﬁxed points ofamapping andcritical points ofthegenerating function isused. Knowing periodic solutions allows us,among other things, toprove the nonexistence ofﬁrst integrals (other than theclassical ones) inmany problems indynamics. Assume, forexample, that onsome level manifold ofknown integrals wediscover aperiodic trajectory which isunstable. Itsseparatrices, ingeneral, form acomplicated network, which weconsidered inAppendix 7. Ifthisphenomenon ofsplitting ofseparatrices isdiscovered, andifwecan show thattheseparatrices arenotcontained inanymanifold oflower dimen- sionthan thelevel manifold weareconsidering, then wecanbesurethatthe system hasnonewﬁrstintegrals. Thecomplicated behavior ofphase curves, which obstructs theexistence ofﬁrstintegrals, canoften bedetected without thehelp ofperiodic solutions byonesimple glance atthepicture, obtained byacomputer, formed bythe intersection ofthephase curves with thesurface ofsection. FInvariance ofgenerating functions Wehave already noted thediscouraging noninvariance ofgenerating functions with respect tothechoice ofacanonical coordinate system ona symplectic manifold. Ontheother hand, werepeatedly used theconnection between ﬁxed points ofamapping and critical points ofthegenerating function. Itturns outthat, although generally thegenerating function isnotin- variantly associated tothemapping, near aﬁxed point there isaninvariant connection. More precisely, suppose wearegiven asymplectic diffeo- morphism ﬁxing some point. Inaneighborhood ofthispoint, wedeﬁne a “generating function ” X—x Y—y _j k k k k (D_2_l2ldX,,+dx,, dl§,+dy,,l '13Aproof ofthisdensity intheC‘topology hasbeen announced byC.Pugh andC.Robinson. [Editors note] 423 Appendix 9:Poincare’s geometric theorem, itsgeneralizations andapplications with thehelp ofsome symplectic coordinate system (x,y).“4 Using another symplectic coordinate system (x’,y’),weconstruct agenerating function CD’ inthesame way. Theorem. Ifthelinearization ofthesymplectic diﬂeomorphism attheﬁxed point hasnoeigenvalues equal to—l,thenthefunctions (Dand<1)’areequiva- lentinaneighborhood oftheﬁxed point, inthesense thatthere isadiﬂeo- morphism g(ingeneral notsymplectic) such that (D(:) =(D’(g(z)) +const. Fortheproof seethearticle: A.Weinstein, Theinvariance ofPoincaré’s generating function forcanonical transformations, Inventiones Mathe- maticae, 16,No.3(1972), 202-214. Itshould benoted that twodiffeomorphisms with generating functions which areequivalent inaneighborhood ofaﬁxed point arenotnecessarily equivalent intheclass ofsymplectic diffeomorphisms (forexample, rotation and rotation through anangle which depends ontheradius, with non- degenerate quadratic parts ofthegenerating function atzero). Since theﬁrstedition ofthisbook hadappeared in1974, thecontent of thisAppendix hasgrown into anew branch ofmathematics: symplectic topology. Todescribe thisdevelopment (triggered bytheconjectures inthis Appendix, which stillremain, forgeneral manifolds, neither proved, nor disproved) onewould need abook longer than thepresent one. Theinterested reader might follow thisdevelopment using the(incomplete) bibliography onpages 503-509. 1“The increase ofthisfunction along anyarcisequal totheintegral oftheform deﬁning the symplectic structure over theband formed bytherectilinear intervals connecting each 3*tint with itsimage. Therefore. thefunction (Disassociated tothemapping invariantly with respect tolinear canonical changes ofcoordinates. 424 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids depending onparameters Several times inthiscourse wehave encountered families ofellipsoids in euclidean space. Forexample, instudying thedependence onparameters of characteristic frequencies ofsmall oscillations, weencountered equipotential surfaces which were ellipsoids ineuclidean space, depending upon thedegree ofrigidity ofthesystem, (themetric ofthespace wasdeﬁned bythekinetic energy). Another example wastheellipsoid ofinertia ofarigid body (the parameter here wastheshape oftherigid body anditsdistribution ofmass). Here wewillconsider thegeneral problem ofdescribing thevalues ofthe parameter forwhich thespectrum ofeigenvalues degenerates, i.e.,thecor- responding ellipsoid becomes anellipsoid ofrevolution. Wenote that the eigenvalues ofaquadratic form oneuclidean space (orthelengths oftheaxes ofanellipsoid) change continuously under continuous changes ofthe parameters ofasystem (the coefficients oftheform). Itseems natural to expect thatinasystem depending ononeparameter, under changes ofthe parameter, atcertain moments oneoftheeigenvalues would collide with another, sothatforthese values oftheparameter thesystem would have a multiple spectrum. Suppose, forexample, thatwewant tomake theellipsoid ofinertia ofa rigid body intoanellipsoid ofrevolution bymovement ofanadjustable mass along anarcrigidly attached tothebody sothatthere isoneparameter at ourdisposal. Thethree major axesa,b,andcwillbecontinuous functions of thisparameter, andatﬁrstglance itseems thatforasuitable value ofthe parameter (p)wecanachieve equality oftwooftheaxes, saya(p)=b(p). It turns out,however, thatthisisnotso,andthatgenerally weneed toattach atleast twoadjustable masses tomake theellipsoid ofinertia anellipsoid of revolution. Ingeneral, amultiple spectrum intypical families ofquadratic forms is observed only fortwoormore parameters, while inone-parameter families ofgeneral form thespectrum issimple forallvalues oftheparameter. Under achange ofparameter inthetypical one-parameter family, theeigenvalues canapproach closely, butwhen they aresufficiently close, itisasifthey begin torepel oneanother. Theeigenvalues again diverge, disappointing the person who hoped, bychanging theparameter, toachieve amultiple spec- trum. Inthisappendix weconsider thereasons forthisseemingly strange be- havior oftheeigenvalues, andwediscuss brieﬂy analogous questions for systems with various groups ofsymmetries. AThemanifold ofellipsoids ofrevolution Consider thesetofallpossible quadratic forms onthen-dimensional eucli- dean space R".This sethasitself anatural structure ofavector space of dimension n(n+l)/2.Forexample, thequadratic forms ontheplane form a three-dimensional space (aform Ax’ +2Bxy +Cyzhasascoordinates the three numbers A,B,andC). 425 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids The positive-deﬁnite forms form anopen region inthis space ofall quadratic forms (forexample, inthecaseoftheplane thisistheinside ofone nappe ofthecone B2=ACofdegenerate forms). Every ellipsoid centered attheorigin deﬁnes apositive-deﬁnite quad- ratic form, forwhich itisthelevel setof1;conversely, thesetoflevel 1ofany positive-deﬁnite quadratic form isanellipsoid. Wecantherefore identify the setsofpositive-deﬁnite quadratic forms andellipsoids centered attheorigin. Inthiswaywegivethesetofellipsoids with center 0inIR"thestructure ofa smooth manifold ofdimension n(n+1)/2(this manifold iscovered byone chart: aregion inthespace ofquadratic forms). Now consider thesetofallellipsoids ofrevolution. Weclaim thatthisset hascodimension 2inthespace under consideration, i.e.,itisgiven bytwo independent equations, rather than oneasitwould seem atﬁrstglance. More precisely, wehave Theorem 1.Thesetofellipsoids ofrevolution isaﬁnite union ofsmooth sub- manifolds ofcodimension 2andhigher inthemanifold ofall ellipsoids. The codimension ofamanifold isthedifference between thedimension oftheambient space andthedimension ofthesubmanifold. PROOF. Weﬁrstconsider anellipsoid inn-dimensional space which hastwo equal axes, andwhose other axes aredistinct. Such anellipsoid isdeﬁned by thedirections ofthedistinct axes, which gives (n_1)+(n_2)+...+2=(lji)2fL__2l different parameters, andalso bythemagnitudes oftheaxes, which gives n—1parameters. Thus thetotal number ofparameters is n2—n—2+2n—2 2 , which istwolessthan thedimension ofthespace ofallellipsoids (which is n(n+1)/2). This count ofparameters also shows that thesetofellipsoids with exactly twoequal axes isamanifold. Asforellipsoids with alarger number ofequal axes, itisclear thatthey form asetofeven smaller dimension. Arigorous proof follows from the following lemma. Lemma. Thesetofall ellipsoids with v2double, v3triple, v4four-fold axes, etc. isasmooth submanifold ofthe manifold ofall ellipsoids, withcodimension 2V2 + 5V3 + 9V4 + Z _ "l" 2)V,'. 426 Appendix l0:Multiplicities ofcharacteristic frequencies, andellipsoids Theproof ofthistheorem reduces tothesame kind ofparameter count as inthespecial case analyzed above (which corresponds tov2=1,v3= ----=0).Thereader caneasily carry outthiscalculation, noting ﬁrst thatthedimension ofthemanifold ofallk-dimensional subspaces inann- dimensional vector space isequal tok(n—k)(since ak-dimensional plane in general position inann-dimensional space canbethought ofasthegraph of amapping from ak-dimensional space toan(n—k)-dimensional space, and such amapping isgiven byarectangular k><(n—k)matrix). EXAMPLE. Consider thecase n=2,i.e.,ellipses intheplane. Anellipse is determined bythree parameters (e.g., thelengths ofthetwoaxes andthe angle giving thedirection ofoneofthem). Thus themanifold ofellipses inthe plane isthree-dimensional, asitmust bebyourformula. Acircle, however, isdetermined byoneparameter (theradius). Thus the manifold ofcircles inthespace ofellipses isalineinathree-dimensional space, andnotasurface asitwould seem atﬁrstglance. This "paradox" becomes, perhaps, clearer from thefollowing calculation. Thequadratic forms Ax: +2Bxy +(“x2with differenteigenvalues formasubmanifold ofthethree-dimensional space with coordinates A,B,andC,given byoneequation it,—/ll=O,where ,l,_2(A, B,C) aretheeigenvalues. However, theleft-hand sideofthisequation isthesum oftwosquares, asisclear from theformula forthediscriminant ofthecharacteristic equation: A=(A +C)’-4(AC-BZ)=(A -C)2+431. Thus thesingle equation A=0determines alineinthethree-dimensional space ofquadratic forms (A=C,B=O),andnotasurface. Asimple consequence ofthefactthatthemanifold ofellipsoids ofrevolu- tionhascodimension 2isthatthismanifold does notdivide thespace ofall ellipsoids (and themanifold ofquadratic forms with amultiple spectrum does notdivide thespace ofquadratic forms), asalinedoes notdivide athree- dimensional space. Therefore, wecanassert notonly thatinanellipsoid in “general position” alltheaxes share different lengths, butalsothatanytwo suchellipsoids canbeconnected byasmooth curve inthespace ofellipsoids con- sisting entirely ofellipsoids withaxes ofdijferent lengths. Furthermore, iftwo ellipsoids ingeneral position areconnected byasmooth curve inthespace ofellipsoids which contains apoint which isanellipsoid ofrevolution, then byanarbitrarily small displacement ofthecurve wecanremove itfrom the setofellipsoids ofrevolution, sothatonthenewcurve allthepoints willbe ellipsoids without multiple axes. One consequence ofwhat wehave saidisasimple proof ofthetheorem that characteristic frequencies increase when therigidity ofasystem is increased. Thederivative ofanon-multiple eigenvalue ofaquadratic form with respect toaparameter isdetermined bythederivative ofthequadratic form inthecorresponding characteristic direction. Iftherigidity isincreased, thepotential energy increases inevery direction, including thecharacteristic 427 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids directions. Thus thecharacteristic frequencies also increase. Hence we have proved thetheorem onthegrowth offrequencies inthecase when it ispossible togofrom theoriginal system toamore rigid system, avoiding multiple spectra. The proof inthepresence ofmultiple spectrum isnow obtained byapassage tothelimit, based onthefactthattheinterior ofthe path from theoriginal system tothemore rigid system canberemoved by anarbitrarily small perturbation from thesetofsystems with multiple spectra. Insummary, wecansaythatatypical one-parameter family ofellipsoids (orquadratic forms ineuclidean space) does notcontain ellipsoids ofrevolu- tion (quadratic forms with multiple spectra). Applying thistoanellipsoid ofinertia weobtain theconclusion above about thenecessity fortwoadjust- able masses. Weturn now totwo-parameter systems. Itfollows from ourcalculations that, inatypical two-parameter system, ellipsoids ofrevolution areen- countered only atisolated points oftheparameter plane. Consider, forexample, aconvex surface inthree-dimensional euclidean space. Thesecond fundamental form ofthesurface determines anellipse inthetangent space atevery point. Therefore, wehave atwo-parameter family ofellipses (which canbetranslated tooneplane bychoosing alocal coordinate system near apoint onthesurface). Wecome totheconclusion that, atevery point ofthe surface except atcertain isolated points, theellipse hasaxesofdifferent lengths. Therefore, onsurfaces ofgeneral form, there aretwoorthogonal ﬁelds ofdirections (the major andminor axes oftheellipses) with isolated singular points. Indifferential geometry these directions arecalled thedirections ofprincipal curvature, andthese singular points are called umbilical points. Forexample, onthesurface ofanellipsoid there arefour umbilical points: theylieontheellipse containing themajor andminor axes, andtwoofthem areclearly visible inthepicture ofthe geodesics onanellipsoid (cf.Figure 207). Inexactly thesame way, inatypical three-parameter family, ellipsoids of revolution areencountered only oncertain lines inthethree-dimensional parameter space. Forexample, ifatevery point ofthree-dimensional eucli- dean space, wearegiven anellipsoid (i.e., asymmetric two-index tensor), then thesingularities oftheﬁelds ofprincipal axes willbe,ingeneral, on certain lines (where twoofthethree ﬁelds ofdirections have discontinuities). These lines, liketheumbilical points inthepreceding example, areofseveral different types. Their classiﬁcation (fortypical ﬁelds ofellipsoids) canbe obtained from theclassiﬁcation ofsingularities oflagrangian projections given inAppendix 12. Inatypical four-parameter family, ellipsoids ofrevolution occur ontwo- dimensional surfaces inthespace ofparameters. These surfaces have no singularities other than transverse intersections atisolated points ofthe parameter space; these values oftheparameters correspond toellipsoids with two(different) pairs ofequal axes. Triple axesappear ﬁrstforﬁveparameters, atisolated points oftheparam- eterspace. Thevalues oftheparameters corresponding toellipsoids with a double axis form athree-dimensional manifold intheﬁve-dimensional 428 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids parameter space with twotypes ofsingularities: transversal intersections of twobranches along some curve andconic singularities atisolated points (not lying onthiscurve), i.e.,atpoints oftheparameter space corresponding to ellipsoids with three equal axes. These conic singularities have thefollowing structure: byintersecting thethree-dimensional manifold ofellipsoids of revolution with afour-dimensional sphere ofsmall radius with center atthe singular point, weobtain twocopies oftheprojective plane. Theresulting em- beddings oftheprojective plane inthefour-dimensional sphere arediffeo- morphic totheembedding given bytheﬁvespherical harmonies ofdegree two onthetwo-dimensional sphere (ﬁvelinear combinations ofthefunctions xixJ-, orthonormal inthespace offunctions onthesphere xf+x§+x§=1, orthogonal totheidentity, giveaneven mapping ofS2intoS4and,therefore, anembedding RP2 —>S‘). Itremains todescribe thebehavior oftheeigenvalues ofaquadratic form inatypical two-parameter family astheparameter approaches asingular point where thetwoeigenvalues coincide. Alittle calculation shows thatthe graph ofthepairofeigenvalues weareconsidering has,over theplane of parameters near thesingular point, theform ofatwo-sheeted cone, whose vertex corresponds tothesingular point, andeach ofitsnappes tooneofthe eigenvalues (Figure 243). \/ /\ Figure 243 Characteristic frequencies ofone- andtwo-parameter families ofoscil- lating systems ofgeneral form Atypical one-dimensional subfamily ofourtwo-dimensional family has theform ofacurve intheplane ofparameters which does notpass through anysingular points. Every one-parameter family which contains asingular point canberemoved from itbyasmall perturbation; theresulting one- parameter family willbeacurve inthespace ofparameters passing near the singular point. Thegraph oftheeigenvalues over acurve ontheplane of parameters passing near asingular point consists ofthose points ofthecone which project onto thiscurve. Therefore, thisgraph near thesingular point is close toahyperbola, resembling apairofintersecting straight lines (apairof straight lines would beobtained ifourone-parameter family passed through thesingular point). 429 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids This discussion ofeigenvalues oftwo-parameter systems ofquadratic forms explains thestrange behavior ofcharacteristic frequencies when a single parameter isvaried: ingeneral (except forcompletely singular cases), when asingle parameter isvaried thecharacteristic frequencies canapproach oneanother butcannot collide; after approaching, they must again gooffin different directions. BApplication tothestudy ofoscillations ofcontinuous media Thegeneral argument above hasnumerous applications inthestudy ofthe dependence onparameters ofthecharacteristic frequencies ofvarious mechanical systems with ﬁnitely many degrees offreedom; however, themost interesting applications may betosystems with inﬁnitely many degrees of freedom, describing oscillations ofcontinuous media. These applications are based onthefactthatthecodimensions ofmanifolds ofellipsoids with given multiplicities ofaxesaredetermined bythese multiplicities anddonotdepend onthedimension ofthespace. Forexample, thecodimension ofthesetofellipsoids ofrevolution inthe manifold ofallellipsoids isequal totwoinaspace ofanydimension; there- fore, itisnatural toassume that intheinﬁnite “manifold” ofellipsoids in inﬁnite-dimensional hilbert space, thesetofellipsoids ofrevolution has codimension 2(and, inparticular, thespace ofellipsoids without multiple axes isconnected). Ofcourse, arguments ofthiskind need rigorous justiﬁcation. Wewillnot, however, occupy ourselves with this,butwewillseewhat conclusions follow from theargument above ifweapply ittotheproblem ofoscillations in continuous media. Thekinetic energy ofacontinuous medium ﬁlling acompact region Dis expressed interms ofthedeviation uofapoint xfrom equilibrium bythe formula T=%ju,2dx. D Fordeﬁniteness, wecantakethemedium tobeamembrane (inthiscasethe region Distwo-dimensional, and thedeviation uone-dimensional). The kinetic energy deﬁnes aeuclidean structure ontheconﬁguration space ofthe problem (i.e.,inthespace offunctions u).Thepotential energy isgiven bythe Dirichlet integral U=%I(Vu)2 dx 1) (from themathematical point ofview these data constitute thedeﬁnition of themembrane). The squares ofthecharacteristic frequencies ofthemembrane arcthe eigenvalues ofthequadratic form Uontheconﬁguration space, whose metric 430 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids isdeﬁned using thekinetic energy. Weassume thatatypical membrane cor- responds toatypical quadratic form (this assumption means transversality of themanifold ofquadratic forms corresponding todifferent membranes to themanifold offorms with multiple eigenvalues). Ifwebelieve inthisprop- ertyofgeneral position, wecome tothefollowing conclusions. 1.Formembranes ingeneral position, allthecharacteristic frequencies are different. Wecangofrom onemembrane ingeneral position toanother byacontinuous path consisting entirely ofmembranes with simple spectra. Furthermore, atypical path connecting anytwomembranes does notcontain even one membrane with amultiple spectrum (except, possibly, theends ofthepath). 2.Byvarying twoparameters ofthemembrane wecanmake twocharacter- isticfrequencies coincide; toobtain atriple frequency, wemust have at ourdisposal ﬁveindependent parameters; forafour-fold frequency we need tenparameters, etc. 3.If,bystarting from amembrane with asimple spectrum andcontinuously deforming it,wepass toanother membrane with asimple spectrum along anypath ingeneral position, then asaresult, thek-thlargest characteristic frequency ofthesecond membrane isalways obtained independently of thepath ofdeformation from thek-thlargest characteristic frequency of theoriginal membrane; continuations ofcharacteristic functions, however, dogenerally depend onthepath ofdeformation (i.e.,bychanging thepath, thesignoftheresulting characteristic function canbechanged). Inparticular, ifbystarting from amembrane with asimple spectrum anddeforming itwedescribe aclosed path inthespace ofmembranes and return totheoriginal membrane, bypassing thesetofmembranes with multiple spectra (which hascodimension 2),then thek-th characteristic frequency returns toitsoriginal value, while thek-thcharacteristic func- tion may change sign. [Editor’s note: Conclusions likethishave been proven byK.Uhlenbeck (Amer. J.Math. 98(1976), 1059-1078)] CTheeﬂect ofsymmetries onthemultiplicity ofthespectrum Amultiple spectrum istheexception insystems ofgeneral form, butit isnotremovable under small perturbations incases when thegiven system issymmetric andthedeformations preserve thesymmetry. Consider, forexample, asystem ofthree identical masses atthevertices ofanequilateral triangle, connected tooneanother andtothecenter ofthe triangle byidentical springs, andcapable ofmoving intheplane ofthe triangle. The system hasrotational symmetry oforder 3.Therefore, there isalinear operator gacting ontheconﬁguration space (which hasdimension 6),whose third power isequal to1and which leaves invariant both the euclidean structure oftheconﬁguration space andtheellipsoid inthecon- ﬁguration space giving thepotential energy. ltfollows thatthisellipsoid must beanellipsoid ofrevolution. Ifwelet gbetheindicated operator ontheconﬁguration space andfavector onthe 431 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids major axisoftheellipsoid, then theaxisinthedirection gﬁisalsoamajor axis(since therotation gtakes theellipsoid toitself). There aretwopossibilities forthevector gtfzeither gﬁ=5,orthevectors tfandgﬁarelinearly independent. Inthesecond case, theplane spanned by thevectors 5andgéconsists entirely ofmajor axes. Therefore, theeigenvalues corresponding tothese axes areatleast double. Thespace spanned bythe three vectors -5,gﬁ,andgztfisinvariant under g.Itiseither twodimensional (inwhich casegactsbya120° rotation) orthree dimensional (inwhich case gactsbythesame rotation around 5+gi+925asanaxis). Inthelatter case, wemay choose thedirection ofthissum foroneoftheprincipal axes ofthe ellipsoid, with thetwoother principal axes inthethree-dimensional space perpendicular toit.Itistherefore possible tochoose theprincipal axesforan ellipsoid which isinvariant under anorthogonal transformation oforder three (inaspace ofanynumber ofvariables), sothateach axisiseither ﬁxed under thetransformation orisrotated by120° inaninvariant plane spanned byit andanother axis(orthogonal toit,aswellastoallother axes) ofthesame length. Inwhat follows, weshall assume that theaxes ofellipsoids andthe directions ofthecorresponding characteristic oscillations have been chosen inthemanner justdescribed. Our argument shows that characteristic oscillations ofasystem with third-order rotational symmetry canbeoftwotypes: those invariant under rotation by120° (gé=5)andthose passing under such arotation toinde- pendent characteristic oscillations with thesame frequency (giandCindepen- dent). Inthesecond case, there actually arise three forms ofcharacteristic oscillations with thesame frequency (5,gﬁ,andgzvf), butonly twoofthem are independent: €+9€+Q’€=0 since thesumofthree vectors ofequal length ontheplane forming angles of 120° isequal tozero. Thenumber ofcharacteristic oscillations ofoursystem isgenerally equal to6.Toﬁndouthow many ofthem areoftheﬁrst(symmetric) andsecond (nonsymmetric) type, wecanusethefollowing argument. Consider the limiting case, when each ofthemasses oscillates independently from the others. Inthiscase, wecanchoose anorthonormal basis oftheconﬁgura- tionspace consisting ofsixcharacteristic oscillations, twoforeach point, for which that point moves andtheother twodonot. Wedenote byiiand 11,thecharacteristic vectors corresponding tothei-thpoint with charac- teristic frequencies aandb,respectively, andletxi,y,becoordinates inthe orthonormal basis 5,,11,.Then thepotential energy canbewritten inthe form U=%(¢1’><% +b2y§)- 432 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids Thesymmetry operator gpermutes thecoordinate axes: git=52 962='53 953=61> 9'11: '12 9'12='13 9'13='11- Wecannow represent oursix-dimensional space astheorthogonal direct sumoftwostraight lines andtwotwo-dimensional planes, invariant under thesymmetry operator g.That is,theinvariant lines aredeﬁned bythe directions ofthevectors €1+€2+é3 andm+r12+n3, andtheinvariant planes aretheir orthogonal complements inthespaces spanned bythevectors 6;and11,,respectively. Theﬁrststraight lineisthe direction ofasymmetric characteristic oscillation with frequency a,andthe second thedirection ofonewith frequency b.Inexactly thesame way, every vector intheﬁrstplane isadirection ofcharacteristic oscillation with fre- quency awhich, under rotation by120°, goes toanindependent oscillation ofthesame frequency; forallvectors inthesecond plane, theoscillation is alsonotsymmetric, with frequency b. Thus, inthisdegenerate case ofthree independent points, there aretwo independent characteristic oscillations ofsymmetric type, and four un- symmetric, ofwhich thelatter aredivided into twopairs. Ineach pair the oscillations have thesame eigenvalue andareobtained from oneanother by rotation oftheplane ofourpoints by120°. Wenow claim thattheconclusion above holds trueforanylawofinter- action between ourpoints iftheinteraction issymmetric, i.e.,ifthepotential energy ofthesystem ispreserved under rotation oftheplane by120°. Infact, decompose the6-dimensional conﬁguration space intoanortho- gonal sumoftheplane ofinvariant vectors ofgandofitsorthogonal comple- ment. The potential energy willdecompose into asum oftwoquadratic forms——one intwovariables, theother infour. Now consider characteristic oscillations inthetwo-dimensional and four-dimensional conﬁguration spaces, with potential energy described above. Thefour-dimensional space decomposes intotwog-invariant planes, orthogonal inthepotential energy metric. Wehave obtained asystem ofsixcharacteristic oscillations having therequired properties. Thus, inasystem ingeneral form ofthree points intheplane with rotational symmetry oforder 3,there arefour different characteristic frequencies, two ofwhich aresimple andtwodouble. Each ofthesimple characteristic fre- quencies corresponds toasymmetric characteristic oscillation, andeach of thedouble ones tothree characteristic oscillations obtained from oneanother byrotation by120° andsumming tozero (sothat only twoofthem are independent). 433 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids PROBLEM. Classify thecharacteristic oscillations ofasystem withthesymmetries ofan equilateral triangle (allowing notonly rotation by120°, butalso reﬂection through thealtitude ofthe triangle). PROBLEM. Classify thecharacteristic oscillations ofasystem whose group ofsymmetries isthe group of24rotations ofthecube. ANSWER. Theoscillations willbeofﬁve types. Byrotations, from each oscillation onecanobtain systems of8,or6,or4,or2,orlindependent oscillations (inthelastcasetheoscillations are entirely symmetric). Remark. Toclassify oscillations insystems withanygroup ofsymmetries, aspecial apparatus hasbeen developed (theso-called theory ofgroup representations). Cf..forexample. Michael Tinkham, Group Theory andQuantum Mechanics. McGraw-Hill. 1964. DThebehavior offrequencies ofasymmetric system under a variation ofparameters preserving thesymmetry Weassume now thatoursymmetric system depends inageneral wayonsome number ofparameters, andthat thesymmetry isnotdisturbed when the parameters arevaried. Then thecharacteristic frequencies ofvarious multi- plicities willalsodepend ontheparameters, andthequestion arises ofwhen thecharacteristic frequencies will collide. Wewill conﬁne ourselves to formulating aresult forthesimplest case ofsystems with third-order rota- tional symmetry (forrotational symmetry ofanyorder n23,theanswer is thesame). Thedetails canbefound inthefollowing articles: V.I.Arnold, Modes and quasi-modes, Functional Analysis and ItsApplications, 6:2 (1972), 94-101; V.N.Karpushkin, Theasymptotic behavior oftheeigen- values ofsymmetric manifolds andthe“most probable” representations of ﬁnite groups, Moscow Univ. Math. Bull. 29(1974), no.2,136-139. Characteristic oscillations ofanysystem with rotational symmetry of order 3aredivided intotwotypes: symmetric oscillations, andoscillations carried byrotation by120°intoindependent ones. Forageneral system with third-order rotational symmetry (without, inparticular, any additional symmetry) allthecharacteristic frequencies oftheﬁrsttype aresimple, and ofthesecond, double. Inaddition, itturns outthatifasystem depends ina general wayononeparameter andissymmetric forallvalues oftheparam- eter, then under variation oftheparameter, thecharacteristic frequencies of symmetric oscillations donotcollide with oneanother, and thedouble characteristic frequencies ofasymmetric oscillations donotsplit. Inaddition, thedouble characteristic frequencies ofasymmetric oscillations donot collide with oneanother under achange ofparameters. However, thechar- acteristic frequencies ofsymmetric andasymmetric oscillations move under changes ofparameter independently from oneanother, sothatfordiscrete values oftheparameter thecharacteristic frequency ofasymmetric oscilla- tion andthe(double) characteristic frequency ofanasymmetric oscillation cancollide (and pass through oneanother). 434 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids Inorder tomake twocharacteristic frequencies ofsymmetric oscillations collide, wemust vary atleast twoparameters; andtomake twocharacteristic frequencies ofasymmetric oscillations collide wemust vary atleast three. Ingeneral, inthetypical family ofsystems with third-order rotational symmetry, forthecollision ofisimple characteristic frequencies (isymmetric oscillations) andjdouble frequencies (junsymmetric oscillations) tooccur, thenumber ofparameters ofthefamily must beatleast (L-;2(i+L2)+jz_ Weapply thistooscillations ofsymmetric membranes. Here wewill assume thatthemembrane isofgeneral form, admits rotation by120°, and corresponds toanellipsoid ofgeneral form inthespace ofellipsoids ofthe conﬁguration space admitting thetransformation oftheconﬁguration space induced bytherotation ofthemembrane. Theexact formulation ofthisassumption isthat, forallmembranes except asetofinﬁnite codimension, themapping from thespace ofsymmetric membranes intothespace ofsymmetric ellipsoids istransverse toeach ofthemanifolds ofellipsoids with agiven number ofmultiple axes. Ifweagree tothisassumption, wecome tothefollowing conclusions about oscillations ofsymmetric membranes. 1.For membranes ofgeneral form admitting rotation by120°, asymp- totically one-third ofthecharacteristic frequencies (counting them with multiplicities) aresimple. andthecorresponding characteristic oscilla- tions admit rotation by120°. Theremaining characteristic frequencies are double; each double characteristic frequency corresponds tothree eigen- functions whose sum iszero andwhich aretaken tooneanother under rotation by120°. 2.Ingeneral one-parameter families ofsuch symmetric membranes, forisolated values oftheparameters there arecollisions ofasingle fre- quency with adouble frequency, butthere arenocollisions ofsingle frequencies with oneanother orcollisions ofdouble frequencies with one another. 3.Theminimal number ofparameters ofafamily ofmembranes forwhich more complicated collisions ofcharacteristic frequencies arerealized (stably with respect tosmall perturbations preserving thesymmetry) is given bytheformula SIM+ vij, where v,-jisthenumber ofpoints ofcollision ofisingle andjdouble frequencies. 435 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids Inparticular, foratypical small deformation ofacircular membrane preserving rotational symmetry oforder 3,athird oftheeigenvalues (corresponding toeigenfunctions with azimuthal part cos3k<p and sin3k(p) immediately disperse. Under further one-parameter deforma- tionthesimple anddouble characteristic frequencies canpassthrough one another, buttwosimple ortwodouble frequencies cannot collide with one another. EDiscussion Thevalue oftheconcepts ofgeneral position andsymmetry lies,inparticular, inthefactthat they allow ustoobtain some information inthose cases where wecannot ﬁnd anexact solution ofaproblem. Inparticular, for almost nomembranes doweknow theforms ofthecharacteristic oscillations. Nevertheless, from general arguments wecansaysomething, forexample, about themultiplicities ofeigenvalues. The study ofhigh-frequency oscillations ofcontinuous media isvery important inmany ﬁelds (optics, acoustics, etc.), andspecial methods have been developed forapproximate determination oftheform ofcharacter- isticoscillations. One ofthese methods (called themethod ofquasi-classical asymptotics) consists ofseeking anoscillation which islocally close toa simple harmonic wave ofshort length, butwhich changes itsamplitude and thedirection ofitsfront from point topoint. Analysis (which wewillnotgointo here) shows that insome cases we canconstruct approximate solutions, with theindicated properties, ofthe equation foreigenfunctions. They areapproximate solutions inthesense thatthey almost satisfy theequation foreigenfunctions (notinthesense that they areclose torealeigenfunctions). Inparticular, ifthemembrane hastheform ofanequilateral triangle with smoothed andstrongly blunted corners, then wecanconstruct anapproxi- mate solution ofthetype described which differs appreciably from zero only inaneighborhood ofoneofthealtitudes ofthetriangle. (Physicists callthis approximate solution thewave analogue ofabeam moving along thealtitude ofthetriangle; thisbeam isastable‘ '5trajectory onabilliard table having theshape ofourmembrane; c.f.thefollowing appendix onshort wave asymptotics). Itfollows from symmetry andgeneral position arguments that typical membranes with rotational symmetry ofthird order have norealcharacter- isticoscillations ofthetype described. Assume thatoneofthecharacteristic "5Thecondition forlinear stability ofabilliard trajectory hastheform ('1'1”'2_')("i —l)("z —l)>0, where Iisthelength oftheinterval ofthetrajectory andr,andrzaretheradii ofcurvature of thewalls atitsends. 436 Appendix 10:Multiplicities ofcharacteristic frequencies, andellipsoids oscillations ofthemembrane isconcentrated near analtitude (but notnear thecenter ofthemembrane). Then, rotating itby120° and240° weobtain three characteristic oscillations with thesame characteristic frequency. These three oscillations areindependent (this follows from thefactthattheir sum isnotzero). Therefore, thecharacteristic frequency hasmultiplicity 3,which does notoccur intypical systems with third-order rotational symmetry. From thisargument itisclear thatattempting toconstruct rigorous high- frequency asymptotics foreigenfunctions isarather hopeless task; what we canhope todoistoobtain approximate formulas foralmost characteristic oscillations. Such analmost characteristic oscillation candiffer verystrongly from realcharacteristic oscillations, butifwegivethemembrane theinitial condition corresponding toit,then foralongtimetheoscillation willresemble astanding wave (characteristic oscillation). Anexample ofanalmost characteristic oscillation isthemotion ofone oftwoidentical pendulums connected byaveryweak spring. If,attheinitial moment, wesettheﬁrstpendulum inmotion andleave thesecond ﬁxed, then foralong time itwillappear thatonly theﬁrstpendulum isoscillating, and theoscillation willbealmost characteristic. Fortrue characteristic oscilla- tions, both pendulums oscillate with thesame amplitude. Theproblem ofconnecting thegeometry ofamembrane withtheproperties ofits character- isticoscillations hasbeen intensively studied inrecent years bymany authors (including H.Weyl. S.Minakshisundaram andA.Pleijel, A.Selberg, .1.Milnor, M.Kac, I.Singer. H.McKean, M.Berger, Y.Colin deVerdiére, J.Chazarain, J.J.Duistermaat, V.F.Lazutkin, A.I.Schnirelman, andS.A.Molchanov). Tothesimplest question, "Can youhear theshape ofadrum?" theanswer turns outtobe negative: there exist non-isometric riemannian manifolds with thesame spectrum. Onthe other hand, several properties ofamanifold canberecovered from theeigenvalues ofthe laplacian andfrom theproperties ofeigenfunctions (forexample, thecomplete setoflengths ofclosed geodesics canberecovered). 437 Appendix ll:Short wave asymptotics From thepoint ofview ofphysical optics, thedescription ofthepropagation oflight ingeometric optics, using rays(i.e.,Hamilton’s canonical equations) orwave fronts (i.e.,theHamilton-Jacobi equation), isonly anapproximation. According totheideas ofphysical optics, light iselectromagnetic waves, and geometric optics isaﬁrst approximation, agood description of phenomena only when thelength ofthewaves issmall compared tothesize oftheobjects being considered. Amathematical version ofthese physical ideas consists ofasymptotic formulas forsolving thecorresponding differential equations—formulas which give better approximations forhigher-frequency oscillations (i.e.,for shorter waves). These asymptotic formulas canbewritten interms ofrays (i.e.,motions insome hamiltonian dynamical system) orfronts (i.e.,solutions oftheHamilton-Jacobi equation). Similar short wave asymptotics exist forsolutions ofmany equations in mathematical physics, describing allwave processes. Indifferent areas of physics and mathematics they areconnected with different names. For example, inquantum mechanics, short wave asymptotics arecalled quasi- classical approximations; theyaredetermined bytheso-called WKBJ method (Wentzel, Kramers, Brillouin, Jeffreys), although these approximations were used much earlier byLiouville, Green, Stokes, Rayleigh andothers. The construction ofshort wave asymptotics isbased ontheidea that, locally, aseries ofalmost strictly sinusoidal waves isobserved ateach place, although theamplitudes ofthese waves andthedirections oftheir fronts change slowly from point topoint. Formal substitution ofafunction ofthis form into thepartial differential equations describing thewave process reduces us(inaﬁrst approximation forwaves ofsmall length) tothe Hamilton-Jacobi equation forwave fronts. Thehigher-order approximations allow ustodetermine aswellthedependence oftheamplitude ofoscillation onthepoint. Ofcourse, theentire procedure requires amathematical foundation. The exact formulation andproof ofthecorresponding theorems arenotatalleasy. Particular difficulty isintroduced by“caustics” (i.e., focal orconjugate points, orturning points). Caustics areenvelopes offamilies ofrays; they canbeseen onawall illuminated byrays reﬂected from some smooth curved surface. Iftherays orthogonal tothewave fronts intersect andform caustics, then near the caustics theformulas forshort wave asymptotics must beslightly changed. Namely, thephase ofoscillations along each rayundergoes astandard dis- continuity (one-fourth ofawave) upon each passage oftheraythrough a caustic. Aprecise description ofallthese phenomena may beconveniently devel- oped interms ofthegeometry oflagrangian submanifolds ofthecorrespond- ingphase space andtheir projections onto theconﬁguration space. Here, caustics areinterpreted assingularities oftheprojection, from phase space toconﬁguration space, ofthat lagrangian manifold which represents a 438 Appendix 11:Short wave asymptotics family ofrays. Thus, thenormal forms ofsingularities oflagrangian pro- jections introduced inAppendix 12supply aclassiﬁcation ofsingularities of caustics formed bysystems ofraysin“general position.” Inthisappendix weintroduce (without proof )thesimplest formulas of short wave asymptotics fortheSchrodinger equation ofquantum mechanics. Amore detailed exposition canbefound inthefollowing places: J.Heading, Introduction tophase integral methods, Methuen C0. Ltd., 1962. (Cf. especially Appendix ll(byV.P.Maslov) intheRussian translation ofHeading’s book, Moscow 1965). V.P.Maslov, Théorie desperturbations etméthodes asymptotiques, Pairs, Dunod, 1972(Russian edition: Moscow University, 1965). V.I.Arnold, Onacharacteristic class entering intoconditions ofquantization, Functional Analy- sisanditsApplications, v.l(1967). L.Hormander, Fourier integral operators. Acta Math. 127(1971), 79-l83. AQuasi-classical approximation forsolutions ofSchrodinger’s equation Schrodinger’s equation foraparticle inaﬁeld with potential energy Uin euclidean space isanequation foracomplex-valued function ¢(q,t): 0 h’ihl=——Ai//+U(q)1//, qelR",telR.ﬁt 2 Here, hissome realconstant which isalsoasmall parameter oftheproblem being considered, andAistheLaplace operator. Weassume thattheinitial condition hastheshort wave form I//I,-0 =<P(q)@“’”"“", where thesmooth function (pisnonzero only inside some bounded region. Wewillﬁnd below anasymptotic (ash—>0)formula forthesolution of Schrodinger’s equation with such aninitial condition. First ofall,weconsider themotion ofaclassical particle intheﬁeld with potential energy U,i.e.,weconsider Hamilton’s equations _6H _ 0H(i=5 t>=—%, Wh¢r<=H=%P2+U(q) in2n-dimensional phase space. Thesolutions ofthese equations determine aphase ﬂow (under some conditions onthepotential, which weassume ful- ﬁlled; these conditions prevent theparticle from going offtoinﬁnity ina ﬁnite time). Weassociate toourshort wave initial condition alagrangian submanifold ofthephase space (i.e.,amanifold whose dimension isequal tothedimension oftheconﬁguration space andonwhich the2-form dp/\dqdeﬁning thesym- plectic structure onthephase space isidentically zero). Namely, wedeﬁne 439 Appendix 11:Short wave asymptotics the“momentum” corresponding toourinitial condition asthegradient of thephase, i.e.,weset ifsi>(q)—aq. Lemma. Foranysmooth function s,thegraph ofthefunction p(q)constructed byitinthephase space R2"={(p,q)}isalagrangian manifold. Conversely, ifalagrangian manifold projects diﬂeomorphicall yonto theq-space (i.e.,it isagraph), then itisgiven bysome generating function s,according tothe formula above. Wedenote thelagrangian manifold constructed from theinitial condition (with thefunction s)byM.After time tthephase ﬂow g’carries themanifold Mtoanother manifold g’M. Thisnewmanifold isalsolagrangian, since the phase ﬂow preserves thesymplectic structure. Forsmall t,thenew lagrangian manifold, liketheold,projects diffeo- morphically onto theconﬁguration space. However, forlarge tthisisnot necessarily true(Figure 244). P A .11 . /g”.u g’-V ____+.______P,-_ l 1 4, Q> (I Figure 244 Transformation oflagrangian manifolds bythephase flow Inother words, several points ofthenewlagrangian manifold mayproject toonepoint Qoftheconﬁguration space. Weassume that there areonly ﬁnitely many ofthese points andthatthey areallnondegenerate (i.e.,thatat each ofthepoints ofthenewlagrangian manifold which project onto Q,the derivative oftheprojection mapping onto theconﬁguration space isnon- degenerate). Thenondegeneracy condition issatisﬁed foralmost allpoints Q.Those exceptional points Q forwhich itisnotsatisﬁed form asetofmeasure zero intheconﬁguration space. Inthegeneral case. thissetisasurface whose dimension isonelessthan thedimension oftheconﬁguration space. This surface, playing theroleofacaustic inourproblem, canitself have complicated singularities. 440 Appendix 11:Short wave asymptotics Thepoints ofthenewlagrangian manifold projecting tothepoint Qarose under thephase ﬂow transformation from several points oftheoriginal lagrangian manifold (constructed from theinitial condition). Inother words, after time t,several trajectories ofclassical particles, with initial conditions belonging totheoriginal lagrangian manifold, arrive atQ. Welet(pj,qj)denote these initial points inthephase space, andSjthe action along thetrajectories ofthephase ﬂow coming from thepoint (pl-,q1-). More precisely, weset S,(Q, t)=s(q,-) +J~tLdd, 0 whereL-"7-v<q>andg"<p..q,-1-ow).qw» Then, ash—>0,thesolution ofSchrodinger’s equation with theoscillating initial condition given bythefunctions sand(phasasymptotic form DQ-1/2 (i/h)-8'-(Q »t—<t»/211»¢(Q,I)=21¢/Pfq,-) 9’’ ’+O01), where ujisaninteger (theMorse index) which willbedeﬁned below. Inorder toexplain thisformula, weﬁrstconsider thecase when thetime interval tissmall. Inthiscase, thesum isreduced toasingle term, since the lagrangian manifold obtained from theoriginal lagrangian manifold bythe phase ﬂow transformation after small time projects diffeomorphically onto theconﬁguration space. Inother words, ofthefamily ofparticles correspond- ingtotheinitial condition forSchrodinger’s equation, only onearrives atQ after thesmall time t. Forsmall t,theMorse index isequal tozero (aswewillseebelow from its deﬁnition). Inthiswaythefunction ¢(Q, t)has,liketheinitial condition, a rapidly oscillating form. Thus, thefunction Sdeﬁning thewave fronts attime tisnone other than thevalue attime tofthesolution oftheHamilton-Jacobi equation, theinitial condition forwhich isgiven bythefunction sdeﬁning thewave front attheinitial moment. Theamplitude ofthewave attime tat thepoint Qisobtained from theamplitudes, attheinitial moment atthe original point, ofthetrajectories coming toQmultiplied byacertain factor. This factor ischosen sothat, under motions oftheparticles corresponding toourinitial conditions, theintegral ofthesquare ofthemodulus ofthe function i//,over aregion ofconﬁguration space ﬁlled with particles, does not change with time. (Here weassume thatattheinitial moment, some region in theconﬁguration space hasbeen selected; then thephase points onthe original lagrangian manifold areselected whose projections onto thecon- ﬁguration space lieinthisregion; their images under theaction ofthephase ﬂow after time tarefound; ﬁnally, theprojections ofthese images onto the conﬁguration space form theregion “ﬁlled with particles attime t.”) 441 Appendix 11:Short wave asymptotics BTheMorse andMaslov indices Thenumber ujisdeﬁned asthenumber offocal points tothemanifold M ontheinterval [0,t]ofthephase curve starting outatthepoint (pl-,q1). Focal points tothemanifold Maredeﬁned asfollows. Wechose thepoint Qsothat, under projection ofthelagrangian manifold obtained from Mat time t,anondegeneracy condition issatisﬁed atthispoint. However, ifwe consider theentire phase curve coming from thepoint (pj,q,-),then atsome moments oftime0between 0andt,thenondegeneracy condition maynotbe 0satisﬁed atthepoint (p(6), q(0)) ofthelagrangian manifold gM.Such points arecalled focal points tothemanifold Malong thisphase curve. Wenote that thedeﬁnitions offocal points toMand theMorse index donotdepend on Schrodinger’s equation, butrelate simply tothegeometry ofthephase ﬂow inthecotangent bundle totheconﬁguration space (ortothecalculus ofvariations, which isthesame thing). Inparticular, asourlagrangian manifold Mwemaytake theﬁber ofthecotangent bundle passing through thepoint (po,qo)(given bythecondition q=qo).lnthiscaseafocal point to Monthephase curve going outfrom (po,qo)iscalled conjugate totheoriginal point (more precisely, theprojection ofthisfocal point onto theconﬁguration space issaidtobeconjugate tothepoint qoalong theextremal intheconﬁguration space starting atqowith momentum po). Intheeven more special case ofmotion along ageodesic onariemannian manifold. afocal point toaﬁber ofthecotangent bundle iscalled conjugate totheinitial point ofthegeodesic along thisgeodesic. Forexample, thesouth poleofasphere isconjugate tothenorth polealong anymeridian. The Morse index ofaninterval ofageodesic, equal tothenumber ofpoints conjugate tothe initial point, plays animportant roleinthecalculus ofvariations. Namely. weconsider the second differential oftheaction asaquadratic form onthespace ofvariations (with ﬁxed end- points) ofthegeodesic wearestudying. Then theindex ofinertia ofthisquadratic form isequal totheMorse index (cf.,forinstance, J.Milnor, Morse Theory, Princeton University Press, 1967). Thus thegeodesic, uptotheﬁrstconjugate point, isaminimum ofthe action, whichjustiﬁes thename “principle ofleast action" forvarious variational principles ofmechanics. Wenote that incalculating theMorse index, thefocal points must be counted with multiplicity (themultiplicity ofafocal point ingeneral position isequal to1). TheMorse index isaparticular case oftheso-called Maslov index, which isdeﬁned independently ofthephase ﬂowforanycurve onalagrangian mani- foldofthecotangent bundle over theconﬁguration space. Consider theprojection ofourn-dimensional lagrangian manifold onto then-dimensional conﬁguration space. This isasmooth mapping ofmani- folds ofthesame dimension. Itcanhave singular points, i.e.,points atwhich therank ofthederivative mapping drops, andinaneighborhood ofwhich theprojection isnotadiffeomorphism. Itturns outthatingeneral thesetofsingular points hasdimension n—1 andconsists ofthe union ofasmooth manifold ofdimension n—1made upof simple singular points atwhich therank drops to1,andaﬁnite setofmani- folds whose dimensions aren—3andsmaller. Here, “ingeneral” means that 442 Appendix 11:Short wave asymptotics these properties canbeattained byanarbitrarily small perturbation ofthe lagrangian manifold, under which itremains lagrangian. Weshould point outthat, among thepieces ofvarious ranks intowhich thesetofsingular points isdivided, there isnopiece ofdimension n—2.After thesimplest singular points, forming amanifold ofdimension n—1,there arethepoints where therank drops bytwo; theyform a manifold ofdimension n-3.Theprojection ofthe setofsingular points onto theconﬁguration space (thecaustic) consists, ingeneral, ofpieces ofalldimensions from Oton—1without omissions. Furthermore, itturns outthat the(n—l)-dimensional manifold ofthe simplest singular points istwo-sided inthelagrangian manifold; thatis,we cancoordinate theorientations ofthenormals atallpoints inthefollowing way. Consider some simple singular point onthelagrangian manifold. Wetake asystem ofcoordinates q1,...,q,,inaneighborhood oftheprojection ofthis point onto theconﬁguration space. Letpl,...,p,,becorresponding coordi- nates intheﬁberofthecotangent bundle. Inaneighborhood ofoursingular point, wecanconsider thelagrangian manifold asthegraph ofthevector function (q1,p2,...,p,,)ofthevariables (pl,qz,...,q,,)(oravector function ofananalogous form inwhich theroleofthedistinguished coordinate is played notbytheﬁrstcoordinate butbyanyoftheremaining coordinates). Singular points near thegiven onearethen deﬁned bythecondition Eiq,/dp, =0.Forlagrangian manifolds ingeneral position, thisderivative changes signupon passing from onesideofthemanifold ofsingular points to theother inourneighborhood ofthesimple singular point. Wewillcallthe sidewhere thisderivative ispositive thepositive side. Wenote thatitisnecessary toprove thatthedeﬁnitions ofpositive direction near different points agree with oneanother. Furthermore, itmust beshown thatthepositive direction near onepoint iswelldeﬁned, i.e.,does notdepend Onthecoordinate system. Allthiscanbedone by direct calculations (cf.thearticle cited above in“Functional Analysis”). Forfurther development ofthese ideas, seeV.I.Arnold, Stunn Theory andSymplectic Geometry, Funct. Anal. Appl. 19(1985). Now theMaslov index ofanoriented curve onalagrangian manifold is deﬁned asthenumber ofpassages from thenegative sideofthemanifold of singularities tothepositive side, minus thenumber ofpassages intheother direction. Inthisweassume thattheends ofthecurve arenonsingular and thatthecurve intersects only themanifold ofsimple singular points andonly with nonzero angles. Having deﬁned theindex forsuch curves, wecandeﬁne itforanarbitrary curve connecting twononsingular points: todothisitis sufficient toapproximate thecurve byonewhich intersects only themanifold ofsimple singular points andonly with nonzero angles. Itcanbeshown that theindex does notdepend onthechoice oftheapproximating curve. PROBLEM. Find theindex ofthecircle p=cost,q=sintoriented bytheparameter t, 03t521:,inthelagrangian manifold pl+qz=1ofthephase plane. Auswnn. +2. 443 Appendix 1l:Short wave asymptotics Finally, theMorse index ofaphase curve inR2"cannowbedeﬁned asthe Maslov index ofacurve inan(n+l)-dimensional lagrangian manifold ina suitable (2n+2)-dimensional phase space. Ascoordinates inthisspace we willtake(po,p;qo,q)(where (p,q)eR2"). Ifwe setqo=randpo=—H(p,q), andletthepoint (p,q)range over then-dimensional lagrangian manifold in R2"obtained from theoriginal after time tbytheaction ofthephase flow, then under change oftthepoints inR2"*2 form an(n+1)-dimensional lagrangian manifold. Thegraph ofthemotion ofaphase point under the action ofthephase ﬂow canbeconsidered asacurve onthis(n+1)-dimen- sional lagrangian manifold. Wecanverify thattheMaslov index ofthisgraph agrees with theMorse index oftheoriginal phase curve. Cindices ofclosed curves The indices ofclosed curves onlagrangian submanifolds ofalinear phase space canalsobecalculated with thehelpofacomplex structure. Inaddition tothesymplectic structure dp/\dqonthelinear phase space R2"={(p,q)}, weintroduce aeuclidean structure (with scalar square p2+qz)and a complex structure, inwhich multiplication byiis IriR’"—> R2” Itinq) =(—q,1>) Z=P+iqC"={Z}- Allthree structures areconnected bytherelation [Xiy]=(IX,y), where thesquare brackets denote theskew-scalar product. Linear transformations ofthephase space preserving any two (and, therefore, allthree) structures arecalled unitary transformations. Such trans- formations take lagrangian planes tolagrangian planes. Every lagrangian plane canbeobtained from anyother (e.g., from the realplane R"given bytheequation q=0)byaunitary transformation. In addition, anytwounitary transformations AandBcarrying therealplane tothesame lagrangian plane differ byaunitary transformation which isa realorthogonal transformation: B=AC, where CIR" =R". Conversely, anypreliminary orthogonal transformation does notchange the image oftheplane under theaction ofaunitary transformation. Wenow note that thedeterminant ofanorthogonal transformation is equal toi1.Therefore thesquare ofthedeterminant ofaunitary transforma- tioncarrying therealplane toagiven lagrangian plane depends only onthe lagrangian plane itself anddoes notdepend atallonthechoice ofunitary transformation. After these preliminary remarks wereturn toourlagrangian manifold andclosed oriented curve lying init.Atevery point ofthecurve, there isa plane tangent tothelagrangian manifold inthesymplectic vector space. The square ofthedeterminant oftheunitary transformation carrying thereal 444 Appendix ll:Short wave asymptotics plane tothistangent plane isacomplex number with modulus one. Asa point moves along ourclosed curve, thiscomplex number changes. After an entire circuit ofthecurve, thesquare ofthedeterminant makes some integral number ofrotations around theorigin ontheplane ofcomplex variables, oriented from 1toi.This integer istheindex oftheclosed curve. Theindices ofclosed curves enter intoasymptotic formulas forstationary problems (characteristic oscillations). Assume that thephase ﬂow cor- responding tothepotential Uhasaninvariant lagrangian manifold lying on theenergy level H=E.Then theequation %Al//=l2(U(q) —E)ll/ hasaseries ofeigenvalues /lo.—>ocwith asymptotic form AN=no+O(u§ 1) if,forevery closed contour yonthelagrangian manifold, wehave thecon- gruence 2 .$§pdqEindy(mod 4). ‘Y Intheone-dimensional case, thelagrangian manifold isacircle, itsindex isequal to2,andtheformula above reduces totheso-called “quantization condition” inflipdq =21r(N+i)- Y Theeigenfunctions corresponding tothese eigenvalues arealsoassociated with lagrangian manifolds, butthisassociation isnotsosimple. Infact, wecannot write down asymptotic formulas foreigenfunctions, butonly forfunctions approximately satisfying theequations of characteristic functions. These functions turnouttobesmall outside theprojection ofthe lagran- gian manifold onto theconﬁguration space. Theasymptotic formulas have singularities near thecaustics formed bytheprojection. Theactual eigenfunctions, however. canbehave entirely differently, atleast iftheeigen- value ismultiple orifthere areeigenvalues close toit(cf.Appendix 10). 445 pm.-vs-oizau-1-_-.~».-Appendix 12:Lagrangian singularities Lagrangian singularities aresingularities ofprojections _oflagrangian mani- folds onto conﬁguration space. Such singularities areencountered in investigating global solutions totheHamilton-Jacobi equation, instudying caustics, focal orconjugate points, inanalyzing thepropagation ofdis- continuities andshock waves inthemechanics ofasolid medium, andalsoin problems ofshort wave asymptotics (cf.Appendix ll). Inorder todescribe lagrangian singularities wemust ﬁrstsayafewwords about singularities ofsmooth mappings ingeneral. Webegin with the simplest examples. ASingularities ofsmooth mappings ofasurface onto aplane Themapping projecting asphere onto aplane issingular ontheequatorial circle (atpoints oftheequator therank ofthederivative drops toone). Asa result, acurve isformed ontheplane ofprojection (theso-called apparent contour) bounding regions inwhich points have different numbers ofpre- images: every point oftheplane inside theapparent contour hastwo pre-images, andevery point outside hasnone. Inmore complicated cases of“apparent contours” there canbemore complicated singularities. Consider, forexample, thesurface given inthree- dimensional space with coordinates (x,y,z)bytheequation (Figure 245) x=yz—z° andthemapping ofprojection parallel tothez-axis onto theplane with coordinates (x,y). Thesingular points oftheprojection form asmooth curve onthesurface (with equation 322=y).However, theimage ofthiscurve onthe(x,y)plane isnotasmooth curve. This image isasemi-cubical parabola with acusp at thepoint (0,0)with equation 27x2 =4y3. Such acurve divides theplane intotwoparts: asmaller part (inside the cusp) andalarger part (outside). Over each point ofthesmaller part there arethree points ofoursurface, andover each point ofthelarger partthere is onlyone. Wenow consider anysmall deformation ofoursurface. Itturns outthat, under projection ofanysurface close toours, theapparent contour will always have asimilar singularity (semi-cubical cusp) atsome point close to thesingularity oftheapparent contour oftheoriginal surface. Inother words, thissingularity isnotremovable byasmall perturbation ofthesurface. Furthermore, inplace ofadeformation ofthesurface, wecanarbitrarily deform themapping itself ofthesurface totheplane (nolonger caring whether itisaprojection), aslong asitremains smooth andthedeformation issmall. Itturns outthat, forthese deformations too,thecusp does notdis- appear butisonly slightly deformed. Theexamples presented here exhaust alltypical singularities ofmappings ofasurface totheplane. Itcanbeshown thatallmore complicated singu- 446 Appendix 12:Lagrangian singularities x -V Figure 245 Whitney’s tuck larities areremovable byasmall perturbation. Therefore, byslightly de- forming anysmooth mapping, wecanalways arrange thatinaneighborhood ofanypoint ofthesurface, themapping willbeeither nonsingular, or structurally similar totheprojection mapping ofasphere onto aplane near theequator, orstructurally similar totheprojection mapping ofthesurface considered above with acubic cusp ontheapparent contour. Thewords “structurally similar to”mean that, onthepre-image surface andtheimage plane, wecanchoose local coordinates (inaneighborhood of ourpoint anditsimage) such thatinthese coordinates themapping willbe written inaspecial way. Namely, thenormal forms towhich themapping ofthesurface totheplane willbereduced inaneighborhood ofpoints ofthe three types indicated above willbe yl=x, yz=x2 (nonsingular point) y,=xf yo=x2 (afold, asontheequator ofthesphere) y,=xlxz —xi‘ yo=x2 (a“tuck” with acusp ontheapparent contour) Here (x1,x2)arethelocal coordinates inthepre-image, and(yl,yo)arethe local coordinates intheimage. Theproof ofthistheorem (itisduetoH.Whitney) anditsmultidimen- sional generalizations canbefound inworks onthetheory ofsingularities of smooth maps, such as V.I.Arnold, Singularities ofsmooth mappings, Russian Math. Surveys 23:1 (1968) 1-44. Symposium onSingularities ofSmooth Manifolds andMaps, Univ. ofLiverpool, 1969-70. Proceedings. Springer, 1971. Seeespecially thearticle ofR.Thom andH.Levine. Golubitsky andGuillemin, Stable Mappings andTheir Singularities, Springer-Verlag, 1973. 447 Appendix 12:Lagrangian singularities BSingularities ofprojection oflagrangian manifolds Wenow consider ann-dimensional conﬁguration manifold, thecorrespond- ing2n-dimensional phase space, andann-dimensional lagrangian submani- fold (i.e., ann-dimensional submanifold onwhich the2-form giving the symplectic structure ofthephase space isidentically zero). Byprojecting thelagrangian manifold onto theconﬁguration space, we obtain amapping ofonesmooth n-dimensional manifold toanother. Atmost points, thismapping isalocal diffeomorphism, butatsome points ofthe lagrangian manifold therank ofthedifferential drops. These points aresaid tobesingular. Under projection ofthesetofsingular points totheconﬁgura- tionspace an“apparent contour” isformed, which iscalled acaustic inthe lagrangian case. Caustics canhave complicated singularities; however, asintheusual theory ofsingularities ofsmooth maps, wecangetridofsingularities which aretoocomplicated byasmall perturbation (here, byasmall perturbation, wemean asmall deformation ofalagrangian manifold inphase space under which thismanifold remains lagrangian). After thisthere remain only thesimplest unremovable singularities, for which wecanwrite outnormal forms andwhich wecanstudy once andforall. When considering problems ingeneral position which donotsatisfy any special properties ofsymmetry, itisnatural toexpect thatonly these simple unremovable singularities willappear. Consider, forexample, thecaustics formed onawall bylight from apoint source reﬂected from some smooth curved surface (here thefour-dimensional phase space isformed bystraight lines intersecting thesurface ofthewallin allpossible directions, andthelagrangian submaiiifold bytherays oflight coming from thesource asthey intersect thewall). Bymoving thesource, we canseethat generally thecaustics have only simple singularities (semi- cubical cusps), while more complicated singularities appear only forspecial, exceptional positions ofthesource. Wewillgive below, forn35,normal forms forsingularities ofthepro- jection ofann-dimensional lagrangian submanifold of2n-dimensional phase space onto ann-dimensional conﬁguration space. There areaﬁnite number ofthese normal forms, andtheir classiﬁcation isrelated (inarather mysteri- ousway) with theclassiﬁcations ofsimple Liegroups, simple degenerate critical points offunctions, regular polyhedra, andmany other objects. For nZ6,thenormal forms ofsome singularities must inevitably contain parameters. Forfurther details thereader isreferred tothearticles: V.I.Arnold, Normal forms forfunctions near degenerate critical points. theWeyl groups of Ak,Dk,E,,,andlagrangian singularities, Functional Analysis andItsApplications 6:4(1972) 254-272. V,I,Arnold. Critical points ofsmooth functions andtheir normal forms. Uspekhi Math Nauk 30:5 (1975). 448 Appendix 12:Lagrangian singularities CTables ofnormal forms oftypical singularities ofprojections oflagrangian manifolds of dimension n35 Wewillusethefollowing notation: (ql,...,q,,)arecoordinates ontheconﬁguration space, (pl,...,p,,)arethecorresponding impulses, sothatpandqtogether form asymplectic coordinate system inthephase space. Wewillgivealagrangian manifold with thehelp ofagenerating function Fbytheformulas q_6F p_ 6F 1apt J 5%", where theindex iruns over some subset of{l,...,n}andj runs over there- mainder of{l,...,n}.That is,i=l,j>lforsingularities denoted inthelist byA,,,andi=1,2,j>2forsingularities denoted byDkandE,,. With thisnotation, oneandthesame expression F(pi,q,-)canbecon- sidered asgiving alagrangian manifold inspaces ofadifferent number of dimensions: wecanaddarbitrarily many arguments qJ-,onwhich Fdoes not actually depend. The listofnormal forms oftypical singularities isnow asfollows: for n=l A13F=Pi Az3F=iPi§ forn=2,inaddition tothetwoabove, there is /4331: =ipi +q2Pi§ forrt=3,inaddition tothethree preceding, there are _ 5A43F—iP1+ q3Pi +qzpi, D4:F=irim ir3+qsri; forn=4,inaddition totheﬁvepreceding, there are A511” =ir?+earl+qsri+qzri. 135:1: =trim ir‘;+qtri+qsri; forn=5,inaddition totheseven preceding, there are Ae:F =if-Di iqspi + ‘l'q2p%7 DGIF=trim ir3+qsré+qari+qsri. E61F=iriir3+q5r1r§+ qtrim +enri- 449 Appendix 12:Lagrangian singularities DDiscussion ofthenormal forms Apoint oftypeA1isnonsingular. Asingularity oftypeA2isafoldsingularity. Ifwetake (pl,qo,...,q,,)ascoordinates onthelagrangian manifold, then theprojection mapping may bewritten as (rnqz,---,qi)—> (i3ri, q21"'9qll)‘ Asingularity oftype A,isatuck with asemi-cubical cusp onthevisible contour. Toconvince ourselves ofthis, itisenough towrite outthecor- responding mapping ofthetwo-dimensional lagrangian manifold tothe plane: (rt,qt)e(i4ri +Zqzri, (12)- Asingularity oftype A4ﬁrstappears inthethree-dimensional case, and thecorresponding caustic isrepresented byasurface inthree-dimensional space (Figure 246) with asingularity called aswallowtail (wealready en- countered thisinSection 46). The caustic ofasingularity oftype D4inthree-dimensional space is represented asasurface with three cuspidal edges (oftype A3),tangent at onepoint; twoofthese cuspidal edges canbeimaginary, sothat there are twoversions ofthecaustic ofD4. .43 .4; /13 /14 /l3 A3 l °+ I/13 /l3 D4 \ O /I3 A3 Figure 246 Typical singularities ofcaustics inthree-dimensional space ELagrangian equivalence Wemust nowsayinwhat sense theexamples mentioned arenormal forms of typical singularities ofprojections oflagrangian manifolds. First ofall,we willdeﬁne which singularities wewillconsider tohave the“same structure.” Aprojection mapping ofalagrangian manifold onto conﬁguration space willbecalled alagrangian mapping forshort. Suppose thatwearegiven two 450 Appendix 12:Lagrangian singularities lagrangian mappings ofmanifolds ofthesame dimension n(thecorrespond- ingn-dimensional lagrangian manifolds lie,ingeneral, indifferent phase spaces which arecotangent bundles oftwodifferent conﬁguration spaces). We saythattwosuch lagrangian mappings arelagrangian equivalent ifthere isa symplectic diffeomorphism oftheﬁrst phase space tothesecond, taking ﬁbers oftheﬁrstcotangent bundle toﬁbers ofthesecond, andtaking theﬁrst lagrangian manifold tothesecond. Thesymplectic diffeomorphism itself is then called alagrangian equivalence mapping. Wenote thattwolagrangian equivalent lagrangian mappings aretaken onetotheother with thehelpofdiffeomorphisms inthepre-image space and theimage space (or,asthey sayinanalysis, arecarried tooneanother bya change ofcoordinates inthepre-image andintheimage). Infact, oursym- plectic diffeomorphism restricted tothelagrangian manifold gives adiffeo- morphism ofthepre-images; adiffeomorphism oftheconﬁguration-space images arises because ﬁbers arecarried toﬁbers. Inparticular, thecaustics ofthetwolagrangian equivalent mappings are diffeomorphic, hence aclassiﬁcation uptolagrangian equivalence implies a classiﬁcation ofcaustics. However, theclassiﬁcation uptolagrangian equiv- alence isﬁner than theclassiﬁcation ofcaustics, since adiffeomorphism of caustics does notingeneral giverisetoalagrangian equivalence ofthemap- pings. Furthermore, theclassiﬁcation uptolagrangian equivalence isﬁner then theclassiﬁcation uptodiffeomorphisms ofthepre-image andimage, since notevery such pair ofdiffeomorphisms isrealized byasymplectic diffeomorphism ofthephase space. Alagrangian mapping considered inaneighborhood ofsome chosen point iscalled lagrangian equivalent atthatpoint toanother lagrangian mapping (also with achosen point), ifthere isalagrangian equivalence oftheﬁrst mapping insome neighborhood oftheﬁrstpoint onto thesecond insome neighborhood ofthesecond point, carrying theﬁrstpoint tothesecond. Wecan now formulate aclassiﬁcation theorem forsingularities of lagrangian mappings indimensions n35. Theorem. Every n-dimensional lagrangian manifold (n35)can,byanarbi- trarily small perturbation intheclass oflagrangian manifolds, bemade into onesuch thattheprojection mapping onto theconﬁguration space willbe lagrangian equivalent atevery point tooneofthelagrangian mappings in thelistabove. Inparticular, atwo-dimensional lagrangian manifold can beputin “general position” byanarbitrarily small perturbation intheclass of lagrangian manifolds, sothattheprojection mapping onto theconﬁguration space (two-dimensional) willnothave singularities other than folds (which canbereduced byalagrangian equivalence tothenormal form A2)ortucks (which canbereduced byalagrangian equivalence tothenormal form A3). 451 Appendix 12:Lagrangian singularities Wenote thatthisassertion about two-dimensional lagrangian mappings does notfollow from theclassiﬁcation theorem forgeneral (non-lagrangian) mappings. Intheﬁrst place. lagrangian mappings make upaveryrestricted class among allsmooth mappings, andtherefore they can(and actually doforn>2)have astypical, singularities which arenottypical for mappings ofgeneral form. Secondly, thepossibility ofreducing amapping tonormal form by diffeomorphisms ofthepre-image andimage does notimply that thiscanbedone using a lagrangian equivalence. Inthisway, thecaustics ofatwo-dimensional lagrangian manifold in general position have assingularities only semi-cubical cusps (and points of transversal intersection). Allmore complicated singularities break upunder asmall perturbation ofthelagrangian manifold, theresulting cusps andself- intersection points ofcaustics areunremovable bysmall perturbations, and areonly slightly deformed. Normal forms ofthesingularities A4,D4,...canbeused inasimilar way forstudying thecaustics oflagrangian manifolds ofhigher dimensions, and alsoforstudying thedevelopment ofcaustics oflow-dimensional lagrangian manifolds, when parameters onwhich themanifold depends arevaried.‘ '6 Other applications oftheformulas ofthissection canbefound inthetheory ofLegendre singularities, i.e.,singularities ofwave fronts. Legendre transforms, envelopes, andconvex hulls (cf.Appendix 4).Thetheories oflagrangian andLegendre singularities have direct application, notonly ingeometric optics andthetheory ofasymptotics ofoscillating integrals, butalsoin thecalculus ofvariations, inthetheory ofdiscontinuous solutions ofnonlinear partial differential equations, inoptimization problems, pursuit problems. etc.R.Thom hassuggested thegeneral name catastrophe theory forthetheory ofsingularities, thetheory ofbifurcations, andtheir applications. '16See,e.g.,V.Arnold, Evolution ofwavefronts andequivariant Morse lemma, Comm. Pure Appl.Math., 1916,No.6. 452 Appendix l3:TheKorteweg—de Vries equation Notallﬁrstintegrals ofequations inclassical mechanics areexplained by obvious symmetries ofaproblem (examples arespeciﬁc integrals ofKepler’s problem, theproblem ofgeodesics onanellipsoid, etc.). Insuch cases, we speak of“hidden symmetry.“ 17 Interesting examples ofsuch hidden symmetry arefurnished bythe Korteweg—de Vries equation ut:6”“): _uxxx' This nonlinear partial differential equation ﬁrst arose inthetheory of waves inshallow water; later itturned outthatthisequation isencountered inawhole series ofproblems inmathematical physics. Asaresult ofaseries ofnumerical experiments, remarkable properties ofsolutions ofthisequation with zero boundary conditions atinﬁnity were discovered: ast—>ooandt—>—oo these solutions decompose into “soli- tons”—waves ofdeﬁnite form moving with different velocities. Toobtain asoliton moving with velocity t",itissufficient tosubstitute thefunction u=<p(x—ct)into equation (1).Then weobtain theequation tp”=3(,a2+up+dfortp (disaparameter). This isNewton’s equation with acubic potential. There isasaddle onthe phase space (tp,<p').Theseparatrix going from thissaddle tothesaddle forwhich (p=0de- termines asolution (,0tending toOasx—>i-ac;itisasoliton. When solitons collide, there isacomplicated nonlinear interaction. However, numerical experiments showed thatthesizes andvelocities ofthe solitons donotchange asaresult ofcollision. And, infact,Kruskal, Zabusky, Lax, Gardner, Green, andMiura succeeded inﬁnding awhole series ofﬁrst integrals fortheKorteweg—de Vries equation. These integrals have theform Is=Ps(u, ...,u“’)dx, where PSisapolynomial. Forexample, itiseasy to verify thatthefollowing areﬁrstintegrals ofequation (1): 12 I_1=Judx I0=fu2dx I,=J.(-u?+u3)dx, "25 5 I2 = —511211” +iH4)dX. Theappearance ofaninﬁnite series ofﬁrstintegrals iseasily explained by thefollowing theorem ofLax.‘ 18Wewilldenote theoperator ofmultiplica- tionbyafunction ofxbythesymbol forthefunction itself, andtheoperator ofdiﬁerentiation with respect toxbythesymbol 6.Consider theSturm- Liouville operator L=-62 +udepending onafunction u(x). Weverify directly: Theorem. TheKorteweg—de Vries equation (1)isequivalent totheequation L2=[L,A],where A=463—3(u5+61.4). "1Theterm “accidental symmetry" isfrequently used inEnglish. [Trans note.] “SLax, P.D.,Integrals ofnonlinear equations ofevolution andsolitary waves. Comm. Pure Appl._'\/lath.21(1968) 467-490. 453 Appendix 13:TheKorteweg—de Vries equation Directly from thistheorem ofLax, wehave Corollary. The operators Lconstructed from asolution ofequation (1)are unitarily equivalent forallt;inparticular, each oftheeigenvalues /1ofthe Sturm—Lionville problem Lf=ifwith zeroboundary conditions atinﬁnity isaﬁrst integral oftheKorteweg—de Vries equation. Gardner, V.E.Zakharov andL.D.Faddeev noted thatequation (1)isa completely integrable inﬁnite-dimensional hamiltonian system, andfound thecorresponding action-angle variables.‘ ‘9Asymplectic structure onthe space offunctions vanishing atinﬁnity isgiven bytheskew-scalar product w2(6w, 6v)=§l(w60—v6w)dx, andthehamiltonian ofequation (l)isthe integral 1,.Inother words, equation (1)canbewritten intheform ofHamil- ton’s equation inthefunctional space offunctions ofx,u=(d/dx)(6I,/ou). Every integral ISgives inthiswaya“higher Korteweg—de Vries equation” ti=QS[u], where Q3=(d/dx)(6Is/ou) isapolynomial inthederivatives u,u',...,uz“1.Theintegrals ISareininvolution, andtheﬂows corresponding tothem onthefunctional space commute. Theexplicit form ofthepolynomials P,andQ3,andalsotheexplicit form oftheaction- angle variables (and therefore ofsolutions ofequation (1)),isdescribed interms ofsolutions of thedirect andinverse problems ofscattering theory with potential u. Theexplicit form ofthe polynomials Q,canalsobeobtained from thefollowing theorem of Gardner, generalizing Lax”s theorem. Inthespace offunctions ofx,weconsider adifferential operator oftheform A=Xp,-0'"_'i, where po=l.andtheremaining coeﬂicients p,arepoly- nomials inuandthederivatives ofllwith respect tox.Itturns outthat, forany5there is anoperator Asoforder 2s+1such thatitscommutator with theSturmeLiouville operator L istheoperator ofmultiplication byafunction [L,AS]:Q,. Theoperator A,isdeﬁned bythese conditions uniquely uptotheaddition oflinear combina- tions ofthe A,withr<s;inthesame way, thepolynomials Q,aredetermined uptotheaddition oflinear combinations ofthepreceding Q,‘s. V.E.Zakharov, A.B.Shabat, L.D.Faddeev, andothers, using Lax’s method andtechniques ofinverse scattering theory, have studied awhole series ofphysically important equations, including theequations u,,—uxx= sinuandit//,+tl/xxit//It/1|2 =0. Investigation oftheproblem with periodic boundary conditions forthe Korteweg—de Vries equation ledS.P.Novikovlzo tothediscovery ofan interesting class ofcompletely integrable systems with aﬁnite number of degrees offreedom. These systems areconstructed inthefollowing way. Consider anyﬁnite linear combination ofﬁrst integrals, I=Zc,I,,_;, andletco=1.Thesetofstationary points oftheﬂow with hamiltonian I “°Zakharov, V.E.andFaddeev, L.D.,The Korteweg—de Vries equation isacompletely integrable hamiltonian system, Functional Analysis andItsApplications, 5:4(I971) 280-287. '20Novikov, S.P.,The periodic problem fortheKorteweg—de Vries equation, Functional Analysis andItsApplications, 8:3(1974) 236—246. 454 Appendix 13:TheKorteweg—de Vries equation onthefunctional space isinvariant under thephase ﬂows with hamiltonians IS,including thephase ﬂow ofequation (1). Ontheother hand, these stationary points aredetermined from the equations (d/dx)(<§I/ou) =0,or51/at =d.The second equation isthe Euler-Lagrange equation forthefunctional I—dI_1, involving derivatives oforder n.Therefore, ithasorder 2nandcanbewritten asahamiltonian system ofequations in2n-dimensional euclidean space. Itturns outthatthishamiltonian system with ndegrees offreedom hasn integrals ininvolution andcanbeintegrated completely with thehelp of suitable action-angle coordinates. Inthisway, weobtain aﬁnite-dimensional family ofparticular solutions oftheKorteweg—de Vries equation depending on3n+lparameters (2nphase coordinates andn+1further parameters c1,...,c,,;d). These solutions have, asNovikov showed, remarkable properties; for example, intheperiodic problem theygivefunctions u(x)forwhich thelinear differential equation with periodic coefficients —X" +u(x)X =11X hasaﬁnite number ofzones ofparametric resonance (cf.Section 25)onthe A-axis. After thisbook waswritten, much work wasdone onthesubjects dis- cussed inthisappendix, inparticular byNovikov, Doubrovin, Krichever, Manakov, Matveev, Its,Dikii, Manin, Drinfeld, Gelfand, Lax, Moser, McKean, Van Moerbeke, Adler, Perelomov, Olshanetskii, andmany others. Among other things, Manakov solved theEuler equations ofarigid body in IR"forarbitrary n:these arecompletely integrable. Formore details seethe forthcoming book byNovikov andhiscollaborators. (Note added byauthor intranslation.) 455 Appendix l4:Poisson structures Along with theclassical Poisson bracket offunctions, onealso encounters more general (degenerate) brackets. Atypical example isthePoisson bracket offunctions ofthecomponents M,oftheangular momentum vector: {F,G}=Z(dF/6M,)(6G/tilt/Ij){Mi, Such degenerate brackets may be considered asfamilies ofordinary Poisson brackets orfamilies ofsympletic manifolds. These families generally have singularities (they arenotfoliations): they consist ofsymplectic manifolds (leaves) ofdifferent dimensions, related tooneanother bythecondition ofsmoothness forthegiven degenerate Poisson bracket structure ontheambient space. (Intheangular momentum example above, theleaves areconcentric spheres and their center atthe origin.) Inthisappendix, weshall present thesimplest elementary properties of Poisson structures onﬁnite-dimensional manifolds. Oneshould keep inmind, though, thatinapplications (especially tothemathematical physics ofcon- tinuous media) onefrequently encounters Poisson structures oninﬁnite- dimensional manifolds. Inthese cases, thesymplectic leaves often (but not always) have ﬁnite dimension orcodimension. AP0iSs0n manifolds APoisson structure onamanifold isaLiealgebra structure onitsspace ofsmooth functions (i.e.,abilinear skew-symmetric operation of“Poisson bracket” onfunctions, satisfying theJacobi identity) such thattheoperator ad,={a,}(contraction ofthePoisson bracket with anyﬁxed function a)is anoperator ofdifferentiation bysome vector ﬁeld6,.Thevector ﬁeld0,,isthen called thehamiltonian vector ﬁeld with hamiltonian function a.Themapping dl—>0,gives ahomomorphism from theLiealgebra offunctions totheLie algebra ofvector ﬁelds. Amanifold with agiven Poisson structure iscalled a Poisson manifold. Two points onaPoisson manifold arecalled equivalent ifthey canbejoined byapath consisting ofsegments ofintegral curves ofhamiltonian vector ﬁelds. Theequivalence classes under thisrelation arecalled theleaves ofthePoisson manifold. Thevalues ofallpossible hamiltonian vector ﬁelds atagiven point ofaPoisson manifold form alinear space which isjustthetangent space of theleafthrough that point. Thus theleaves aresmooth manifolds, butthey areingeneral notclosed, andthey have different dimensions. Theclassical (explicitly described byS.Liein1890, butessentially con- sidered already byJacobi) example ofaPoisson manifold isthedual space of a(ﬁnite-dimensional) Liealgebra. Theelements ofthealgebra itself may be considered aslinear functions onthisspace. ThePoisson structure isdeﬁned asanextension oftheLiealgebra structure from thisﬁnite-dimensional sub- space totheentire space ofsmooth functions onthedual oftheoriginal Lie algebra. Such anextension exists andisunique: ifwl, ...,0),,isabasis ofthe 456 Appendix 14:Poisson structures original Liealgebra, then la,biroisson IZ(5a/awiliab/5wjl[wi, wj]Lie' Inthisexample, theleaves aretheorbits oftheco-adjoint representation of theunderlying Liegroup inthedual ofitsLiealgebra. Every leafofaPoisson manifold carries anatural symplectic structure (closed nondegenerate 2-form), deﬁned inthefollowing way. Consider the values oftwohamiltonian vector ﬁelds atapoint oftheleaf.Thevalue ofthe 2-form onthispairofvectors isdeﬁned tobethevalue ofthePoisson bracket ofthehamiltonian functions atthegiven point (thisvalue depends only onthe twovectors andnotonthechoice ofhamiltonian functions). Thefactthatthe form isclosed ontheleaffollows from theJacobi identity; nondegeneracy comes from thefactthat, ifthederivative ofevery function byagiven tangent vector iszero, then thevector itself must bezero. The phase ﬂow ofevery hamiltonian vector ﬁeld preserves thesymplectic structures ontheleaves. Thus, theleaves ofaPoisson manifold areeven dimensional, and the manifold may beconsidered asaunion ofsympletic manifolds (generally of different dimensions), whose symplectic structures arecoordinated bythe condition thatthePoisson bracket ontheambient space besmooth. Forexample, theco-adjoint orbits ofSO(3) (spheres centered attheorigin) may beorganized according tolocal Darboux coordinates: intheneighbor- hood ofanynonzero point, thePoisson structure insuitable local coordinates takes theform {x,y}=1,{x,2}={y,z}=0.Thisnormal form forthePoisson structure onthespace ofangular momenta isconvenient incarrying outthe process ofelimination ofthenodes inthemany-body problem (seeSection III.5.5 ofthepaper: V.I.Arnol’d, Small denominators and problems of stability ofmotion inclassical andcelestial mechanics, Russian Math. Surveys l8,No.6(1963), 85-191). Jacobi realized that the(classical) Poisson brackets oftheﬁrstintegrals ofanyhamiltonian system could beconsidered asaPoisson structure (this structure isdiscussed inSection VI.l.3 oftheauthor’s paper cited above). Theconstruction ofaPoisson structure onthedual space ofaLiealgebra leads toanewLiealgebra. This construction may then berepeated, leading toawhole series ofnew(inﬁnite-dimensional) Poisson structures. More gen- erally, suppose thatoneisgiven anyPoisson structure onamanifold. Then thespace offunctions onthatmanifold carries thestructure ofaLiealgebra. This implies thatthedual space ofthisfunction space carries itsown Poisson structure. Elements ofthisdual space may beinterpreted asdistribution den- sities ontheoriginal manifold. Thus, thespace ofdistributions onaPoisson manifold (forexample, onasymplectic phase space) hasanatural Poisson structure. Thisstructure makes itpossible toapply thehamiltonian formalism toequations ofVlasov type, which describe theevolution ofdistributions of particles inphase space under theaction ofaﬁeld which isconsistent with the particles themselves. 457 Appendix 14:Poisson structures BPoisson mappings Amapping from onePoisson manifold toanother iscalled aPoisson mapping ifitisconsistent with thePoisson structures, i.e.,ifforanytwofunctions onthesecond manifold, thePoisson bracket oftheir pullbacks totheﬁrst manifold coincides with thepullback oftheir Poisson brackets. Forexample, theembedding ofeach symplectic leafinaPoisson manifold isaPoisson mapping. The cartesian product oftwoPoisson manifolds hasanatural Poisson structure, forwhich theprojection oneach factor isaPoisson mapping (the Poisson bracket offunctions pulled back from different factors iszero). S.Lieshowed thatevery Poisson manifold islocally (intheneighborhood ofapoint where thedimension ofthesymplectic leaves islocally constant, for example, intheneighborhood ofageneric point, where therank islocally maximal) decomposible intotheproduct ofasymplectic leafandacomple- mentary space onwhich allPoisson brackets arezero. Onsuch aneighborhood, onemayintroduce coordinates p,-,q,-,c,such that pandqhave theusual symplectic Poisson brackets, while thePoisson bracket ofeach c,-with anyfunction isequal tozero. Inphysics, thecoordinates piand q,arecalled Clebsch variables,‘2‘ while thecfsarecalled Casimir functions. Clebsch introduced hisvariables forthehamiltonian description ofthehydro- dynamics ofideal ﬂuids, while Casimir considered thecenter oftheLiealgebra offunctions onthedual space ofagiven Liealgebra. The dimension ofthesymplectic leafthrough anongeneric point ofa Poisson manifold islessthan thatfornearby generic points. Intheneighbor- hood ofsuch apoint, thePoisson manifold may stillberepresented asthe product ofaneighborhood ofthepoint initssymplectic leafandaneighbor- hood ofadistinguished point insome Poisson manifold ofcomplementary dimension. Inother words, onaminimal transverse manifold toasymplectic leafthere arises a(unique uptodiffeomorphism) local Poisson structureéthe so-called transverse Poisson structure (cf.A.Weinstein, Thelocal structure of Poisson manifolds, J.Diff. Geom. 18(1983), 523—557).1“ Inthetransverse structure, thePoisson brackets ofallfunctions arezero atthedistinguished point (which may betaken astheorigin ofacoordinate system). TheTaylor series forthese brackets begin with {XI-, :ZLCL-Xk +..., 12‘Translator’s note: Theterm Clebsch variables isalsoused torefer tocanonical coordinates onasymplectic manifold which projects onto (rather than embedding into) aPoisson manifold. 1”Warning: AsA.B.Givental’ hasnoted, Theorem 3.1inthispaper isincorrect. (Translator’s note: Forfurther discussion, seeA.Weinstein, Liealgebras andPoisson structures, Astérisque, hors série (1985), 257-271.) 458 Appendix 14:Poisson structures where c,-‘fjarethestructure constants ofaﬁnite-dimensional Liealgebra (the linearized transverse structure). Anatural question arises: Isitpossible toannihilate thehigher order terms intheTaylor series byasuitable change ofcoordinates? Thequestion oftheform oftransverse structures wasalready raised bythe author inSection VI.l.3 ofthepreviously cited article. Ifthelinearized algebra issemisimple andthePoisson structure isanalytic, then onecaneliminate thehigher order terms oftheTaylor series byan analytic change ofcoordinates: J.Conn, Linearization ofanalytic Poisson structures, Annals ofMath. 119(1984), 577—60l. Ananalogous result istrue fortheC°°case, when thelinearized algebra isofcompact type: J.Conn, Linearization ofC°°Poisson structures, Annals ofMath. (1985). A.Weinstein, along with hisearlier proof ofananalogous result forformal series, expressed theconjecture thatsemisimplicity wasanecessary condition fortheannihilation ofnonlinear terms. Thestudy ofsingularities ofPoisson structures intheplane (or,more generally, structures with symplectic leaves ofcodimension 2)leads, however, toadifferent conclusion. CPoisson structures intheplane From thepoint ofview ofdifferential geometry, aPoisson structure isgiven byasmooth bivector ﬁeldonamanifold. Infact,thePoisson brackets ateach point associate anumber toeach pair ofcotangent vectors. Therefore they deﬁne asection ofthesecond exterior power ofthetangent bundle, i.e.,a bivector ﬁeld. TheJacobi identity expresses asortof“closedness” ofthisbivector ﬁeld. Onatwo-dimensional manifold, thisclosedness condition isautomatically satisﬁed everywhere, sothatevery smooth bivector ﬁeld ontheplane gives a Poisson structure. This circumstance allows onetoapply totheclassiﬁcation ofPoisson structures intheplane theusual considerations ofgeneral position (transversality, etc.). Interms ofcoordinates x,y,abivector ﬁeld may be expressed intheform f(6,/\6,),where fisasmooth function. Thecorre- sponding Poisson structure isdeﬁned bythecondition (1) {Xiy}=f(X,y)- APoisson structure ontheplane mayalsobegiven byadifferential 2-form dxAdy/f.This form, likethebivector ﬁeld, isinvariantly connected with the Poisson structure; however, unlike thebivector ﬁeld, ithaspole singularities along thecurve f=0.Theleaves inthiscasearethepoints ofthecurve f=0 andtheconnected components ofthecomplement ofthiscurve intheplane. Points ofthecurve f=0arecalled singular points ofthePoisson structure. Intheneighborhood ofanonsingular point, anyPoisson structure intheplane may beputintothenormal form {x,y}=1. Thefollowing diagram shows thebeginning ofthehierarchy ofsingularities ofPoisson structures ontheplane intheneighborhood ofasingular point. 459 Appendix 14:Poisson structures A0‘ o/iii" A2‘ Ag‘ A4“ Ag‘ A5‘ A91‘ As H v::*"<1>2<vs-':<1>#<De"-~ E6<—E‘-1-<—-E3. Each letter inthediagram represents aPoisson structure which, insuitable local coordinates with origin atthesingular point under consideration, can bewritten intheform {x,y}=f,where thefunction fisgiven byTable 1. Table 1 A0 A21¢ A‘it—1 Dgib 2 2k+l x2 iylk xly iylk-1 x+ — y y 1+ay"" 1+ax+by" Dim E6 E? Es x2y+y2x x3+xy3 Ilia "“">" W "W5 Theorem. Given aPoisson structure onatwo-dimensional manifold, itiseither reducible inaneighborhood ofeach point tooneofthenormal forms inTable 1,oritbelongs toasetofcodimension 8inthespace ofPoisson structures. Thus, ageneric Poisson structure may bereduced inaneighborhood of each point tothenormal form {x,y}=l(nonsingular point) or{x,y}=y (point oftype A0). Inageneric one-parameter family, oneencounters for special values oftheparameter structures ofthetype A1:{x,y}=b(x2 iyz), bab0;intwo-parameter families oneﬁnds A2,etc. Remark 1.Inthetwo-dimensional case, thesetofallPoisson structures forms alinear space, sothatonemay speak ofageneric structure orfamily of structures (having inmind astructure [family] belonging tosome open dense subset ofthespace ofstructures [families]). Theproblem ofclassifying generic Poisson structures inthree ormore dimensions isnotuniquely posed, since thesetofallsuch structures does notform asingle manifold (one may ﬁnd components of“different dimensions,” asintheclassiﬁcation ofLiealgebras). Remark 2.The structure {x,y}=yoftype A0isthestandard Poisson structure onthedual space oftheLiealgebra ofthegroup ofaffine transforma- tions oftheline.This structure wasconsidered in1965, inconnection with the 460 Appendix l4:Poisson structures study oftheEuler equations forleft-invariant metrics ongroups (inthis case—-the Lobachevskii metric onahalf-plane), atwhich time itwasalready realized thatthestructure isstable andislocally equivalent toanystructure ofthe form {x,y}=y+---,where thedotsdesignate higher order terms. This (evident) observation contradicts thepreviously mentioned conjecture ofA. Weinstein, according towhich thepossibility ofremoving anyhigher order terms byaformal change ofcoordinates wascharacteristic ofthelinear Poisson structures onthedual spaces ofsemisimple Liealgebras. Remark 3.Theparameters a,binthetable above aremoduli (invariants depending continuously onthestructure). More precisely, structures equivalent toagiven onearefound only aﬁnite number oftimes astheparameters are varied. Therational functions inTable lmay bereplaced bypolynomials, butit isnotvery convenient todoso.Thenumber ofmoduli inthenumerator is onelessthan thenumber ofirreducible components ofthecurve f=0.This is notmerely acoincidence. One invariant ofaPoisson structure ontheplane istheresidue constructed from theform dx/\dy/f (initially, oneconstructs aresidue-form oneach component, then itsresidue attheorigin). Thesumof theresidues corresponding toallthecomponents iszero. Therefore the number ofmoduli is1lessthan thenumber ofcomponents. DPowers ofvolume forms Theclassiﬁcation ofPoisson structures ontheplane may beconsidered as theclassiﬁcation ofdifferential forms ofthetype f(dx/\dy)“, where fis asmooth (orholomorphic) function. More generally, itisnatural toconsider forms ofthetype (2) f(dx)“ =.f(xlv":sXn)(d-X1 /\/\dX..)°'. where atisaﬁxed number, generally complex. Theclassiﬁcation ofsuch forms andtheir deformations intheone-dimensional case, recently carried outby V.P.Kostov, revealed theroleofresonance values ofat(certain negative rational numbers). Forexample, theresonance case n=l,at=—lcorresponds totheclassi- ﬁcation ofthesingularities andtheir bifurcations forvector ﬁelds ontheline, i.e.,singular points ofdifferential equations at=v(x)andtheir bifurcations in ﬁnite-parameter families. Ageneric one-parameter family may bereduced by asmooth (holomorphic) change oftheparameter andasmooth (holomorphic) change ofthevariable x,depending smoothly (holomorphically) onthe parameter, totheform >2=x2+e+c(e)x3. (For kparameters, thecorre- sponding form isX=x"+1 +e1x"“ + +2,,+c(a)x2"*‘.) Thenonresonance casewasstudied byS.Lando forallnandat:heshowed thatalmost every versal deformation ofthefunction fdeﬁnes, after multiplica- tionby(dx)"‘, aversal deformation oftheform, aslong asatisnotaresonance value. 461 Appendix 14:Poisson structures Thecaseoi=—1,which isinteresting inconnection with Poisson structures, isgenerally aresonance case. Instead ofpowers ofvolume fomis, asin(2), wemay consider thedifferential forms fﬂdx> B=1/ai whose classiﬁcation isobviously equivalent. The hypersurface f=0isinvariantly connected with theform (3).The classiﬁcation therefore begins with thereduction tonormal form ofthesingu- larity manifold f=0.Thebeginning ofthehierarchy ofsingular points of hypersurfaces isknown. Insuitable local coordinates, ahypersurface isgiven byoneoftheequations inthefollowing list: A“: Di:xixiixr‘ ext:---¢x5=0. #24;iXi‘+‘iX§i'"iX§=0, #20; E6: xi‘+x§ix§i-~ix§=O; E7: x{+x,x§ix§i---ix,‘,’=0; E3: x:1‘+x§ix§i---ix§=0. After wehave brought thehypersurface intonormal form, theclassiﬁcation oftheforms (2)or(3)comes down toclassifying forms ofthetype (4) f”l1(Xi,---,X,.) dxi M0)it0, where f=0isthegiven equation ofthesingularity hypersurface andhis asmooth (holomorphic) function which remains tobeputinnormal form. EThequasi-homogeneous case Weshall consider here thecaseinwhich thesingularity hypersurface f=0is quasi-homogeneous (this condition holds forthecases A,D,E). Deﬁnition. Afunction fiscalled quasi-homogeneous ofweight p,with weights w,attached tothevariables x,-,ifitisaneigenfunction with eigenvalue pfor thequasi-homogeneous Euler vector ﬁeld e(oriszero): af=pf, where e=Zwix,-(6/6x5). Aquasi-homogeneous polynomial iscalled nondegenerate ifthecritical point 0hasﬁnite multiplicity (i.e.,itisCisolated). From here on,wewilltake the weights w,tobepositive numbers. Theorem. Letfbeanondegenerate quasi-homogeneous polynomial ofweight 1. Then thediﬂerentialform fﬂhdx(where dx=dxl/\---/\dx,,andhisaholo- morphic function onaneighborhood of0)maybereduced byabiholomorphic coordinate change inaneighborhood ofzerototheform ff’(1+¢)dx,1*.-.'~ere ¢isaquasi-homogeneous polynomial ofweight —[i—o,0=wl+ +w,,. 462 _4 Appendix 14:Poisson structures Theweight of¢ischosen sothattheweight oftheform f"¢dxiszero. Ananalogous theorem istrueforsmooth h(andsmooth coordinate changes), except thatintherealcaseonemust replace 1+¢>byi1+<15. EXAMPLE 1.IfBispositive, then ¢E0,sothat thecomplex form reduces toflldx. More generally, ¢EOifthe(possibly complex) number Bisnotanegative rational number: inthiscase, anonzero quasi-homogeneous polynomial of weight —B—0does notappear. Ifthepolynomial f(orjust itsquasi- homogeneity type w)isﬁxed, then theresonance values ofBform aﬁnite set ofarithmetic progressions inthenegative rationals (fortheremaining B, ff’hdxreduces totheform flldx). EXAMPLE 2.IfB=—1,then themonomials occurring in¢maybeenumerated bytheinterior integral points oftheNewton diagram off.Themonomial x"‘=x["‘...x,§""corresponds tothepoint (ml+l,...,m,, +1)ofthediagram (i.e.,theexponent oftheform x'"dx). EXAMPLE 3.Suppose thatB=—1,n=3,andfisoneoftheA,D,Epolynomials introduced above, deﬁning asimple singularity. Calculating weights, weﬁnd that —B—0<0;therefore ¢E0,from which weobtain: Corollary I.Theform withpolesingularity d h(x,y,2)dxAyAdz, M0)¢0, f(X,y!Z) where fisoneofthepolynomials A,D,E,may bereduced totheform dxAdyAdz/fbyaholomorphic (smooth) change ofcoordinates. Inexactly thesame wayforanynZ3,afactor h(x,,...,x,,)which does not vanish attheorigin canbeconverted tounity. Corollary 2.Asimple form (i.e.,onenothaving moduli) ofthetypedxlA---A dx,,/f(x,,. ..,x,,),where fisaholomorphic (smooth) function near theorigin andn>2,maybereduced byacoordinate change inaneighborhood ofthe origin toanormal form inwhich fiseither 1oroneoftheA,D,Epolynomials. Corollary 3.Asimple (nothaving moduli) n-vector ﬁeld inn-dimensional space (n>2)islocally equivalent toanormal form f-(61A A6,),where fis either 1oroneofthe A,D,Epolynomials; 6,,=6/6x,,. Corollary 4.Forlg6,ingeneric l-parameter families ofn-vector ﬁelds on n-dimensional space (n>2),theﬁeld inaneighborhood ofeach point andfor each value oftheparameters isequivalent tooneofthesimple ﬁelds inthe preceding corollary. 463 Appendix 14:Poisson structures Corollary 5.Forl56,ingeneric l-parameter families offorms dxAdyAdz/ f(x,y,z),oneﬁnds onlyforms which intheneighborho locally equivalent tooneofthefollowing 24types: dxAdyAdz x2+y2iZ2’dxAdyAdz dxAdyAdz 1 7 x 7 dxAdyAdz x2yiy3+z2’dxAdyAdz dxAdyAdz x4iy2—_tz2’ x5+_V2iz2’ dxAdyAdz dxAdyAdz dxAdyAdz x2y_|_y4iZ2i x'r+y2iz2> x2yiys‘+z2’odofeach point are dxAdyAdz x3+y2iz2’ dxAdyAdz X6iy2iz2’ dxAdyAdz x3+y“iz2' Forn=2andB=—1,thetheorem may beapplied inthefollowing way. Corollary 6.Letfbeanondegenerate quasi-homogeneous polynomial ofweight 1withargument weights wl,W2.Then theform ﬂ§ ;,(()())¢() f(X,y) where hisasmooth (holomorphic) function inaneighborhood of0,canbe reduced byasuitable smooth (holomorphic) coordinate change toaform in which h=if+ql,where ¢isaquasi-homogeneous polynomial ofweight 1—w,—W2 Correspondingly, bivector ﬁelds and Poisson structures may belocally reduced totheform f(X,y)(@..A<9.) _f(X,y):1+¢<><.y> ’{"”’}‘ i1+¢(><.y)' Calculating theweights ofthesimple singularity types oftwovariables, weobtain Table 1from thelastcorollaA,D,Eforfunctions ry.Forexample, for A,wehave wl=W2=theweight ofqbequals 0,andso¢isconstant. Thedimension ofthespace ofequivalence classes offorms hdxAdy/f, where h(t))ale0andfisaﬁxed nondegenerate quasi-homogeneous polynomial, equals thedimension ofthespace ofquasi-homogeneous pol FVarchenko’s theorem A.N.Varchenko hasproven aseries ofgeneralizatio theorem. Here weshall describe thesimplest ofthese.ynomials ofweighto. nsofthepreceding 1.Letfbeaquasi-homogeneous polynomial ofweight 1inthevariables x1,...,x,, withweights w,,...,w,,.Suppose that, forsome setIofmulti-indices, theresidue classes ofthemonomials x’generate (asavector space) thefactor algebra ofthealgebra offormal power series <lI[[x,, ...,x,,]]/(6f/6x,,. ..,6f/6x,,). 464 Appendix 14:Poisson structures Theorem. Every germ fllhdxisequivalent toagerm oftheform f/‘(l + Z/l.,,,,,x"‘f‘) dx,where thel’sarenonnegative integers andthem’sareele- ments ofIsuch thattheweight ofeach form f"x'"f' dxisequal tozero. 2.Wedeﬁne thedegree ofnon-quasi-homogeneity ofthegerm ftobethe dimension ofthefactor space (f,6f/dx,, ...,6f/6x,,)/(6f/6x,,..., 6f/6x,,). Theorem. Foralmost allB,thenumber ofmoduli oftheform fﬂhdx,A Adx,, (forﬁxed Bandfandarbitrary h,h(0)950)isequal tothedegree of non-quasi-homogeneit yofthegerm f.Theexceptional (resonance) values of Bconsist ofaﬁnite number ofarithmetic progressions ofnegative rational numbers, with diﬂerence —l.Inparticular, foranyBZ0,thenumber of moduli equals thedegree ofnon-quasi-homogeneity. 3.EXAMPLE. ForB=0,weobtain: Corollary. Thenumber ofmoduli oftheform hdx(h(0) aé0),relative tothe group ofdiﬂeomorphisms preserving thegerm off,equals thedegree of non-quasi-homogeneity off(equal tozero, ifthegerm offisequivalent toa quasi-homogeneous one). 4.Intheresonance cases, theresult ismore complicated. EXAMPLE. Letn=2,B=—1(Poisson structures intheplane). Theorem. Thenumber ofmoduli foragerm ofaPoisson structure with given singular curvef =0equals thedegree ofnon-quasi-homogeneity ofthegerm offaugmented byonelessthanthenumber ofirreducible components ofthe germ ofthecurve f=O. Inresonance cases, thenumber ofmoduli behaves inarather regular wayalong each arithmetic progression with difference —l.Namely, when B decreases by1thenumber ofmoduli increases (notnecessarily strictly), but itsmaximal value does notexceed (foranyB>—n)the“nonresonant” value (i.e.,thedegree ofnon-quasi-homogeneity off)bymore than thenumber ofJordan blocks associated with theeigenvalue em" ofthemonodromy operator ofthefunction f. GPoisson structures andperiod mappings Aninteresting source ofPoisson structures isprovided bytheperiod mappings ofcritical points ofholomorphic functions (A.N.Varchenko and A.B. Givental’, Mapping ofperiods andintersection form, Funct. Anal. Appl. 16, (1982), 83-93). Period mappings allow onetotransfer tothebase ofaﬁbre bundle certain structures which liveonthe(co)homology spaces oftheﬁbres. APoisson 465 Appendix 14:Poisson structures structure onthebase arises inthisway from theintersection form inthe middle-dimensional homology oftheﬁbres, when thisform isskew-symmetric. Period mappings aredeﬁned bythefollowing construction. Suppose that oneisgiven alocally trivial ﬁbration. Associated tosuch aﬁbration arethe bundles (over thesame base) ofhomology andcohomology oftheﬁbres with complex coefficients. These bundles arenotonly focally trivial, butthey are locally trivialized inacanonical way(theinteger cycles inaﬁbre areuniquely identiﬁable with integer cycles inthenearby homology ﬁbres). Aperiod mapping isdeﬁned asasection ofthecohomology bundle. Suppose now thatoneisgiven, onthetotal space ofadifferentiable ﬁbre bundle, adifferential form which isclosed oneach ﬁbre. Theperiod mapping ofthisform associates toeach point ofthebase thecohomology class ofthe form ontheﬁbre over thispoint. Ifoneisgiven avector ﬁeld onthebase oftheﬁbration, then any(smooth) period mapping maybedifferentiated along thisvector ﬁeld, andthederivative isagain aperiod mapping. Infact, neighboring ﬁbres ofthecohomology bundle areidentiﬁed with oneanother bytheabove-mentioned “integer” local trivialization, soasection may beconsidered (locally) asamap intooneﬁbre andmay bedifferentiated asanordinary (vector-valued) function. Suppose nowthatthebase isacomplex manifold having thesame complex dimension astheﬁbres ofthecohomology bundle. Aperiod mapping iscalled nondegenerate ifitsderivatives along anyC-independent vectors ateach point arelinearly independent. Inother words, aperiod mapping isnon- degenerate ifthecorresponding local maps from thebase totypical ﬁbres are diffeomorphisms. Thederivative ofanondegenerate period mapping thus allows ustomap thetangent bundle ofthebase isomorphically onto thecohomology bundle. The dual isomorphism goes from thehomology bundle tothecotangent bundle ofthebase. This isomorphism transfers tothebase anyadditional structures carried bythehomology groups. Suppose that theﬁbres ofouroriginal bundle are(real) oriented even dimensional manifolds, andconsider their homology inthemiddle dimension. Inthiscase, thehomology ofeach ﬁbre carries abilinear form: theindex of intersection. This form issymmetric ifthedimension oftheﬁbre isamultiple of4;otherwise, itisskew-symmetric. Theform isnondegenerate iftheﬁbre is closed (i.e.,compact andwithout boundary); otherwise, itmaybedegenerate. Weshall suppose below that weareinthesituation where theform is skew-symmetric. Inthissituation anondegenerate period mapping induces aPoisson structure onthebase. Infact,theisomorphism described above, between thecotangent spaces ofthebase and thehomology groups oftheﬁbres (carrying their skew-symmetric intersection forms), deﬁnes askew-symmetric bilinear form onpairs ofcotangent vectors. The Poisson bracket oftwofunctions onthe base isdeﬁned asthevalue ofthisform onthedifferentials ofthefunctions. This bracket deﬁnes aPoisson structure (ofconstant rank) onthebase. 466 Appendix 14:Poisson structures _AJ AI 47 Figure 247 Poisson structure andtheswallowtail This isobvious from thefactthat thelocal identiﬁcation ofthebase with thecohomology ofthetypical ﬁbre, given bytheperiod mapping, provides the base with local coordinates whose Poisson brackets areconstant.123 Varchenko andGivental’ observed thatifoneconstructs, inthewayjust described, using ageneric 1-form, aPoisson structure onthecomplement of thediscriminant locus inthebase ofaversal deformation ofacritical point ofafunction oftwovariables, then thisstructure may beholomorphically extended across thediscriminant locus. (One may replace thediscriminant locus above bythewave front ofatypical singularity.) Weshall limit ourselves heretothesimplest examples ofPoisson structures arising inthisway. Consider thethree-dimensional space ofpolynomials C3={x4+/llxz + 2.2x+23}with coordinates ilk.Thepolynomials with multiple roots form therein thediscriminant surface (aswallowtail; seeFigure 247). The Poisson structures arising from period mappings may bereduced (bydiffeomorphisms preserving theswallowtail) tothefollowing form: the symplectic leaves aretheplanes /12=const., andtheir symplectic structures areofthe form dI.,AdI.3. The ﬁbration ofinterest here isformed bythecomplex curves {(x,y): yz=x4+/llxz +2.2x+2.3},andtheperiod mapping isgiven by,forexample, theform ydx. (See V.I.Arnold, A.N.Varchenko, S.M.Gusein-Zade, Singularities ofDiﬂerentiable Mappings, Vol.2:Monodrom yandtheAsympto- ticsofIntegrals, Birkhéiuser, 1988, §l5, orUspekhi Mat. Nauk 40,no.5 (1985).) '23Inthecasewhere theintersection form issymmetric, theanalogous construction deﬁnes on thebaseaﬂatpseudo-riemannian (possibly degenerate) metric. 467 Appendix 14:Poisson structures ThePoisson structures ontheswallowtail space which arise from period mappings may becharacterized locally among allgeneric structures bythe following property: thelineofself-intersections ofthetailliesentirely inone symplectic leaf.Therequired genericity condition isthatthetangent planes at theorigin tothesymplectic leafandtheswallowtail donotcoincide. Every smooth function which isconstant along thelineofself-intersections ofthe tail,andwhose derivative along thesymplectic leafattheorigin isnonzero, maybereduced inaneighborhood oftheorigin, byadiffeomorphism preserv- ingthetail,totheform 12+const.; also, afamily ofholomorphic symplectic structures intheplanes /12=const. may bereduced totheform d/ll/\dig byaholomorphic local diffeomorphism ofthree-dimensional space which preserves theswallowtail aswellasthefoliation bytheplanes. One may conjecture more generally that those Poisson (inparticular, symplectic) structures onthebase ofaversal deformation ofasingularity, in- duced from theintersection form byaninﬁnitesimally stable period mapping, may becharacterized (uptodiffemorphisms preserving thebifurcation set)by anatural condition ontherank oftherestricted Poisson structure tothestrata ofthediscriminant locus. The“natural condition” inthethree-dimensional example above isthat theline ofself-intersections oftheswallowtail be contained inasymplectic leaf.Infour-dimensional space, ananalogous role would apparently beplayed bythecondition thatacertain submanifold be lagrangian, namely, themanifold ofpolynomials having twocritical points with critical value zero inthesymplectic space ofpolynomials x5+/l1x3 + /12x2 +/13x+/14(theranks ofthesymplectic structure onthetangent spaces totheother strata may alsobeimportant). 468 Appendix 15:Onelliptic coordinates Asystem ofJacobi’s elliptic coordinates isassociated toeach ellipsoid in euclidean space. These coordinates make itpossible tointegrate theequations ofgeodesics onthegiven ellipsoid, aswellascertain other equations, such as theequations ofmotion forapoint onasphere under theinﬂuence ofaforce with quadratic potential, orforapoint onaparaboloid under theinﬂuence ofauniform gravitational ﬁeld. These facts suggest that, even onaninﬁnite-dimensional Hilbert space, there should beaclass ofintegrable systems associated toeach symmetric operator. Tostudy these systems, itisnecessary toextend thetheory ofelliptic coordinates totheinﬁnite-dimensional case. Todothis,itisﬁrstnecessary to express theﬁnite-dimensional theory ofconfocal quadric surfaces incoordinate freeform. Inthetransition totheinﬁnite-dimensional case, symmetric operators on ﬁnite-dimensional euclidean spaces must bereplaced byself-adjoint operators onHilbert spaces. Since theelliptic coordinates arenotreally connected with theoperator itself, butrather with itsresolvent, theunboundedness ofthe original operator (which might be,forexample, adifferential operator) does notpresent aserious obstacle. Insome cases, theelliptic coordinates onHilbert space obtained from a self-adjoint operator form acountable sequence; however, when theoperator hasacontinuous spectrum, thecoordinates form acontinuous family. Inthis case, thetransformation from theoriginal point oftheHilbert space (thought ofasafunction space) tothecontinuous family ofelliptic coordinates ofthe point may beconsidered asanonlinear mapping between function spaces. This mapping, byanalogy with theFourier transform, might becalled the Jacobi transform: theoriginal function istransformed intoafunction which expresses theelliptic coordinates interms ofsome continuous “index.” (More precisely, theresult ofthetransform isameasure onthespectral parameter axis.) Thestudy ofthefunctional analytic properties andtheinversion ofthe Jacobi transform willprobably beaccomplished before toolong. Following anexposition ofthegeneral theory ofelliptic coordinates, we shall describe below some oftheapplications ofthese coordinates topotential theory. This appendix isbased onthefollowing papers bytheauthor. Some remarks onelliptic coordinates, Notes oftheLOMI Seminar (volume dedicated toL.D.Faddeev onhis50thbirthday), 133(1984), 38-50. Integrability ofhamiltonian systems associated with quadrics (after J. Moser), Uspekhi 34,no.5,214. Some algebro-geometrical aspects oftheNewton attraction theory, Pro- gress inMath. (I.R.Shafarevich volume), 36(1983), 1-4. Magnetic analogues ofthetheorem ofNewton andIvory, Uspekhi 38, no.5(1983), 145-146. Further details onbackground material fortheresults inthisappendix may befound inthefollowing papers. 469 Appendix 15:Onelliptic coordinates R.B.Melrose, Equivalence ofglancing hypersurfaces, Invent. Math. 37 (1976), 165-191. J.Moser, Various aspects ofintegrable Hamiltonian systems, in;J.Gucken- heimer and S.E.Newhouse, eds. “Dynamical systems”, CIME Lectures, Bressanone, Italy, June 1978, Cambridge, Mass., Birkhéiuser, Boston, 1980, pp.233-289. V.I.Arnold, Lagrangian manifolds with singularities, asymptotical ofrays, andunfoldings oftheswallowtail, Funct. Anal. Appl. 15(1981). V.I.Arnold, Singularities invariational calculus, J.Soviet Mathematics 27 (1984), 2679-2713. A.B.Givental’, Polynomial electrostatic potentials (Seminar report, in Russian), Uspekhi Mat. Nauk 39,no.5(1984), 253-254. V.I.Arnold, OntheNewtonian potential ofhyperbolic layers, Selecta Math. Sovietica 4(1985), 103-106. A.D.Vainshtein andB.Z.Shapiro, Higher-dimensional analogs ofthe theorem ofNewton andIvory, Funct. Anal. Appl. 19(1985), 17-20. AElliptic coordinates andconfocal quadrics Elliptic coordinates ineuclidean space aredeﬁned with theaidofconfocal quadrics (surfaces ofdegree two). Thegeometry ofthese quadrics isobtained from thegeometry ofpencils ofquadratic forms ineuclidean space (i.e.,from thetheory ofprincipal axesofellipsoids orfrom thetheory ofsmall oscillations) byapassage tothedual space. Deﬁnition 1.Aeucildean pencil ofquadrics (resp. quadratic forms) inaeuclidean vector space Visaone-parameter family ofsurfaces ofdegree two %(A,1x, x)=1 (resp. forms AA),where A,1=A—AE (E=“identity”), andwhere Aisasymmetric operator A:V—>V*, A*=A. Deﬁnition 2.Aconfocal family ofquadrics inaeuclidean space Wisafamily ofquadrics dual tothequadrics ofaeuclidean pencil inW*: %(A;1t.t)=1- Thus, quadrics which areconfocal tooneanother form aone-parameter family, butthequadratic forms deﬁning thefamily donotdepend linearly on theparameter. EXAMPLE. Thefamily ofplane curves which areconfocal toagiven ellipse consists ofallthose ellipses andhyperbolas with thesame foci.InFigure 248, 470 Appendix 15:Onelliptic coordinates 1 Figure 248 Aconfocal family andthecorresponding euclidean pencil thecurves ofaconfocal family areshown ontheleft,andthecurves ofthe corresponding euclidean pencil areshown ontheright. Theelliptic coordinates ofapoint arethevalue oftheparameter Aforwhich thecorresponding quadrics ofaﬁxed confocal family pass through thepoint. Weﬁxanellipsoid ineucildean space with allitsaxes ofdifferent lengths. Theorem l(Jacobi). Through each point ofann-dimensional euclidean space there pass nquadrics confocal toagiven ellipsoid. Smooth confocal quadrics intersect atright angles. PROOF. Each point other than 0inourspace corresponds toanaﬂine hyper- plane inthedual space, consisting ofthose linear functionals whose value is Iatthegiven point. Interms ofthedual space, Theorem 1means thatevery hyperplane notpassing through 0inann-dimensional euclidean space is tangent toprecisely nofthequadrics inaeuclidean pencil, andthevectors from 0tothepoints oftangency arepairwise orthogonal (Figure 248,right). Theproof oftheproperty ofeuclidean pencils juststated isbased onthe factthat theaforementioned vectors deﬁne theprincipal axes ofthequa- dratic forms B=%(Ax, x)—%(l,x)2,where (l,x)=1istheequation ofthe hyperplane. Asamatter offact, onaprincipal axisofanyquadratic form B,corre- sponding totheproper value ll,theform B—/lEreduces to0aiong with its gradient. The vanishing ofthisform atthepoint ofintersection ofthe principal axisandthehyperplane means thatthepoint ofintersection lieson thequadric §(Ax, x)=1,while thevanishing ofthegradient means thatthe quadric andthehyperplane aretangent atthepoint. [1 Theorem 2(Chasles). Given afamily ofconfocal quadrics inn-dimensional euclidean space, alineingeneralposition istangent tort—1diﬂerent quadrics inthefamily, andtheplanes tangent tothequadrics atthepoints oftangenc y arepairwise orthogonal. 471 Appendix 15:Onelliptic coordinates PROOF. Weproject thequadrics intheconfocal family along apencil ofparallel lines onto thehyperplane perpendicular tothepencil. Each quadric deﬁnes an apparent contour (thesetofcritical values oftheprojection ofthequadric). Foraprojection whose direction isingeneral position, theapparent contour isaquadric (i.e.,asurface ofdegree two) intheimage hyperplane. Here weneed alemma. Lemma. Theapparent contours ofthequadrics inaconfocal family form themselves aconfocal family ofquadrics. PROOF. Onpassage tothedual, sections become projections andviceversa. Theapparent contours oftheprojections ofconfocal quadrics along apencil ofparallel lines aretherefore dual tothesections ofthedual quadrics bya hyperplane passing through theorigin. Thesections ofthequadrics inaeuclidean pencil byahyperplane through 0form aeuclidean pencil ofquadrics inthehyperplane. The lemma now follows byduality. El Returning totheproof ofTheorem 2,weapply thelemma above tothe projections along thelineinthestatement ofthetheorem. According tothe lemma, theapparent contours oftheprojections oftheconfocal quadrics in Theorem 2form aconfocal family ofquadrics inahyperplane. ByTheorem 1, n—1ofthese apparent contours passthrough each point, where theyintersect atright angles. This completes theproof ofTheorem 2. E] Theorem 3(Jacobi and Chasles). Given ageodesic onaquadric Qinn- dimensional space, there isasetofn—2quadrics confocal toQsuch thatall thetangent lines tothegeodesic arealsotangent tothequadrics intheset. PR00r (Beginning). Weconsider themanifold oforiented lines ineuclidean space. This manifold hasanatural symplectic structure asthemanifold of characteristics inthehypersurface p2=1inthephase space ofafreeparticle moving under itsown inertia inoureuclidean space. (The characteristics onahypersurface inasymplectic manifold arethe integral curves oftheﬁeldofcharacteristic directions, i.e.,theﬁeldofdirections which areskew-orthogonal tothetangent spaces ofthehypersurface. Inother words, thecharacteristics ofthehypersurface arethephase curves forany hamiltonian ﬂow whose hamiltonian function vanishes toﬁrstorder onthe hypersurface. Thesymplectic structure onthemanifold ofcharacteristics onahyper- surface inasymplectic manifold isdeﬁned insuch awaythattheskew-scalar product ofanytwovectors tangent tothehypersurface isequal totheskew- scalar product oftheir projections inthemanifold ofcharacteristics. Note, ﬁnally, that thenotion ofcharacteristics isequally welldeﬁned for 472 Appendix 15:Onelliptic coordinates anysubmanifold ofasymplectic manifold onwhich theinduced 2-form has constant nullity. Thecharacteristics then have dimension equal tothatnullity, andthemanifold ofcharacteristics stillinherits asymplectic structure.) lj Lemma A.Each characteristic ofthemanifold oflines tangent toagiven hypersurface ineuclidean space consists ofallthelines tangent toasingle geodesic onthehypersurface. PROOF orLEMMA A.Foreﬁiciency ofexpression, wewillidentify thecotangent vectors toeuclidean space withtangent vectors byusing theeuclidean structure, sothatouroriginal phase space isrepresented asthespace ofvectors based atpoints ofeucildean space (i.e.,momenta areidentiﬁed with velocities). The unitvectors tothegiven hypersurface form asubmanifold ofoddcodimension (equal to3)inphase space. Thecharacteristics ofthissubmanifold deﬁne the geodesic ﬂow onthehypersurface. Themap which assigns toeach vector thelineinwhich itliestakes the codimension 3submanifold justdescribed tothemanifold oflines tangent tothehypersurface. Under thismapping, characteristics aretransformed to characteristics (with respect tothesymplectic structure onthespace oflines). This proves thelemma. [I] [Remark. Thepreceding argument may beeasily extended tothefollowing general situation, ﬁrstconsidered byMelrose. LetYandZbeapairofhy- persurfaces inasymplectic manifold Xwhich intersect transversally along a submanifold W.Weconsider themanifolds ofcharacteristics BandCofthehy- persurfaces YandZtogether with thecanonical quotient ﬁbrations Y—>—>B andZ—>->C;themanifolds BandCinherit symplectic structures from X. Intheintersection W,there isadistinguished hypersurface (ofcodimension 3inX)consisting ofpoints atwhich therestriction toWofthesymplectic structure onXisdegenerate. This hypersurface ZinWmay alsobedeﬁned asthesetofcritical points ofthecomposed mapping WQ>Y ->->B (or WC->Z ->->Cifonewishes). These objects form thefollowing commutative diagram: 2n-l (/ X2"‘\ 2n—1 Yl \ Wilt:-2‘/Yzj B2n—2/ J$232»-2 \ z2n—3/ The analogue toLemma Ainthissituation istheassertion that the characteristics ontheimages ofthemappings Z—>BandZ—>Caretheimages ofoneandthesame curve onZ(namely, thecharacteristics ofZconsidered asasubmanifold ofthesymplectic manifold X). Lemma Aitself isthespecial caseoftheassertion above inwhich X=R2" 473 Appendix 15:Onelliptic coordinates (thephase space ofafreeparticle inR”),thehypersurface Yconsists oftheunit vectors (given bythecondition p2=1,i.e.,alevel surface ofthehamiltonian forafreeparticle), andthehypersurface Zconsists ofthose vectors which are based atthepoints ofthegiven hypersurface inR".Inthiscase, Bisthe manifold ofalloriented lines ineuclidean space, andZisthemanifold ofunit vectors tangent tothehypersurface. Themapping Z->Bassigns toeach unit vector thelinewhich contains it.Themanifold Cisthe(co)tangent bundle of thegiven hypersurface. Z1—>Cistheembedding intothisbundle ofitsunit sphere bundle (inother words, theembedding ofalevel surface ofthekinetic energy, i.e.,thehamiltonian formotion constrained tothehypersurface). Itisalways useful tokeep thediagram above inmind when oneisdealing with constraints insymplectic geometry.] PROOF orTHEOREM 3(Middle). Wesuppose given asmooth function on euclidean (conﬁguration) space whose restriction toacertain linehasanon- degenerate critical point. Inthissituation, thefunction willalsohave acritical point when restricted toeach nearby line;i.e.,oneach nearby line,there will beanearby point where thelineistangent toalevel surface ofthefunction. Thevalue ofthefunction atthecritical point isthusafunction (deﬁned locally) onthespace oflines. Wecallthisfunction oflines theinduced linefunction (from theoriginal point function). II] Lemma B.Iftwopoint functions ineuclidean space aresuch thatthetangent planes totheir level surfaces areorthogonal atthepoints where agiven line istangent tothese surfaces (these points being ingeneral different forthe twofunctions), thenthePoisson bracket oftheinduced linefunctions iszero atthegiven line(considered asapoint inthespace oflines). PROOF orLEMMA B.Wecalculate thederivative ofthesecond induced line function along thephase ﬂow whose hamiltonian istheﬁrstinduced function. Thephase curves fortheﬁrstinduced function, which lieonitslevel surfaces, arethecharacteristics ofthose surfaces. Alevel surface fortheﬁrstinduced function consists ofthose lines which aretangent toasingle level surface of theﬁrst point function. Each characteristic ofthissurface, according to Lemma A,consists ofthelines which aretangent toasingle geodesic onthe level surface oftheﬁrstpoint function. Foraninﬁnitesimally small displacement ofapoint onageodesic in asurface, thetangent linetothegeodesic rotates (uptoinﬁnitesimal quantities ofhigher order) intheplane spanned bytheoriginal tangent andthenormal tothesurface. Byhypothesis, thetangent plane tothelevel surface ofthe second function atthepoint where thissurface istangent toourlineis perpendicular tothetangent plane ofthelevel surface oftheﬁrstfunction. Therefore, under theabove-mentioned inﬁnitesimally small rotation, theline remains tangent tothesame level surface ofthesecond function (upto inﬁnitesimals ofhigher order). Itfollows thattherateofchange ofthesecond 474 Appendix 15:Onelliptic coordinates induced function under theaction ofthephase ﬂow given bytheﬁrstiszero attheelement inquestion ofthespace oflines, which proves Lemma B. [I PROOF orTHEQREM 3(End). Weﬁxalineingeneral position inIR".According toTheorem 2,thislineistangent ton1quadrics intheconfocal family, at n—1points. Weconstruct intheneighborhood ofeach ofthese points a smooth function, without critical points, whose level surfaces arethequadrics ofourconfocal family. Weﬁxoneofthese quadrics (the“ﬁrst”) andconsider thehamiltonian system onthespace oflines whose hamiltonian function istheﬁrstinduced linefunction. Each ofitsphase curves onaﬁxed level surface oftheham- iltonian function consists ofthetangent lines toonegeodesic ofthatquadric (Lemma A).Theremaining induced functions have zero Poisson bracket with thehamiltonian, byLemma B(since theplanes tangent totheconfocai surfaces atthepoints where theytouch onelineareorthogonal, byTheorem 2). Thus alltheinduced functions areﬁrstintegrals forthehamiltonian system generated byanyoneofthem. Since thelines tangent toageodesic ontheﬁrst quadric form aphase curve oftheﬁrstsystem, alltheinduced functions take constant values onthiscurve. That proves Theorem 3,aswellasthefollowing result. [I Theorem 4.Thegeodesic ﬂow onacentral surface ofdegree 2ineuclidean space isacompletely integrable system inthesense ofLiouville (i.e.,ithasasmany independent integrals ininvolution asithasdegrees offreedom). Remark. Strictly speaking, weproved Theorem 3only forlines ingeneral position, buttheresult extends bycontinuity totheexceptional cases (in particular, toasymptotic lines ofourquadrics). Inthesame way, Theorem 4 wasinitially proved justforquadrics with unequal principal axes, butpassage toalimit extends theresult tomore symmetric quadrics ofrevolution (aswell astononcentral “paraboloids”). BMagnetic analogues ofthetheorems ofNewton andIvory Elliptic coordinates make itpossible toextend Newton’s well-known theorem onthegravitational attraction ofasphere tothecase ofattraction byan ellipsoid. Deﬁnition. Ahomeoidal density onthesurface ofanellipsoid Eisthedensity ofalayer between Eandaninﬁnitely nearby ellipsoid which ishomothetic toE(with thesame center). Thefollowing isawell-known result. Ivory’s Theorem. Aﬁnite mass, distributed onthesurface ofanellipsoid with homeoidal density, does notattract anyinternal point; itattracts every 475 Appendix 15:Onelliptic coordinates external point thesame wayasifthemass were distributed with homeoidal density onthesurface ofasmaller confocal ellipsoid. The attraction inIvory’s theorem isdeﬁned bythelawofNewton or Coulomb: inn-dimensional space, theforce isproportional tor“"(aspre- scribed bythefundamental solution ofLaplace’s equation). Newton’s theorem onthe(non)attraction ofaninternal point carries over tothecase ofahyperbolic homeoidal layer andtothecase ofanattracting mass distributed onalevel hypersurface ofahyperbolic polynomial ofany degree. (Apolynomial ofdegree m,f(x,,...,x,,) iscalled hyperbolic ifits restriction toanylinethrough theorigin hasallitsroots real.) Ahomeoidal charge density onthezero hypersurface f=0ofahyperbolic polynomial isdeﬁned asthedensity ofahomogeneous inﬁnitesimally thin layer between thehypersurfaces f=0andf=a->0(thesigns ofthecharges being chosen sothatsuccessive ovaloids have opposite charges). [Ahomeoidal charge does notattract theorigin (noranyother point within theinnermost ovaloid), andthisproperty ispreserved ifthecharge density is multiplied byanypolynomial ofdegree atmost m—2. Generalization: Ifahomeoidal charge density ismultiplied byanypolynomial ofdegree m—2+r,thenthepotential inside theinnermost ovaloid isaharmonic polynomial ofdegree r(A.B.Givental’, 1983).] When oneattempts toﬁndaversion forhyperboloids ofIvory’s theorem ontheattraction ofconfocal ellipsoids, itturns outthat anessential roleis played bythetopology ofthehyperboloids. When passing tohyperboloids ofdifferent signatures, onemust consider, instead ofhomeoidal densities, harmonic forms ofdifferent degrees, andinstead oftheNewton orCoulomb potential, thecorresponding generalized forms-potentials given bytheBiot- Savart law. Inthesimplest nontrivial case ofahyperboloid ofonesheet inthree- dimensional euclidean space, theresult isasfollows. Thehyperboloid divides space into twoparts: “internal” and“external,” thelatter being nonsimply connected. Weconsider elliptic coordinate curves from thesystem whose level surfaces arethequadrics confocal tothegiven hyperboloid. The elliptic coordinate curves onourhyperboloid, which areobtained byintersecting with theconfocal ellipsoids (closed lines ofcurvature on thehyperboloid), arecalled theparallels ofthehyperboloid. Theorthogonal curves, obtained byintersection with thetwo-sheeted hyperboloids, arecalled themeridians. Although theelliptic coordinate system hassingularities (oneach symmetry plane ofthequadrics inthefamily), thehyperboloid issmoothly ﬁbred by theparallels (dilfeomorphic tothecircle) andmeridians (diffeomorphic to theline). Theregion inside thehyperboloidal tube isalsosmoothly ﬁbred bymeri- dians (orthogonal totheellipsoids intheconfocal family), while theam:-Ilar 476 Appendix 15:Onelliptic coordinates Figure 249 Magnetic ﬁelds generalizing thetheorems ofNewton andIvory region outside thehyperboloid issmoothly ﬁbred byparallels (orthogonal to thehyperboloids oftwosheets). Theorem. Acurrent with asuitable density, ﬂowing along themeridians ofa hyperboloid, produces amagnetic ﬁeld which iszeroinside thehyperboloidal tube, while theﬁeld intheannular exterior region isdirected along the parallels. Acurrent with asuitable density, ﬂowing along theparallels ofa hyperboloid, produces amagnetic ﬁeld which iszero intheexterior annular region, while theﬁeld inside thehyperboloidal tube isdirected along the meridians. (SeeFigure 249.) Thecurrent densities giving risetosuch magnetic ﬁelds, which generalize thehomeoidal charge densities onellipsoids, maybedescribed inthefollowing way. There areassociated toeach family ofconfocal quadrics inthree- dimensional euclidean space two“focal curves”: anellipse andahyperbola. (SeeFigure 250.) Thefocal ellipse istheboundary ofthelimiting ellipsoid of thefamily inwhich theshortest axisshrinks tozero; thefocal hyperbola arises inasimilar wayfrom thehyperboloids ofoneortwosheets. Figure 250 Focal ellipse andfocal hyperbola 477 l l l 1w l F lAppendix 15:Onelliptic coordinates Wedeﬁne ahomeoidal density onafocal ellipse inthefollowing way. To begin weconsider anynonplanar parallel, deﬁned asthenonplanar inter- section ofanellipsoid with ahyperboloid ofonesheet. Ahomeoidal density onthisparallel isdeﬁned asthedensity onaninﬁnitesimally thin “wire,” obtained byintersecting thelayer between thegiven ellipsoid andahomothetic oneinﬁnitesimally nearby with thelayer between thegiven hyperboloid and ahomothetic oneinﬁnitesimally close by,both homotheties being taken with respect tothecenter oftheconfocal family. Wenormalize thishomeoidal density ontheparallel insuch awaythatthemass oftheentire parallel isequal to1. Now weconsider thefocal ellipse asalimit ofnonplanar parallels. Itturns outthat thenormalized homeoidal densities ontheparallels have awell- deﬁned limit astheparallels approach thefocal ellipse. This limiting density iscalled thehomeoidal density onthefocal ellipse. Thehomeoidal density onafocal hyperbola isdeﬁned inananalogous way. Wemay now describe thecurrent densities referred toas“suitable” inthe theorem above onmagnetic ﬁelds. Thesurface ofahyperboloid ofonesheet isﬁbred over thefocal ellipse (theﬁbre over apoint isthemeridian which lies onthesame hyperboloid oftwosheets asthatpoint). Theﬂuxofthemeridianal current suitable forthetheorem, through anycurve onthehyperboloid, equals theintegral ofthehomeoidal density form onthe focal ellipse over theprojection ofthatcurve onto thefocal ellipse (along the hyperboloids oftwosheets). Thedensity oftheﬂow along theparallels isinduced inananalogous way from thehomeoidal density onthefocal hyperbola. Remark. Themagnetic ﬁeld oftheparallel ﬂow with theindicated density, inside thehyperboloidal tube, coincides outside each confolal ellipsoid (upto sign) with thenewtonian orcoulombian ﬁeld produced byacharge which is distributed with homeoidal density onthatellipsoid?“ Inexactly thesame way, themagnetic ﬁeld intheannular domain outside thehyperboloid ofonesheet coincides (uptosign), intheregion between the sheets ofeach confocal hyperboloid oftwosheets, with thecoulombian ﬁeld produced bytwoequal charges with opposite signs distributed onthetwo sheets ofthehyperboloid with homeoidal density (O.P.Shcherbak). Theresults formulated above have recently been extended byB.Z.Shapiro andA.D.Vainshtein tohyperboloids ineuclidean spaces ofanynumber of dimensions. Forahyperboloid inIR",dilfeomorphic toS"><IR’,aharmonic k-form isconstructed ontheexterior region (diffeomorphic totheproduct of S"with ahalf-space) andaharmonic l-form isconstructed ontheinterior. Thecorresponding homeoidal densities aredeﬁned onthefocal ellipsoid with codimension kandthefocal hyperboloid oftwosheets with codimension ‘“This isactually thedensity with which acharge willdistribute itself onthesurface ofa conducting ellipsoid. 478 Appendix 15:Onelliptic coordinates lbythesame limiting procedure that wedescribed above fork=I=1, using theintersections oflayers between inﬁnitesimally close andhomothetic quadrics. Noncomputational proofs ofthese geometric theorems areunknown, even forthespecial caseofmagnetic ﬁelds inthree-dimensional space. Remark. Thepresence ofdistinguished harmonic forms onhyperboloids andintheir complementary domains suggests that onemight trytoﬁnd ﬁltrations, analogous tothose arising inthetheory ofmixed Hodge structures, inspaces ofdifferential forms onnoncompact (and possibly even singular) algebraic andsemialgebraic realmanifolds. 479 Appendix 16:Singularities ofraysystems Thesimplest example ofaraysystem isthesystem ofnormals toasurface in euclidean space. Inaneighborhood ofasmooth surface, itsnormals form asmooth ﬁbration, butatsome distance from thesurface various normals begin tointersect one another (Figure 251). Thecomplicated ﬁgures which arethereby formed were already investigated byArchimedes, buttheir fulldetails were notrevealed until thediscovery in1972 oftherelation between singularities ofraysystems andthetheory ofgroups generated byreﬂections. This relation, forwhich there isnoevident apriori reason (and which isas surprising as,say,therelation between theproblems oftangents andareas), hasturned outtobeapowerful instrument forthestudy ofcritical points of functions. By1978, ithadbecome clear thatthetheory ofreﬂection groups alsogoverns thesingularities oftheHuygens evolvents. Huygens (1654) discovered thattheevolvent ofaplane curve hasacusp singularity ateach point where itmeets thecurve (Figure 252). Evolents of plane curves andtheir higher-dimensional generalizations arewave fronts onmanifolds with boundary. Singularities ofwave fronts, likethose ofray systems, areclassiﬁed interms ofreﬂection groups. While rays andfronts onmanifolds without boundary arerelated tothe Weyl groups intheA,D,andEseries, singularities ofevolvents aredescribed bythegroups oftypes B,C,andF(theones with double connections intheir Dynkin diagrams). Theremaining reﬂection groups (I2(p),H3,H4)continued forsome time to have novisible relation tothetheory ofsingularities. This situation changed inthefallof1982 when itwasdiscovered thatthesymmetry group H3ofthe icosahedron governs thesingularities ofevolvent systems intheneighborhood ofinﬂection points ofplane curves. Theappearance oftheicosahedron ataninﬂection point ofacurve looks asmystical astheicosahedron inKepler’s lawofplanetary distances. Butthe presence oftheicosahedron hereisnotanaccident: upon theinvestigation in 1984 ofmore complicated systems ofraysandfronts, theremaining group H4 appeared. Weshall giveinthisappendix abn'ef description ofthetheory ofsingularities ofraysystems. Further details may befound inthefollowing references: V.I.Arnold, Singularities ofraysystems, Russian Math. Surveys 38(1983). V.I.Arnold, Singularities invariational calculus, J.Soviet Math. 27 (1984), 2679-2713. O.V.Lyashko, Classiﬁcation ofcritical points offunctions onamanifold with singular boundary, Funct. Anal. Appl. l7(1983), 187-193. O.P.Shcherbak, Singularities offamilies ofevolvents intheneighborhood ofaninﬂection point ofthecurve, andthegroup H3,generated byrelections, Funct. Anal. Appl. 17(1983), 301-303. A.N.Varchenko andS.V.Chmutov, Finite irreducible groups, generated byrelections, aremonodromy groups ofsuitable singularities, Funct. Anal. Appl. 18(1984), 171-183. 480 4 Appendix 16:Singularities ofraysystems Figure 251 Acaustic astheenvelope ofrays Figure 252 Anevolvent ofacurve V.I.Arnold, Singularities ofsolutions ofvariational problems (Seminar report, inRussian), Uspekhi Mat. Nauk 39,no.5(1984), 256. O.P.Shcherbak, Wave fronts andreﬂection groups. Russian Math. Surveys, 43,no.3(1988). Itogi Nauki iTechniki, Sovremennye Problemy matematiki, Noveishie dostijenia, Moscow, VINITI, vol.33(1988). English translation: J.Sov. Math. 27(1984). Many oftheresults which wewilldescribe concern such simple geometric objects thatitissurprising thattheywere notalready known inclassical times. Forinstance, thelocal classiﬁcation ofprojections ofgeneric surfaces in three-dimensional space wasnotdiscovered until 1981. Thenumber ofequi- valence classes ofgerms ofprojections turned outtobeﬁnite—namely 14: neighborhoods ofpoints ongeneric surfaces canhave that many different appearances when viewed from different points inspace. ASymplectic manifolds andraysystems 1.Thespace oforiented lines ineuclidean space may beidentiﬁed with the(co)tangent bundle ofthesphere (Figure 253), anditthereby obtains a symplectic structure. 2.More generally, weconsider anyhypersurface inasymplectic manifold. Theskew-orthogonal complement toitstangent space ateach point iscalled 481 Appendix 16:Singularities ofraysystems Figure 253 Thespace oforiented lines ineuclidean space thecharacteristic direction. Theintegral curves oftheﬁeld ofcharacteristic directions onahypersurface arecalled characteristics. Themanifold ofchar- acteristics inherits asymplectic structure from theoriginal manifold. 3.Inparticular, themanifold ofextremals ofageneral variational problem carries asymplectic structure. 4.Weconsider thespace ofbinary forms (homogeneous polynomials in twovariables) ofaparticular odddegree. Thegroup oflinear transformations oftheplane actsonthiseven dimensional linear space. Uptomultiplication byaconstant, there isaunique nondegenerate skew-symmetric form onthis space which isinvariant under theaction ofthegroup SL(2) oflinear trans- formations with determinant equal to1.This fonn gives anatural symplectic structure onthemanifold ofbinary forms ofeach odddegree. 5.Thebinary forms inxandyforwhich thecoefficient ofx2"*1 isunity form ahypersurface inthespace ofallforms. Themanifold ofcharacteristics of thishypersurface isnaturally identiﬁed with themanifold ofmonic polynomials ofeven degree x2"+ inx.Wehave thereby deﬁned anatural symplectic structure onthisspace ofpolynomials. 6.Theone-parameter group oftranslations along thex-axis preserves the symplectic structure justintroduced. Thehamiltonian function forthisgroup isaquadratic polynomial (found already byHilbert (1893)). Themanifold ofcharacteristics foranylevel surface ofthishamiltonian function may be identiﬁed withthemanifold ofmonic polynomals ofdegree 2k—1inxforwhich thesumoftheroots iszero. Thus wehave anatural symplectic structure on thisspace ofpolynomials. BSubmanifolds ofsymplectic manifolds Therestriction ofasymplectic structure toasubmanifold isaclosed 2-fonn, butitisnotnecessarily nondegenerate. Forsubmanifolds ineuclidean space there is,inaddition totheintrinsic geometry, anextensive theory ofextrinsic curvatures. Insymplectic geometry, thesituation issimpler: 482 Appendix 16:Singularities ofraysystems Theorem (A.B.Givental’, 1981). Therestriction ofthesymplectic form toa germ ofasubmanifold inasymplectic manifold determines thegerm uptoa symplectic diﬁfeomorphism oftheambient manifold. Anintermediate theorem, inwhich oneusesthevalues ofthesymplectic form atallvectors based onthesubmanifold, notjustthose tangent toit,was proved earlier byA.Weinstein (1971). Unlike Weinstein’s theorem, Givental’s theorem makes itpossible toclassify generic submanifold germs insymplectic manifolds: itissuﬂicient tousetheclassiﬁcation ofdegenerate symplectic structures obtained byJ.Martinet (1970) andhissuccessors. EXAMPLES. 1.Ageneric two-dimensional surface insymplectic space issym- plectically diffeomorphic inaneighborhood ofeach point with thesurface p2=pf,p3=q3= =0(inDarboux coordinates). 2.Onfour-dimensional submanifolds, oneﬁnds stable curves ofelliptic and hyperbolic Martinet singular points with normal forms P2=P1Pa iqiqz +q§/6. P3=0, P4=q.t="'=0- [The ellipticity orhyperbolicity ofasingular point isdetermined bythe nature ofthedynamical system invariantly attached tothesubmanifold. The divergence-free vector ﬁelds inthree-dimensional space which arise have entire curves ofsingular points. Theclassiﬁcation ofsingular lines turns out tobelesspathological than theclassiﬁcation ofsingular points (which is almost asdifficult asallofcelestial mechanics).] This concludes adescription oftheﬁrststeps inthetheory ofsymplectic singularities onsmooth manifolds. CLagrangian submanifolds inthetheory ofraysystems Werecall thatalagrangian submanifold isasubmanifold ofsymplectic space onwhich thesymplectic structure pulls back tozeroandwhich hasthehighest possible dimension consistent with thisproperty (equal tohalfthedimension oftheambient manifold). EXAMPLES. 1.Each ﬁbre ofacotangent bundle islagrangian. 2.Themanifold of alloriented normals toasmooth submanifold (ofanydimension) ineuclidean space isalagrangian submanifold ofthespace oflines. 3.Themanifold ofall polynomials x2"'+ divisible byx"‘islagrangian. Alagrangianﬁbration isaﬁbration allofwhose ﬁbres arelagrangian. EXAMPLES. l.Thecotangent ﬁbration islagrangian. 2.The Gauss ﬁbration from thespace oflines ineuclidean space totheunitsphere ofdirections is lagrangian. 483 Appendix 16:Singularities ofraysystems Alllagrangian ﬁbrations ofaﬁxed dimension arelocally (onaneighbor- hood ofapoint inthetotal space) symplectically diffeomorphic. Alagrangian mapping istheprojection ofalagrangian submanifold tothe base ofalagrangian ﬁbration, i.e.,atriple V->E—>B,where theﬁrstarrow isanimmersion onto alagrangian manifold and thesecond arrow isa lagrangian ﬁbration. EXAMPLES. 1.Agradient mapping ql—>@S/(761 islagrangian. 2.The normal mapping which maps each normal vector ofasubmanifold ineuclidean space toitstipislagrangian. 3.The Gauss mapping which takes each point ofa transversely oriented hypersurface ineuclidean space totheunit vector at theorigin inthedirection ofthenormal islagrangian. (The corresponding lagrangian manifold consists ofthenormals themselves.) Anequivalence oflagrangian mappings isaﬁbre-preserving symplectic diffeomorphism ofthetotal spaces oftheﬁbrations which takes theﬁrst lagrangian manifold tothesecond. Thesetofcritical values ofalagrangian mapping iscalled acaustic. The caustics ofequivalent mappings arediffeomorphic. EXAMPLE. Thecaustic ofthenormal mapping ofasurface istheenvelope of thefamily ofnormals, i.e.,thefocal surface (surface ofcenters ofcurvature). Every lagrangian mapping islocally equivalent toagradient (ornormal, orGauss) mapping. Thesingularities ofgeneric gradient (ornormal, orGauss) mappings arethesame asthose forarbitrary generic lagrangian mappings. Thesimplest ofthese areclassiﬁed bythereﬂection groups A,,,Dk,E6,E7,E8 (seeAppendix 12). EXAMPLE. Weconsider amedium ofdust particles moving inertially, with their initial velocities forming apotential ﬁeld. After time t,theparticle atx moves tox+t(dS/dx). Wethereby obtain aone-parameter family ofsmooth mappings R3—>R3. These mappings arelagrangian. Infact,apotential ﬁeld ofvelocities gives alagrangian section ofthecotangent bundle. The phase ﬂow ofNewton’s equations preserves thelagrangian property. Forlarge t,though, ourlagrangian manifold isnolonger asection: itsprojection onthebase develops singular- ities. Thecaustics ofthecorresponding lagrangian mappings areplaces where thedensity ofparticles hasbecome inﬁnite?“ According toYa.B.Zel’dovich (1970) ananalogous model (taking intoaccount gravity andtheexpansion of 125Therelation between caustics anddust-like media wasﬁrstdiscovered byLifshitz, Sudakov, andKhalatnikov: seethesurvey byE.M.Lifshitz andI.M.Khalatnikov, Investigations in relativistic cosmology, Adv. Phys. 12(1963), 185. 484 agmlmEmoﬁzg'6m§=O:m2on_mgusmi\3\\“Omix_\\_“__“N7MW \\\\_\__>\\_ggixﬁAM“in“?\\\§\\\\§E,_>\_\\\\“@qrQQ 0:28EODBUG<ﬁg0;“;h Appendix 16:Singularities ofraysystems theuniverse) describes theformation oflarge scale nonhomogeneities inthe distribution ofmatter intheuniverse. According tothetheory ofLagrange singularities, thenewborn caustics have theform ofelliptic saucers (Figure 254)(after time tfrom themoment of birth, asaucer haslength oforder t1/2,depth oforder t,andthickness oforder t3/2). Thebirth ofasaucer corresponds toA3.Themetamorphoses ofcaustics which occur ingeneric one-parameter families oflagrangian mappings are shown inFigure 255(V.I.Arnold, Wave fronts evolution andequivariant Morse lemma, Comm. Pure Appl. Math. 6(1976), 319-335). Theorem (1972). The germs ateach point ofgeneric lagrangian mappings between manifolds ofdimension 55aresimple (i.e.,having nomoduli) and stable. Thesimple stable germs oflagrangian mappings areclassiﬁed bythe reﬂection groups A,D,E,inawaywhich willbeexplained below. DContact geometry andsystems ofrays andwave fronts Werecall thatacontact structure onanodddimensional smooth manifold isanondegenerate ﬁeld oftangent hyperplanes. The speciﬁc condition of nondegeneracy isinessential here, since near generic points, allgeneric hy- perplane ﬁelds onmanifolds ofaﬁxed odd dimension arediffeomorphic (Darboux’s theorem forcontact structures, Appendix 4). EXAMPLES. 1.Themanifold ofcontact elements ofasmooth manifold consists ofallitstangent hyperplanes. Therateofchange ofacontact element belongs tothecontact structure ifandonly iftherateofchange ofthepoint ofcontact (i.e.,thepoint where thehyperplane istangent tothemanifold) belongs tothe contact element itself. 2.Themanifold of1-jets offunctions y=f(x)hasa contact structure dy=pdx(p=(if/ﬁx forthe1-jet ofafunction f). Theextrinsic geometry ofasubmanifold ofcontact space islocally deter- mined bytheintrinsic geometry (Givental’s theorem oncontact structures). Integral submanifolds ofacontact structure arecalled Legendre (or legendrian) submanifolds ifthey have thelargest possible dimension. EXAMPLES. 1.Thesetofallcontact elements tangent toaﬁxed submanifold (ofanydimension) isaLegendre submanifold. 2.Inparticular, allcontact elements atagiven point form aLegendre submanifold (aﬁbre ofthebundle ofcontact elements). 3.Thesetofallthe1-jets ofasingle function isaLegendre submanifold inthespace of1-jets. Aﬁbration iscalled aLegendre ﬁbration ifitsﬁbres areLegendre submanifolds. EXAMPLES. 1.Theprojective cotangent ﬁbration (attaching each contact ele- ment toitspoint ofcontact) isLegendre. 2.Theﬁbration of1-jets offunctions over the0-jets (forgetting thederivative) isLegendre. 486 Appendix 16:Singularities ofraysystems AllLegendre ﬁbrations ofaﬁxed dimension arelocally contact diffeo- morphic (inaneighborhood ofapoint inthetotal space oftheﬁbration). The projection ofaLegendre submanifold onthebase ofaLegendre ﬁbration iscalled aLegendre mapping. Theimage ofaLegendre mapping is called itsfront. EXAMPLES. 1.TheLegendre transformation: Ahypersurface inprojective space maybelifted tothespace ofcontact elements ofprojective space asaLegendre submanifold. Themanifold ofcontact elements ofprojective space isalso ﬁbred over thedual projective space. (The ﬁbration assigns toeach contact element theplane containing it.)This isaLegendre ﬁbration. Theprojection ofthelifted Legendre submanifold maps itonto thehypersurface which is projectively dual totheoriginal one. Thus, theprojective dual ofasmooth hypersurface isthefront ofaLegendre mapping. 2.Frontal mappings: Laying outasegment oflength toneach normal toahypersurface ineuclidean space, weobtain aLegendre mapping whose front isequidistant from thegiven hypersurface. Every Legendre mapping islocally equivalent toaLegendre transfonna- tion, aswellastoafrontal mapping. Thetheory ofLegendre singularities thus coincides exactly with thetheory ofsingularities ofLegendre transformations andoffrontal mappings. Equivalence, stability, andsimplicity ofLegendre mappings aredeﬁned justasthelagrangian case. Theorem (1973). Thegerms, atallpoints, ofgeneric Legendre mappings between manifolds ofdimension $5aresimple andstable. Thesimple andstable germs ofLegendre mappings areclassiﬁed bythegroups A,D,E1their fronts arelocally diﬂeomorphic (inthecomplex domain) tothemanifolds ofnon- regular orbits ofthecorresponding reﬂection groups. EXAMPLE. Theonly singularities ofatypical wave front inthree-dimensional space are(semicubic) cuspidal curves (A2)and“swallowtails” (A3,Figure 256; nearsuch apoint, thefront isdiﬂeomorphic tothesurface formed bythepoly- nomials with multiple roots inthespace ofpolynomials x4+axz+bx+c). Figure 256 Singularities ofwave fronts 487 Appendix 16:Singularities ofraysystems Ofcourse, there may alsobetransverse intersections ofbranches offronts of thetypes justdescribed. Remark. Therealforms ofsimple singularities offronts mayalsobedescribed interms ofreﬂection groups. E.Looijenga hasshown thattherealcomponents inthecomplement ofasimple germ ofafront may beidentiﬁed with the conjugacy classes ofinvolutions (elements oforder 2)inthenormalizer of thereﬂection group, conjugacy being taken with respect tothereﬂection group itself. (SeeE.Looijenga, Thediscriminant ofarealsimple singularity, Compositio Math. 37(1978), 51-62.) EApplications ofcontact geometry tosymplectic geometry Alllagrangian singularities may beobtained from Legendre singularities, if onerealizes thelatter byprojections ofLegendre submanifolds ofthespace ofl-jets offunctions onto thespace of0-jets. Ifoneforgets thevalue ofeach function, thespace ofl-jets isprojected onto phase space (i.e.,thecotangent bundle); aLegendre submanifold intheﬁrstspace projects toalagrangian submanifold inthesecond. Inparticular, thecaustic ofalagrangian mapping istheimage ofthecuspidal edge ofthefront ofaLegendre mapping under a projection with one-dimensional ﬁbres. Theorem (O.V.Lyashko, 1979). Allholomorphic vector ﬁelds transverse tothe front ofasimple singularity arelocally equivalent under holomorphic dijfeo- morphisms preserving thefront. EXAMPLE. Ageneric vector ﬁeld intheneighborhood ofthemost singular point ofaswallowtail {x4+axz+bx+c=(x+d)2 isequivalent, by aholomorphic diffeomorphism preserving theswallowtail, tothenormal form 8/dc(Figure 257). The reduction ofvarious objects tonormal form, byadiffeomorphism preserving awave front orcaustic, isabasic technique forstudying the geometry ofsystems ofrays andfronts. Forinstance, thestudy ofthemeta- Q J Figure 257 Thenormal form ofavector ﬁeldattheswallowtail 488 Appendix 16:Singularities ofraysystems <1AismElla-inE9lie»<3 Figure 258 Perestroikas ofwave fronts morphoses ofmoving wave fronts isbased onthefollowing result, which is “dual” totheprevious one. Theorem (1976). Allgeneric holomorphic functions equal tozero atthemost singular point ofasimple singularity ofafront arelocally equivalent under holomorphic diffeomorphisms which preserve thefront. EXAMPLE. Inaneighborhood ofthemost singular point ofaswallowtail, a generic function maybereduced, byadiffeomorphism preserving theswallow- tail,tothenormal form a. This theorem isaspecial caseoftheequivariant Morse lemma. Itisapplied inthefollowing way. Theinstantaneous wave fronts together form a“large front” inspace-time. “Time” isafunction onspace-time. Wereduce this function tonormal form byadiffeomorphism which preserves thefront, and wethereby obtain anormal form forthemetamorphoses oftheinstantaneous fronts. The metamorphoses offronts in[R3areshown inFigure 258. The problem ofdescribing themetamorphoses ofcaustics ingeneric one-parameter families (Figure 255)issolved inexactly thesame way. Inthiscase, thetime function isreduced tonormal form byatransformation ofspace-time which preserves the“large caustic.” Ifthedimension ofspace-time isnolarger than 4,then allthesingularities ofthelarge caustic areoftypes AandD. Thecaustics oflagrangian singularities intheAseries differ from thewave fronts intheAseries only byashiftof1unitintheindex. Thesame istherefore truefortheir metamorphoses. Thecaustics intheDseries arenotthesame asthefronts. Thenormal forms forageneric timefunction intheneighborhood ofacaustic singularity oftype Dwere found byV.M.Zakalyukin (1975). Thetopological normal forms for 489 Appendix 16:Singularities ofraysystems thetime function areespecially simple: Caustic Real case Complex case D; it,+A2 A,+A, DI ,1,112,1, +1, /l,+12 D2k+1 iii /ii D210 kZ3 41i/lz 41+ 42 Here, thelarge caustic Duisthesetof/lforwhich 9r'(-, /l)hasadegenerate critical point, where .9"(x,/l)=ixixl +%xi"‘ +%xi“2 +---+ﬁ.,,_2x, +2/lax, (,u24). Thereduction tonormal form ofthegerm ofthetime function isaccom- plished byalocal homeomorphism ofthespace RF‘ (C“"), which preserves thelarge caustic andwhich issmooth everywhere except at0(V.I.Bakhtin, 1984). J.Nye(1984) hasnoticed thatnotallmetamorphoses ofcaustics andfronts may berealized bythemotion ofafront under anequation ofeikonal (or Hamilton—Jacobi) type. Forexample, thecaustic ofaraysystem cannot have theform of“lips” with twocusps (although thisispossible forlagrangian caustics). Thepoint isthattheinclusion ofalagrangian orLegendre manifold inthehypersurface given byaHamilton—Jacobi oreikonal equation imposes topological restrictions onthecoexistence, andthus onthemetamorphoses, ofsingularities, even though theindividual singularities may berealized on hypersurfaces. This isnamely thecasewhen thelevel surface ofthehamiltonian islocally nondegenerately convex inthemomentum variables. The vector ﬁelds generating thediffeomorphisms preserving afront are those which aretangent toit.The study ofthese vector ﬁelds leads toan unusual “convolution” operation ontheinvariants ofareﬂection group. Toapair ofinvariants (functions ontheorbit space) weassociate anew invariant—the scalar product ofthegradients ofthefunctions (pulled back from theorbit space totheoriginal euclidean space). Thelinearization ofthisoperation deﬁnes asymmetric bilinear mapping from each cotangent space oftheorbit space intoitself. Theorem (1979). Thelinearized convolution ofinvariants ofareﬂection group isisomorphic asabilinear operation totheoperation onthelocal algebra of thecorresponding singularity given bytheformula (p,q)r—>S(p-q), where S=D+(2/h)E, DisEuler’s quasi-homogeneous derivation, andhisthe Coxeter number. In1981, A.N.Varchenko andA.B.Givental’ (who alsoproved thetheorem above fortheexceptional groups) found afar-reaching generalization ofthis 490 Appendix 16:Singularities ofraysystems result. They replaced theeuclidean structure bytheintersection form of theunderlying period mapping, which arises from afamily ofholomorphic differential forms ontheﬁbres oftheMilnor ﬁbration ofaversal family of functions. Anondegenerate intersection form deﬁnes (depending ontheparity ofthenumber ofvariables) either alocally ﬂatpseudo-euclidean metric with astandard singularity ontheLegendre front orasymplectic structure which extends holomorphically tothefront. EXAMPLE. Thespace ofmonic polynomials withodddegree andsumofthe roots equal tozero acquires yetanother symplectic structure. Relative tothis structure, thesubmanifold ofpolynomials withthemaximal number ofdouble roots turns outtobelagrangian. When theintersection form isindeﬁnite, thesymplectic structure isreplaced byaPoisson structure (seeAppendix 14). FTangential singularities Theﬁrstapplications ofthetheory oflagrangian andLegendre singularities, around which thetheory itself developed (~1966), concerned short wave asymptotics intheform oftheasymptotics ofoscillatory integrals. Asurvey ofthese applications (including thedetermination ofuniform estimates for oscillatory integrals when saddle points meet, thecalculation ofasymptotics using Newton polyhedra, theconstruction ofmixed Hodge structures, appli- cations tonumber theory andthetheory ofconvex polyhedra, andestimates oftheindex ofsingular points ofvector ﬁelds andthenumber ofsingular points ofalgebraic surfaces) may befound inthebook: V.I.Arnold, A.N.Varchenko, andS.M.Gusein-Zade, Singularities of Diﬂerentiable Mappings, Vol. II,Monodromy andAsymptotics ofIntegrals, Moscow, Nauka, 1984. English translation: Birkhauser, 1988. andinthepaper V.I.Arnold, Singularities ofraysystems, Proceedings oftheInternational Congress ofMathematicians, August 16-24, 1983, Warsaw. Here weshall present other applications ofthetheory oflagrangian and Legendre singularities tothestudy oftheconﬁgurations ofprojective mani- folds andtangential planes ofvarious dimensions. Oneisledtosuch problems from variational problems with one-sided constraints (such astheobstacle problem), aswellasfrom thestudy ofNekhoroshev’s exponent ofroughness forunperturbed hamiltonian functions (seeAppendix 8). Weconsider ageneric surface inthree-dimensional projective space (Figure 259). Thecurve ofparabolic points (p)divides thesurface intoadomain of elliptic points (e)andadomain ofhyperbolic points (h);thelatter domain contains thecurve ofinﬂection points oftheasymptotic lines (f),with its 491 Appendix 16:Singularities ofraysystems P. P hh1» f ° f Figure 259 Projective classiﬁcation ofpoints ofasurface points ofbiinﬂection (b),self-intersection (c),andtangency totheparabolic curve (t), From thisclassiﬁcation ofpoints, onemayderive both estimates ofcurva- tureexponents andthefollowing classiﬁcation ofprojections. Theorem (O.A.Platonova andO.P.Shcherbak, 1981). Every projection from apoint outside ageneric surface in[RP3islocally equivalent ateach point of thesurface totheprojection along lines parallel tothex-axis ofasurface z=f(x,y),where fisoneofthefollowing 14functions: x,x2,x3+xy,x3ixyz,x3+xy3,x“+xy, x“+xzy-l-xy2,x5 ix3y+xy,x3 ixy4,x“ +xzy+xy3,x5 +xy. Byaprojection wemean here adiagram V—>E->Bconsisting ofan embedding andaﬁbration; anequivalence ofprojections isthen a3x2 commutative diagram whose vertical arrows arediffeomorphisms. Theonly singularities oftheprojection from ageneric center arefolds and Whitney tucks. Thetucks appear when theprojection isalong anasymptotic direction. Theremaining singularities arevisible only from special points. The ﬁniteness ofthenumber ofsingularities ofprojections (and therefore thenumber ofsingularities ofapparent contours) wasnotobvious before theresult above wasobtained, since there isacontinuum ofinequivalent singularities for generic three-parameter families ofmappings from asurface totheplane. Theregions ofspace from which thegeneric surface hasadifferent appear- ance, aswellasthecorresponding views ofthesurface, areshown inFigure 260 (forthemost complicated cases). The hierarchy oftangential singularities becomes more comprehensible when itisreformulated interms ofsymplectic andcontact geometry. R. Melrose (1976) observed thattherays tangent toasurface aredescribed by apair ofhypersurfaces insymplectic phase space: oneofthem, p2=1,is deﬁned bythemetric; theother isdeﬁned bythesurface. Asigniﬁcant partofthegeometry ofasymptotic lines maybereformulated interms ofthispairofhypersurfaces. Inthisway, wemay transfer concepts from thegeometry ofsurfaces tothemore general case ofarbitrary pairs of 492 SmCtSySyaYf0Se_n_lYMUgn_1S61_mdnWPA wOg______mHOm____oE8Uzmmg2:__oMgcobwag2;gmaim;QQ\\_QQwA4 \\\\_J_J_/_/I___\W\\__)_\__\‘\\\‘\“\\W‘____\\ gv\2“Nwk_ \l|| \\1‘\\\\\\\\___ \3_/I‘T‘,H\\|__Q_Q__I_\\1WV\‘(EQ ‘K‘\N1‘@1\g N‘3/K3'\‘_I__/_\RV_Q. 33’“_,\__, lKg‘Q+3*+1“NuFHV;I_k_\ 3 ‘v m\3/3. L__H_.%_N6_v\_EiM m l Appendix 16:Singularities ofraysystems hypersurfaces insymplectic space, andthereby usethegeometric intuition gained from surface theory tostudy general variations problems with one- sided phase constraints. LetYandZbehypersurfaces inthesymplectic space Xwhich intersect transversely along asubmanifold W.Projecting YandZonto their manifolds ofcharacteristics, weobtain thehexagonal diagram r.Z'X'\.> l\l"/T./ Xu,\;/V inwhich Zisthecommon manifold ofcritical points fortheprojections ofW onUandV. EXAMPLE. LetXbethe{q,p}phase space forafreeparticle ineuclidean space (qistheposition oftheparticle, pitsmomentum). Yisthemanifold ofunit vectors (p2=1).Zisthemanifold ofvectors attheboundary (qbelongs to ahypersurface F).Then Uisthemanifold ofrays, Visthetangent bundle of theboundary F,Wisthemanifold ofunitvectors attheboundary, andZis theunittangent bundle oftheboundary. Ifaunittangent vector totheboundary isnotasymptotic, then both ofthe projections W—>UandW->Vhave foldsingularities atthispoint. Each of them deﬁnes aninvolution onWwhich ﬁxes Z. EXAMPLE. There aretwoinvolutions, aandr,onthemanifold oftangent vectors along aconvex plane curve W(Figure 261). Their product isBirkhoff’s billiard mapping (1927). Using pairs ofinvolutions, Melrose found alocal normal form forpairs of hypersurfaces insymplectic space which areinthesituation justdescribed. (This wasfortheC°°case; intheanalytic case, oneusually obtains divergent Figure 261 Thetwoinvolutions generating thebilliard mapping 494 Appendix 16:Singularities ofraysystems series, justasinthetheory ofEcalle (1975) andVoronin (1981) onresonant dynamical systems.) Formore complicated singularities (forexample, near asymptotic direc- tions), pairs ofhypersurfaces have moduli. Forthetwosimplest singularity types after thefold, itispossible toputinnormal form (atleast for- mally) thepairconsisting oftheﬁrsthypersurface anditsintersection with thesecond. This allows ustostudy, inaneighborhood ofanasymptotic or biasymptotic unittangent vector totheboundary, themapping which assigns theraycontaining ittoeach unitvector attheboundary. Thecritical values ofthismapping inthesymplectic space oflines aredescribed bythefollowing result, since themanifold oftangent rays islocally diffeomorphic near a biasymptotic raytotheproduct ofaswallowtail andaline. Theorem (1981). Allthegeneric symplectic structures intheneighborhood ofa point inthedirect product ofaswallowtail andalinear space areformally diffeomorphic bylocal diﬂeomorphisms preserving theproduct structure. GTheobstacle problem Weconsider anobstacle bounded byasmooth surface ineuclidean space. Theobstacle problem consists ofthestudy ofthesingularities ofthefunction deﬁned outside theobstacle whose value ateach point isthelength ofthe shortest path remaining outside theobstacle andjoining thepoint toaﬁxed initial set.This variational problem onamanifold with boundary isunsolved even inthree-dimensional space. Each minimizing path consists ofsegments ofstraight lines andsegments ofgeodesics onthesurface oftheobstacle (Figure 262). Weconsider therefore asystem ofgeodesics onthesurface oftheobstacle, orthogonal toaﬁxed front. Thesystem ofallraystangent tothese geodesics forms alagrangian variety inthesymplectic space oflines, justasanysystem ofextremals foravaria- tional probiem. Butwhile inanordinary variational problem thislagrangian variety isasmooth manifold (even atcaustics), thelagrangian variety arising intheobstacle problem hassingularities. From thelasttheorem (inthe previous section), oneobtains: Figure 262 Anextremal oftheobstacle problem 495 Appendix 16:Singularities ofraysystems .4 l4 Figure 263 Theopen (“unfurled”) swallowtail Corollary (1981). Thelagrangian variety ofrays inageneric obstacle problem hasasemicubic cuspidal edge along each asymptotic rayandasingularity diffeomorphic toanopen swallowtail ateach biasymptotic ray. Theopen swallowtail isthesurface inthefour-dimensional space ofmonic polynomials x5+Ax3+Bxz+Cx+Dformed bythepolynomials with triple roots. Differentiation ofthepolynomials turns theopen swallowtail into anordinary one;when theswallowtail isopened, thecuspidal edge isretained, buttheself-intersection disappears (Figure 263). Theorem (1981). Inthegeneric motion ofawave front, thecuspidal edges of theinstantaneous fronts sweep outanopen swallowtail infour-dimensional space-time (over theusual swallowtail caustic). Theorem (O.P.Shcherbak, 1982). Consider ageneric one-parameter family of space curves andsuppose that,forsome value oftheparameter (time), oneof thecurves hasapoint ofdouble ﬂatness (oftype 1,2,5).Then theprojective duals ofthese curves form asurface inspace-time which islocally diffeo- morphic totheopen swallowtail. Theopen swallowtail istheﬁrstmember ofawhole series ofsingularities. Consider, inthespace ofmonic polynomials x"+Z,x"_1 +---+/1,,_1, theset ofpolynomials with aroot ofﬁxed comultiplicity k,(x-—a)""‘(x" +--'). Differentiation ofpolynomials preserves thecomultiplicity ofroots. Theorem (A.B.Givental’, 1981). Thesequence ofsetsofpolynomials ofﬁxed comultiplicity becomes stabilized asthedegree grows, beginning withdegree n=2k+1(i.e.,when theself-intersections areeliminated). EXAMPLE. Theopen swallowtail istheﬁrststable variety over theordinary swallowtail. Theappearance ofswallowtails intheobstacle problem wasaxiomatized byGivental’ (1982) inhistheory oftriads. 496 Appendix 16:Singularities ofraysystems Deﬁnition. Asymplectic triad (H,L,l)consists ofasmooth hypersurface Hin asymplectic manifold andalagrangian submanifold Lwhich istangent toH toﬁrstorder along ahypersurface IofL. Thelagrangian variety generated bythetriad istheimage ofLinthe manifold ofcharacteristics ofthehypersurface H. EXAMPLE 1.Consider, intheproblem ofbypassing anobstacle with boundary FcR",thedistance along geodesics from aninitial front asafunction s:F-+R.Themanifold Lconsisting ofallextensions ofthel-form dsfrom F toIR",together with thehypersurface H:p2=1,forms atriad. Thelagrangian variety generated bythistriad isprecisely thevariety ofrays tangent tothe geodesics inoursystem ofextremals onF. EXAMPLE 2.Inthesymplectic manifold ofmonic polynomials 3*“=x“+ i.1x"'1 + +Z4with even degree d=2m,thepolynomials divisible byx"‘ form alagrangian submanifold L. Consider thehamiltonian fortranslation along thex-axis. [This polynomial in2.isequal to h=Z(-1)*a»"<W<1’, 1+j=d,37""=4%/ax‘. The hypersurface h=Oistangent tothelagrangian submanifold Lalong thesubspace lofpolynomials divisible byx"'“, thus forming atriad. The lagrangian variety generated bythistriad isanopen swallowtail ofdimension m—1(thesetofpolynomials x“*‘ +a1x"_3 + +a,,_2 having aroot of multiplicity greater than halfthedegree).] Theorem (A.B.Givental’, 1982). Thetriads inExample 2arestable. Every germ ofageneric triad isdiffeomorphic toagerm ofatriad inExample 2. Corollary. Thevariety ofraystangent tothegeodesics inthesystem ofextremals ofageneric obstacle problem islocally symplectically diffeomorphic toa lagrangian open swallowtail. Incontact geometry, there aretwokinds ofLegendre varieties associated toobstacle problems: varieties ofcontact elements offronts and varieties ofl-jets oftime functions. Theﬁrstofthese arediffeomorphic tolagrangian open swallowtails; thesecond arediffeomorphic tocylinders over the ﬁrst. EXAMPLE. Consider theproblem ofbypassing anobstacle intheplane which isbounded byacurve with aninﬂection point. Thefronts, which arethe evolvents ofthecurve, have twokinds ofsingularities: ordinary cusps (oforder 3/2)onthecurve itself andsingularities oforder 5/2onthetangent line through theinﬂection point (Figure 264). Over points oftheboundary curve, 497 Appendix 16:Singularities ofraysystems Figure 264 Theevolvents ofacubical parabola theLegendre variety isnonsingular, while over points onthetangent line through theinﬂection point ithasacuspidal edge oforder 3/2. Theorem (1978). Inthespace ofcontact elements totheplane, ﬁbred over the plane itself, thesurface consisting ofthecontact elements oftheevolvents of ageneric curve near apoint ofinﬂection islocally equivalent byaﬁbre- preserving diﬂeomorphism tothesurface consisting ofallpolynomials with multiple roots inthespace ofpolynomials x3+axz+bx+c,ﬁbred into lines parallel totheb-axis. This surface (Figure 265), together with thesurface c=0representing the contact elements along theboundary curve, forms avariety which isdiffeo- morphic tothesetofirregular orbits forthereﬂection group B3.This observa- tionledtothetheory ofboundary singularities (1978). EXAMPLE (I.G.Shcherbak, 1982). Consider ageneric curve onasurface in three-dimensional euclidean space. Atcertain points, thedirection ofthecurve coincides with principal curvature directions ofthesurface. Itfollows from thetheory oflagrangian boundary singularities that theWeyl group F4is ba Tangent through the inﬂection point Curve! Figure 265 Thesurface ofcontact elements oftheevolvents 498 Appendix 16:Singularities ofraysystems A1 A5 A5 (. F4 ! % iC]. B: B1 Figure 266 Thecaustic’s singularity F4 connected with each such point: thefocal points ofthesurface (A2), focal points ofthecurve (A’2), andnormals tothesurface atpoints ofthecurve (B2) together form anF4caustic near thecenter ofcurvature (Figure 266). Wewillnotdwell here onthetheory ofboundary singularities, butitis worth mentioning the“Lagrange duality“ relating afunction anditsrestric- tiontotheboundary (uptostable equivalence): thismay bethought ofas amodern version oftheLagrange multiplier rule(I.G.Shcherbak, 1982). Returning toinﬂection points ofplane curves, weconsider thegraph ofthe multiple-valued time function inanobstacle problem. Thelevel curves ofthis function aretheevolvents oftheobstacle boundary. Therefore, thegraph of thisfunction hastheform (shown inFigure 267)ofasurface with twocuspidal edges (oforders 3/2and5/2). When Ishowed thissurface toA.B.Givental’, herecognized O.V.Lyashko’s drawing ofthesingular orbit Zofthegroup H3(symmetries oftheicosahedron). Givental’s conjecture wassoon veriﬁed: 3/2 /’A H» H1 *’ \' L 11111-01; 5/2 A: i 4w—* Figure 267 Thediscriminant ofH3 499 Appendix 16:Singularities ofraysystems Theorem (O.P.Shcherbak, 1982). Thegraph ofthe(multiple-valued) time function intheproblem ofbypassing anobstacle bounded byageneric plane curve isformally diffeomorphic near aninflection point ofthecurve tothe variety Z. Theproof ofthistheorem uses: Theorem (O.V.Lyashko, 1981). Thevariety Eisdiffeomorphic tothevariety ofpolynomials x5+ax‘+bxz+chaving amultiple root. Lyashko’s theorem describes thevariety ofsingular orbits forthegroup H3 astheunion ofthetangents tothecurve (t,t3,t5),while Shcherbak’s theorem applies toanycurve oftheform (t+o(t),t3+o(t3), t5+o(t5)). Thesame singularity appears onageneric front atthepoint oftangency of aasymptotic raywith thebounding surface ofanobstacle inR3. Finally, wedescribe avariational problem leading tothesingularity H4 (after O.P.Shcherbak). Thegroup H4consists ofthesymmetries ofaregular polyhedron inR4.Its 120vertices lieonS3zSU(2) andform thebinary icosahedral group (the binary group being theinverse image ofthesymmetry group oftheicosahedron under thedouble covering S3—>SO(3)). Consider theproblem ofbypassing anobstacle bounded byasmooth surface inthree-dimensional euclidean space. Theextremals beginning ata ﬁxed point outside theobstacle generate apencil (one-parameter family) of geodesics onthesurface. Atimefunction isthedistance from aﬁxed initial manifold (e.g., apoint) along stationary (not necessarily minimizing) paths consisting ofarcsofgeodesics andtheir tangents, considered asa(multiple- valued) function oftheterminal point inspace (solution oftheHamilton~Jacobi equauon) Theorem (O.P.Shcherbak, 1984). Forageneric obstacle, thegraph ofthetime function atapoint which isfocalfor thepencil along anasymptotic tangent ataparabolic point ofthesurface islocally diffeomorphic tothevariety Zof singular orbits ofthegroup H4. Anexplicit parametrization ofZis: (a,b2/2 +ac,c2/2+ab3,b5/5 +c3/3+absc). Thegroup H4isrelated toafour-dimensional subspace ofthebase space oftheversal deformation ofE8(this connection isexplained inRemark 7,§9 ofthepaper byV.I.Arnold, Indices ofsingular points of1-forms onmani- folds with boundary, convolution ofinvariants ofreﬂection groups, and singular projections ofsmooth surfaces, Russian Math. Surveys 34:2 (1979), l~42). 500 a=OAppendix 16:Singularities ofraysystems Hg a<O 1A, .‘ /41 2l H_ HIHZI Figure 268 Thecaustic’s singularity H4 ll4-.3,eii1Q . \ A \ H H-1 H.! ab)A,1 AtA: Ht HL a iIn A; H3 Figure 269 Thefront’s perestroika H4 Appendix 16:Singularities ofraysystems Corresponding tothisfour-dimensional subspace, there isanembedding ofthelocal algebra D4intothelocal algebra E8,which induces ontheformer thesame grading which isgiven bytheconvolution ofinvariants ofH4. O.P.Shcherbak hasshown that thisrelationship establishes yetanother description ofthevariety ofsingular orbits ofH4: Theorem. Consider those values of/lforwhich thecurve x5+y3+/l1x3y+ /l2x3 +33y+/l4=0issingular. Oneoftheirreducible components ofthis three-dimensional hypersurface inA-space isdiffeomorphic tothevariety of singular orbits ofthegroup H4. Thecaustic andthree typical sections ofthevariety ofsingular orbits ofH4 areshown inFigures 268and269. SeeO.P.Shcherbak, Wavefronts and reﬂection groups, Russian Math. Surveys, 43(1988). 502 Bibliography ofSymplectic Topology Arnold, V.I.Surunepropriété topologique desapplications globalement canoniques delamécanique classique. C.R. Acad. Sci.Paris 261(1965), 3719-3722. Arnold, V.I.Onacharacteristic classentering thequantisation conditions. Funct.Anal. Appl. 1:1(1967), 1-14. Arnold, V.I.Acomment on“Surunthéoreme delageometric”. In:Izbrannye trudy A. Puankaré. Moscow, Nauka, 1972, vol.2,pp.987-989. Arnold, V.I.Lagrange andLegendre cobordisms. Funct.Anal. Appl. 14:3(1980), 1-13; 14:4(1980), 8—17. Arnold, V.I.TheSturm theorems andsymplectic geometry. Funct. Anal. Appl. 19 (1985), 251-259. Arnold, V.I.First steps insymplectic topology. Russ. Math.Surv. 41:6(1986), 1-21. Arnold, V.I.Onfunctions withmildsingularities. Funct.Anal. Appl.23:3(1989), 1—10. Arnold, V.I.,andGivental, A.B.Symplectic geometry. In:Dynamical Systems-4 (Enc. ofMath. Sc.vol.4).Berlin-Heidelberg-New York, Springer, 1990, pp.l—136. Arnold, V.I.Surlespropriétés topologiques desprojections lagrangiennes engeo- métrie symplectique descaustiques. Preprint 9320, CEREMADE, Université Paris- Dauphine, 1993, pp.1—9(Cahiers deMathématiques delaDécision, 14/6/93). Arnold, V.I.Some remarks onsymplectic monodromy ofMilnor ﬁbration. In:Progress inMath.,A.FloerMemorial Volume. Basel-Boston, Birkhéiuser, 1993. Arnold V.I.Invariants andPerestroikas ofPlane Fronts. Trudy Steklov Math. Inst., Russ. Acad. ofSc.,Moscow 1994, 81pp. Arnold, V.I.Ontopological properties ofLegendre projections incontact geometry ofwave fronts. Algebra andAnalysis. S.Petersbourg Math.J.6:3(1994). Arnold, V.I.Symplectic geometry andtopology. In:Trends andPerspectives inModern Mathematics. Cambridge Univ. Press, 1994(Preprint MIT, 1993, 68pp). Arnold, V.I.Invariants anddiscriminants ofplane curves andofthefronts of Legendrian curves. J.B.Lewis Memorial Lectures, Rutgers, 1993, 106pp;(toappear: Providence, AMS, 1994). Arnold, V.I.,Ed.Singularities andCurves (Advances inSov.Math), Providence, AMS, 1994. 503 Bibliography ofSymplectic Topology Atiyah. M.New invariants of3-and 4-manifolds. ln:The .\!utlieniutit'al Heritage of H.Weyl. Durham. NC.1987(Sympos. Pure Math. vol.48].Providence. AMS, 1988, pp.285-289. Audin, M.Quelques calculs encobordisme lagrzmgien. Ann. Inst. Fourier 35:3(1985), 159- 194. Audin, M.Coborrlisms ilininiersions Lugrungi'enm>s atLt‘_t]t'llll!‘lt’!lH(’.\‘ (Truvaux en Cours, vol.20).Hermann. 1987. 203pp. Audin, M.Fibres normaux d'immersion endimension double. points doubles d‘immer- sions lagrangiennes etplongements totalement reels. Comm. Math. Heir. 6311988). 593-623. Audin, M.Hamiltoniens periodiques surlesvarietes symplectiques compactes de dimension 4.In:L€('I. Notes inMath. 1416. Berlin-Heidelberg-New York. Springer, 1990, pp.1-25. Audin, M.The Topology ofTorus Action onSymplectic Marti/olrls. Basel, Birkhiiuser. 1991. Banyaga, A.Surlastructure dugroupe desdiﬂéomorphismes quipreservent une forme symplectique. Comm. Math. Helv. S3(1978), 174-227. Bennequin, D.Entrelacements etequations dePfaﬂ. Astérisque 107-108 (1983). 83- 161. Bennequin, D.Quelques remarques simples surlarigidite symplectique. ln:Géométrie Symplectique etdeContact: Autour duThéoréme dePoincare-Birkhoff, P.Dazord andN.Desolneux-Moulis, eds.Paris, Herman, 1984, pp.1-50. Bialy M.L., andPolterovich, L.V.Lagrangian singularities ofinvariant toriofhamil- tonian systems with twodegrees offreedom. Invent. Math. 97:2(1989), 291 303. Bialy, M.,andPolterovich, L.Hamiltonian diﬂeomorphisms andLagrangian distribu- tions. Geom. Funct. Anal. 2(1992), 173-21. Bialy, M.,andPolterovich, L.Optical Hamiltonian Functions. Preprint 1992, 20p. Boothby W.M., andWang, H.C. Oncontact manifolds. Ann. Math. 68(1958), 721-734. Calabi, E.Onthegroup ofauthomorphisms ofasymplectic manifold. In:Problems in Analysis (Symposium inhonour ofS.Bochner). Princeton Univ. Press, 1970, 1-26. Chaperon, M.Quelques questions degéometrie symplectique [d‘apres, entre autres, Poincare, Arnold, Conley etZehnder], Seminaire Bourbaki 1982-83. Astérisque 105-106 (1983), 231-249. Chaperon, M.Uneidéedutype“géodésiques brisées” pour lessystemes hamiltoniens. CR. Acad. Sci.Paris 298(1984), 293-296. Chaperon, M.Anelementary proof oftheConley-Zehnder theorem insymplectic geometry. In:Dynamical Systems andBifurcations, B.L.J. Braaksma, H.W. Broer, F. Takens, eds.(Lecture Notes inMath. 1125) Berlin-Heidelberg-New York, Springer, 1985, 1-8. Chaperou, M.Familles géneratrices. Cours al’ecole d’été Erasmus deSamos (1990), Publication Erasmus, 1993. Chekanov, Yu.V. Legandrova teorija Morsa. Uspekhi Mat. Nauk 42:4(1987), 139- 141. Chekanov, Yu.V. Caustics ingeometrical optics. Funct.Anal. Appl. 20(1986), 223-226. Chekanov, Yu.V. Lagrangian toriinasymplectic vector space andglobal symplec- tomorphisms. Bochum Preprint 169,1993, 13p(toappear inMath. Z). Conley, C.,andZehnder, E.TheBirkhoff-Lewis ﬁxed point theorem andaconjecture ofV.I.Arnold. Invent. Math. 73(1983), 33-49. McDulT, D.Thestructure ofrational andruled symplectic 4-manifolds. JAMS 3:1 (1990), 679-712. 504 Bibliography ofSymplectic Topology McDuﬂ, D.Elliptic Methods inSymplectic Geometry. Bull. Amer. Math.Soc.23(1990), 311-358. McDufT, D.Symplectic manifolds with contact-type boundaries. Invent. Math. 103 (1991), 651-671. McDuﬁ", D.Blow-ups andsymplectic embeddings indimension 4.Topology 30(1991), 409-421. McDuﬁ', D.Singularities ofJ-holomorphic curves. J.Geom. Anal. 3(1992), 249- 266. McDu1T, D.Notes onruled symplectic 4-manifolds. Preprint, 1992 (toappear inTrans. Amer. Math.Soc.). McDuﬂ", D.,andPolterovich, L.Symplectic packing andalgebraic geometry. Preprint, 1992. McDu1T, D.Remarks ontheuniqueness ofsymplectic blowing-up. Proceedings of1990 Warwick Symposium, Camb_ Univ. Press, 1993. McDuﬁ", D.,andSalamon, D.Notes onJ-holomorphic curves. Stony Brook preprint, 1993. McDuﬁ", D.,andTraynor, L.The4-dimensional symplectic camel andrelated results. (London Math. Soc.Lect. Notes Series). Cambridge Univ. Press (toappear). McDu1T, D.,andSalamon, D.Symplectic Topology (inpreparation). Duistermaat, J.J.OntheMorse index invariational calculus. Adv. Math. 21(1976), 173-195. Duistermaat, J.J.Onglobal action-angle variables. Comm. Pure Appl.Math. 33(1980), 687-706. Ekeland, I.,andHofer, H.Symplectic topology andHamiltonian dynamics. Math.Z. 200(1988), 355-378. Ekeland, I.,andHofer, H.Symplectic topology andHamiltonian dynamics II.Math. Z.203(1990), 553-567. Eliashberg, J.Rigidity ofsymplectic andcontact structures, Preprint, 1981. Eliashberg, Ya.Cobordisme dessolutions derelations dilferentielles. In:Sem. Sud- Rhodenien deGeom., tome 1,P.Dazord andN.Desolneux-Moulis, eds.Hermann, 1984, pp.17-32. Eliashberg, Y.Thecomplexiﬁcation ofcontact structures ona3-maifold. Usp. Math. Nauk 6:40(1985), 161-162. Eliashberg, Y.Classiﬁcation ofovertwisted contact structures on3-manifolds. Invent. Math. 98(1989), 623-637. Eliashberg, Y.Filling byholomorphic discsanditsapplications. In:Geometry ofLow- Dimensional Manifolds, Vol. 2,S.K. Donaldson andC.B. Thomas, eds.(London Math. Soc.Lect. Notes Ser.151)Cambridge Univ. Press, 1990, pp.45-67. Eliashberg, Y.,andGromov, M.Convex symplectic manifolds. Proceedings ofSympo- siainPure Mathematics, E.Bedford etal.(eds), 52:2(1991), 135-162. Eliashberg, Y.,andPolterovich, L.Biinvariant metrics onthegroup ofHamiltonian diffeomorphisms. Preprint, 1991. Eliashberg, Y.New invariants ofopen symplectic andcontact manifolds. J.Amer. Math. Soc.4(1991), 513-520. Eliashberg, Y.,andRatiu, T.Thediameter ofthesymplectomorphism group isinﬁnite. Invent. Math. 103(1991), 327-340. Eliashberg, Y.Onsymplectic manifolds withsome contact properties. J.Diff.Geometry 33(1991), 233-238. Eliashberg, Y.,andHofer, H.Unseen symplectic boundaries. Preprint, 1992, 16pp. 505 Bibliography ofSymplectic Topology Eliashberg, Y.,andPolterovich, L.Unknottedness ofLagrangian surfaces insym- plectic 4-manifolds. Preprint, 1992, 9pp. Eliashberg, Y.,andPolterovich, L.Newapplications ofLuttinger’s surgery. Preprint, 1992, 12pp. Eliashberg, Y.Contact 3-manifolds twenty years since J.Martinet’s work. Ann. Inst. Fourier 42(1992), 165-191. Eliashberg, Y.,andHofer, H.Anenergy-capacity inequality forthesymplectic holonomy ofhypersurfaces ﬂatatinﬁnity. Preprint, 1992, 8pp. Eliashberg, Ya.Topology of2-knots inR‘andSymplectic Geometry (Progress inMath., A.Floer memorial volume). Boston-Basel, Birkhauser, 1993. Eliashberg, Y.Legendrian andtransversal knots intight contact 3-manifolds. In: Topological Methods inModern Mathematics. Houston, Publish orPerish, 1993, pp.171-193. Eliashberg, Y.Classiﬁcation ofcontact structures on[R3.Duke Math.J.Intern. Math. Res.Not. N°3(1993), 87-91. Eliashberg, Y.,andHofer, H.Towards thedeﬁnition ofsymplectic boundary. Preprint, 1993. Floer, A.Proof oftheArnold conjecture andgeneralizations tocertain Kaehler manifolds. Duke Math.J.53(1986), 1-32. Floer, A.Morse theory forLagrangian intersections. J.DijfGeom. 28(1988), 513-547. Floer, A.Theunregularized gradient ﬂowforthesymplectic action. Comm. PureAppl. Math. 41(1988), 775-813. Floer, A.Arelative Morse index forthesymplectic action. Comm. Pure Appl. Math. 41(1988), 393-407. Floer, A.Aninstanton invariant for3-manifolds. Comm. Math. Phys. 118:2 (1988), 215-240. Floer, A.Witten’s complex ininﬁnite dimensional Morse theory. J.Dijf Geom. 30 (1989), 207-221. Floer, A.Cuplength estimates forLagrangian intersections. Comm. Pure Appl. Math. 42(1989), 335-356. Floer, A.Symplectic ﬁxed points andholomorphic spheres. Comm. Math.Phys. 120 (1989), 575-611. Floer, A.,andHofer, H.Symplectic homology I:Open SetsinC".Preprint, 1992. Floer, A.,Hofer, H.,andWysocki, K.Applications ofSymplectic Homology I. Preprint, 1992. Fortune, B.,andWeinstein, A.Asymplectic ﬁxed point theorem forcomplex projective spaces. Bull. Am.Math. Soc.12:1(1985), 128-130. Fuchs, D.B. Maslov-Arnold characteristic classes. Sov.Math.Dokl. 9(1968), 96-99. Ginzburg, V.L.Calculation ofcontact andsymplectic cobordism groups. Topology 31:4(1992), 757-762. Ginzburg, V.L.,andKhesin, B.A.Steady ﬂuidﬂows andsymplectic geometry. Preprint IHES, October 1992, 20pp.(toappear in:J.Geom. Phys.). Giroux, E.Convexité entopologie decontact. Comm. Math. Helvet. 66(1991), 637- 677. Givental, A.B. Lagrangian imbeddings ofsurfaces andtheopen Whitney umbrella. Funct. Anal. Appl. 20:3(1986), 35-41. Givental, A.B. Periodic mappings insymplectic topology. Funct. Anal. Appl. 23:4 (1989), 287-300. 506 Bibliography ofSymplectic Topology Givental, A.B.Nonlinear generalization oftheMaslov index. In:Singularity Theory andItsApplications, V.Arnold, Ed.(Advances inSoviet Math., vol.1),Providence, AMS, 1990, pp.71-103. Givental, A.B.Asymplectic ﬁxed point theorem fortoric manifolds. In:Progress in Math. (A.Floer memorial volume), Boston-Basel, Brkhiiuser, 1993. Gray, J.W.Some global properties ofcontact structures. Ann.Math. 69(1959), 421- 450. Gromov, M.Partial Differential Relation. Berlin-Heidelberg-New York, Springer, 1996. Gromov, M.Pseudo-holomorphic curves insymplectic manifolds. Invent. Math. 82 (1985), 307-347. Guillemin, V.,andSternberg, S.Birational equvalence insymplectic category. Invent. Math. 97(1989), 485-522. I-Iarlamov, V.,andEliashberg, Y.Onthenumber ofcomplex points ofarealsurface inacomplex surface. Proc. LITC-82 (1982), 143-148. Hofer, H.,andZehnder, E.Anewcapacity forsymplectic manifolds. In:Analysis Et Cetera, Boston, Academic Press, 1990, 405-428. Hofer, H.Onthetopological properties ofsymplectic maps. Proc. Roy. Soc. Edinburgh, Ser.A.115(1990), 25-38. Hofer, H.Symplectic Invariants. In:Proceedings oftheInternational Congress of Mathematicians inKyoto, 1990. Berlin-Heidelberg-New York, Springer, 1991. Hofer, H.Symplectic capacities. In:Durham Conferences, S.K.Donaldson andC.B. Thomas, Eds.London Math. Soc.1992. Hofer, H.,andSalamon, D.Floer homology andNovikov rings. Preprint, 1992. 39pp. Hofer, H.Estimates fortheenergy ofasymplectic map. Comment. Math. Helv. 68 (1993), 48-72. Kazarian, M.E. Umbilical Characteristic Number ofLagrangian Mappings of3- dimensional Pseudooptical Manifolds. Preprint, Ruhr-Univ. Bochum, 1993, 12pp. Kuksin, S.Inﬁnite-Dimensional Symplectic Capacities andaSqueezing Theorem for Hamiltonian PDE’s. Preprint Forschungsinstitut fiirMathematik ETH Ziirich, August 25,1993. Lalonde, F.,andSikorav, J.-C. Sous-variétés Lagrangiennes exactes desﬁbres cotan- gents. Comm. Math. Helvet. (1991), 18-33. Lalonde, F.Isotopy ofSymplectic Balls andStructure ofIrrational Ruled Symplectic 4-Manifolds. Preprint, 1992. Lalonde, F.Isotopy ofSymplectic Balls, Gromov’s Radius andtheStructure ofRuled Symplectic 4-Manifolds. Preprint, 1992. Lalonde, F.,andMcDuﬂ, D.TheGeometry ofSymplectic Energy. Preprint #1993/6 IMS SUNY Stony Brook, June 1993, 26pp. Laudenbach, F.,andSikorav, J.-C. Persistence d’intersection avec lasection nulle aucours d’une isotopic hamiltonienne dans unﬁbrecotangent. Invent. Math. 82:2 (I985), 349-358. Laudenbach, F.,andSikorav, J.C.Disjonction hamiltonienne etlimites desous-varietés lagrangiennes. Preprint, Centre deMath., Ecole Polytechnique, Septembre 1993. Lee, Yng-Ing. Nonlagrangian limits ofLagrangian discs. Duke Math. J.Int.Math. Research Notes, N2,1993. Luttingcr, K.Lagrangian Tori inIR‘.Preprint, 1992. 507 Bibliography ofSymplectic Topology Lutz, R.Structures decontact surlesﬁbres principaux encercles dedimension 3.Ann. Inst. Fourier 3(1977), 1-15. Martinet, J.Formes decontact surlesvarietes dedimension 3.In:Lect. Notes inMath., Berlin-Heidelberg-New York, Springer, 1971, pp.142-163. Meckert, C.Formes decontact surlasource connexe dedeux varietes decontact. IRMA, Strasbourg, 1980. Moser, J.Onthevolume elements ofamanifold. Trans. Amer. Math. Soc.120(1965), 286-294. Oh,Y.-G. Asymplectic ﬁxed point theorem onT2"><CP". Math. Z.203:4 (1990), 535-552. Polterovich, L.New invariants ofembedded totally realtoriandoneproblem of Hamiltonian mechanics. In:Methods ofQualitative Theory andtheTheory of Bifurcations, Gorki, 1988, pp.84-90. Polterovich, L.Strongly optical Lagrange manifolds. Math. Notes Ac.Sc.USSR 45 (1989), 152-158. Polterovich, L.Symplectic Displacement Energy forLagrangian Submanifolds. Preprint, 1991. Polterovich, L.Thesurgery ofLagrange submanifolds. Geom. Funct. Anal. 2(1991), 213-246. Polterovich, L.TheMaslov class ofLagrange surfaces andGromov’s pseudoholomor- phiccurves. Trans. Amer. Math. Soc.325(1991), 241-248. Rabinowitz, P.Critical points ofindeﬁnite functionals and periodic solutions of differential equations. In:Proceedings ICM Helsinki 1978. Acad. Sci.Fennica, Helsinki, 1980, pp.791-796. Sato, H.Remarks concerning contact manifolds. Tohoku Math.J.29(1977), 577-584. Siegel, C.L. Symplectic geometry. Amer. J.Math.65:1(1943). Sikorav, J.C.Problémes d’intersections etdepoints ﬁxesengeometric hamiltonienne. Comm. Math. Helvet. 62:1(1987), 62-73. Sikorav, J.-C.Rigidite symplectique dans lecotangent deT".Duke Math.J.59(1989), 227-231. Sikorav, J.-C.Systémes Hamiltoniens ettopologie symplectique. Pisa,ETSEditrice, 1990. Sikorav, J.-C.Quelques proprietes desplongements lagrangiens. Preprint, 1990. Tabachnikov, S.L.Calculation ofthegeneralized Bennequin invariant ofaLegendrian curve from thegeometry ofitsfront. Funct. Anal. Appl. 22:3(1988), 246-248. Tabachnikov, S.Around fourvertices. Russian Math.Surveys 45:1(1990), 229-230. Tabachnikov, S.Geometry ofLagrangian andLegendrian 2-Web. Preprint, Arkansas Univ., 1992, 22pp. Traynor, L.Symplectic Embedding Trees forGeneralized Camel Spaces. Preprint 034-93 MSRI Berkeley, January 1993, 19pp. Traynor, L.Symplectic Packing Constructions. Preprint, October 1993, 20pp. Vasil’ev, V.A. Characteristic classes ofLagrangian andLegendre manifolds dual to singularities ofcaustics andwave fronts. Funct. Anal. Appl. 15(1981), 164-173. Vasil’ev, V.A. Self-intersections ofwave fronts andLegendre (Lagrangian) charactristic numbers. Funct. Anal. Appl. 16(1982), 131-133. Vassiliev, V.A. Lagrange andLegendre Characteristic Classes. New York, Gordon and Breach, 1988. Vasil’ev, V.A. Topology ofspaces offunctions having nocomplicated singularities. Funct. Anal. Appl. 23:4(1989), 24-36. 508 Bibliography ofSymplectic Topology Weinstein, A.Symplectic manifolds andtheir lagrangian submanifolds. Adv. Math. 6 (1971), 329-346. Weinstein, A.Lectures onsymplectic manifolds. C.B.M.S.Regional Conf. Ser.inMath. vol.29,Providence, AMS, 1977. Weinstein, A.Periodic orbits forconvex hamiltonian systems. Ann. Math. 108(1978), 507-518. Weinstein, A.Onthehypotheses ofRabinowitz’s periodic orbit theorems. J.Diﬂ.Eq. 33(1979), 353-358. Weinstein, A.Contact surgeries andsymplectic handle bodies. Hokkaido Math. J.20 (1991), 241-251. Viterbo, C.Capacites symplectiques etapplications. Seminaire Bourbaki, n°714, Astérisque 177-178 (1989), 345-362. Viterbo, C.Anewobstruction toembedding Lagrangian tori.Invent. Math.100(1990), 301-320. Viterbo, C.Plongement lagrangiens etcapacites symplectiques destores dans R2".C.R. Acad. Sci.Paris, Sér.I,Math. 311(1990), 487-490. Viterbo, C.Symplectic topology asthegeometry ofgenerating functions. Math.Ann. 292(1992), 685-710. 509 Acceleration 7 Action 60 Action-angle variables 280 Action function 253 Action variables 280, 2/81, 283 Adiabatic invariant 297, 413 Adjoint representation ofagroup 320 Afﬁne space 4 Angular momentum 30,46,323,328 Apocenter 35 Atlas 78 equivalence 78 symplectic 229 Averaging principle 291 Basic forms 167 Betti number 199 Birkhoff normal form forahamiltonian 386 foratransformation 388 Boundary ofachain 186 Canonical transfonnation 239 free 259,266 inﬁnitely small 269 Caustic 448,484 Center ofmass 46 Chain 185Index Characteristic 235,256,369,472 path length 312 Chart 77 Charts, compatibility of78 Chasles’ theorem 471 Chebyshev polynomial 27 Circulation 187 Closed form 196 Closed system 44 Coadjoint representation ofa group 320,457 Cocycle ofaLiegroup, two- dimensional 372 Codimension ofamanifold 426,430 Cohomology 199 class ofaLiealgebra 372 Commutator 208,211 Complex structure 224 Conﬁguration space 8 ofasystem withconstraints 77 Conjugate direction 251 Conservation ofcirculation, lawof 332 Conservation ofenergy, lawof16,22, 207 Conservative force ﬁeld 28,29,42 Conservative system 13,22,48 Constraint holonomic 77 ideal 92 511 Index Contact diffeomorphism 359 element 349 element, oriented 359 form 356 hamiltonian function 363 hyperplane 354 plane 356 structure 349,353,486 vector ﬁeld 360 Contactiﬁcation ofasymplectic manifold 368 Coordinate, cyclic 61,67 Coordinates elliptic 471 generalized 60 Coriolis force 130 Cotangent bundle 202,320 space 202,320 vector 202 Covariant derivative 308,310 Curl 194 ofatwo-dimensional velocity ﬁeld 333 Curvature tensor 307 Cycle I97 Cyclic coordinate 61,67 D’Alembert-Lagrange principle 92 Darboux’s theorem 230 forcontact structures 362 Degrees offreedom 80 Density,homeoidal 475 Derivative covariant 308, 310 inadirection 208 ﬁsherman’s 198 Lie 198 ofamap 82 Detuning 391 Diffeomorphism homologous tothe identity 419Distance between simultaneous events 5 Divergence 188 Effective potential energy 34 Eigenvalues ofthehamiltonian 381 Ellipsoid ofinertia 139,425 Ellipsoid ofrevolution 425 Elliptic coordinates 471 Elliptical transformation 388 Energy effective potential 34 kinetic 15,48,84 non-mechanical 49 potential ll,15,48.84 total 16,22,48,66 Equilibrium position 16,94,98 Euclidean space 5 structure 5,322 Euler angles 148, 149 Euler equations 143 Euler-Lagrange equation 58 Euler’s equation forageneralized rigid body 325 Events 5 simultaneous 5 Evolution 293 Exterior derivative ofaform 189 forms 163-166 monomials 168 multiplication 170 product 166,170 Extremal 57 conditional 92 Factorization ofaphase ﬂow 325 ofconﬁguration space 379 Fermat’s principle 249 Fiber lying over x81 Field Differential equations, ﬁrstorder non- axially symmetric 43 linear partial 369 Differential forms 174-181 Differential operator 208 Dimension ofamanifold 78 Discrete subgroup 275 512central 29,42,60 ofnondegenerate hyperplanes 353 reduced 378 right-invariant 214 Flux ofaﬁeld through asurface 187 Focal point toamanifold 442 Force 13,44 centrifugal 130 constraint 91 coriolis 130 external 45 generalized 60 inertial 94,129 inertial, ofrotation 130 internal 44 ofinteraction 44,48 Form basic 167 closed 196 nonsingular 235 Foucault pendulum 132 Frequencies independent 286 ofaconditionally periodic motion 286 relation among 289 Frequency deviation 391 Front 487 Functional 55 differentiable 56 Functions ininvolution 272 Galilean coordinate system 6 group 6 space 6 structure 5 transformation 6 Galileo’s principle ofrelativity 3 Galin’s theorem 384 Gardner’s theorem 454 Generalized velocities 60 Generating function 259 invariance of423 Geodesic ﬂow 313 oforiented contact elements 360 Group ofparallel displacements 4 Hamiltonian ﬂow 204 function 65,203,270,381 vector ﬁeld 203Index Hamilton-Jacobi equation 255, 260 Hamilton’s canonical equations 65, 236, 241 Hamilton’s principle ofleastaction 59 Hermitian-orthonormal basis 343 Hermitian scalar product 343 Hermitian structure ofcomplex projective space 343 Holonomic constraint 77 Homeoidal density 475 Homology 199 Homotopy formula 198 Huygens’ principle 250 Huygens’ theorem 250 Indicatrix 249 Inertia ellipsoid 139 Inertia operator 136,323 Inertia tensor 323 Inertial coordinate system 3 Inertial force 94,129 Integrability condition foraﬁeldofhyperplanes 352 Frobenius 350 Integral ofaform over achain 186 Integral invariant 206 relative 207 Integration ofdifferential forms 181 Invariant tori 401 nonresonant 402 resonant 402 Involutivity 63 Isotropic plane ofasymplectic space 222 lsovorticial ﬁelds 332 Ivory’s theorem 475 Jacobi equation 310 Jacobi identity 208, 211 Jacobi’s theorem 260,471 Jordan blocks, nonremovable 382 Kahler manifold 347 Kahler metric 347 513 Index Kepler’s law 31,32 Kepler’s problem 38 Kinetic energy 15,48,84 Kolmogorov’s theorem 405 Korteweg—de Vries equation 453 Lagrange’s equations 60,65 Lagrangian equivalence ofmappings 450 function 60 manifold 439 mapping 450,484 plane ofasymplectic space 222 singularity 446 system 83 system, non-autonomous 86 Laplace vector 413 Lax‘s theorem 453 Legendre ﬁbration 367,486 involution 366 manifold 365 mapping 487 singularity 367 submanifold 365 transformation 61,366,487 Liapunov stability 115 Liealgebra 208, 319 ofaLiegroup 213 ofﬁrstintegrals 217 ofhamiltonian functions 214 Poisson structure ondual 457 ofvector ﬁelds 211 Liebracket 213 Liegroup 213, 319 Linearization ofasystem 100, 101 Liouville’s theorem 69 onintegrable systems 271 Lissajous ﬁgure 24-27 Lobachevsky plane 303 Locally hamiltonian vector ﬁeld 218 Manifold connected 78 embedded 80 Kahler 347 lagrangian 439 Legendre 365 parallelizable 135 514riemannian 82 symplectic 201 Mapping ataperiod 115 Maslov index 442 Maupertuis’ principle ofleast action 245 Moment ofavector withrespect toan axis 43 Moment ofinertia withrespect toan axis 138 Momentum 45 generalized 60 Morse index 442 Motion conditionally periodic 285,413 inagalilean coordinate system 7 inamoving coordinate system 124 translational 124 Moving coordinate system 123 Neighborhood ofapoint ofa manifold 78 Newton’s equation 8 Newton’s principle ofdeterminacy 4 Noether’s theorem 88 Normal slowness ofafront 251 Null plane ofasymplectic space 222 Null vector ofaform 235 Nutation 152, 158 Obstacle problem 495 One-parameter group of diffeomorphisms 21,208 Optical pathlength 251 Orthogonal group 225 Oscillations characteristic 104 phase 397 small 102 Parallel translation ofavector ona surface 301, 302 Parametric resonance 119,225 Pericenter 35 Period mappings 466 Phase curve 16 ﬂow 21,68 flow, locally hamiltonian 218 plane 16 point 16 space 22,68 space, reduced 219 velocity vector ﬁeld 16 Poincare-Cartan integral invariant 237 Poincare’s lemma 197 recurrence theorem 71 relative integral invariant 238 Poinsot’s theorem 145 Point ofcontact 354. 356 Poisson action ofaLiegroup 372 bracket 211,214 manifold 456 vector 379 Poisson’s theorem 216 Polyhedron, singular k-dimensional 184 Polynomial. reﬂexive 226 Potential energy ll,15,48,84 Procession 153. 158 Principal axes 138 Projection, natural 81 Projective space, complex 343 Quadratic hamiltonian 381 eigenvalues of381 Quadric 470 Quasi-homogeneous function 462 Ray 251 Rayleigh's theorem 336 Reﬂexive polynomial 226 Regular point ofthespace ofangular momenta 328 Relative equilibrium 379 Resonant terms 391 Riemannian curvature 304 curvature inatwo-dimensional direction 308 manifold 82 metric 82 metric, left-invariant 322, 329 metric, right-invariant 329 Right translation 214Index Rigid body 133 _ Rigidity ofasystem 110 Rotation 124 Scalar product 5 Schroedinger equation 439 Sectorial velocity 32 Singularity lagrangian 446 Legendre 367 tangential 491 Skew-orthogonal complement 219 vectors 219 Skew-scalar product 219, 375 Soliton 453 Space average 286 Space ofsimultaneous events 5 Splitting ofseparatrices 394 Stability 99 Liapunov 115 strong 117 Stationary coordinate system 124 flow 331 group 275 rotation 145,328 Steiner’s theorem 141 Stokes‘ formula 192 Stokes‘ lemma 233 multidimensional 234 Stream function 333 Subalgebra 217 Swallowtail 258,368,450,467,487, 495 Symplectic atlas 229 basis 220 coordinate system 221 group 221 linear transformation 221, 225 linear transformation, stable 227 linear transformation, strongly stable 227 structure 201 structure ofcomplex projective space 345 structure,linear 219 structure ofaprojective algebraic manifold 346 515 Index Symplectic (cont.) structure ofspaces of polynomials 482 triad 497 vector space 219 Symplectiﬁcation ofacontact manifold 356 ofacontact vector ﬁeld 361 System closed 44 isoenergetic integrable 403 mechanical 7 natural 84 nondegenerate integrable 290 with onedegree offreedom 15 with twodegrees offreedom 22 Tangent bundle 81 space 80 vector toamanifold 81 Theorem ontheaverages 286 Three-body problem. restricted 415 Time 5 average 286 interval 5 Top fast 155 Lagrange’s 148 rapidly thrown 158 sleeping 154 symmetric 148 Tori, invariant 401 Track ofachain under homotopy 204 Trajectory 7 Transverse subspaces 224 Two-body problem 49 516Unitary group 225 transformation 444 Variation 56 Vector ﬁeld ofgeodesic variation 310 hamiltonian 203 locally hamiltonian 218 Velocities addition of125 generalized 60 Velocity 7 angular 125 ﬁrstcosmic 41 second cosmic 12 sectorial 32 Virtual variations 92 Vortex lines 233 tube 233.235 Vorticity ofatwo-dimensional velocity ﬁeld 332 Wave front 249 velocity ofmotion of251 Williamson‘s theorem 382 Work ofaﬁeld 28 ofaforce 28 World 5 lines 7,8 points 5 Young duality 64 Young’s inequality 64 Graduate Texts inMathematics 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93continued from page it WELLS. 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