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Whittaker Analytical Dynamics 2nd ed 1917 (classical mechanics)

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Classic Cambridge University Press treatise by E.T. Whittaker (Edinburgh), second edition, 1917, with an introduction to the problem of three bodies. The contents list covers kinematics, Lagrange's equations, integration principles, particle and rigid-body problems, vibrations, non-holonomic and dissipative systems, least action, Hamiltonian systems, and contact transformations. This is a downloaded book, not Phil's own writing.

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EE: : etic : Pecote caeteaepakcare htiy isxipoe%ain‘BiaieEtaegsosae:BsisePostLEGSBealtink:Pa NieeeHa0aa .ii Z?AeSieRS-“a 5ee+=Hothoeae iestr:ayeesialiteae| |ideebpaecotegia| eeeasas a| ieeePaty!seas| ameceEeSe4 ;peape,APantySsat i.eithaem:4 tihedigpeFisens1 WeA)scoalaptiepi"eaLp|| ‘aei, BePalb%;ce? oaHeraRaefig JemSeespel| :RicheyaeastGataySngDae renpeelsGeyThs5202: oiere **, BLNoh te f 4 ImM4aaoN gies Ss=ey aae‘1aged AGa Aero, %..Bia 0)3eae oSeG see me I SyRs ie) aa a ees 4 | ATREATISE ONTHE ANALYTICAL DYNAMICS OFPARTICLES ANDRIGID BODIES CAMBRIDGE UNIVERSITY PRESS C.F.CLAY, MANAGER Soulum: FETTER LANE, E.G. 100PRINCES STREET lirfu gorft:G.P.PUTNAM SSONS Bombag,Calcutta antoJftafcras: MACMILLAN AND CO., LTD. Toronto: J.M.DENTAND SONS, LTD. Til 1CMARUZEN-KABUSHIKI-KAISHA Allrightsreserved ATREATISE ONTHE ANALYTICAL DYNAMICS OFPARTICLES ANDRIGID BODIES; WITH ANINTRODUCTION TOTHE PROBLEM OFTHREE BODIES , BY E.T.WHITTAKER Hon. Sc.E).(Dubl.);F.R.S. Professor ofMathematics intheUniversityofEdinburgh SECOND EDITION CAMBRIDGE : ATTHKUNIVERSITY PRESS 1917 J 11n r Uf irstEdition 1904 Second Edition 1917 PREFACE TOTHESECOND EDITION TNrevisingthisbook forasecond edition, Ihave endeavoured togive- references to,and insome cases accountsof,thenumerousoriginal researches inDynamicswhich havebeenpublished byvariousinvestigators since the first editionappeared.Ihave moreover added some historical matter, andrewritten manysections. Itisnotnecessarytospecifythese indetail, butperhapsImaymention that thenewexplanationofthe transformation-theoryofDynamicsin125sprangfrom adesire todo justicetotheearliestgreatwork ofHamilton sgenius:thatthechangesin 69(themotion ofabodyabout arixedpointunder noforces) arose from myopinionthattheJacobian functions arepreferabletotheWeierstrassian inthenumericalcomputations:andthat Ishould have liked togive afullerproofofSundman stheorem(181),butthoughtitbetter togive onlysuchanaccount asmight impelthereader toconsult MrSundman s own accessible andreadable memoir. Iwishagaintorecordmyobligationstothe staff oftheCambridge UniversityPress. E.T.WHITTAKER. EDINBURGH, August,1916. a3 CONTENTS CHAPTER I. KINEMATICAL PRELIMINARIES. SECTION PAGE 1.Thedisplacementsofrigid bodies 1 2.Euler stheorem onrotations about apoint...... 2 3.Thetheorem ofRodrigues andHamilton....... 3 4.Thecompositionofequal andopposite rotations aboutparallelaxes . 3 5.Chasles theorem onthemost general displacementofarigidbody. . 4 6.Halphenstheorem onthecomposition oftwogeneral displacements. 5 7.Analytic representationofadisplacement 6 8.Thecompositionofsmall rotations 7 9.Euler sparametric specificationofrotations round apoint... 8 10.TheEulerian angles 9 11.Connexion oftheEulerian angles with theparameters ,?;, ,^. 10 12.Theconnexion ofrotations withhomographies:theCayley-Klein parameters11 13.Vectors 13 14. Velocity andacceleration;their vectorial character 14 15.Angular velocity;itsvectorial character 15 16.Determination ofthecomponentsofangular velocityofasysteminterms oftheEulerian angles, andofthesymmetrical parameters. . 16 17.Time-flux ofavector whose componentsrelative tomovingaxes aregiven. 17 18. Specialresolutions ofthevelocity andacceleration 18 MISCELLANEOUS EXAMPLES 22 CHAPTER II. THEEQUATIONS OFMOTION. Contents vii SECTION PAGE 26.Lagrangesform oftheequations ofmotion ofaholonomic system. . 34 27. Conservative forces;thekineticpotential 38 28.Theexplicitform ofLagrangesequations 39 29.Motion ofasystem which isconstrained torotate uniformly round anaxis . 40 30.TheLagrangian equationsforquasi-coordinates 41 31. Forces derivable from apotential-function which involves thevelocities . 44 32. Initial motions .... . . . . . . . . . 45 33. Similarityindynamical systems 47 34.Motion with reversed forces 47 35.Impulsivemotion 48 36.TheLagrangian equationsofimpulsive motion...... 50 MISCELLANEOUS EXAMPLES... 51 CHAPTER III. PRINCIPLES AVAILABLE FORTHEINTEGRATION. 37.Problems which aresoluble byquadratures 52 38.Systems withignorable coordinates . . . . . . . . 54 39. Special cases ofignoration ;integralsofmomentum andangular momentum 58 40.Thegeneral theorem ofangular momentum...... 61 41.Theenergy equation 62 42.Reduction ofadynamical problemtoaproblem with fewerdegreesof freedom, bymeans oftheenergy equation 64 43. Separationofthevariables;dynamical systemsofLiouville stype. . 67 MISCELLANEOUS EXAMPLES 69 CHAPTER IV. THESOLUBLE PROBLEMS OFPARTICLE DYNAMICS. 44.Theparticlewith onedegreeoffreedom;thependulum. . . . 71 45.Motion inamoving tube.......... 74 46.Motion oftwointeractingfreeparticles. 76 47. Central forces ingeneral:Hamilton stheorem 77 48.Theintegrablecases ofcentral forces;problems soluble interms ofcircular andellipticfunctions.......... 80 49.Motion under theNewtonian law 86 50.Themutual transformation offields ofcentral forceand fields ofparallel force"93 51.Bonnet stheorem............ 94 52.Determination ofthemost generalfieldofforce under which agiven curve orfamily ofcurves canbedescribed 95 53.Theproblemoftwocentres ofgravitation 97 54.Motion onasurface 99 55.Motion onasurface ofrevolution;cases soluble interms ofcircular and ellipticfunctions........... 103 56.Joukovskystheorem........... 109 MISCELLANEOUS EXAMPLES 111 vii.i Contents CHAPTER V. THEDYNAMICAL SPECIFICATION OFBODIES. SECTION PAGK 57. Definitions; . . 117 58.Themoments ofinertia ofsomesimple bodies -.118 59. Derivation ofthemoment ofinertia about anyaxiswhen themoment of inertia about aparallelaxisthrough thecentre ofgravityisknown 121 60.Connexion between moments ofinertia withrespecttodifferent setsofaxes throughthesameorigin....... 122 61.Theprincipal axes ofinertia;Cauchysmomenta!ellipsoid. . .124 62. Calculation oftheangular momentum ofamoving rigidbody. . .124 63. Calculation ofthekineticenergyofamoving rigidbody. . . .126 64.Independenceofthemotion ofthecentre ofgravity andthemotion relative toit 127 MISCELLANEOUS EXAMPLES 129 CHAPTER VI. THESOLUBLE PROBLEMS OFRIGID DYNAMICS. 65.Themotion ofsystems withonedegreeoffreedom;motion round afixed axis, etc. 131 66.Themotion ofsystems withtwodegreesoffreedom 137 67. Initial motions 141 68.Themotion ofsystems with threedegreesoffreedom.... 143 69.Motion ofabody about afixedpoint under noforces . . . .144 70. Poinsot skinematicalrepresentation ofthemotion;thepolhode and herpolhode............ 152 71.Motion ofatoponaperfectly rough plane ;determination oftheEulerian angle 6............. 155 72.Determination oftheremaining Eulerianangles, andoftheCayley-Klein parameters ;thespherical top......... 159 73.Motion ofatoponaperfectly smooth plane163 74.Kowalevski stop............ 164 75.Impulsive motion 167 MISCELLANEOUS EXAMPLES.. . , 169 CHAPTER VII. THEORY OFVIBRATIONS. 76. Vibrations aboutequilibrium ... . . . . . . - 177 77.Normal coordinates........... 178 78. Sylvesterstheorem ontherealityoftheroots ofthedeterminantal equation 183 79. Solution ofthedifferentialequations ;-theperiods ;stability. . .185 80.Examplesofvibrations aboutequilibrium187 81. Effect ofanew constraint ontheperiodsofavibrating system. . 191 82.Thestationarycharacter ofnormal vibrations 192 83. Vibrations about steady motion . . . . 193 84.Theintegrationoftheequations. . . . . . .19;"&gt; 85.Examplesofvibrations aboutsteadymotion...... 203 86. Vibrations ofsystems involving movingconstraints -07 MISCELLANEOUS EXAMPLES 208 Content* ix CHAPTER VIII. NON-HOLONOMIC SYSTEMS. DISSIPATIVE SYSTEMS. PAGESECTION 87.Lagrangesequationswithundetermined multipliers 88.Equationsofmotion referred toaxesmovinginanymanner . 89. Applicationtospecialnon-holonomic problems217 90. Vibrations ofnon-holonomic systems 91. Dissipative systems;frictional forces 92. Resistingforces which dependonthevelocity 93. Rayleighsdissipation-function 94. Vibrations ofdissipative systems 95.Impact 96. Loss ofkinetic energyinimpact 97.Examplesofimpact....... MISCELLANEOUS EXAMPLES...... CHAPTER IX. THEPRINCIPLES OFLEAST ACTION ANDLEAST CURVATURE. 98.Thetrajectoriesofadynamical system245 99.Hamilton sprincipleforconservative holonomic systems.... 245 100. TheprincipleofLeast Action forconservative holonomic systems. .247 101. Extension ofHamilton sprincipletonon-conservative dynamical systems 248 102. Extension ofHamilton sprinciple andtheprincipleofLeast Action to non-holonomic systems.......... 249 103. Arethestationary integralsactual minima? Kinetic foci . . .250 104. Representationofthemotion ofdynamical systems bymeans ofgeodesies.253 105. Theleast-curvature principleofGauss andHertz..... 254 106. Expressionofthecurvature ofapathinterms ofgeneralisedcoordinates 256 107. Appellsequations258 108. Bertrand stheorem 260 MISCELLANEOUS EXAMPLES 261 CHAPTER X. HAMILTONIAN SYSTEMS ANDTHEIR INTEGRAL-INVARIANTS. 109. Hamilton sform oftheequationsofmotion . . . . . .263 110. Equations arising from theCalculus ofVariations..... 265 111. Integral-invariants 267 112. The variationalequations 268 113. Integral-invariantsoforder one......... 269 114. Relative integral-invariants 271 115.Arelative integral-invariant which ispossessed byallHamiltoniansystems 272 116.Onsystemswhichpossesstherelativeintegral-invariant|2pS&lt;/. .272 117. Theexpressionofintegral-invariantsinterms ofintegrals. . .274 118. Thetheorem ofLieandKoenigs 275 119.The lastmultiplier 276 Content* 120. Derivation ofanintegral from twomultipliers279 121.Applicationofthe lastmultipliertoHainiltonian systems;useofa single knownintegral280 122.Integral-invariants whose order isequaltotheorder ofthesystem.283 123. Reduction ofdifferential equationstotheLagrangian form . . .284 124. (a.seinwhich thekinetic energyisquadraticinthevelocities . .285 MISCELLANEOUS EXAMPLES . ... 286 CHAPTER XL THETRANSFORMATION-THEORY OFDYNAMICS. 125. Hamilton sCharacteristic Function andcontact-transformations . .288 126. Contact-transformations inspaceofanynumber ofdimensions . .292 127. The bilinear covariant ofageneraldifferential form.... 296 128. The conditions foracontact-transformation expressed bymeans ofthe bilinear covariant 297 129. Theconditions foracontact-transformation interms ofLagrangesbracket- expressions............ 298 130. Poisson sbracket-expressions ......... 299 131. Theconditions foracontact-transformation expressed bymeans ofPoisson s bracket-expressions ........ 300 132. Thesub-groupsofMathieu transformations andextended point-transforma tions............. 301 133. Infinitesimal contact-transformations ........ 302 134.Theresulting newview ofdynamics304 135. Helmholtz sreciprocal theorem......... 304 136. Jacobi stheorem onthetransformation ofagiven dynamical systeminto anotherdynamical system......... 305 137. Representationofadynamical problem byadifferential form . . .307 138. TheHarniltonian function ofthetransformed equations.... 309 139. Transformations inwhich theindependentvariable ischanged. .310 140.New formulation oftheintegration-problem ...... 310 MISCELLANEOUS EXAMPLES . ... 311 CHAPTER XII. PROPERTIES OFTHEINTEGRALS OFDYNAMICAL SYSTEMS. 141. Reduction oftheorder ofaHarniltonian system byuseoftheenergy integral............. 142. Hamilton spartialdifferential equation....... 143. Hamilton sintegralasasolution ofHamilton spartialdifferential equation 144. Theconnexion ofintegrals with infinitesimal transformations admitted by thesystem............ 318 145. Poisson stheorem............ 320 146. TheconstancyofLagrangesbracket-expressions321 147. Involution-systems322 Contents xi SECTION PAGE 148. Solution ofadynamical problem when halftheintegralsareknown .323 149. Levi-Civita stheorem. 325 150. Systemswhichpossess integralslinear inthemomenta.... 328 151. Determination oftheforces acting onasystemforwhich anintegralis known 331 152. Applicationtothecase ofaparticle whose equationsofmotion possessan integral quadraticinthevelocities....... 332 153. General dynamical systems possessing integrals quadraticinthevelocities .335 MISCELLANEOUS EXAMPLES . 336 CHAPTER XIII. THEREDUCTION OFTHEPROBLEM OFTHREE BODIES. 154. Introduction 339 155. The differential equationsoftheproblem340 156. Jacobi sequation............ 342 157. Reduction tothe12th order, byuseoftheintegralsofmotion ofthecentre ofgravity343 158. Reduction tothe8thorder, byuseoftheintegralsofangular momentum andelimination ofthenodes........ 344 159. Reduction tothe6thorder 347 160. Alternative reduction oftheproblemfrom the18th tothe6thorder .348 161. Theproblemofthree bodies inaplane 351 162. The restricted problemofthree bodies . 353 163. Extension totheproblemofnbodies....... 356 MISCELLANEOUS EXAMPLES , 356 CHAPTER XIV. THETHEOREMS OFBRUNS ANDPOINCARE. 164. Bruns theorem (i)Statement ofthetheorem 358 (ii) Expressionofanintegralinterms oftheessential coordinates of theproblem 358 (iii)Anintegral must involve themomenta...... 359 (iv)Only oneirrationality canoccur intheintegral.... 360 (v)Expressionoftheintegralasaquotientoftwo realpolynomials.361 (vi) Derivation ofintegrals from thenumerator anddenominator of thequotient. . . 362 (vii)Proof that (does notinvolve theirrationality.... 366 (viii) Proof that &lt;isafunctiononlyofthemomenta andtheintegrals ofangular momentum 371 (ix) Proof that &lt;isafunction of 7",Z,J/,N 374 (x)Deduction ofBrims theorem,forintegrals which donotinvolve t376 (xi) Extension ofBrims result tointegrals which involve thetime .378 xii Contents 165. Poincare stheorem (i)Theequationsofmotion oftherestricted problemofthree bodies .380 (ii) Statement of Poincare":stheorem....... 381 (iii) Proof that 4&gt;isnotafunction ofIff, 381 (iv) Proof that 4&gt;cannot involve thevariables q,q%. . .382 (v)Proof thattheexistence ofaone-valuedintegralisinconsistent with theresult of(iii)inthegeneralcase...... 383 (vi)Removal oftherestrictions onthecoefficients B m&gt; t,,,2. . .384 (vii) Deduction ofPoincare stheorem 385 CHAPTER XV. THEGENERAL THEORY OFORBITS. 166. Introduction 386 167. Periodic solutions 386 168. Poincare snormal variables foraperiodicorbit 387 169.Acriterion forthediscoveryofperiodicorbits . . . . . .388 170. Lagrangesthreeparticles 390 171.StabilityofLagrangesparticles:periodicorbits inthevicinity. .394 172. The differentialequationofthenormaldisplacementfrom anorbit .395 173. Kortewegstheorem........... 397 174. Theindex ofstability 398 175. Characteristic exponents.......... 400 176. Propertiesofthecharacteristic exponents....... 401 177. Attractive andrepellent regionsofafield offorce 403 178. Applicationoftheenergy integraltotheproblemofstability. . .406 179.Applicationofintegral-invariantstoinvestigationsofstability. . .407 MISCELLANEOUS EXAMPLES . . 107 CHAPTER XVI. INTEGRATION EYTRIGONOMETRIC SERIES. 180. Theneed forseries which convergeforallvalues ofthetime;Poincare s series............. 410 181. Theregularisationoftheproblemofthree bodies . . . . . 411 182. Trigonometricseries 412 183. Removal ofterms ofthe firstdegree from theenergy function . .413 184. Determination ofthenormal coordinates byacontact-transformation .414 185. Transformation tothetrigonometric form ofIf . . . . .417 186. Othertypesofmotion which lead toequationsofthesame form . .419 187. Removal ofaperiodictermfromH........ 420 188. Removal offurtherperiodicterms from // 4-2-2 189. Reversion totheoriginal coordinates 423 MISCELLANEOUS EXAMPLES . . 424 INDEX OFAUTHORS QUOTED 4^5 INDEX OFTERMS EMPLOYED . 428 CHAPTER I KINEMATICAL PRELIMINARIES 1.Thedisplacements ofrigidbodies. ThenameAnalytical Dynamicsisgiventothatbranch ofknowledge inwhich themotions ofmaterial bodies, considered asdue tothemutual interactions ofthebodies, arediscussed bytheaidofmathematicalanalysis. Itisnatural tobeginthisdiscussionbyconsideringthevariouspossible typesofmotion inthemselves, leavingoutofaccount foratime thecauses towhich theinitiation ofmotion maybeascribed;thispreliminary enquiry constitutes thescience ofKinematics. Theobjectofthepresent chapteris toestablish anumber ofkinematical theorems which willberequiredinthe restofthework. Kinematics isinitselfanextensivesubject,foracomplete account ofwhich thestudent isreferred totreatises dealing exclusively withit,e.g.that ofKoenigs (Paris, 1897). In what follows weshall confine ourattention totheorems which areofutilityintheappli cations ofKinematics toDynamics. Weshallsaythat amaterialbodyisrigidwhen themutual distance of every pairofspecified pointsinitisinvariable, sothat thebody doesnot expandorcontract orchangeitsshapeinanyway,althoughitmaychange itsposition with reference tosurrounding objects. Ifarigid bodyismoved from onepositiontoanother, thechangeof positioniscalled adisplacementofthebody. Certainspecial kinds of displacementhave receivedspecific names; thus, ifthepositioninspace ofevery pointofthebodywhich liesonsomestraightlineLisunchanged, thedisplacementiscalled arotation about thelineL;ifthepositionin spaceofsomepointPofthebodyisunchanged,thedisplacementiscalled arotation about thepointP;and ifthe linesjoiningthe initial and final positionsofeach ofthepointsofthebodyareasetofparallel straightlines oflength I,sothat theorientation ofthebodyinspaceisunaltered, the displacementiscalled atranslationparalleltothedirectionofthelines, through adistance I. w.D. 2 Kinematical Preliminaries[CH.i 2.Elder stheorem onrotations about apoint*. Consider arigid body,oneofwhosepointsismade immoveable bysome attachment;supposethat thebodyisfree toturnabout thispointinany manner, andletanytwopossible configurationsofthebodybetaken :for convenience weshall callthese theconfiguration Pandtheconfiguration Q. Weshallnowshew that itispossibletobringthebodyfromtheconfiguration Ptotheconfiguration Qbysimply rotatingitabout some definite line throughthefixedpoint,i.e.thatarotation about apointisalways equivalent toarotation about alinethroughthepoint. Toestablish thisresult (which was firstgiven byEuler),denote thefixed point by0;letOA,OBbethepositions,intheconfiguration P,oftwolines throughthefixedpointwhich arefixed inthebodyandmove with it;let OA,OBbethepositionsofthesame lines intheconfiguration Q.Draw theplanewhich isperpendiculartotheplaneAOA and bisects theangle .ACM/: anddraw alsotheplanewhich isperpendiculartotheplaneBOB andbisects theangleBOB. Let00bethelineofintersection ofthese two planes, supposingthem tobenotcoincident;iftheyarecoincident, we denoteby00thelineofintersection oftheplanesOAB andOAB . Thenclearlyineither casetheline00 isrelated tothelinesOA,OB inexactlythesamewayasitisrelated tothelinesOAandOB;that isto say,theanglesAGOandBOG arerespectively equaltotheanglesAOGand BOG. Itfollows that ifthesystemOABC isrotated about insuch away thatthelinesOAandOBcome intothepositions OAandOBrespectively, thenOG will retain itsposition unchanged. The lineOC istherefore unaffected bythedisplacementinquestion,andsothedisplacement can berepresented byarotationthroughsomeangleround 06;whichproves thetheorem. When abodyiscontinuously movinground oneofitspoints,which is fixed inspace,thedisplacementfrom itspositionattime ttoitspositionat time t+At,can,byEuler stheorem, beobtained byrotatingthebodyabout some definite linethroughthefixedpoint.Thelimiting positionofthis line,when theinterval Atfisindefinitely diminished,iscalled theinstantaneous axisofrotation ofthebodyatthetime t. When abodyiscontinuously moving round oneofitspoints,which isfixed, thelocus oftheinstantaneous axis inthebodyisacone,whose vertex isatthefixedpoint: the locus oftheinstantaneous axis inspaceisalsoaconewhose vertex isatthefixedpoint. Shew thattheactual motion ofthebody canbeobtained bymakingtheformer ofthese cones (supposedtoberigidlyconnected with thebody)rollonthelatter cone(supposedto befixed inspace). (Poiusot.) Asimilarproofshews thatifanytwopositions ofaplane figureinthe same planearegiven,thedisplacement fromonepositiontotheother canbe *NoviComment. Petrop.xx.(1776), p.189, 25. 2-4] Kinematical Preliminaries 3 regardedasarotation about somepointintheplane. Thispointiscalled thecentreofrotation. When thebodyisregardedascontinuously moving,thesmalldisplace ment fromonepositiontothepositionwhich succeeds itafteraninfinitesimal interval oftime cantherefore beaccomplished byarotation round apoint; thispointiscalled theinstantaneous centreofrotation. Example1.Alamina moves inanymanner initsplane. Prove thatthelocus atany instant ofpoints which areatinflexions oftheirpathsisacircle, which touches theloci inthelamina andinspaceofthecentre ofinstantaneous rotation.(Coll. Exam.) Example2.Arigidbodyintwodimensions issubjected successivelytotwo finite displacementsinitsplane.IfD2bethelinejoining thecentres ofdisplacement, and if DIbethelinewhich isbroughtintotheposition D2byhalfthe firstdisplacement (i.e. byrotation throughhalftheangle), and ifD3bethepositiontowhichD2isbrought by halftheseconddisplacement, shew thatthecentre ofthetotal displacementoftherigid bodyistheintersection ofDIand Z&gt;3.(Coll. Exam.) 3.Thetheorem ofRodrigues andHamilton*. Anytwosuccessive rotations about afixedpointcanbecompoundedinto asinglerotationbymeans ofatheorem, whichmaybestated asfollows : Successive rotations about three concurrent linesfixedinspace, throughtwice theangles oftheplanes formed bythem, restore abodytoitsoriginal position. For letthelines bedenoted byOP,OQ,OR. Draw.Op,Oq,Orper pendiculartotheplanes QOR,ROP,POQ respectively. Then ifabodyis rotatedthrough tworight anglesaboutOq,andafterwardsthrough tworight angles about Or,thepositionofOP isonthewhole unaffected, while Oqis moved totheposition occupied byitsimageinthe lineOr;the effect is therefore thesame asthat ofarotation roundOPthroughtwice theangle between theplanesPRandPQ,which wemaycalltheangleRPQ. It follows that successive rotations round OP,OQ,ORthroughtwice theangles RPQ,PQR,QRP, respectively,areequivalenttosuccessive rotationsthrough tworight anglesabout thelinesOq,Or,Or,Op,Op,Oq;butthe latter rotations willclearlyonthewholeproducenodisplacement; which establishes thetheorem. 4.Thecomposition ofequal andoppositerotations aboutparallelaxes. Acase ofspecialinterest isthat inwhich abodyissubjectedinturn to tworotations ofequalamount inoppositesenses about twoparallelaxes. Inneitherdisplacementisanypointofthebody displacedinadirection paralleltotheaxes, and this istherefore true ofthe totaldisplacement. Moreover, ifanylinebetaken inthebodyinaplane perpendiculartothe *0.Rodrigues, Journ. deMath. v.(1840), p.380;Hamilton, Lectures onQuaternions, 344; theproof heregivenisduetoBurnside, ActaMath. xxv.(1902). 12 4 Kinematical Preliminaries[CH.i axes, this line inthe firstdisplacementwillbeturnedthroughanangle equaltotheangleofrotation, andintheseconddisplacementwillbeturned backthroughthesameangle;soitsfinalpositionwillbeparalleltoits original position;whichevidentlycanbethecase foreverylinewithout exception, onlywhen the totaldisplacementisequivalenttoasimple translation. Itfollows thattwosuccessiveequal andoppositerotations about parallelaxes areequivalenttoatranslation inadirectionperpendicularto theaxes; or,inother words, arotation about anyaxis isequivalentto arotationthroughthesameangle aboutanyaxisparalleltoit,togetherwith asimple translation inadirectionperpendiculartotheaxis. The converse ofthis,namelythetheorem thatarotationofarigid bodyabout any axis, precededorfollowed byatranslation inadirection perpendiculartotheaxis, aretogether equivalenttoarotationofthebody about aparallel axis, isalso true, being essentiallythesame astheresult stated in2,thatanydisplacementinaplanecanberegardedasarotation round somepointintheplane. Byconsideringtheangle between the initial and finalpositionsofanylinewhich isperpendiculartotheaxisand moves with thebody,weseethattheanglesofrotation round thetwoaxes areequal. 5.Chasles theorem onthemostgeneral displacement ofarigid body*. Weshallnow,considerdisplacementsofamoregeneralcharacter. Itis evident that afreerigid bodycanbemoved fromanyoneselected con figuration PinspacetoanyotherQbyfirstmovingsome selectedpointof thebodyfrom itspositionintheconfiguration Ptoitspositioninthe configuration Q,each oftheotherpointsofthebodybeing moved byasimple translationparalleltothis(sothatthebodyisoriented inthesamewayafter theoperationasbefore), andsecondly rotatingthebodyabout thispointinto theconfiguration Q.ByEuler stheorem, this latteroperationcanbe performed bysimply rotatingthebodyabout alinethroughthepoint ;so weseethat themostgeneral displacement ofarigid bodycanbeobtainedby first translatingthebody,andthenrotatingitabout aline. Weshallnowshew that thelineabout which therotation takesplace can besochosen, thatthemotionoftranslation isparalleltothis line. For letA betheinitialpositionofanypointofthebody, andBthepositiontowhich thispointisbrought bythemotion oftranslation. LetAKbethe line through Aparalleltothelineround which therotation takesplace, and letKbethefoot oftheperpendicular fromBonAK. Then themotion of translation canevidentlybeaccomplishedintwostages,the first ofwhich isatranslationparalleltothe lineabout which therotation takesplace, *Mozzi, Discorso matematicosopra ilrotamento momentaneo delcorpi, Naples, 1763; Cauchy, Exercices deHath. n.(Paris, 1827), p.87;Oeuvres, (2)vn.p.94;Chasles, Bulletin Unit: des Sciences(Ferussac),xiv.(1830), p.321;Comptes Rendus deVAcad. xvi.(1843), p.1420. 4-6] Kinematical Preliminaries 5 bringingthepointAtotheposition K,and thesecond ofwhich is atranslationperpendiculartothelineabout which therotation takesplace, bringingthepointKtothepositionB.Butby 4,thesecond translation, togetherwith therotation which followsit,aretogether equivalentto asimplerotation about anew axisparalleltothe first one. Iftherefore any pointonthis axisbetaken asbase-point,thewholedisplacementcanbe accomplished byatranslation ofthebody paralleltoacertain linethrough thispoint, togetherwith arotation about this line; this establishes the theorem. Thiscombination ofatranslation andarotation round alineparallelto thedirection oftranslation iscalled ascrew;theratio ofthedistance of translation totheangleofrotation iscalled thepitchofthescrew. Itis clear that inascrewdisplacement,theorder inwhich thetranslation and rotation takeplaceisindifferent. 6.Halphenstheorem onthecomposition oftwogeneral displacements. Halphen hasshewn* howtodeterminegeometricallytheresultant ofanytwoscrew- displacementsasascrew-displacement. LetAvandA2denote theaxes ofthetwoscrews, and -4]2theircommonperpendicular. LetBIbethelinewhich isbroughttotheposition A12byhalfthe firstdisplacement (i.e.halfthetranslation, androtation throughhalftheangle), and letB2bethelineto whoseposition A^isbrought byhalfthesecond displacement;letCdenote thecommon perpendiculartothelines BIandB2.Halphensresult isthat theaxisoftheresultant screw-displacementisC,and thedisplacementistwice thatwhichbringstheline1tothe position B2. For letDIandD2belinessuch that halfthegiven displacementswillbringA12tothe position DIandD2totheposition A12respectively, and let C"bethecommonperpendicular toDIandD2. Thefigure thus obtained, andthatwhich isobtained from itbyrotatingitthrough two right angles aboutAi2,evidentlycoincide;whence wehave therelations : Intercept made onB1byA{andC=Intercept made on Z&gt;tbyAIand C", Intercept made onB2byA2andC=Intercept made onD2byA2andC, Intercept made onCbyBIandB2=Intercept made on C"byDIandD2, Angle between theplanesAiB 1,BtC=Angle between theplanes AIDltDiC , Angle between theplanesA2B2,B2C=Angle between theplanes A2D2,D2C, Angle between B1andB2 =Angle between DlandD2. Itfollows thatthescrew aboutA1bringsCtotheposition of C"produced,theinter section ofB1andCbeing broughttothepositionoftheintersection ofD^andC;and then thescrew aboutA2bringsC"tothepositionofCproduced,theintersection of D2andCbeing brought totheintersection ofB2andC;soCistheaxisoftheresultant screw, andtheamount ofthetranslation istwice theintercept made onCby^andB2. Also thelineBI,which bythe firstscrew isbroughttotheposition Z&gt;j,isbythesecond broughttoaposition making thesameangle withB2thatB2makes withBl;andtherefore *Nouvelles Annales deMath.(3)i.p.298(1882). Theproof given here isduetoBurnside, Mesa, ofMath. xix. p.104(1889). 6 Kinematical Preliminaries[CH.I therotation oftheresultant screw istwice theangle between BzandB.This establishes Halphenstheorem. Example. Shew thatanyinfinitesimal displacement ofarigidbody canbeobtained bythecompositionoftwoinfinitesimal rotations roundlines, andthatoneofthese lines canbearbitrarilychosen. 7.Analytic representation ofadisplacement. Weshallnowseehowanydisplacementofarigidbodycanberepresented analytically. LetrectangularaxesOxyzbetaken, fixed inspace:these willbesupposed toformaright-handed system,i.e.iftheaxes aresoplacedthatOzisdirected vertically upwards andOyisdirected tothenorthern horizon, thenOxwill bedirected totheeast. Letthedisplacement considered beequivalenttoa rotationthroughanangle&&gt;about alinewhosedirection-anglesare(a, /3,7), andwhichpasses throughapointAwhose coordinates are(a, b,c),together with atranslationthroughadistance dparalleltothis line. Theangle&&gt; must betaken with itsappropriate sign,thesignbeing positive when theline (a,/3,7)beingdirectedvertically upwards,therotation from thesouthern horizon tothenorthern isroundbythe east. LetthepointPwhose coordinates are(x,y,z)bebrought bythedisplacementtothepositionofthe pointQ(X,Y,Z);and letthepointPbebrought bythetranslation alone tothepositionofthepointR(, 77, );thenwehaveevidently =x+dcosa, 77=y+dcosft, %=z+dcos7. LetKbethefoot oftheperpendicular fromR(orQ)ontheaxis of rotation, and letLbethefootoftheperpendicular fromQonKR. Then wehave Xg=projectionofthebroken lineRLQ ontheaxisOx, itbeingunderstood thatprojections have theirappropriate signs,sothatthe projectionofalineABontheaxisofa;is(XBXA),not(XAXB). Now theprojectionofKRontheaxisOxis fa(projectionofAKontheaxisOx) or |-a-cosa{(-a)cosa+(T?-b)cos/3+(f-c)cos7], andasRL=(1-cosco)KR, itfollows that theprojectionofRLonthe axisOx is -(1-costu)[f-a-cosa{(-a)cosa+(17-b)cos@+(f-c)cos7}]. Moreover, thelineLQisnormal totheplaneRKA, and itsdirection-cosines arethereforeproportionaltothequantities ( c)cos/3 (77 6)cos7, (a)cos7 ( c)cosa, (?? b)cosa(a)cos/3, 6-8]Kinematical Preliminaries 7 andsince thesum ofthesquaresofthese threequantities,divided bythe expression {(-a)2+0?- &)2+(-c)2 },representsthequantity sirfRAK, it follows that thesum ofthethree squaresisequaltoKR1 ,andthethree quantitiesthemselves aretheprojectionsontheaxes ofalength+KR measured alongthe lineLQ. SinceLQ=KRsinw,theprojectionofLQ ontheaxisOxistherefore +sin &){( c)cosft (r) b)cos7}. Onconsideringaspecial case, e.g.supposingthattheaxisofrotation isthe axis Oz,weseethattheupper signiscorrect;andthuswehave Z-=-(1-cos &&gt;){(- )-cos2a(-a) cosacos/3(r) b)cosacos7( c)} +sin to{cos /3( c)cos7(77. 6)j. Substitutingforrj,their values interms ofx,y,z,wehave X=x+dcosa(1-cosw){(x a)sin2a cosacos(3(y b)cosacosy(z c)} +sin to(cos/3(z c)cosy(y &)}. Similarly wehave F=y+dcos/3-(1-cos&)){(y-6)sin2 /3 -cos/3cosy(z c)cos/3cosa(x a)} 4-sin ft)[cos7(#a)cosa.(z c)} and Z=z+dcos7-(1cosw){(z-c)sin27 cos7cos a.(xa)cos7cos/3(y b}} +sin &)(cosa.(y b)cosft(x a)}. Theseequations givethenew coordinates X,Y,Zinterms ofthe coordinates x,y,zoftheoriginal positionofthepointandthequantities which define thedisplacement. 8.Thecomposition ofsmall rotations. We shallnowapplythelast result tothecase inwhich therotation is infinitesimal, theaxis ofrotationpassing throughtheoriginandtherebeing nomotion oftranslation. We shall write 8&gt;|rforco,whereS-v/risasmall quantitywhosesquarecanbeneglected.Theequationsofthelast article nowbecomeX=x+(zcos /3ycos7)8-v/r, -F=y+(xcos7zcosa)S-^r, Z z+(ycosaxcos/3)8^. Butthese aretheequationswhich weshould obtain ifwesuccessively (in anyorder) subjectedthebodytoinfinitesimal rotations cosa .tyabout Ox, cos/3.8-^raboutOy,andcos7.8^about Oz. Itfollows thatanysmall rota tionStyabout alineOK isequivalenttosuccessive small rotationsSty.cosKOx 8 Kinematical Preliminaries[CH.i about Ox, Si/r.cosKOy aboutOy,and&$.cosKOz about Oz,where Ox,Oy,Oz areanythreemutually perpendicular lineswhich intersect OKinoneofits points, 0. 9.Entersparametric specification ofrotations round apoint*. Theanalytic expressionsforthetranslationalpartofadisplacement are, aswehave seen, extremely simple; buttheexpressionsfortherotational partarenotsosimple, andthese willnowbefurther considered.Suppose then that arigidbodyisrotatedthrough anangle&&gt;about alinethrough theorigin, whosedirection-anglesarea,ft,7.By 7,thecoordinates (X,Y,Z)ofthenewpositionofapoint whoseoriginal coordinates were (x,y,z)aregiven bytheequations (X=x2sina\w(xsin2aycosacosftzcosacos7) +2sin-iwcos\w{zcosftycos7), iF=y2sin2 |a)(ysin2 ftzcosftcos7xcosftcosa) +2sini&)cos^o)(xcos7zcosa), =z2sin2G)(sin2yxcos7cosaycos7cos/3) +2sin^cocos ^&&gt;(ycosaxcos/3). Now introduceparameters,77, ,^,definedbytheequations =cosasina&gt;, 77=cos/3sin|&&gt;,=cos7sini&&gt;,x= cos&&gt;; these parameters evidently satisfytheidentical relation r+7?2+r+ %2=i, andtheaboveequationscanbewritten intheform Iftherefore thecoordinate axes aredenotedbyOXYZ, and ifmoveable axeswhichoriginallycoincide with these arebroughtintotheposition Oxyz bythegiven rotation, thedirection-cosines ofthetwo setsofaxes with reference toeach other aregiven bythefollowing scheme : X Y Z *NoviComment. Petrop.xx.(1776), p.208, 6sqq. 8-rlO]Kinematical Preliminaries 9 Itisreadilyseen thattheparameters (", 17", f", x"),correspondingtotheresultant of twosuccessive displacements ( ,r,,f,*)and(, 77,f,x),aregiven bytheequations x"= xx-^f-w -f- These formulae (which were discovered independentlyatdifferent times byGauss, Rodrigues, Hamilton, andCayley) reallyconstitute thetheorem forthemultiplication of quaternions.For^,,?;,may beregardedasthecomponentsofaquaternion* X+i+y+)wherei,j,ksatisfytheequations i*=j*=k*=-\, ij=-ji=k, jk=-kj=i,ki=-ik=j; andtheabove formulae arethen allcomprehendedinthesingle equation X"+Zi+jj+t"k= (X+&+W+M(X+&+JJ+*) Thereader who isacquaintedwith quaternionswillobserve that the effect ofthe rotation onanyvectorpistoconvert itinto thevector qpq1 ,whereqdenotes the quaternion x+&+n}+&"&gt;thequaternionitself isnottherotational operator. 10.TheEulerianangles. Themostpracticallyuseful ofthevarious methods ofparametrically representingthedisplacementofarigidbodyduetoarotation round afixed pointislikewise due toEulerf:ithasthedisadvantageofbeing unsym- metrical, but isotherwiseverysimpleandconvenient. Let bethefixedpointround which therotation takesplace,and let OXYZ bearight-handed systemofrectangularaxes_fixedinspace.Let Oxyzberectangularaxes fixedrelativelytothebodyandmovingwith it, andsuch thatbefore thedisplacementthetwosetsofaxesOXYZ andOxyz arecoincident inposition.LetOKbeperpendiculartotheplane zOZ, drawn sothat ifOZ isdirected tothevertical andtheprojectionofOz perpendiculartoOZisdirected tothesouth, thenOK isdirected totheeast. Denote theangles zOZ,YOK,yOK by6, &lt;, -^r,respectively:these are known asthethree Eulerian angles definingthepositionoftheaxesOxyz with reference totheaxesOXYZ. Inorder tofindthedirection-cosines ofOx,Oy,Oz,withrespecttoOX, weobserve thatthese areequaltotheprojectionsonOx,Oy,Oz,respectively, ofaunitlengthmeasuredalongOX.Now thisunitlengthhasprojections cos$alongOLand sin&lt;along OK,whereOL istheintersection ofthe planesXOY andZ0z\ butalengthcos&lt;alongOLhasprojectionscos &lt;f&gt;sin6 along Ozand cos$cos6along OM,whereOM istheintersection ofthe planes xOyandZOz; andalengthcos cosalongOMhasprojections cos$cos6cosA|Talong Oxand cos&lt;cos6sin-fyalong Oy;also, alength sin &lt;alongOKhasprojections sin&lt;sim/r along Oxand sin (f&gt;cosi/r *Thisquaternionwillhave itstensor equal tounity. tNoviComment. Petrop.xx.(1776), p.189. 10 Kinemat icalPreliminaries[CH.I along Oy.Hencefinallytheprojections onOx,Oy,Ozrespectivelyofthe unitlength measured onOXare Icos d&gt;cos6cos&sind&gt;sin-vjralonsfOx I a , r \cos &lt;pcosasiny-sin$costyalong Oy, \cos (/&gt;sin6alongOz. Proceedinginthisway,weobtain forthedirection-cosines ofthetwosetsof axesOXYZandOxyzwithrespecttoeachother thefollowing scheme : X Y Z 11.ConnexionoftheEulerianangleswith theparameters,77, ^. The relations between theEulerianangles 6, &lt;, \|randtheparameters f/j&gt;%f9maybeobtained bycomparingtheschemes ofdirection- cosines which havebeengivenin 9and10;theymayhowever beobtained directlyasfollows : LetOXYZandOxyzbethefixed axesandtheaxes derived from these bytherotation wround alineOR,whosedirection-anglesare(a, ft,7). Draw asphereofunit radius with thepointascentre, sothatplanes passing throughintersect thesphereingreat circles, and lines intersect the sphereinpoints. Then inthespherical triangle RZz, thesides are7,7,0, andtheangleatRis o&gt;;whence wehave therelation sn=sin7sin| Moreover, letvdenote theangleRZY, soRZz=ITT&lt; v.Then the arcRZ isbroughttotheposition Rzbysuccessive rotations&lt;about Z, 6about thepoleofZz,and-v/rabout z;butthe firstofthese transforms RZ intoanarcmakinganangle \-rr (f&gt;-v+ &lt;/&gt;orITT vwith Zz,atZ;the second rotation transforms this intoanarcmakingthesameangle ^TTv with Zz,butpassing throughz ;andthethird rotation transforms itintoan arcmakinganangle ^TTv+tywith Zz,atz.But thisanglemust beequal toTTRzZ, orTTRZz, orTT(^TT &lt;j&gt; v),or\tr+ &lt;/&gt;-fv;sowehave v=TTv or 10-12]Kinematical Preliminaries 11 Hence, since inthespherical triangleRZX thesides are a,7,^TT,andthe angleatZis-|TTvor(TT -v^+$),wehave cosa=sin7sin\(-v/r $). Substitutingforsin7from theequation already found, thisgives cosasin\w=sin^6sin|(^ $), or =sin\Qsin|(i/r&lt;). Similarlyfrom thespherical triangle RZYwehave cosft=sin7cos(i/r 0), andagain eliminatingsin7,wehave cosftsin^tu=sin cos(i|r 0), or?;=sini#cosi(^ &lt;). Moreover, sincewehaveshewn that inthespherical triangle RZz the sides are7,7,0,andtheanglesare(rr ^r &lt;), (?r i/r &lt;),&&gt;,wehave therelations COSift)=COS-i0COS-J(-V/r +0), and sin |&&gt;cos7=cos^0sin%(-v/r+&lt;), orx=cos1cos|(\/r+0), =cos^sinl (-v/r+0). Thefourparameters g,77,^areAusexpressed intermsoftheEulerian angles 6, &lt;f&gt;,tybytherelations (f=sin1sini(-^ -&lt;/&gt;), 77=sin^0cos|(i|r ^&gt;), |f=cos1sin(i^+ (/&gt;), 1^=cos^cosY(^+0). 12.7%econnexion ofrotations withhomographies ;jtheCayley- Kleinparameters. Consider nowasphere, onthesurface ofwhich anyfigures (which weshall call8}are drawn. Letthesefigures bestereographically projected onaplane (e.g.bytaking the highest pointofthesphere asvertex ofprojection andthetangent-planeatthelowest pointofthesphere astheplane):weshall calltheprojected figures P.Now letthe sphere berotated through adefiniteangle about some axisthroughitscentre, sothatthe figures onitssurface areshifted tonewpositions:letthefiguresintheirnewpositions be calledS;and letthestereographic projections ofthefigures 8(with thesame vertex and planeofprojection asbefore) becalledP .Thencorrespondingtotherotation ofthe sphere, whichchanges 8toS,wehave atransformationintheplane, which changes the figuresPintothefiguresP .Weshallnowexamine thistransformation moreclosely. IfoneofthefiguresPisacircle intheplane, weknow thatthecorresponding figureS must beacircle traced onthesphere,sincebystereographic projection acircle ischanged intoacircle: therefore Smust alsobeacircle; andhencePmust -also beacircle. Thusweseethat thetransformations oftheplane, whichcorrespondtorotations ofthe sphere, must besuch astochange anycircle intheplane intoanother circle intheplane. 12 Kinematical Preliminaries[CH.I Itmaybeshewn* thatanytransformation ofthiskindmayberepresented analytically inthefollowing way: Letz=x+yJ 1,where xandyaretherectangular coordinates ofanypointinthe plane ;sothat tothispoint therecorresponds adefinite value ofthecomplexvariable z. Similarlyletz=x+i/\/-1,where xandyrefer tothepoint intowhich thepoint (x,y) ischanged bythetransformation. Thenanyone-to-onetransformation oftheplane,which changesallcircles intocircles^, maybedefined byanequation ofthetype ,_= wherea,6,c,dare(realorcomplex} constants; orelsebyatransformation ofthis latter kindcombined withareflexion inoneoftheaxesofcoordinates. Atransformation represented byanequationofthetype ,_az+b cz+d iscalled ahomographic transformation, orhomography.Itappears therefore thathomo graphies inaplane correspondtorotations ofasolidbodyabout afixedpoint,insuch a waythat iftwohomographies correspond respectivelytotworotations, thehomography compounded ofthesecorrespondstotherotation compoundedofthetworotationsJ. Weshallnow seehowtheconnexion between rotations andhomographies may be represented analytically. Letusreplace theparameters ,77,f,^bynewparameters a,/3,y,8,defined bythe equations /3--y _/3+y q-_a+8 *"~ 9rl~9,~ 9--&gt;X~~a4 Z* 2* Z sothatthey areconnected with theEulerianangles d,(j&gt;,^bytheequations . .y=ism-.e =cos-.e2 These" Cayley-Klein"parameters clearly satisfytherelation aS/3y=1; andreplacing thequantities ,77, ,^inthescheme ofdirection-cosines givenin 9by their values interms ofa,/3,-y,8,wehave forthevalues ofthedirection-cosines interms ofa,/3,y,8thefollowing scheme : X Y Z *Cf.L.K.Ford,Anintroduction tothetheory ofautomorphic functions (London, 1915). fAstraight line istoberegarded asaparticular kind ofcircle. JKlein, Math. Ann. ix.(1875), p.183; Cayley, Math. Ann. xv.(1879), p.238. 12,13]Kinematical Preliminaries 13 Itmay readilybeshewn that theparameters (a", ft", y", 8")correspondingtothe resultant oftwosuccessive displacements (a ,/3,y,8)and(a, ft,y,8)aregiven bythe equations a"=aa+yt3, ft"=aft+ft8 , y"=ya+8y, 8"=yft +88 . These equations shew thatthetransformation ,a"z+ff y"z+8" istheresult ofperforminginsuccession thetwosubstitutions ,az+ft ,az+ftJ=t *,,andz=-^jyz+8 yz+8 andtheconnexion between rotations andhomographic transformations isthus evident analytically. OneadvantageoftheCayley-Klein parameters,ascomparedwith theparameters (|, r),f,x),isthattheyretain some ofthesimplicityofthequaternion calculus, while using the*J1ofordinary algebrainstead ofthei,j,kofHamilton squaternions. Example1.Let(6, &lt;,\^)denote theEulerianangles. Supposethatapointinspace which iscarried about with theaxesOxyzhasthevectorialangles (0lt t)(referredtothe fixed axesOXYZ) before themotion, and(di,fa)after themotion. Denotinge1^1tan|dt byf1}and**tan\6{by /,shew that ^=f!"**008*0- sin$6 "fje-^s Example2.Iffrom theequations thequantities X-p,X22 ,XiX2areformed, and ifthese quantities areregarded asumbral symbols and thequantities X^,X^,X1X2,%i2 ,x22 ,xlx2arereplaced by-Y+iX, Y+iX, Z,-y+ix,y+ix, z,respectively, shew thattheequations obtained are f-Y+iX= a2 (-y+ix]+2apz+/32(y+ix\ Y+iX=y2(-y+ix]+2y8z+82 (y+ix), (Z=ay(-y+ix)+(a8+py)z+p8(y+w), andthat these arethethreeequations connectingthecoordinates(X,Y,Z)ofapoint referred totheaxesOXYZ with itscoordinates(x,y,z)referred totheaxesOxyz. .Example3.If y+ix:y+ix :z=\\ :1 : and -Y+iX :Y+iX:Z=\l\1:1 :i shew that and\^= y 13. Vectors. Wenowproceedtoconsider the essential features involved inthe displacement bysimpletranslation ofarigid body. Theoperationoftranslation initself, consideredapartfrom thebody translated, evidently possesses thefollowing properties: 14 Kinematical Preliminaries[OH.i 1. Itcanbespecified completely byanyoneoftheequal andparallel lines ofspacewhich have agiven length (viz.thedistance ofthetranslation) andgivendirection(viz.thedirection ofthetranslation) ;since such aline furnishes allthedatawhich describe theoperation. 2. IfABbeoneofthese lines, andACDE...KB beabroken line joiningitsextremities, then theoperation represented byAB isequivalent tothesumoftheoperations represented byAC,CD,DE,..,KB. Theseproperties1and2arepossessed byalargenumber ofoperations andquantitiesother than theoperationoftranslation;anoperationor quantitywhichpossesses them iscalled avectorquantity. By2,avectorAB isequivalenttothesum ofthree vectors AK,KL,LB, respectively paralleltothreegiven rectangular axes,andformingabroken linejoiningthepointsAandB.These three vectors arecalled thecom ponentsofthevectorABalongthegivenaxes. IfIbethelengthand(a,ft,7) thedirection-anglesofAB,thelengthsofthecomponentvectors areclearly (Icosa,Zcos/3,Icosy), beinginfacttheprojectionsofABontheaxes. Asinglevector which isequivalenttoanynumber ofgivenvectors is called their resultant. Ifavector isconceived asvaryingindependenceonaparameter (e.g.the time),thedifference between thevectorscorrespondingtoanytwovalues of theparameterisalsoavector, andhence therate ofchangeofthevector withrespecttotheparameterisalsoavector, whosecomponentsarethe rates ofchangeofthecorresponding components.This iscalled theflux ofthevector withrespecttotheparameter. 14.Velocity andacceleration;their vectorial character. Consider nowabodywhich isbeing continuouslytranslated(thoughnot necessarily alwaysinthesamedirection) withoutanychangeoforientation. Itstotal translation toanytime tisavectorquantity,andhence therate at which thischangeswith thetime,i.e.itstime-flux, isalsoavectorquantity, which iscalled thevelocityofthebody;ifx,y,zarethecoordinates referred tofixed axes ofanypointfixed inthebodyandmovingwithit,then thecom ponentsofthevelocityreferred tothese axes aretherates ofchangeofx,y,z, i.e.arex,y,z(where dotsdenote differentiations withrespecttothetimet}. Similarlytherate ofchangeofthevelocityisagainavector, whose componentsarex,y,z(twodotsindicatingsecond derivatives withrespect tothetime) ;thisvector iscalled theacceleration ofthebody. Itisclear that ifPandQaretwomoving points,thevector which representsthetranslation (orvelocity,oracceleration) ofQisthesum ofthe vector whichrepresentsthetranslation(orvelocity,oracceleration, asthe casemaybe)ofPandthevector whichrepresentsthetranslation (orvelocity, oracceleration)ofQrelative toP,i.e.ofQreferred toaxeswhoseorigin moves with P,andwhose directions areinvariable. 13-15] Kinematical Preliminaries 15 15.Angular velocity:itsvectorial character. Consider next abodywhich isrotating continuously about aline. Let6 denote theangleturnedthroughatanytime t:then Qrepresents thespeed ofturningatthetime t.Iffromanypointonthelineround which the rotation takesplaceasegment whoselength represents6ismeasuredalong the line, thissegmentwillevidentlyfurnish acomplete specificationofthe nature oftherotation attheinstantt,or(asitisgenerally expressed)of theangular velocityofthebody. The direction inwhich thesegmentis measured from thebase-pointistobeconnected with thesense ofrotation bytheusual convention, namelythatwhen thesegmentisdirectedvertically upwards therotation from thesouthern horizon tothenorthern isround by the east. Anangular velocityisthereforerepresented byalineofdefinitelength and direction. Nowby 8,ifabodyone ofwhosepointsisfixed experiencesasmall rotationStyround anylineOK, thisdisplacementis equivalenttosuccessive small rotationsStycosaround Ox,Stycosftround Oy,andStycos7round Oz,where Ox,Oy,Ozareanythreemutually perpendicularlinespassing through and(a, fi,7)arethedirection-angles ofOKwith reference toOxyz. From this itisclear thatwecanregard an angular velocity represented byalength tymeasured onOKasequivalent toangular velocitiesrepresented bylengths tycosa,tycosj3,tycos7, measuredalong Ox,Oy,Oz,respectively. But this isessentiallythefundamentalpropertyofvectors, andcanbe expressed bythestatement that angular velocities can beresolved and compounded accordingtothevectorial law. Itmust beobserved however thatanangular velocitydoes not fulfil all theconditions which enter intothedefinition ofavector, foranangular velocity about one line isnotequivalenttoanangular velocityofthesame magnitude about aparallelline.Angular velocity must therefore beregarded asavector which islocalisedalongadefinite line. Example. Arightcircular coneofsemi-verticalangle$rollswithoutsliding onaplane.Tofind itsinstantaneous axisofrotation, and todetermine itsangular velocity about this axisintermsoftheangular velocity ofthelineofcontact intheplane. Since allpoints ofthegenerator which isincontact withtheplane areinstantaneously atrest(forthere isnosliding), thisgeneratoristheinstantaneous axis ofrotation of thecone. Leto&gt;denote theangular velocity oftheconeabout thisgenerator, and let6 denote theangular velocityofthelineofcontact intheplane. Then themotion ofthe axis ofthecone canberepresented byanangular velocity6round thenormal tothe plane, andthewhole motion ofthecone iscompounded ofthistogether witharotation round theaxisofthecone. Itfollows thatthecomponent ofangular velocityofthecone about alinethrough thevertex oftheconeperpendicular totheaxis is6cos/3 ;but thismustequal theresolvedpartof cointhisdirection, which iswsin/3. Wehave therefore w=6cot/3, which istherequired relation between o&gt;and 6. 16 Kinematical Preliminaries [CH.I 16.Determination ofthecomponents ofangular velocity ofasystemin terms oftheEulerian angles, andofthesymmetrical parameters. Thepositionatanytime ofarigid bodywhich iscontinuously moving about afixedpointismostconvenientlydescribed bytakingtwo sets ofrectangular axes, ofwhich onesetOXYZ arefixed inspace,while the other setOxyzarefixedrelativelytothebody,andmove with it;the positionofthebodybeing thenspecified bythethree Eulerianangles 0, &lt;/&gt;,^, which define thepositionoftheaxesOxyz relativelytotheaxesOXYZ. We shallnowdetermine thecomponents, alongthemoving axes, ofthe angular velocityofthebodyatanyinstant. LetOKdenote thelineofintersection oftheplanesXOY andxOy;the angular velocityofthesystemisevidently compoundedofangularvelocities 6about OK, &lt;j&gt;about OZ,and-^rabout Oz.Ofthese, the firstcanbe replaced accordingtothevectorial lawbyangularvelocities 6sini/raboutOx and6costyabout Oy;andthesecond canberesolved into&lt;sin6cos\|r about Ox, &lt;f&gt;sin sin-fyaboutOy,and &lt;j&gt;cos6about Oz.Sofinallyif G&gt;I}eo2,o&gt;3denote thecomponentsofangular velocityofthebodyabout the axes Ox,Oy,Oz,respectively,wehave 1a)l=6sin -Jr &lt;j&gt;sin6cosilr, a , a , &&gt;2=vcosy&lt;6sm t&gt;siny, i a ft)3=y-|-(pCOS c7. From these expressionswecanatoncededuce thevalues oftwj,o&gt;2,o&gt;3 interms ofthesymmetrical parameters,??,,%,of9;forwehave cf&gt;=- d Similarlywehave "4?==^,~H$^.&gt; andwehave cosd=2v2+-+%a . Substitutingthese values intheequation&&gt;3=ty+cos#,wehave The values ofo^and &lt;w2canbeatonce obtained from thisbythe principleofsymmetry;andthuswehave thecomponentsofangular velocity given bytheequations 16,17] Kinematical Preliminaries 17 17.Time-flux ofavector whose componentsrelative tomoving axesare given. Suppose nowthatavectorquantityisspecified byitscomponents f,77, atanyinstant twith reference totheinstantaneouspositionofaright-handed systemofaxesOxyzwhich arethemselves inmotion :and letitberequired tofindthevector whichrepresentstherateofchangeofthegivenvector. Let &&gt;!,&&gt;2,&)3denote thecomponentsoftheangular velocityofthe system Oxyz,resolvedalongtheinstantaneouspositionoftheaxes Ox,Oy,Oz themselves. The time-flux ofthegivenvector isthe(vector) sum ofthetime-fluxes ofthecomponents,?/, ,takenseparately.But ifweconsider thevector, itisincreased inlengthtog+%dtintheinfinitesimal interval oftime dt, andatthesame time isturnedbythemotion oftheaxes, sothat(owingto theangular velocityroundOy)itisdisplaced throughanangleco2dtfrom its position intheoriginal plane zOx, inthedirectionawayfrom Oz,and also (owingtotheangular velocityround Oz)itisdisplaced through anangle (osdtfrom itspositionintheoriginal plane xOy,towardsOy.Thecoordinates ofitsextremityattheendoftheinterval oftime dt,referred tothepositions oftheaxes atthecommencement oftheinterval dt,aretherefore(neglecting infinitesimals oforderhigherthan thefirst) andsothecomponentsofthevector whichrepresentsthetime-flux ofare ,&lt;"3 ,- 2- Similarlythecomponentsofthevectors whichrepresentthetime-fluxes ofthevectors?;and arerespectively &&gt;3?7, and o&gt; Adding these, wehavefinallythecomponents ofthetime-flux ofthegiven vector intheform This result canbeimmediately appliedtofind thevelocity and acceleration ofapointwhose coordinates(x,y,z)attime taregivenwith reference toaxesmovingwithanangular velocity whosecomponents along theaxes themselves attime tare (a&gt; 1,&&gt;2,&&gt;3). Forsubstitutingintheabove formulae, weseethatthecomponentsof thevelocityare xyw-s+zwz,yzo&gt;!+xa)3, z w.D. 18 Kinematical Preliminaries[CH.I Nowapplyingthesame formulae tothecase inwhich thevector whose time-flux issoughtisthevelocity, wehave thecomponentsoftheaccelera tion ofthepointintheform --(x yco3+za}2)(t&gt;3(y ZW-L+xo)s)+o)2(zxwz+2/&&gt;i), -y-(^#o&gt;2-I-ya&gt;i)ft&gt;2(# 2/ Ctt Inthecase inwhich themotion takesplaceinaplane,which wemay take astheplane Oxy,there willbeonlytwocoordinates(x,y},andonlyone componentofangular velocity, namely 6,where 6istheanglemade bythe movingaxes with theirpositionsatsome fixedepoch;thecomponentsof velocityaretherefore(putting z,wl, 2eachequaltozero intheabove expressions) xy6andy+xO, andthecomponentsofacceleration are x-2y6-yO-x62andy+2x6+x0-yfc. Example.Prove that inthegeneralcase ofmotion ofarigidbodythere isateach instant onedefinite pointatafinitedistance which regardedasinvariably connected with thebodyhasnoacceleration attheinstant, providedtheaxis ofthebodysscrewing motion benotinstantaneously stationaryindirection.(Coll. Exam.) 18. Specialresolutionsofthevelocity andacceleration. The results obtained inthe last article enable ustoobtain formulae, which arefrequentlyofuse,relatingtothecomponentsofthevelocity and acceleration ofamoving pointinvariousspecialdirections. (i) Velocity andacceleration inpolarcoordinates. Letthepositionofapointbedefinedbyitspolarcoordinatesr,6,0, connected with thecoordinates (X, Y,Z)ofthepointreferred tofixed rectangularaxesOXYZ bytheequations X=rsin6cos &lt;, Y=rsin6sin (j), Z=rcos6; and let itberequiredtodetermine thecomponentsofvelocityand acceleration ofthepointinthedirection oftheradius vector r,inthe direction which isperpendiculartorand liesintheplane containingrand- OZ(this planeisgenerallycalled themeridianplane), and inthedirection perpendiculartothemeridianplane;these three directions arefrequently described asthedirections ofrincreasing,6increasing,and (f&gt;increasing, 17,18]Kinematical Preliminaries 19 respectively.Take alinethroughtheorigin 0,paralleltothedirection of increasing,asamovingaxisOx;andtake alinethrough 0,paralleltothe direction of &lt;pincreasing,asaxisOy,andalineparalleltothedirection ofr increasingasaxis Oz.The three Eulerianangleswhich determine the positionofthemovingaxesOxyzwith reference tothefixed axesOXYZ are (6, d),0);so(16)thecomponentsofangular velocityofthesystem Oxyz, resolved alongtheaxes Ox,Oy,Ozthemselves, are eoj= sin 6,&&gt;2=0,&lt;w3=(pcos 6. The coordinates ofthemoving point,referred tothemoving axes, are (0, JO,r);and soby17thecomponentsofvelocityofthepointresolved paralleltothemovingaxes are rd, r&lt;j&gt;sin6, r, and thecomponentsofacceleration inthedirections of6increasing, increasing,andrincreasing (again usingtheformulae of17)are -T-(r0)r&lt;2sin6cos+r0,orr0+2r0 rd]2sin cos0, ,a. . . ld ]-r-(rd)sin0)+fcf&gt;sin+r66cos0,or =-Q-j-(r2sin2 6d&gt;), !dt rsin at Iand rr&* rd)2sin26. Ifthemotion ofthepointisinaplane, wecantaketheinitial lineinthis planeasaxisOz,andthequantitiesdenoted byrand inthese formulae becomeordinary polarcoordinates intheplane;since(pisnow zero, the componentsofvelocityandacceleration inthedirections ofrincreasingand increasingare (r, rd), and (rr02 ,r0+2f0). (ii) Velocity andacceleration incylindricalcoordinates. Consider now apointwhosepositionisdefinedbyitscylindrical coordinates z,p,$,connected with thecoordinates (X,Y,Z)ofthepoint referred tofixedrectangularaxesOXYZ bytheequations X=pcos d),^psin $&gt;Z=z\ and letitberequiredtofindthecomponentsofthevelocityandacceleration ofthepointinthedirectionparalleltotheaxisofz,inthedirection ofthe linedrawn from theaxisofztothepoint, perpendiculartotheaxisofz,and inthedirectionperpendiculartothese two lines. These three directions are generallycalled thedirection ofzincreasing,thedirection ofpincreasing, andthedirectionof &lt;pincreasing;andthecoordinate &lt;piscalled theazimuth ofthepoint. Inthiscasewetakemovingaxes Ox,Oy,Ozpassing throughtheorigin andparallel respectivelytothedirections ofpincreasing, &lt;/&gt;increasing, andz 22 20 Kinematical Preliminaries[CH.i increasing. Thecomponentsofangular velocityofthesystem Oxyz,resolved alongtheaxesOxyz themselves, areclearly COj==0, d&gt;2==0, 61)3^(D, andthecoordinates ofthemoving point,referred tothemoving axes, are (p,0,2).Itfollows by17thatthecomponentsofvelocityofthepointin these directions are andthecomponentsofacceleration are (iii) Velocity andacceleration inarc-coordinates. Anotherapplicationoftheformulae of17istothedetermination ofthe componentsofvelocityandacceleration ofapointwhich ismovinginany wayinspace,resolvedalongthetangent, principal normal, andbinormal to itspath. Consider firstthecase ofaparticle movinginaplane:andtake lines throughafixedpoint 0,parallel respectivelytothetangentandinward normal tothepath,asmovingaxesOxandOy.These axes arerotating round withangular velocity (ft,where&lt;istheangle made bythetangent tothepathwithsome fixed line intheplane.Ifvdenotes thevelocityof thepoint,sthearcofthepathdescribed attimet,andptheradius of curvature ofthepathatthepoint,wehave ds ds v p= dtdtp andtheangular velocityoftheaxescantherefore bewritten intheformv/p. Since thecomponentsofthevelocity paralleltothemovingaxes are (v,0),itfollows from 17thatthecomponentsoftheaccelerationparallelto / v\thesame axes are(v,v.-}.Since V pJ ._dvdsdv dv dt dtds ds itfollows thattheacceleration ofthemoving pointinthedirection ofthe dv tangenttoitspathiav-j-,andtheacceleration inthedirection oftheinward normal is . P Now thevelocityofamoving pointisdeterminedbytheknowledgeof twoconsecutivepositionsofthemoving point,andtheacceleration istherefore determined bytheknowledgeofthree consecutivepositions;soeven ifthe pathofthepointisnotplane,itcan forthepurposeofdeterminingits acceleration atanyinstant beregardedasmovingintheosculating planeof 18]Kinematical Preliminaries 21 itspath,since thisplanecontains three consecutivepositionsofthepoint. Hence thecomponents ofacceleration ofthepoint,inthedirections ofthe tangent, principal normal, andbinormal toitspath,are dv v2 dsp (iv) Acceleration alongtheradius andtangent. Theacceleration ofapointwhich isinmotion inaplanemaybeexpressed inthefollowing form*; letrbetheradius vector tothepointfrom afixed originintheplane,ptheperpendicularfromtheoriginonthetangenttothe path,sthearcofthepathdescribed attimet,ptheradius ofcurvature of thepathatthepoint,and vorsthevelocityofthepointattime t;and let hdenote theproduct pv.Then theacceleration ofthepointcanberesolved intocomponents alongtheradius vector totheorigin and ~ralongthe tangenttothepath. Fortheacceleration canberesolved intocomponents vdv/ds alongthe tangentand vz jpalongthenormal;nowavectorFdirected outwardsalong theradius vector canberesolved intovectors Fp/ralongtheinward normal andFdr/ds alongthetangent,soavector v2 /palongtheinward normal canbe rv2rv2dr resolved into inwards along theradius vector and T-along thetangent. pp ppds Theacceleration istherefore equivalenttocomponents dv.rv2dr pp rv2 and-inwards along theradius vector. pp The latter component is- ,andtheformer canbewritten pp 1dv2v2dp Id(vz p*)hdh 2dspds 2p2ds p2ds which establishes Siacci sresult. Example1.Determine themeridian, normal, andtransversecomponents oftheaccelera tionofapoint movingonthesurface oftheanchor-ringv-pH -r-along thetangent,dsonds LetPbethepoint (6,$),and let bethecentre oftheanchor-ring andCthecentre ofthemeridian cross-section onwhichPlies.Thepolarcoordinates ofCrelative to are(c, &lt;p),andthepolarcoordinates ofPrelative toCare(a,0,$) ;sothecomponents ofacceleration ofCrelative to are ctj)transverse and c&lt;p2outwards from theaxis,i.e. c&lt;2sin6alongthenormal, andc02cos6alongthemeridian. *Dae toSiacci, Attidella R.Ace. diTorino, XIY. p.750. 22 Kinematical Preliminaries[CH. Thecomponentsofacceleration ofPrelative toCare fad- a&lt;j&gt;2sinQcos6 along themeridian, -j-(sin2 . &lt;) transverse. sin6dt t&lt;j&gt;2sin2normal. Thusfinally thecomponentsofacceleration ofPinspace are ad(c+asin6}2cos6 alongthemeridian, a62 a&lt;j)2sin26 c&lt;f&gt;2sin6normal, and cd)+-. -.-y-(sin26 . d&gt;) transverse,sm6dt Example2.Ifthetangential andnormalcomponents oftheaccelerationofapoint movinginaplaneareconstant, shew that thepointdescribes alogarithmic spiral. Inthiscase v-r-=a, where aisaconstant,ds so i)2=as. Also =c,where cisaconstant, P so s=Cp,whereCisaconstant, ors=C-j-r, where (f&gt;istheanglemade bythetangent withafixed line. Integratingthisequation, wehave whereAandBareconstants :andthis istheintrinsic equation ofthelogarithmic spiral. Example3.Tofindtheaccelerationofapoint which describes alogarithmic spiral with constant angular velocityabout thepole. BySiacci stheorem, thecomponents ofacceleration are 5-along theradius vectorP3P and 5-=-along thetangent: but iftoistheconstant angular velocity, wehaveh= u&gt;r2- p2ds sothecomponentsofacceleration are co2 ?" .2o)2r3dr =and 5-j-.pAp jras ;inthespin acceleration variesdirectlyastheradius vector.7" 7* CiT Since - ,- ,and-5-areconstant inthespiral, weseethateach ofthesecomponents of MISCELLANEOUS EXAMPLES. 1.Iftheinstantaneous axisofrotation ofabodymoveable about afixed pointisfixed inthebody, shew that itisalsofixed inspace,i.e.themotion isarotation round afixed axis. 2.Apointisreferred torectangularaxes Ox,Oyrevolving about theorigin with angular velocity to;ifthere beanacceleration tox=a,y=0,ofamount ?i2 &lt;o2x(distance), shew that thepathrelative totheaxes canbeconstructed bytaking (i)apoint i]Kinematical Preliminaries 23 x=ri2a/(n2-l), (ii)auniform circular motion with angular velocity (n-l)coabout this, and(iii)auniform circular motion with angular velocity (n+l)a&gt;,butintheopposite sense, about this last. (Coll. Exam.) 3.Thevelocityofapoint movinginaplaneistheresultant ofavelocityvalongthe radius vector toafixedpoint andavelocityvparalleltoafixed line. Prove that the corresponding accelerations are dv .vv ,dv vv^+-cos0, and-s+- t 6beingtheangle thattheradius vector makes withthefixed direction. (Coll. Exam.) 4.Apointmoves inaplane, and isreferred toCartesian axesmaking angles a,/3with afixed lineintheplane, wherea,/3aregivenfunctions ofthetime. Shew thatthecom ponentvelocities ofthepoint are x-xdcot(/3-a)-y$cosec(/3-a), y+y$cot-a)+xdcosec(/3-a), andobtainexpressionsforthecomponentaccelerations. (Coll. Exam.) 5.Apointismovinginaplane:Qisthelogarithmoftheratio ofitsdistances from twofixedpointsintheplane, and&lt;istheangle between them :also2kisthedistance between thefixedpoints. Shew thatthevelocityofthepointis (Coll. Exam.). cosh6-cos(f) 6.Ifintwodifferent descriptionsofacurve byamoving point,theproductofthe velocities atcorresponding placesinthetwodescriptionsisconstant, shew that the accelerations atcorresponding placesinthetwodescriptionsareasthesquaresofthe velocities, andthat their directions make equal angleswith thenormal tothecurve, in opposite senses. (J-vonVieth.) 7.Apointismovinginaparabolaoflatusrectum 4,andwhen itsdistance from the focus isr,thevelocityisv;shew that itsacceleration iscompoundedofaccelerations R andN,alongtheradius vector andnormalrespectively,where R=v^ ,N=^, %-(v*r). (Coll. Exam.)dr2r%drv 8.Shew that iftheaxes ofxandyrotate with angularvelocitiescoi,o&gt;2respectively, and^istheangle between them,thecomponent accelerations ofthepoint (x,y)parallel totheaxesare xxo)i2(x&i+2#1)cot \|/- (?/o&gt;2+ 2/o&gt; 2)cosec\^, andyyo&gt;22+(.coi+2.o&gt;1)cosec\|f+(yw2+2?/o&gt;2)cot\/r. (Coll. Exam.) 9.Thevelocityofapointismadeupofcomponents u,vindirections making angles 0,withafixed line. Prove thatthecomponents /,/inthese directions oftheaccelera tionofthepointwillbegiven by f=uu6cotx~v$cosecx, f=v+udcosecx+ v&lt;}&gt;cotx, Xbeingtheinclination ofthetwodirections. Being given thatthelinesjoining amoving pointtotwofixedpoints arer,sinlength and6,&lt;pininclination tothelinejoining thetwo fixedpoints, determine theacceleration ofthepointinterms of,o&gt;,therates ofincrease of0, (f). (Coll. Exam.) 24 Kinematical Preliminaries|_CIL 10. IfA,B,Cbethree fixedpoints, andthecomponentvelocities ofamoving pointP along thedirections PA,PB,PCbeu,v,andw;shew thattheaccelerations inthesame directions are /1cosAPB 1cosAPC\ andtwosimilarexpressions. (Coll. Exam.) 11.Themovement ofaplane lamina isgiven bytheangular velocitya&gt;andthecom ponentvelocitiesu,voftheoriginresolved alongaxes Ox,Oytraced onthelamina. Find thecomponentvelocities ofanypoint (x,y]ofthelamina. Shew thattheequations ,. -T-tan1 (- "-)=o&gt; at represent circular locionthelamina;onebeing thelocus ofthosepoints which arepassing cusps ontheir curve lociinspace andtheother being thelocus ofthecentres ofcurvature oftheenvelopesinspace ofallstraightlines ofthelamina.(Coll. Exam.) 12.Shew thatwhen apointdescribes aspace-curve,itsacceleration canberesolved intotwocomponents,ofwhich oneactsalongtheradius vector from theprojectionofa fixedpoint ontheosculating plane, andtheotheralong thetangent;andthat these are respectively L^ P3P TdT T*qdq fi~ds+ p*~d7J wherepistheradius ofcurvature, qthedistance ofthefixedpoint from itsprojection on theosculating plane,randparethedistances ofthisprojectionfrom themoving point andthetangent, Tisanarbitraryfunction (equaltotheproductofpandthevelocity) and sisthearc.(Siacci.) 13.Acircle, astraight line,andapointlieinoneplane, andtheposition ofthepoint isdetermined bythelengthstofitstangenttothecircle andpofitsperpendicular to the line. Prove that,ifthevelocityofthepointismade upofcomponents u,vinthe directions oftheselengths and iftheirmutual inclination be6,thecomponent accelerations willbe uuvcosd/t, v+uv/t. (Coll. Exam.) 14.Aparticle moves inacircular arc. Ifr,rarethedistances oftheparticle atP from theextremities A,Bof&fixed chord, shew thattheaccelerations alongAP,BPare respectively dv vv,,dv vv . -j-Hivr-f cosa),andrH ;(rrcosa),dt rrxdt rr^ wherev,varethevelocities inthedirections ofr,r,andaistheangleAPB. Apoint describes asemicircle under accelerations directed totheextremities ofa diameter, which areatanypoint inverselyastheradii vectoresr,rtotheextremities of thediameter. Shew thattheaccelerations are 4a4F24a472 where aistheradius ofthecircle andVthevelocityofthepoint paralleltothediameter. (Coll. Exam.) i]Kinematical Preliminaries 25 15.Themotion ofarigidbodyintwodimensions isdefined bythevelocity (u,v}of oneofitspointsCand itsangular velocity&lt;o.Determine thecoordinates relative toCof thepoint7ofzerovelocity, andshew thatthedirection ofmotion ofanyother pointPis perpendiculartoPI. Find thecoordinates ofthepointJofzero acceleration, andexpresstheacceleration of Pinterms ofitscoordinates relative toJ. (Coll. Exam.) 16.Apoint onaplaneismoving with constantvelocity Vrelative toit,theplaneat thesame timeturning round afixed axisperpendiculartoitwithangular velocityo&gt;.Shew thatthepathofthepointisgiven bytheequation rand6beingreferred tofixed axes,andabeingtheshortest distance ofthepointfrom the axisofrotation. (Coll. Exam.) 17.Theacceleration ofamoving pointQisrepresentedatanyinstant by&lt;oa,where isafixedpointandadescribes uniformly acircle whose centre is .Prove that the velocityofQatanyinstant isrepresented byOp,where isafixedpointandpdescribes acircleuniformly;anddetermine thepathdescribed byQ. (Camb.Math. Tripos, PartI,1902.) 18.Apointmoves alongthecurve ofintersection oftheellipsoid ~g+W+-=1and222 thehyperboloid ofonesheet-^-+?/ &gt;.+"2^T=1 &gt;and itsvelocit yatthePointwhere d~"AOAC-*~~A thecurve meets thehyperboloid oftwosheets,+T-- l~~~?-=1*sJ1a2-/ib21pc2 /i where hisconstant. Prove thattheresolved partoftheacceleration ofthepoint along thenormal totheellipsoidis h2abc(nX) /rin. -7=.. (Coll. Exam.) 19.Arigidbodyisrolling withoutsliding onaplane, andatanyinstant itsangular velocity hascomponents coj,o&gt;2alongthetangenttothelines ofcurvature atthepoint ofcontact, and o&gt;3along thenormal :shew thatthepointofthebodywhich isatthepoint ofcontact hascomponentaccelerations where RI,R%aretheprincipalradii ofcurvature ofthesurface ofthebodyatthepoint ofcontact. (Coll. Exam.) CHAPTER II THEEQUATIONS OFMOTION 19.Theideasofrestandmotion. Intheprevious chapter wehavefrequentlyused theterms"fixed"and "moving"asappliedtosystems. Solongasweareoccupied withpurely kinematicalconsiderations, itisunnecessarytoenter into theultimate significance ofthese words; allthat ismeantis,thatweconsider the displacement ofthe"moving" systems,sofarasitaffects theirconfigurationwithrespecttothesystems which arecalled"fixed," leavingononesidethe question ofwhat ismeantbyabsolute" fixity." When however wecome toconsider themotion ofbodies asduetospecific causes, thisquestion cannolongerbedisregarded. Inpopular languagetheword "fixed" isgenerally used ofterrestrial objectstodenote invariableposition relative tothesurface oftheearth at theplace considered. Buttheearth isrotatingonitsaxis,andatthesame timerevolving round theSun, while theSun inturn, accompanied byall theplanets,ismovingwith alarge velocity along some notvery accuratelyknown direction inspace.Itseemshopeless therefore toattempttofind anything which canbereally considered tobe "at rest." Inthenineteenthcenturyitwassupposed-that theaether ofspace (the vehicle oflight andofelectric andmagnetic actions) was(apart from small vibratory motions) stagnant, and sowascapableofprovidingabasis for absolutefixity. Butthisdoctrine hasbeensubvertedbythemodernPrinciple ofRelativity*, which asserts thateven inthedomain ofelectromagnetic phenomenaitisimpossibletodistinguish absolute restfromastate ofuniform translatorymotion common toallthemembers ofasystem. Accordinglyindynamics, although whenwespeakofthemotion ofbodies wealways implythatthere issome setofaxes, orframe ofreferenceasitmaybecalled, with reference towhich themotion isregardedastaking place, and towhich weapplytheconventional word" fixed," yetitmust notbesupposed thatabsolutefixityhastherebybeen discovered. When weareconsidering Cf.Whittaker sHistory oftheTheoriesofAether andElectricity,ch.xii.(London, 1910) ; orConwaysRelativity (London, 1915). 19,20] TheEquations ofMotion 27 themotion ofterrestrial bodies atsomeplaceontheearth ssurface, weshall take theframe ofreference tobefixed with reference totheearth, and itis then found that thelawswhich willpresentlybegivenare sufficient to explainthephenomenawith asufficientdegreeofaccuracy;inother words, theearth smotion doesnotexercise asufficient disturbinginfluence tomake itnecessarytoallow foritseffects inthemajorityofcases ofthemotion of terrestrial bodies. Itisalsonecessarytoconsider themeaningtobeattached totheword " time," which intheprevious chapterstood merelyforanyparameter varying continuouslywith theconfigurationofthesystemsconsidered. The PrincipleofRelativityreveals thegreatdifficulties thatattendanyattempt toelucidate theidea oftime :inparticular,itisbynomeanseasytodefine simultaneity,i.e.toexplainwhat ismeant bysayingthattwoevents at differentpointsofspace happen"atthesame time." However, asystemof time-measurement which isintelligiblefrom thepointofview ofordinary instrumental work, andwhich issufficient forourpresent purpose,isthe following:wesupposethattheangle throughwhich theearth hasrotated on itsaxis(measured with reference tothefixed stars, whose small motions we can forthispurpose neglect),intheinterval between twoevents, measures thetimeelapsedbetween theevents inquestion.Thisangularmeasure can beconverted intotheordinarymeasure interms ofmean solar hours, minutes, andseconds attherateof360degreesto24x365/366hours. 20.Thelawswhich determine motion*. Considering nowthemotion ofterrestrialobjects, andtakingtheearth as theframe ofreference, itisnatural tobegin byinvestigatingthemotion ofa verysmall materialbody,orparticleasweshall callit,whenmovinginvacuo andentirelyunconnected withsurrounding objects. Thepathsdescribed by such aparticleunder various circumstances ofprojection maybeobserved, andthemethods ofthepreceding chapterenable us,from theknowledge thusacquired,tocalculate theacceleration oftheparticleatanypointofany particularobservedpath.Itisfound that forallthepathstheacceleration isofconstant amount, and isalwaysdirectedverticallydownwards. This acceleration isknown asgravity,and isgenerallydenotedbytheletter g;its amountis,inGreat Britain, about 981centimetrespersecondpersecond. Theknowledgeofthisexperimentalfact istheoreticallysufficient to enable ustocalculate thepathofanyfree terrestrialparticleinvacuo, when thecircumstances ofitsprojectionareknown :theactual calculation willnot begiven here, asitbelongsmoreproperlytoalaterchapter. Thecase ofmotion which isnext insimplicityisthat oftwoparticles which areconnectedtogether byanextremely lightinextensible thread, and *Thelaws ofmotion areduetoNewton :Principia, p.12(ed.1687). ^8 TheEquations ofMotion[OH.n arefree tomove invacuo attheearth ssurface. Solongasthethread is slack, eachparticle moves with theaccelerationgravity, justasiftheother were notpresent.Butwhen thethread istaut, thetwoparticlesinfluence each other smotion. Wecannowasbefore observe thepathofoneofthe particles, andhence calculate theaccelerationbywhich atanyinstant its motion isbeingmodified. Wetherebyarrive attheexperimental fact,that thisacceleration canberepresentedatany instantbytheresultantoftwo vectors, ofwhich onerepresentstheacceleration gand theother isdirected alongtheinstantaneousposition ofthethread. Theinfluence ofoneparticleonthemotion oftheother consists there fore insuperposingontheacceleration due togravityanother acceleration, which actsalongthe linejoiningtheparticlesandwhich iscompounded withgravity accordingtothevectorial lawofcompositionofaccelerations. Denotingtheparticles byAandB,wecanatanyinstant calculate, from the observedpaths,themagnitudesoftheaccelerations/! and/ 2thusexerted by BonAandbyAonBrespectively;and thiscalculationimmediately yields theresult that theratiooff\to/2does notvary throughoutthemotion. On investigatingthemotions which result from various modes ofprojection,at varioustemperatures etc.,weareledtotheconclusion that thisratio isan invariablephysicalconstant ofthepair ofbodiesAandB*. Onconsideration ofthemotion ofmorecomplex systemsitisfound that theexperimentallawsjuststated canbegeneralisedsoastoform acompletebasis for alldynamics, whether terrestrial orcosmic. This generalised statement isasfollows: Ifanysetofmutuallyconnectedparticles areinmotion, theacceleration with which anyoneparticle moves isthe resultant oftheacceleration withwhich itwould moveifperfectly free,and accelerations directedalongthelinesjoiningittotheotherparticles which constrain itsmotion. Moreover, totheseveralparticles A,B,C,...,numbers f^A)m n&gt;m c&gt;canbeassigned, such that theaccelerationalongABdue tothe influence ofBonAistotheaccelerationalongBAdue totheinfluence ofAonBintheratiom^-.m^. The ratiosofthesenumbers mA ,mB,...are invariablephysicalconstantsoftheparticles. Theevidence forthetruth ofthisstatement istobefound intheuniversal agreementofthecalculations based on it,such asthosegivenlater inthis book, with theresults ofobservation. Itwillbenoticed thatonlytheratios ofthenumbers m^,m$,me, ...are determined bythelaw;itisconvenient totakesome definiteparticle Aas astandard, callingittheunitofmass, andthen tocallthenumbers m^jm^, ?7ic/mA ,...themasses oftheotherparticles mB,me, *The ratio isin iact,equal totheratio oftheweight ofBtotheweight ofA;theratio of theweights oftwo terrestrialbodies, asobserved atthesame place ontheearth ssurface, isa perfectlydefinitequantity, anddoesnotvarywith theplace ofobservation. 20,21]TheEquations ofMotion 29 Themass ofthecompound particleformed byunitingtwoormoreparticles isfound tobeequaltothesum ofthemasses oftheseparate particles. Owingtothisadditivepropertyofmass,wecanspeakofthemass ofafinite bodyofanysizeorshape;and itwillbeconvenient totake asourunitof mass themass ofthey^th partofacertainpieceofplatinumknown asthe standard kilogramme;thisunit willbecalled agramme,andthenumber representingtheratio ofthemass ofanyother bodytothisunitmass is called themassofthebodyingrammes. 21. Force. Wehave seen that ineverycase oftheinteraction oftwoparticles Aand B,themutual influence consists ofanacceleration fAonAandanacceleration fiionB,these accelerations beingvectors directedalongABandBArespec tively,andbeing inversely proportionaltothemassesmAandmB.Itfollows thatthevectorquantity mAfAisequaltothevector quantity msfB ,buthas thereverse direction. ThevectormAfAiscalled theforceexerted bythe particle Bontheparticle A;andsimilarlythevectormBfBiscalled theforce exerted bytheparticleAontheparticleB. With thisterminology,thelawofthemutual action ofaconnected systemofparticlescanbestated intheform :theforcesexerted oneach other byevery pair ofconnected particlesareequalandopposite.This isoften called theLawofAction andReaction. Ifthevarious forces which actonaparticle Aasaresult ofitsconnexion with otherparticlesarecompounded accordingtothevectorial law,the resultant forcegivesthetotal influence exerted bythem ontheparticle A; this force divided bymAistheacceleration induced inAbytheother particles;andtheresultant ofthisacceleration andtheacceleration which the particle Awould have ifentirelyfree(duetosuch causes asgravitation)is theactual acceleration withwhich theparticle Amoves. Ingeneral,ifanaccelerationrepresented byavector/isinduced in aparticleofmassmbyanyagency,thevector mfiscalledthe/orce*dueto thiscauseactingontheparticle;andtheresultant ofalltheforces due to variousagenciesiscalled the totalforce actingontheparticle.Itfollows that if(X,Y,Z)arethecomponents paralleltofixedrectangularaxes ofthe total forceactingontheparticleatanyinstant, and(x,y,z)arethecom ponentsoftheacceleration withwhich itspathisbeingdescribed atthat instant, thenwehave theequations mx=X,my=Y,mz=Z. Twoother terms which arefrequentlyusedmayconvenientlybedefined atthispoint. *Force isthevismatrix ofNewton sPrincipia,i.def. 8. 30 TheEquations ofMotion[OH.n Theproductofthenumber whichrepresentsthemagnitudeofthecom ponentofagivenforceperpendiculartoagivenlineLandthenumber whichrepresentstheperpendicular distance ofthelineofaction oftheforce from thelineLiscalled themoment oftheforceabout thelineL. Ifthethreecomponents (X,Y,Z)oftheforceactingonasinglefree particlearegivenfunctions ofthecoordinates(x,y,z)oftheparticle, they aresaid todefine &field offorce. 22. Work. Consider nowanysystemofparticles, whose motion iseitherquitefreeor restricted bygiven connexions between theparticles,orconstraints due to otherparticles which arenotregardedasforming partofthesystem.Letm bethemass ofanyoneoftheparticles,whose coordinates referred tofixed rectangularaxes inanyselectedconfigurationofthesystemare(x,y,z) ;and let(X, Y,Z}bethecomponents, paralleltotheaxes, ofthetotal force actingontheparticleinthisconfiguration. Let(x+Sx,y+By,z+8z)bethecoordinates ofanypoint verynear to thepoint (x,y,z),such that thedisplacementoftheparticlemfromone pointtotheother doesnotviolateanyoftheconstraints(forinstance, ifmis constrained tomove onagiven surface, thetwopoints must bothbesituated onthesurface). Then thequantity X&K+1%+ZSz iscalled thework* done ontheparticle mbytheforcesactingonitinthe infinitesimaldisplacement from theposition (x,y,z)totheposition (x+Bx,y+Sy,z+8z). Thisexpression canevidentlybeinterpreted physicallyasbeingthe productofthedistancethroughwhich theparticleisdisplaced andthecom ponentoftheforce(X,Y,Z)alongthedirection ofthisdisplacement. Since forcesobeythevectorial lawofcomposition,thesum ofthecom ponentsinagivendirection ofanynumber offorcesacting together ona particleisequaltothecomponentinthisdirection oftheir resultant :and hence thework donebyaforce which actsonaparticleinagiven displace ment isequaltothesumofthequantities ofwork done inthesamedisplacement byanysetofforcesintowhich thisforce canberesolved. Suppose nowthat inthecourse ofamotion ofthesystem,theparticlem isgradually displacedfromanyposition (which wecan call itsinitialposition) tosome otherpositionatafinite distance from the first(which wecan call itsfinal position). Thework doneontheparticle bytheforces which acton *Newton defined theActio Agentis astheproductofthevelocity intothecomponentofforce along thedirection ofmotion;itisevidently thetime-flux ofthework done. Cf.Principia,i. p.25(ed.1687). 21-23]TheEquations ofMotion 31 itduringthisfinite displacementisdefined tobethesum ofthequantities ofwork done inthesuccessive infinitesimaldisplacements bywhich wecan regardthefinitedisplacementasachieved. Thework done inafinite dis placementisthereforerepresented bytheintegral dxvdydz\ , as (LS (is/ where theintegrationistaken between the initial and finalpositions along thearcsdescribed inspace bytheparticle during displacement. These definitions cannowbeextended tothewhole setofparticleswhich form thesystemconsidered;thesystem being initiallyinanygivencon figuration,weconsideranymode ofdisplacingthevariousparticlesofthe systemwhich isnotinconsistent with theconnexions andconstraints; the sumofthequantitiesofworkperformedonalltheparticlesofthesystemin thedisplacementiscalled thetotalwork done onthesysteminthedisplace mentbytheforces which actonit. 23.Forces which donowork. There arecertain classes offorces whichfrequentlyoccur indynamical systems,andwhich arecharacterised bythefeature thatduringthemotion theydonowork onthesystem. Amongthesemaybementioned 1.The reactions offixed smooth surfaces: theterm smoothimplies thatthereaction isnormal tothesurface, andtherefore ineach infinitesimal displacementthepointofapplicationofthereaction isdisplacedinadirection perpendiculartothereaction, sothatnowork isdone. 2.The reactions offixedperfectly roughsurfaces: thetermperfectly rough impliesthatthemotion ofanybodyincontact with thesurface isone ofpure rollingwithoutsliding,andtherefore thepointofapplicationofthe reaction is(tothe first order ofsmallquantities)notdisplacedineach infinitesimal displacement,sothatnowork isdone. 3.Themutual reaction oftwoparticles which arerigidlyconnected together:forif(x1}yltz^)and(#2,y2,z2)arethecoordinates oftheparticles, and(X,Y,Z)arethecomponentsoftheforce exertedbythe firstparticle on thesecond, sothat(X, Y,Z)arethecomponentsoftheforce exerted bythesecondparticleonthe first, thetotalwork donebythese forces in anarbitrary displacementis X(8x 2Sx,)+Y(Sy 2Si/i)+Z(8z2&2-A\v 3*/ \*/ Butsince thedistance between theparticlesisinvariable, wehave Cjj/ \2I/. \2_]_/ \2\ A 32 TheEquations ofMotion[CH.n andsince theforce actsinthedirection ofthelinejoiningtheparticles, we have X:Y:Z=(x 2- a?,):(y.2-y,):(z2- z,). Combiningthelasttwoequations, wehave X(8x 2-&O+F(S// 2-SyJ+Z(8z 2-82,)=0, andtherefore nowork isdone intheaggregate bythemutual forces between theparticles. 4.Arigid bodyisregardedfrom thedynamical pointofview asan aggregateofparticles,soconnectedtogetherthat theirmutual distances are invariable. Itfollows from 3thatthereactions between theparticles which arecalled intoplayinorder that thiscondition maybesatisfied(ormolecular forces astheyarecalled, todistinguish them from external forces such as gravity) do,intheaggregate,nowork inanydisplacementofthebody. 5.Thereactions atafixedpivotabout which abodyofthesystem can turn, oratafixedhinge,oratajointbetween twobodies ofthesystem,are similarlyseen tobelongtothecategoryofforces which donowork. Inestimatingthetotal work donebytheforcesactingonadynamical systeminanydisplacementofthesystem, wecanthereforeneglectallforces oftheabove-mentionedtypes. 24.Thecoordinates ofadynamical system. Anymaterialsystemisregardedfrom thedynamical pointofview as constituted ofanumber ofparticles, subjecttointerconnexions andcon straints ofvarious kinds;arigid body being regardedasacollection of particles,which arekeptatinvariable distances from each otherbymeans ofsuitable internal reactions. When theconstitution ofsuch asystem (i.e.theshape, size,andmass of thevariouspartsofwhich itiscomposed,andtheconstraints which acton them)isgiven,itsconfigurationatanytime canbespecifiedinterms ofa certain number ofquantitieswhichvarywhen theconfigurationisaltered, andwhich willbecalled thecoordinates ofthesystem; thus, theposition ofa singlefreeparticleinspaceiscompletelydefined byitsthreerectangular coordinates (x,y,z)with reference tosome fixed setofaxes;theposition of asingle particlewhich isconstrained tomove inafixednarrow tube,which has theform ofatwisted curve inspace,iscompletely specified byonecoordinate, namelythedistance smeasuredalongthearcofthetube totheparticle from some fixedpointinthetubewhich istaken asorigin;thepositionofarigid body,oneofwhosepointsisfixed, iscompletelydetermined bythree co ordinates, namelythethree Eulerianangles 6, &lt;,-v/rof10 ;thepositionof twoparticleswhich areconnected byatautinextensiblestringcanbedefined byfivecoordinates, namelythethreerectangularcoordinates ofoneofthe 23-25] TheEquations ofMotion 33 particlesandtwoofthedirection-cosines ofthestring (since when these five quantitiesareknown, thepositionofthesecondparticleisuniquelydeter mined) ;and soon. Example.State thenumber ofindependent coordinatesrequiredtospecify the configurationatanyinstant ofarigidbodywhich isconstrained tomove incontact with agivenfixedsmooth surface. We shallgenerallydenotebynthenumber ofcoordinatesrequiredto specifytheconfigurationofasystem, and shallsupposethesystems con sidered tobesuch that nisfinite. Thecoordinates willgenerallybedenoted byql}q2,...qn.Ifthesystem containsmovingconstraints(e.g.ifitconsists ofaparticlewhich isconstrained tobeincontact with asurface which in turn ismade torotate with constantangular velocity round afixedaxis), itmaybenecessarytospecifythetime tinaddition tothecoordinates &lt;?i&gt; &lt;?2, qn &gt;inorder todefinecompletelyaconfigurationofthesystem. Thequantities q1}g.2,...qnarefrequentlycalled thevelocitiescorresponding tothecoordinatesql}q2,...qn. Aheavyflexiblestring,freetomove inspace,isanexampleofadynamical system which isexcluded bythelimitation thatnistobefinite; fortheconfiguration ofthe string cannot beexpressedinterms ofafinitenumber ofparameters. 25.Holonomic andnon-holonomicsystems. Itisnownecessarytocallattention toadistinction between twokinds ofdynamical systems,which isofgreat importanceintheanalytical discussion oftheirmotion :thisdistinction maybeillustratedbyasimple example. Ifweconsider themotion ofasphereofgiven radius, which isconstrained tomove incontact with agivenfixedplane, which wecantake astheplane ofxy,theconfigurationofthesphereatanyinstant iscompletely specified byfivecoordinates, namelythetworectangular coordinates(x,y)ofthe centre ofthesphere andthethree Eulerianangles 0, &lt;f&gt;,fyof 10,which specifytheorientation ofthesphere about itscentre. Thesphere cantake upanyposition whatever, solongasitisincontact with theplane;thefive coordinates(x,y,6, &lt;f&gt;, -v/r)cantherefore haveanyarbitraryvalues. Ifnowtheplaneissmooth, thedisplacement fromanyposition, defined bythecoordinates(x,y,6,$,i/r),toanyadjacent position, definedbythe coordinates(x+8x,y+By,9+B0, 4- B&lt;f&gt;, "f+B-^r),whereBac,By,80, B&lt;f&gt;,8^ arearbitrary independent infinitesimalquantities,isapossible displacement, i.e.thespherecanperformitwithoutviolatingtheconstraints ofthesystem. But iftheplaneisperfectly rough,this isnolongerthecasewhen 8x,By,86, 8$, 8-^rarearbitrary;fornow thecondition that thedisplacementofthe pointofcontact iszero(tothe first order ofsmallquantities) must be satisfied, and thisimplies that thequantities 8x,By,80, B&lt;f&gt;,8^areno longer independent, butaremutually connected(infact,theymust besuch w.D. o 34 TheEquations ofMotion[CH.n astosatisfytwonon-integrablelinearequations) ;sothat inthecaseofthe sphereontheperfectly rough plane,adisplacement represented byarbitrary infinitesimal changesinthecoordinates isnotnecessarilyapossibledis placement. Adynamical systemforwhich adisplacement represented byarbitrary infinitesimal changesinthecoordinates isingeneralapossible displacement (asinthecase ofthesphereonthesmoothplane)issaid tobeholonomic; when thiscondition isnotsatisfied (asinthecaseofthesphereontherough plane)thesystemissaid tobenon-holonomic. If (&7i&gt; &/2,fyn)arearbitraryinfinitesimal increments ofthecoordinates inadynamical system,these willdefine apossible displacementifthesystem isholonomic, while fornon-holonomicsystemsacertain number, saym,of equationsmust besatisfied between them inorder thattheymaycorrespond toapossible displacement.Thenumber (nm)iscalled thenumberof degrees offreedomofthesystem.Holonomicsystemsaretherefore charac terised bythefactthatthenumber ofdegreesoffreedom isequaltothe number ofindependentcoordinatesrequiredtospecifytheconfiguration ofthesystem. 26.Lagrangesform oftheequations ofmotionofaholonomicsystem*. Weshallnowconsider themotion ofaholonomicsystemwithndegrees offreedom. Let(qltq2,...qn)bethecoordinates whichspecifythecon figurationofthesystematthetime t. Let niitypifythemass ofoneoftheparticlesofthesystem, and let (xi, 2/t,Zi)beitscoordinates, referred tosome fixed setofrectangularaxes. These coordinates ofindividualparticlesare(from ourknowledgeofthe constitution ofthesystem)known functions ofthecoordinatesq1}q2,...qnof thesystem,andpossiblyoftalso;letthisdependencebeexpressed bythe equations Let(Xi, Yi,Zi)bethecomponentsofthe total force(external and molecular) actingontheparticlemt- ;then theequationsofmotion ofthis particleare mi ar.i=Xit niiyi=Yi}m^=Z{. Multiplythese equations by 8/i ?fc*b dqr dgr dqr *Lagrange, Mecanique Analytique (1788), Seconde Partie, Section iv.Theequations were firstsuggestedinoneofhisearlier papers,Miscell. Taurin. n.(1760). 25,26]TheEquations ofMotion 35 respectively,addthem, andsum foralltheparticlesofthesystem. We thushave 5- fdfi4.;;a&,yd^\?(y9/i,T^8&lt;^4.72,Wi xig+Vi5hZi^]**2,[JLfI-if5t-A dqrJ i\dqrl dqr^ dqrJ where thesymbol Sdenotes summation over alltheparticlesofthesystem; thiscanbeeither anintegration (iftheparticlesareunited intorigid bodies) orasummation overadiscreteaggregateofparticles. Butwehave a^J^fZL^^^ *f^r 7i /-r &lt;-&lt;(,~r .._.. SO Xi^ Xi ^T" dqr dqr _d_(. 3jci\_.^/a ~dt(XidqJXidt\d dq _di.a*A./~Xi Xi_ dtidqji\dqidqr1 dq2dqr andtherefore wehave Now thequantity i2*n(#+#+#) representsthesum ofthemasses oftheparticlesofthesystem,each multiplied byhalf thesquareofitsvelocity;this iscalled theKinetic Energyofthesystem*. From ourknowledgeoftheconstitution ofthe system,thekineticenergycanbecalculated fasafunction of &lt;?i, &lt;?a, qn,q\, &lt;J2,qn,t; weshall denote itby T(qltq2,...qn,qltq.2,...qn,t), andshallsupposethatI7isaknown function ofitsarguments.Since *Themass ofaparticle multiplied bythesquare ofitsvelocity was called thevisvivaby Leibnitz(Acta erud., 1695). tThemethods ofperformingthiscalculation forrigid bodies aregiveninChapter V. 32 36 TheEquations ofMotion[CH.u andijiand Z{arelikewise linear functions ofql}q2,...qn,weseethatTisa quadraticfunction ofqlyq2&gt;...qn;ifthefunctions/, &lt;/&gt;,^donotinvolve the timeexplicitly (asisgenerallythecase ifthere arenomovingconstraints inthesystem),thequantities x,y,zarehomogeneouslinear functions of ql}q2,...qn&gt;andthenTisahomogeneous quadraticfunction ofqltq2,...qn. From thedefinition itfollows that thekinetic energyofasystemisessentially positive; Tistherefore apositive definite quadratic form ingltq2,... &lt;jn,andsosatisfies theconditions that itsdiscriminant andtheprincipalminors ofeveryorder ofits discriminant arepositive. Wehavethus derived from theequationsofmotion theequation dftT\dr_y( YdA+Vd-^+7d-+i\ dt(dqj dqr~i(i dqr+i dq&gt;^" 9gJ andtheexpressionontheleft-hand sideofthisequationdoesnotinvolve the individualparticlesofthesystem, exceptinsofarastheycontribute tothe kinetic energyT.Wehavenowtoseeiftheright-handsideoftheequation canalsobebroughttoaform inwhich theindividualityoftheseparate particlesislost. Forthispurpose,consider thatdisplacementofthesysteminwhich the coordinateqrischangedtoqr+Sqr,while thecoordinates q1,q.2,...q,-i,qr+i,---qn andthetime(sofarasthis isrequiredforthespecificationofthesystem)are unaltered. Since thesystemisholonomic, thiscanbeeffected without violatingthe constraints. Inthisdisplacement,thecoordinates ofthe particle m;arechangedto andtherefore thetotalwork done inthedisplacement byalltheforces which actontheparticlesofthesystemis 1*~\I-*1f~\* I "!. *-\ 09V ogvd&lt; Now oftheforces which actonthesystem,there areseveral kinds which donowork. Amongthese are,aswasseen in 23, 1.Themolecular forces which actbetween theparticlesoftherigid bodies contained inthesystem: 2.Thepressuresofconnecting-rodsofinvariablelength,thereactions atfixedpivots,andthetensions oftaut inextensiblestrings: 3.The reactions ofanyfixed smooth surfaces orcurves with which bodies ofthesystemareconstrained toremain incontact;orofperfectly rough surfaces, sofarasthese canenter intoholonomicsystems: 26] TheEquations ofMotion 37 4.The reactions ofanysmooth surfaces orcurves withwhich bodies ofthesystemareconstrained toremain incontact, when these surfaces or curves areforced tomove insomeprescribedmanner;forthedisplacement considered above ismade onthesuppositionthatt,sofarasitisrequiredfor thespecificationofthesystem,isnotvaried,i.e.thatsuch surfaces orcurves arenotmovedduringthedisplacement;sothat this case reduces tothe preceding. The forcesactingonthesystem,other than these which donowork, are called theexternal forces. Itfollows thatthequantity 9 /&lt;4.V8(k4-Z-d^* i^--rJLi= -pAI-= dqr dqr oqr istheworkdonebytheexternal forces inthedisplacementwhichcorresponds toachangeofqrtoqr+$qr,theother coordinates beingunaltered. This is aquantitywhich (from ourknowledgeoftheconstitution ofthesystem,and oftheforces atwork)isaknown function ofqltq.2&gt;...qn,t;weshall denote itby Qr(qi,q2,qn ,t)8q r. Wehave therefore dfdT\_3T__ n dt(dqj dqr~ ^" Thisequationistrue forallvalues ofrfrom 1toninclusive;wethus havenordinarydifferentialequationsofthesecond order, inwhichqi,q 2,...q n arethedependentvariables and tistheindependentvariable;asthenumber ofdifferentialequationsisequaltothenumber ofdependent variables, the equationsaretheoreticallysufficient todetermine themotion when the initial circumstances aregiven. Wehavenowarrived ataresult which may bethus stated : LetTdenote thekineticenergy ofadynamical system, and let Qi&?i+Qz$qa+...+Qn$qn denote thework donebytheexternalforcesinanarbitrary displacement (&?!, 8q2,...8qn),sothat T,QltQ2,...Qno,re,fromourknowledge ofthe constitutionofthesystem, knownfunctions ofqltq2,...qn,qltq2,...qn,t; then theequations which determine themotionofthesystem maybewritten ddT\ dT These areknown asLagranges equations ofmotion. Itwillbeobserved thattheunknown reactions(e.g.oftheconstraints) donotenter intothese equations. Thedetermination ofthese reactions forms aseparate branch of mechanics, which isknown asKineto-statics* :sowecansaythatinLagranges equationsthekineto-statical relations oftheproblemarealtogether eliminated. *Cf.Heun, Deutsche Math. Ver. ix.(Heft 2)(1900), p.1. 38 TheEquations ofMotion[CH.n 27. Conservativeforces:theKinetic Potential. Certain fields offorcehave thepropertythattheworkdonebytheforces ofthe field inadisplacementofadynamical systemfromoneconfiguration toanotherdepends onlyontheinitial and finalconfigurationsofthesystem, beingthesame whatever bethesequenceofinfinitesimaldisplacements by which thefinitedisplacementiseffected. Gravityisaconspicuous exampleofafield offorce ofthischaracter;thework done bygravityinthemotion ofaparticleofmassmfrom oneposition ataheight hto anotherposition ataheight kabove theearth ssurface ismg(k-k), and thisdoesnot dependinanywayonthepathbywhich theparticleismoved from oneposition tothe other. Fields offorce ofthistypearesaid tobeconservative. Lettheconfigurationofanydynamical system bespecified byn coordinatesql}q2,...qn.Choose someconfigurationofthesystem, say that forwhich . q,-=a r, (r= 1,2,...n), asastandardconfiguration;then iftheexternal forcesactingonthesystem areconservative, thework donebythese forces inadisplacementofthe system from theconfiguration (q1}q2,...qn)tothestandardconfigurationisa definite function ofqltq^,...qn,notdependingonthemode ofdisplacement. Let thisfunction bedenotedbyV(q}, &lt;?2,...qn);itiscalled thePotential Energy*ofthesystemintheconfiguration (qltq2,...qn).Inthiscasethe work donebytheexternal forces inanarbitrary displacement (%, Sq2,...Bqn) isevidently equaltotheinfinitesimal decrease inthefunction V,corresponding tothedisplacement,i.e.isequaltothequantity 9F dV. 8F,~~5~&lt;7i~~s~~ #2 5oqn ; dql dq2 dqn Lagrangesequationsofmotion therefore taketheform____ dt\dqr)3qr dqr Ifweintroduce anewfunction Lofthevariablesqltq2,...qn,qlt...qn,t, definedbytheequationL=T-V, thenLagrangesequationscanbewritten d/dL\ dL -7iUT- 15=0, (r=1.2,...n).dt\dqrjdqr *ThePotential-function wasintroduced byLagrangein1773 (Oeuvres,vi.p.335). The name Potential isduetoGreen(1828). 27,28] TheEquations ofMotion 39 The function Liscalled theKinetic Potential, orLagrangian function; thissinglefunctioncompletely specifies,sofarasdynamical investigations areconcerned, aholonomicsystemforwhich theforces areconservative. 28.Theexplicit form ofLagranges equations. We shallnowshewhowthesecond derivatives ofthecoordinates with respecttothetime canbefoundexplicitlyfromLagrangesequations. Lettheconfigurationofthedynamical systemconsidered bespecified by coordinatesq1}q2,...qn;weshallsupposethat theconfigurationcanbe completely specifiedinterms ofthese coordinates alone, withoutt,sothat thekineticenergyofthesystemisahomogeneous quadraticfunction of qltq2,...q n.Aswas seen in 26,this isalwaysthe casewhen the constraints areindependentofthetime, butnot ingeneral when the constraints have forced motions(asforinstance inthecase ofaparticle constrained tomove onawirewhich ismade torotate inagiven way). Supposethen thatthekineticenergyis nnT=|22,akiqkqi,k=n=i where aH=a{k,andwhere thecoefficients&lt;%areknown functions of TheLagrangian equationsofmotion forthesystemare dtdq dq or or*vJi^V.""*i?iika(r=l,2,...n), (r=l, 2,...w), n w I"/^vj~lv c*c1 \" * \ rt / io \ZOnqt+2,2,\qiqm=Qr, (r=1,2,...n), s=i ;=im=i Ir where thesymbol 1,which iscalled aChristoffelssymbol*,denotes the expression 1/8$; r9$mr_5$;m\ 2V3gm 9^ 9gr/ \dqmdqldqr Theseequations, beinglinear intheaccelerations, canbesolved forthe quantities qs.Infact, letDdenote thedeterminant U-ll $12 $13 $21 $22 $23 $31 $32 *Itwasintroduced byChristoffel, JournalfilrMath. LXX.(1869), and isofimportance inthe theory ofquadratic differential forms. 40 TheEquations ofMotion[CH.n and letArsbetheminor ofarsinthisdeterminant.Multiplythenequations oftheabovesystem byAlv,A2v ,...Anv ,respectively, andaddthem: re- w memberingthatthequantity 2Arvar8iszerowhen sisdifferent fromv,and hasthevalueDwhen sisequaltov,wehave nnn |~^~|n Thisequationistrue for allvalues ofvfrom 1toninclusive; and thesenequations,inwhichq\,q2,...qnaregiven explicitlyasfunctions of ql}q2,...qn&gt;qltq2,...qn,canberegardedasreplacing Lagrangesequations ofmotion. 29.Motionofasystem which isconstrained torotateuniformly round an axis. Inmany dynamical systems, somepartofthesystemiscompelled byan externalagencytorevolve with constantangular velocity wround agiven fixed axis;themotion ofabead onawirewhich ismade torotate inthis wayisasimple example. Thereis,aswehave seen, noobjectiontothe directapplicationofLagrangesequationstosuch cases, provided thesystem isholonomic; but itisoften more convenient touseatheorem which we shallnowobtain, andwhich reduces theconsideration ofsystemsofthiskind tothat ofsystemsinwhich noforced rotation about thegivenaxistakes place. Suppose that,independentlyoftheprescribed motion round theaxis, the systemhasndegreesoffreedom, sothat ifthegivenaxis istaken asaxisof z,andanyplane throughthis axisandturning with theprescribed angular velocityistaken astheplane fromwhich theazimuth &lt;/&gt;ismeasured, the cylindrical coordinates ofanyparticlemofthesystemcanbeexpressedin terms ofncoordinatesqltq2,...,qn,theseexpressions notinvolvingthetime t. Then ifthekineticenergyofthesystemintheactual motion beT,and ifthe work donebytheexternal forces inanarbitraryinfinitesimaldisplacement beQi&qi+QSq,+...+Qn&qn,where QltQ2,...,Qnwillbesupposedto depend onlyonthecoordinatesqltq2,...,qn,and ifthekineticenergyof thesystem when theforcedangular velocityisreplaced byzerobedenoted byTltwehave Now thequantity2?wr2willbeafunction ofqltq2,...,qn,which is determinedbyourknowledgeoftheconstitution ofthesystem:denote itby W.Thequantity 2rar2will alsobeaknown function ofq1}q2,...,qni 28-30] TheEquations ofMotion 41 qi,...,qn,beinglinear inq1}q2,...,qn,itwillbezeroif,when coiszero, the motion ofevery particlehasnocomponentinthedirection of &lt;/&gt;increasing; while ifnisequaltounity,sothat there isonlyonecoordinateq,itwillbe theperfectdifferential withrespecttotofafunction ofq:these arethetwo cases ofmostfrequent occurrence, andweshall include them bothbyas- dY sumingthat^m?^2 &lt;isoftheform -T-,whereYisagivenfunction ofthe CLT&gt; coordinatesqltq2,...,qn. Wehave therefore andtheLagrangian equations d/dT\ dT canbewritten intheform ddT,\ ddY\ dT, ddY\ dW dt( r, ,,..,. rjdqr dqr^ Theseequations shew that, subjecttotheassumption already mentioned, themotion isthesame asiftheprescribed angular velocity were zero,and thepotential energy were tocontain anadditional term^mr2a)2 .Inthis way,bymodifyingthepotential energy, weareenabled topass from a system which isconstrained torotate about thegivenaxis toasystemfor which this rotation does nottakeplace. Thetermcentrifugal forcesis sometimes used oftheimaginaryforces introduced inthiswaytorepresent theeffect oftheenforced rotation. 30.TheLagrangian equations forquasi-coordinates. Intheform ofLagrangesequations givenin 26,thevariables aren coordinatesq1}q2,...,qn,andthetimet;theknowledgeofthesequantities, together with aknowledgeoftheconstitution ofthesystem,issufficient to determine thepositionofanyparticleinanyconfigurationofthesystem, whichmaybeexpressed bysayingthatq1}q.2,...,qnaretrue coordinates of thesystem. Weshallnow findtheformwhich istakenbytheequations when thevariables used arenolongerrestricted tobetrue coordinates of thesystem*. Consider asystem denned byntrue coordinatesqltq2,...,qn,the kineticenergy beingTandthework donebytheexternal forces ina *Particular cases ofthetheorem ofthis article wereknown toLagrange andEuler: the general form oftheequations isdue toBoltzmann (Wien. Sitzungsbcrichte, 1902) andHamel (Zcitschrift fiirMath. u.Phys. 1904). 42 TheEquations ofMotion[OH.n displacement (8q l}8q2,...,%)being&%+Q28q,+...+Qnfyn&gt;sothatthe Lagrangian equationsofmotion ofthesystemare Let &&gt;i, o)2,..., tonbe ?iindependentlinear combinations ofthevelocities &lt;?i&gt;fy,,qn,definedbytherelations a&gt;r=alrqj+a2rq2+...+anrqn (r=l, 2,...,w)...(2), where,or21,...,annaregiven functions ofqltqz,...,qn;and letdir^,dir^ ...,d7rnbenlinear combinations ofthedifferentialsdqltdq2,...,dqn,defined bytherelations d7rr=alrdq l+ 2rd&lt;?2+ ...+ctnrdqn (r=1,2,...,), where thecoefficients aarethesame asintheprevioussetofequations. These lastequations would beimmediately integrableiftherelations C$ C^ ^~=V^were satisfied forallvalues ofK,r,andm,andinthat casevariablesoqm oqK irrwould exist which would betrue coordinates;weshall nothowever suppose theequationstobenecessarily integrable,sothat dir^, cfor2,...,d7rn willnotnecessarilybethedifferentials ofcoordinates 7ra,7r2,..., irn;weshall callthequantities d^, dir^, ..., d-rrndifferentials ofquasi- coordinates. Suppose that therelations(2),when solved forql}q.2,...,qn,givethe equations q&lt;=/3Klwl+/3K2&)2+...+/3Knwn (r=I,2,...,n)...(3). MultiplyingtheLagrangian equations (1)by&r,@2i.,..., /8Br ,respectively, andadding, weobtain theequation (d3T\ d Now^Q KSqKistheworkdonebytheexternal forces onthesysteminan arbitrary displacement, so2l/3KrQKS7rristhework done inadisplacement inwhich allthequantitiesSTTarezero,exceptS?rr.Iftherefore thework donebytheexternal forces onthesysteminanarbitraryinfinitesimal dis placement (Swx, S?r2,...,S7rn)isn^TTj+II2S7r2+...+Un87Tn,W6have 2/3 \d -f^_^l =n K" (dt\dqj dqK)~ Bymeans ofequations (3)wecaneliminateq1}q2,...,qnfrom the function T,sothatTbecomes afunction ofwl,&&gt;2,...,wniqlyq2,...,qn(we supposeforsimplicitythat tisnotcontainedexplicitlyinT) ;letthisform ofTbedenotedbyT. O/TTrl^PThenwehave-=2rCL*. 30] TheEquations ofMotion 43 andtherefore d/df dTdaKSdT ^--j- sj sdcosatdqK But *5&lt;ftKraKSiszero orunity accordingasrisdifferent from, orequal K to,s:sowehave d/^UXS*d*sdTSflaT-TT dt(fo r)+7."25"5i .5 Wealsohave dT__dT v8f a&&gt;g_af^vaf8ams. -a o~r**oo"i "r^^ Qmy oqKoqK sdwsdqKdqK smd(osdqK * Now Sy8Kr^ ,or2=s*5 -,would represent ^ if?rrwere atrue coordinate;weshall denote itbythesymbol=whether TT,.isatrue OTT,- coordinate ornot. Also theexpression depends onlyontheconnexion between thetrue coordinates andthedif ferentials ofthequasi-coordinates, and isindependentofthenature or motion ofthedynamical systemconsidered :weshall denote thisexpression by7rz-Wehave therefore 5rgii=-- ==Ur (r=1,2,...,ri).at\0r/ si Od)sC7Tr These nequations aretheequations ofmotion expressedintermsofthe quasi-coordinates ;when thequasi-coordinatesaretrue coordinates, the ^ ^ quantities 7^are allzero, since theconditions 5-^=-,.-aresatisfied, and oqmoqK theequations reduce totheordinary Lagrangian equations ddT\ dT Example. Arigidbodyisfreetoturnabout oneofitspoints 0,which isfixed, so that thecoordinates ofthebody canbetaken tobethethree Eulerianangles 6, &lt;,^, which(10)specify theposition ofaxesOxyz,fixed inthebodyandmoving withit,with reference toaxesOXYZ fixed inspace. Letanarbitrary displacement (80,8$,8-^)ofthe bodybeequivalent totheresultant ofsmall rotations ST^, STT,,8*3round Ox,Oy,Oz, respectively,sothatdiri, dir.2,o?7r3canbetaken asthedifferentials ofquasi-coordinates: let0)1,o&gt;2,o&gt;3bethecomponents about theaxesOxyzoftheangular velocityofthebody 44 TheEquations ofMotion[CH.n atany instant, sothatdiri,dir2,d-n-3arethe differentials ofquasi-coordinates corre sponding respectivelytothevelocitiesi, 2,&&gt;3.Shew that theequationsofmotion of thebody are (-(-}- Idt \o&lt;ai/ Id/dT\dT dTdT ~ 1-&lt;BO^ ^r=IIi, ar a?7ay ffl23--h i3--o=H3, a&gt;3/ 00)!OQ)2OTs where 7*isthekineticenergyofthebody, expressedinterms of a&gt;1?a&gt;2,o&gt;3,6,0, *//;n:,n2,n3arethemoments about theaxes Ox,Oy,Oz,respectively,oftheexternal forces ,ar. ofd&af a&lt;/&gt;afa^acting onthebody ;and stands for^-^-- j.,J21+i-i. OTT,. otOTT,.0&lt;pVTrrdy07rr 87^Itwillappearlater thatTdepends onlyona1}o)2,&&gt;3,sotheterms =arezero. uirr 31. Forces derivable from apotential-function which involves the velocities. Incertain cases theconceptionofapotential-energyfunction canbe extended todynamical systemsinwhich theactingforcesdependnotonly ontheposition butonthevelocities andaccelerations ofthebodies. Forconsider adynamical system whoseconfigurationisspecified by coordinatesq1}q2,...,qn,andsupposethat thework donebytheexternal forces inanarbitrary displacement (Bq lt8q2,...,8qn)is Then ifQrcanbeexpressedintheform dvd/dv jr- -r^-.- (r=1,2,... ,n\ dqrdt\dqrj whereVisagivenfunction ofqltq2,...,qn,qlt...,qn,theLagrangian equa tions ofmotion are _ _ dt(dqj dqr~ dqrdt(dq r) and ifakineticpotential Lbedefinedbytheequation L=T-V, theequations takethecustomaryform dfdL\ dL -T,U-.-)-a-=0dt\dqrjdqr The function Vcanberegardedasageneralised potential energy function. Anexampleofsuch asystemisfurnishedbythemotion ofa 30-32]TheEquations ofMotion 45 particle subjecttoWeber selectrodynamiclawofattraction* toafixed point,theforceperunitmassactingontheparticle being 1/f-2__2r f&gt; r2\ c2 where risthedistance oftheparticlefrom thecentre offorce :inthiscase thefunction Visdefined bytheequation Example.Iftheforces Qlt^2,...,Qnofadynamical system which isspecified by coordinatesqlyq2,...,q narederivable from ageneralised potential-function F,sothat _- oqrdt shew thatQl}Q2,...,QHmust belinear functions ofq\, q-2,...,qn,satisfyingthen(2n relations Onthegeneralconditions fortheexistence ofakinetic potentialofforces, reference maybemade to. Helmholtz, JournalfiirMath., Vol. c.(1886). Mayer, Leipzig. Berickte, Vol. XLVIII. (1896). Hirsch, Math. Annalen, Vol. L.(1898). 32. Initial motions. The differentialequationsofmotion ofadynamical systemcannot in generalbesolved inafinite form interms ofknown functions. Itishow ever always possible (exceptinthevicinityofcertainsingularitieswhich need notbeconsidered here)tosolve asetofdifferentialequations bypower- series,i.e.toobtain forthedependentvariablesqlfq2,...,qnexpressionsof thetype */22J2I2 2 ? thecoefficients a,b,...caninfactbeobtained bysubstitutingthese series in thedifferentialequations, andequatingtozerothecoefficients ofthevarious *W.Weber, Annalen d.Phys.LXXIII.(1848), p.193. Cf.Whittaker sHistory oftheTheories of-Aether andElectricity, pp.226 231. 46 TheEquations ofMotion[OH.n powersoft;theexpansionswillconvergeingeneralforvalues oftwithin some definite circle ofconvergenceinthetf-plane. Itisplainthat these series willgiveanyinformation which maybe requiredabout theinitial character ofthemotion (tbeing measured from the commencement ofthemotion),sinceiisthe initial value ofqlt6Xisthe initial value ofql9andsoon.Thismethod ofdiscussingthe initial motion ofasystemisillustrated bythefollowing example. Example.Consider themotion ofaparticleofunitmass, which isfreetomove ina plane andinitiallyatrest,andwhich isacted onbyafield offorcewhose components paralleltofixed rectangularaxes atanypoint (x,y}are(X,Y) ;and letitberequiredto determine theinitial radius ofcurvature ofthepath. Let(#+,y+*i)bethecoordinates ofanypoint adjacenttothe initialpoint (.z;y), sothat,T)mayberegardedassmallquantities ;then theequations ofmotion are .#).3Y(x,y} 8y Iftherefore weassume forand77theexpansions (itisnotnecessarytoinclude terms oflower order than t2 ,since thequantities ,TJ, ,T) areinitially zero), andsubstitute inthese differentialequations, wefind,oncomparing thecoefficients ofvarious powersoft,therelations y),6=0,c- Thepathoftheparticle nearthepoint(Xy)istherefore given bytheseries where denotes thequantity \&. Now ifthecoordinates|and77ofanycurve areexpressedinterms ofaparameter u, theradius ofcurvature atthepointuisknown tobe sotheradius ofcurvature correspondingtothezerovalue ofu,forthecurvegiven bythe above expressions,is (x~+r)x-(x*-+Y^\Y Voxdy) \ctxdy) andthis istherequired radius ofcurvature ofthepathoftheparticleattheinitialpoint. 32-34] TheEquations ofMotion 47 33. Similarityindynamical systems*. Ifanysystemofconnectedparticles andrigidbodies isgiven,itis possibletoconstruct anothersystem exactlysimilar toit,butonadifferent scale. Ifnowthemasses and forces inthetwosystems, which wecan call thepattern andmodelrespectively,bear certain ratios toeach other, the workingsofthetwosystemswillbesimilar, though possiblyatspeedswhich arenotthesame butbearaconstant ratio toeach other. Tofindtherelation between thevarious ratios involved, letthelinear dimensions ofthemodel andpatternbeintheratiox :1,letthemasses of corresponding particlesbeintheratioy:1,lettherates ofworkingbeinthe ratio z :1,sothatthetimeselapsedbetweencorresponding phasesareinthe ratio 1 :z,and lettheforces beintheratiow.\. Then foreachparticle we haveanequationofmotion oftheform mx=X; soifmisaltered intheratioy:1,xisaltered intheratio xz* :1,andXis altered intheratiow :I,wemust have w=xyz*, andthis istherequiredrelation between thenumbers x,y,z,w. Example.Iftheforcesacting arethose duetogravity, wehavew=y, andconse quently xy2=I,sothattherates ofworking areinverselyasthesquare roots ofthelinear dimensions. Ifthe forcesacting arethemutualgravitations oftheparticles, every particle attracting everyotherparticle withaforceproportional totheproduct ofthemasses and theinverse square ofthedistance, wehavew=y2/x2 ,sothattherates ofworking arein theratioy\:x%. 34.Motion with reversedforces. Aspecialcase ofsimilarityisthat inwhich theratiowhasthevalue 1. Wehave seenthatthemotion ofanydynamical system which issubjected toconstraintsindependentofthetime, andtoforces whichdepend onlyon thepositionsoftheparticles,isexpressed bytheLagrangian equations d/8T\ dT where thekineticenergyI7 isahomogeneous quadratic function ofthe velocitiesq\,ja,...,qn,involvingthecoordinatesqltq2,...,qn,inanyway, andQisafunction ofql,q2,...,qnonly. Introduce anewindependent variable definedbytheequation T=it, where i=V1, and letaccents denote differentiations withregardto r.Then since *Newton, Principia, Book n.Sect. 7,Prop.32. 48 TheEquations ofMotion[CH.n d/dT\,dT dt\M r) a&lt;Tarehomgeneous ofdegree-2indt,theaboveequations dfc^\ dbeCmedrdq;)-^=-$&gt;&lt;r=1,2, ...,), where isthesame function ofq,,qz,...,qn}qlt...,qnthatTisof 3i, &&gt; ..-,qn, qi,q*, -.., n- But ifT(instead oft)benowinterpretedasdenotingthetime, these last equationsaretheequationsofmotion ofthesamesystem whensubjectedto thesame forces reversed indirection. Moreover, ifa1}a2,...,an,ft,/32,..., narethe initial values ofqltq,, ...,qn&gt;qltq2&gt;...,qn&gt;respectively,inany particularcaseofthemotion oftheoriginal system, then al}a2,...,an,-ifr, i/32&gt;..., iftn willbethecorresponding quantitiesinthetransformed problem. Wethushave thetheorem that inanydynamical system subjected toconstraintsindependent ofthetimeand toforces whichdepend onlyonthe position oftheparticles,theintegrals oftheequations ofmotion are stillreal iftbereplaced by*J-Itand theinitial velocities &, ...,/3nby-V^l&,-V-l/32,...,-V-l{jnrespectively; andtheexpressions thusobtainedrepre sent themotion which thesamesystem would haveif,with thesame initial conditions, itwere acted onbythesameforces reversed indirection. 35.Impulsive motion. Incertain cases(e.g.inthecollision ofrigid bodies) thevelocities ofthe particlesinadynamical systemarechangedsorapidlythatthetimeoccupied intheprocess may,foranalytical purposes, bealtogether neglected. Thelawswhichgovern theimpulsive motion ofasystembearaclose analogytothose whichapplyinthecase ofmotion under finite forces :they canbeformulated inthefollowing way*. Thenumber whichrepresents themass ofaparticle, multiplied bythe vector whichrepresentsitsvelocityatany instant, isavectorquantity (localisedinalinethroughtheparticle) which iscalled themomentum of theparticleatthatinstantf; thethreecomponents paralleltorectangular- axesOxyzofthemomentum ofaparticleofmassmatthepoint (x,y,z)are therefore (mx,my,mz).Ifanynumber ofparticles form adynamical system, thesum ofthecomponentsinanygiven direction ofthemomenta ofthe particlesiscalled thecomponentinthat directionofthemomentumofthe system. Theimpulsive changesofvelocityinthevariousparticlesofa connectedsystemcanberegardedastheresult ofsudden communications ofmomentum totheparticles. The effect ofanagency which causesimpulsive motion inthesystem *They were involved inthediscovery ofthelaws ofimpact in1668byWallis andWren, Phil. Trans. No.43,pp.864, 867. tMomentum isthequantitas motus ofNewton sPrincipia, Book i.Def. 2.Theideacanbe traced back toDescartes. 34,35] TheEquations ofMotion 49 willbemeasured bythemomentum which itwould communicate toasingle freeparticle.Iftherefore(?/0)v,w)arethecomponentsofvelocityofa particleofmass m,referred tofixed axes inspace,before theimpulsive communication ofmomentum totheparticle, and if(u, v,w)arethecom ponentsofvelocityoftheparticleafter theimpulse,then thevectorquantity (localisedinalinethroughtheparticle) whose componentsare m(u-w),m(vv),m(ww) representstheimpulse actingontheparticle. Forthediscussion oftheimpulsivemotion ofaconnectedsystemof particles,itisclearly necessarytohavesomeexperimentallawanalogousto thelawofAction andReaction offinite forces;such alaw iscontained in thestatement that the totalimpulse actingonaparticle ofaconnected systemisequaltotheresultantoftheexternalimpulseontheparticle (i.e.the impulse communicatedbyagenciesexternal tothesystem, measuredbythe momentum which theparticle wouldacquireiffree) together withimpulses directedalongthelines whichjointhisparticletotheotherparticles which constrain itsmotion; and themutually inducedimpulses between twoconnected particles areequalinmagnitude andoppositeinsign. Ifweregardthecomponentsofanimpulseasthetime-integralsofthe componentsofanordinaryfinite force which isvery largebutactsonlyfor averyshort time, thelawjuststatedagreeswith thelawofAction and Reaction forfinite forces. Change ofkineticenergy due toimpulses. Thechangeinkineticenergyofadynamical system whoseparticles areacted onbya givensetofimpulses maybedetermined inthefollowing way. Letanimpulse /,directedalongalinewhose direction-cosines referred tofixed axes of reference are(A,^, j/),becommunicated toaparticleofmass m,changingitsvelocity from v,inadirection whose direction -cosines are(Z ,M,NO),tov,inadirection whose direction-cosines are(L,J/,JV).Theequationsofimpulsive motion are Multiplying theseequations respectively by $(vL+vQL),$(vM+vM),and andadding, wehave Thechangeinkineticenergyoftheparticleisthereforeequaltotheproductofthe impulse andthemean ofthecomponents, before and after theimpulse, ofthevelocityof theparticleinthedirection oftheimpulse. Now consideranydynamical system ofconnectedparticles andrigid bodies, towhich given impulses arecommunicated;applying thisresult toeachparticle ofthesystem, and summing, weseethat thechangeinthekineticenergy ofthesystemisequaltothesumofthe impulses appliedtoit,eachmultiplied bythemeanofthecomponents, before andafterthe communicationoftheimpulse, ofthevelocity ofitspoint ofapplication inthedirectionofthe impulse. Inthisresult wecanclearly neglect theimpulsive forces between themolecules ofanyrigidbodyofthesystem. W.D. 50 TheEquations ofMotion[CH.u 36.TheLagrangian equations ofimpulsivemotion. Theequationsofimpulsivemotion ofadynamical systemcanbe expressedinaform*analogoustotheLagrangian equationsofmotion for finite forces, inthefollowing way. Let(X i}Yi,Zi)bethecomponentsofthe totalimpulse (external and molecular) appliedtoaparticle m;ofthesystem,situated atthepoint (#,-, yi,z^.Theequationsofimpulsivemotion oftheparticleare m{(an-xio)=Xiymt(yt-yio)=Ti}m{(z{-z^)=Z{, where(x^,yio,z^)and(xi}yi}z^denote thecomponentsofvelocityofthe particlebefore and after theapplicationoftheimpulse. If &lt;?!,^2,...,qndenote thenindependentcoordinates interms ofwhich theconfigurationofthesystemcanbeexpressed, wehave therefore ^ 1/ \ """I ,/ \vst ,/ eYv2m,- s(xt- asfe)= h(yi- y*&gt;)^-+(&izit})5 9gv (JiXj ??/* OZ i^T -*t^T^f^ ;, cqr oqr cqr/ where thesummation isextended over alltheparticlesofthesystem. Now informingthesummation ontheright-handsideofthisequation, itisseen asin26that themolecularimpulses betweenparticlesofthe systemcanbeomitted :thequantity dqrl dqr^ dqr canthereforereadilybefound when theexternalimpulsesareknown: we shall denote itbythesymbol Qr.Wehaveconsequently 2m, (&t-4-o) +(yt- 2/io)+(*t-^)=Qr. t ( oqr oqr dqr) But asin26wehave da^_dxi. "bxi__d .. dq,.~dq rVi dqr-dq r(* andsimilarly .dxi__9^,. *dqr~dqU whereqroandqrdenote thevelocities ofthecoordinateqrbefore and after theimpulse respectively. Thus if *Due toLagrange, Mec. Anal.(2e ed.),n.p.183. 36] TheEquations ofMotion 51 denotes thekineticenergyofthesystemafter theimpulse,theabove equationcanbewritten intheform dqr\dq r/Q /rlY7\ /}T1 where(~ )denotes thequantity correspondingto^-,butrelatingtothe \oqr/ooc[r instant before theimpulse. Similarequationscanbefound fortherestofthecoordinatesq1}g2 &gt;&gt;qnl andthusweobtain thesetofnequations fdT\ which areknown astheLagrangian equations ofimpulsivemotion. These arealgebraical equationsforthedetermination ofqltq2,...,qnin terms ofqw,qw,...,qm;theyarenotdifferentialequationsliketheLagrangian equationsofmotion for finite forces, since thesecond derivates ofthe coordinates withrespecttothetimedonotenter. MISCELLANEOUS EXAMPLES. 1.Tworigid bodies movinginspaceareconstrained onlybyataut inextensiblestring joining agiven pointofonebodytoagiven point oftheother, andoneofthebodies is constrained torollwithoutsliding onagivenfixed surface. Howmany degreesoffreedom hasthesystem, andhowmany independent coordinates arerequiredtospecifyitscon figuration? 2.Apointisreferred tocurvilinear coordinatesa,b,c,andthesquare ofits velocityis 2T=Aa?+Bb2+Cc2+2Fbc+2Gcd+2Hdb. Shew thatp,q,r,thecomponentaccelerations inthedirections ofthetangentstothe coordinatelines, aregiven bythree equations ofthetype d_fiT\ dT ..H dt\dd j(dfF\dT TT (~* 3-. )----=pJA+-,-_q+-T--,r.(Coll.Exam.)oaj da JB2v/6 3.Aparticle which isfreetomove inspaceisinitiallyatrestattheorigin, and isin afieldofforcewhose components (X,F,Z)atanypoint (x,y,z)aregivenbytheexpansions JT=a+bx+ quadratic andhigher terms inx,y,z; Y= cx+quadratic andhigher terms inx,y,z; Z=dx2+cubic andhigher terms inx,y,z. Find theradii ofcurvature andtorsion oftheorbit attheorigin. 42 CHAPTER III 37.Problems which aresolublebyquadratures. Thedetermination ofthemotion ofaholonomicdynamical systemwith afinitenumber ofdegreesoffreedom hasinthepreceding chapterbeen shewn todependonthesolution ofasetofordinarydifferentialequations. Ifndenotes thenumber ofdegreesoffreedom, and(q1}q2,...,qn)arethe coordinatesspecifyingtheconfigurationofthesystematthetimet,then thesetofequationsconsists ofndifferentialequations, each ofthesecond order, withqltq2,...,q nasdependentvariables and tasindependentvariable. This setofequationsissaid tobeoforder 2n,theorderbeingdefined to bethesum oftheorders ofthehighestderivates ofthedependentvari ablesoccurringintheequations.Itisawell-known result ofthetheory ofordinarydifferentialequationsthatthenumber ofarbitraryconstants of integrationinthesolution ofasetofdifferentialequationsisequalto theorder ofthesystem;whence itfollows that there are2nconstantsof integrationinthegeneralsolutionofaholonomic dynamical problemwith ndegrees offreedom. Nowanygivensetofdifferentialequationsoforder kcanbereduced totheform -^=Xr(ae1}x2,...,xk,t}, (r=1,2,...,k\ whereXl}X2,...,Xkareknown functions oftheirarguments, bytakingas new variables(x1,x2,...,xk)theoriginal dependentvariablestogetherwith their derivates upto(but notincluding)thehighestderivatesoccurringin theoriginalsetofequations. Thuse.g.thesetofequations (where QlandQ2areanyfunctions ofthearguments indicated) which isof order 4,canbereduced totheset dxl_dx2_dx3_f), x dx4_^/ ~T7 ^3&gt; ~~rr ^4&gt;~jT^Jl\&ltMS* ^3&gt;&4/I ~~JT ^2\&lt-%2&gt; fiat^4)5 37] Principlesavailable fortheintegration53 bytaking #1=01, a*=&, #3=0i, ^4=02- Theform -^=Xr(x1,x2,...,# fc&gt; (r=l,2,...,&) maytherefore beregardedasthetypicalform forasetofdifferential equationsoforder k. rlf Ifafunction f{x l,x2,...,x k,t)issuch that~-iszerowhen(#,,#2&gt; &gt;#*) tt-t areanyfunctions oftwhatever whichsatisfythese differentialequations,the equation /(#i,x2,...,#, t)=Constant iscalled anintegralofthesystem. Thecondition that agivenfunction / mayfurnish anintegralofthesystemiseasilyfound;fortheequation dfldt=gives ^-4+*-4+...+~-xk+57=0, ox-i cx2 dxk dt or VY4-^Tj L.8/Y^8^-o&lt;*1+o-^2+ +5^A+57=^; d^j oa;2 dxk dt andthisrelation must beidenticallysatisfied inorder that theequation f(x i,oc2,...,aek,t)=Constant maybeanintegralofthesystemofdifferentialequations. Sometimes thefunction /itself(asdistinct from theequation /=constant)iscalled an integral ofthesystem. Thecompletesolution ofthesetofdifferentialequationsoforder kis furnishedbykintegrals frOn 2, ,afc,t)=ar, (r=1,2,...,k), where altaz,...,akarearbitrary constants, providedtheseintegralsare distinct, i.e.nooneofthem isalgebraicallydeducible from theothers. For letthevalues of as1}xz,...,xk,obtained from theseequationsasfunctions of t,a,! ,a2,...,ak,be xr= &lt;,.(!, a,,...,ak,t), (r--l, 2,...,k}; then if(a-j,x2,...,xk)areanyparticularsetoffunctions oftwhichsatisfy thedifferentialequations,itfollows fromwhat hasbeen saidabove that bygivingtothearbitrary constants arsuitable constant values wecanmake theequations /r(ar,,x2,...,a!k,t)=ar (r=l, 2,...,k) true forthisparticularsetoffunctions(xl}x2,...,xk);andtherefore this set offunctions(xlyxZ)...,xk)willbeincluded amongthefunctions defined by theequationsxr= &lt;j&gt;r.Thesolution ofadynamical problemwithndegrees offreedom maytherefore beregardedasequivalenttothedetermination of 2wintegralsofasetofdifferentialequationsoforder 2n. 54Principles available fortheintegration [OH.in Thus thedifferential equation ?=-?&gt; which isofthesecond order, possessesthetwointegrals f ?2+?2=,, Itan-1?_;= a I ? where!and a.2arearbitrary constants. Onsolvingtheseequationsforqandq,wehave andtheseequations constitute thesolution ofthedifferentialequation. Themoreelementarydivision ofdynamics,with which thisandthe immediately succeeding chaptersareconcerned, isoccupiedwith thedis cussion ofthosedynamical problemswhich canbesolvedcompletelyin terms oftheknownelementaryfunctions ortheindefiniteintegralsofsuch functions. These aregenerallyreferred toasproblemssolublebyquadratures. Theproblemsofdynamicsarenotingeneralsolublebyquadratures ;andin those cases inwhich asolution byquadraturescanbeeffected, there must alwaysbesomespecialreason forit,infactthekineticpotentialofthe problem must have somespecialcharacter. Theobjectofthepresent chapteristodiscuss thosepeculiaritiesofthekineticpotential which are mostfrequentlyfound inproblemssolublebyquadratures, andwhich infact aretheultimateexplanationofthesolubility. 38.Systemswithignorablecoordinates. Wehave seen(27)that themotion ofaconservative holonomic dynamical systemwithndegreesoffreedom, forwhich thecoordinates are &lt;?!,22,...,qnandthekineticpotentialisL,isdeterminedbythedifferential equations dfdL\dL -T.U-T-)-5=0, (r=1,2,...,n).at\dqrJdqr 7)TThequantity ^-r-isgenerallycalled themomentumcorrespondingtothe oqr coordinate qr. Itmayhappenthatsome ofthecoordinates, sayqlyqz,...,qk,arenot explicitlycontained inL,althoughthecorrespondingvelocitiesql,q%,...,qk aresocontained. Coordinates ofthiskind aresaid tobeignorable orcyclic; itwillappearinthefollowing chaptersthat thepresenceofignorable coordinates isthemostfrequently-occurringreason forthesolubilityof particular problems byquadratures. TheLagrangian equationsofmotion whichcorrespondtothekignorable coordinates are 37,38] Principlesavailable fortheintegration 55 andonintegration,these canbewritten where/3j,/32,..., /3&areconstants ofintegration.These lastequationsare evidently kintegralsofthesystem. Weshallnowshewhowthese kintegralscanbeutilised toreduce the order ofthesetofLagrangiandifferentialequationsofmotion*. kdLLetRdenote thefunction LSqr~-r- .Bymeans ofthekequations / r=\ v$r f|-A,(r-1, 2,...,*),. wecanexpressthekquantities qly &lt;j2,...,qk,which arethevelocities cor respondingtotheignorable coordinates, interms of qk+i&gt; qk+2&gt;&gt;&lt;ln, &lt;ik+i, qk+2,,Qn,@i,$*, -&gt;ftk\ weshallsupposethat inthiswaythefunction Risexpressedinterms ofthe latter setofquantities. Now letSfdenote theincrementproducedinanyfunction /ofthequan titiesqk+l ,qk+2 ,...,qn,fr,q2,...,qn(orofthequantities qk+1)qk+2,...,qn, qk+1,...,qn,&, 2,...,$0byarbitraryinfinitesimalchanges Bqt+1 ,Sq k+2&gt; ..., Bqn,Bq1}...,Sqninitsarguments. Thenwehave ar v3L\=b(L2,qr), r=\oqr/ bythedefinition ofR.But and since Wehave therefore r=k+ivqrJ r=k+lOq rJr=\ andsince theinfinitesimalquantities occurringontheright-handsideofthis equationarearbitrary andindependent,theequationisequivalenttothe / *Thetransformation which follows isreally acase oftheHamiltonian transformation, which isdiscussed inChapter X;itwashowever firstseparately given byEouth in1876,andsomewhat laterbyHelmholtz. 56Principles available fortheintegration [CH.in systemofequations ;= , (r=k+l, k+2,...,n),dqrdqr dLdR Substitutingthese results intheLagrangian equationsofmotion, we have ^Lf^\_^-nft7mI &lt;"\ I"~~O ^J \*V l"J-JA/~| .... 7fc).ac\oqr/oqr NowEisafunctiononlyofthevariablesqk+l ,qk+2,...,q n,qk+l ,...,q n, andtheconstants &, y32, ,ftk-sothis isanewLagrangian system of equations, which wecanregardasdefininganewdynamical problem with only (nk)degreesoffreedom, thenewcoordinatesbeing qk+1 ,qk+2,...,qn, andthenew kineticpotential beingR.When thevariablesqk+l ,qk+2 ,...,qn havebeen obtained interms oftbysolvingthisnewdynamical problem, the remainder oftheoriginal coordinates, namely ql}q2,...,qk&gt;canbeobtained from theequations Hence adynamical problem withndegrees offreedom, which haskignoralle coordinates, canbereduced toadynamical problem which hasonly (nk) degrees offreedom. Thisprocessiscalled theignoration ofcoordinates. The essential basis oftheignoration ofcoordinates isinthetheorem thatwhen the kineticpotential doesnotcontain oneofthecoordinatesqrexplicitly, althoughitinvolves thecorresponding velocity q,.,anintegral ofthemotion canbeatonce written down, namely^=constant. This isaparticular case ofamuch more general theorem which willbegiven later, totheeffect thatwhen adynamical system admits aknown infinitesimal contact-transformation, anintegral ofthesystem can.beimmediately obtained. Iftheoriginal problem relates tothemotion ofaconservativedynamical systeminwhich theconstraints areindependentofthetime,wehave seen that itskineticpotential Lconsists ofapart (the kineticenergy) which is ahomogeneous quadratic function ofqltq., ...,qn,andwhich involves qk+i, qk+z,,qninanyway,together with apart (thepotential energywith sign reversed) which involvesqk+1,qk+2 ,...,qnonly. But inthenew dynamical system which isobtained after theignorationofcoordinates, the kineticpotential Rcannot bedivided intotwopartsinthisway:infact,R willingeneral contain terms linear inthevelocities. Andmoregenerallywhen(ashappens very frequentlyinthemore advancedpartsofDynamics) thesolution ofonesetofLagrangiandifferentialequationsismade todepend 38] Principlesavailable fortheintegration57 onthat ofanother setofLagrangiandifferentialequationswith asmaller number ofcoordinates, thekineticpotentialofthisnewsystemisnot necessarilydivisible intotwogroupsoftermscorrespondingtoakinetic and apotential energy. Weshallsometimes usetheword natural todenote those systemsofLagrangian equationsforwhich thekineticpotentialcontains onlyterms ofdegrees2and inthevelocities, andnon-natural todenote thosesystemsforwhich thiscondition isnot satisfied. Asanexampleoftheignorationofcoordinates, consider adynamical systemwithtwo degrees offreedom, forwhich thekinetic energyis andthepotential energyis wherea,6,c,daregiven constants. Itisevident thatq^isanignorable coordinate, since itdoesnotappear explicitlyinT V. ThekineticpotentialofthesystemisorV. andtheintegral correspondingtotheignorable coordinate is whereftisaconstant, whose value isdetermined bytheinitial circumstances ofthemotion. Thekineticpotentialofthenewdynamical system obtained byignoringthecoordinate andtheproblemisnowreduced tothesolution ofthesingle equation d(3R dt\dq or Asthis isalinear differential equation with constantcoefficients, itssolution canbe immediately written down :itis whereAand eareconstants ofintegration,tobedeterminedbytheinitial circumstances ofthemotion. Thisequation gives therequired expressionofthecoordinate &lt;?2mterms ofthetime :thevalue ofqiinterms oftcanthenbededuced from theequation whichgives 8in2 andsocompletes thesolution ofthesystem. 58Principles available fortheintegration [CH.in 39.Special cases ofignoration; integrals ofmomentum andangular momentum. We shallnow considerspeciallythetwocommonesttypesofignorable coordinates indynamical problems. (i)Systems possessing anintegral ofmomentum,. Letthecoordinates ofaconservative holonomicdynamical systemwith ndegreesoffreedom beql}q2&gt;...,qn;and letTbethekineticenergyofthe system, andVthepotential energy,sothat theequationsofmotion ofthe systemare d(dT\dT_dV~-~~ Supposethatoneofthecoordinates, sayql}isignorable, andmoreover is such thatanalteration ofthevalue ofq1byaquantity I,theremaining coordinatesqz,qs,...,qnbeing unaltered, correspondstoasimpletranslation ofthewholesystem throughadistance Iparalleltoacertain fixed direction inspace;weshall take thistobethedirection ofthe#-axis inasystemof fixedrectangularaxes ofcoordinates. Sinceq-iisanignorable coordinate, wehave theintegral O/TT -z-r=Constant, dql andweshallnow discuss thephysical meaningofthisequation. Wehave O/TT O ab-*Sb*miW+*f+if &gt; where thesummation isextended over alltheparticlesofthesystem, dT^f.dAi .difi .r$-M^T* &lt;+v-^+z-d^} bv826 as,y&lt; 3ft*^/y^ ^ ,i 9#o 9vi 9^j=Zmtiki, since inthiscase^-=1,^=0,^=0. dql oqi dq1 NowSmgfej represents (35)thecomponent paralleltothe -axis of themomentum ofthesystemofparticles m^,andconsequentlythis isthe rirr physical meaningofthequantity.inthepresentcase. 0Ji ~\m i The integral ^=Constant 9g, can therefore beinterpretedthus :When adynamical systemcan be translated asifrigidinagivendirection withoutviolatingtheconstraints, and thepotential energyistherebyunaltered(thewayinwhich thekinetic energy dependsonthevelocities isobviouslyunaltered bythistranslation, so 39] Principlesavailable fortheintegration59 thecorrespondingcoordinate isignorable),then thecomponent paralleltothis direction ofthemomentum ofthesystemisconstant, This result iscalled thelawofconservation ofmomentum*, andsystems towhich itappliesaresaid topossess anintegral ofmomentum. (ii)Systems possessing anintegral ofangular momentum. Again takingasystemwith coordinatesq1}q2,...,qnandkinetic and potential energies TandVrespectively,letusnowsupposethat the coordinateqlisignorable,andmoreover issuch thatanalteration ofqlby aquantity a,theother coordinates remaining unchanged, correspondsto asimplerotation ofthewholesystem throughananglearound agiven fixed line inspace:weshall take this line astheaxis ofzinasystem offixedrectangularaxes ofcoordinates. Sinceqlisanignorable coordinate, wehave theintegral ?irr ;r-r-=Constant ............................. (1), 9?i andwehave todetermine thephysical interpretationofthisequation. Wehave asbefore dT /^ .dyi,.dzA [j=-- r-yt^-+Zi= , VdqiL 9?i dqj where thesummation isextended over alltheparticlesofthesystem.But ifwewrite Xi^ncosfa, yi= rism&lt;f&gt;i, wehave dfa=dq l, d%i dxi.~= ~ riSm&gt;i=:~ i .. and therefore 3-r-=2mf(-dayt+yiXi)...........................(2). oql Now ifrdenote thedistance ofany particleofmassmfrom agiven straightline atany instant, and if &lt;adenote theangular velocityofthe particleabout theline,theproductmr*to iscalled theangular momentum oftheparticle about the line. Let beany point, and letP,Pbetwoconsecutivepositionsofthe moving particle,theinterval oftime between thembeingdt.Then the *Thishasbeen evolvedgradually from theobservation ofNewton, Principia, Book i.iutrod. toSect. XL,that ifanynumber ofbodies areacted ononlybytheirmutual attractions, their common centre ofgravity willeither beatrest, ormove uniformlyinastraight line. 60Principles available fortheintegration [CH.m angular momentum about anylineOKthroughisclearlythelimiting value oftheratio ??z -7-xTwice thearea oftheprojectionofthetriangle OPP on aplane perpendiculartoOK, soif(I,m,n)arethedirection-cosines ofOKand if(X,p,v)arethedirection- cosines ofthenormal tothetriangle OPP,weseethat theangular momentum aboutOK isequaltotheproductof(l\+mp+nv)intothe angular momentum about thenormal totheplaneOPP .Itisevident from thisthat iftheangular momenta ofaparticle about anythreerectangular axesOxyzatanytime areA,,A2,A3respectively,then theangular momentum aboutanylinethrough whose direction-cosines referred tothese axes are (I,m,n}isIh-i+mh2+nhs;wemay expressthisbysayingthatangular momenta about axesthrough apointarecompounded accordingtothevectorial law. Theangular momentum ofadynamical system about agivenaxis is defined tobethesum oftheangular momenta oftheseparate particlesof thesystem about thegivenaxis;inparticular,theangular momentum of asystemofparticles typified byaparticleofmass m,whose coordinates are (xi,yiyZi},about theaxis ofzisSm^r^,where andthesummation isextended over alltheparticlesofthesystem;this expressionfortheangular momentum ofasystemcanbewritten intheform 2m* (ijiXi-x^i), i andoncomparingthiswithequation (2)wehave theresult that theangular rlTmomentumofthesystem considered, about theaxisofz,is~-r d^i Theequation (1)impliestherefore that theangular momentum ofthe system about theaxisofzisconstant :andwehave thefollowingresult : When adynamical system canberotated asifrigid round agivenaxiswith outviolatingtheconstraints, and thepotential energyisthereby unaltered, the angular momentumofthesystemabout thisaxis isconstant. This result isknown asthetheorem ofconservationofangular momentum*. Example. Asystemofnfreeparticlesisinmotion urider theinfluence oftheir mutual forces ofattraction, these forces beingderived from akineticpotential V,which contains thecoordinates andcomponentsofvelocityoftheparticles,sothattheequations ofmotion oftheparticlesare .._ard/3F\ mrXr~ e *Keplerslaw,thattheradius from thesuntoaplanet sweeps outequal areas inequal times, wasextended byNewton toallcases ofmotion under acentral force :from thisthegeneral theorem ofconservation ofangular momentum hasgradually developed. 39,40] Principlesavailable fortheintegration61 shew thatthese equations possesstheintegrals / 9F\ 2(mrxr+^- J=Constant, / 9F\ 2(m.,yr+K-i Constant, r\ yJ 2(mrzr+^r- )=Constant, rV czr/ ( 3F 9F)2\mr(yrzr-zryj+yr- ?.--zr ^.-j=Constant, 9Fan 2\mr(zrxr-x,.zr)+zrJTT--xr3\=Constant, r (^oxrczr) ( 3F 9F! 2\mr(xryr-yrxr)+xrC~yrgj}-=Constant, which mayberegardedasgeneralisationsoftheintegralsofmomentum andangular momentum. (Levy.) 40.Thegeneraltheorem ofangularmomentum. Theintegralofangularmomentum isaspecialcase ofamoregeneral result, whichmaybeobtained inthefollowing way. Consider adynamical systemformed ofanynumber offreeorconnected andinteracting particles:iftheyaresubjectedtoanyconstraints other than themutual reactions oftheparticles,weshallsupposetheforces duetothese constraints tobecounted amongtheexternal forces. Takeanylinefixed inspace,andchoose oneofthecoordinates which specifytheconfigurationofthesystem (sayq^tobesuch thatachangein q1}unaccompanied byanychangeintheother coordinates, impliesasimple rotation ofthesystemasifrigidround thegiven line,throughanangle equal tothechangeinqlfWesupposetheconstraints tobesuch that this isa possible displacementofthesystem. TheLagrangian equationforthecoordinateqlis dtdq andthisreduces to since thevalue ofq:(asdistinguishedfromq^cannot haveanyeffect on a/77T^T thekineticenergy,andthereforejmust bezero.Now -^-ristheangular momentum ofthesystemabout thegiven line;andQiS^jisthework done onthesystem bytheexternal forces inasmalldisplacement 8q1}i.e.asmall rotation ofthesystemabout thegivenlinethroughanangle 8q1,fromwhich itiseasilyseen thatQtisthemoment oftheexternal forces about thegiven line.Wehave therefore theresult that therateofchange oftheangular 62 Principlesavailable fortheintegration [CH.in momentumofadynamical systemabout anyfixedline isequaltothemoment oftheexternalforcesabout this line. Thelawofconservation ofangular momentumobviouslyfollows from thiswhen themoment oftheexternal forces iszero. Similarlywecanshew that therateofchange ofthemomentum ofa dynamical system paralleltoanyfixeddirection isequaltothecomponent, paralleltothis line,ofthetotal externalforces actingonthesystem. Forimpulsivemotion itiseasytoestablish thefollowing analogous results : Theimpulsiveincrementofthecomponent ofmomentumofasysteminany fixeddirection isequaltothecomponentinthisdirectionofthetotal external impulses appliedtothesystem. Theimpulsiveincrementoftheangular momentumofasystemround any axis isequaltothemoment round thataxisoftheexternalimpulses appliedto thesystem. 41.TheEnergy equation. Weshallnowintroduce anintegralwhichplaysagreat partindynamical investigations,andindeed inallphysical questions. Inaconservative dynamical systemletqi}q2,...,qnbethecoordinates and letLbethekineticpotential:weshallsupposethat theconstraints areindependentofthetime, sothatLisagivenfunction ofthevariables qi,#2,&gt;qn,q\, q-2,&gt; &lt;jnonly,notinvolvingtexplicitly. Weshall not, at first, restrict Lbyanyfurther conditions, sothat thediscussion willapply tothenon-natural systemsobtained afterignorationofcoordinates, aswell as tonatural systems. Wehave dL ..9Z .dL ~T~=q?D^~"riOroat r=\uqrr=i dqr " ..dL .didL\ =2,qr;r-r-+2qr-r,Urr- ,bytheLagrangian equations r=ivqrr=i at\oqrJ d/.dL^ Integrating, wehave n *q, r=lv&lt;Jr where hisaconstant. Thisequationisanintegralofthesystem, and iscalled theintegral of energyorlawofconservation ofenergy*. *Galileo wasacquainted with thefactthat thevelocityofaparticle sliding down aninclined plane from restdepends onlyonthevertical height through which ithasdescended. From this elementary particular case theprinciple wasgradually evolved byHuygens, Newton, John andDaniel Bernoulli, andLagrange. 40,41] Principlesavailable fortheintegration63 Wehave seen that innaturalsystems,inwhich theconstraints donot involve thetime, thekineticpotential Lcanbewritten intheformTV, whereT(thekineticenergyofthesystem)ishomogeneousandofdegree2 inthevelocities, whileVisafunction ofthecoordinatesonly.Inthis case, therefore, theintegralofenergy becomes n 2r ,^.OljTh=Zq r^L r=i oqr =2TT+V, sinceTishomogeneousofdegree2inqlt Itfollows that inconservative naturalsystems,thesumofthekinetic and potential energiesisconstant. This constant value hiscalled thetotalenergy ofthesystem. This latter result can alsobeobtaineddirectlyfrom theelementary equationsofmotion. Forfrom theequationsofmotion ofasingle particle, namelymixi=Xi ,mtyi=Yt,m{^=Z{ , wehave Em* (xixi+yiiji+ZiZi)=2(XiXi+Yiiji+Z^t), where thesummation isextended over alltheparticlesofthesystem,or d .SImt(x?+y?+zf)=2(Xdx+Ydy+Zdz}, i i sothattheincrement ofthekineticenergyofthesystem,inanyinfinitesimal partofitspath,isequaltothework donebytheforcesactingonthesystem inthispartofthepath,andtherefore isequaltothedecrease inthepotential energyofthesystem. Thesumofthekinetic andpotential energiesofthe systemistherefore constant. Theequationofenergy d .$m(&+f+p)=Xdx+Ydy+Zdz (whereforsimplicity wesupposethesystemtoconsist ofasingle particle)istruenot onlywhen(#,y,z)denote coordinates referred toanyfixedaxes, butalsowhenthey denote coordinates referred toaxeswhich aremoving withanymotion oftranslation inafixed direction with constantvelocity. For let(, ;,f)denote thecoordinates oftheparticle referred toaxes fixed inspace andparalleltothemoving axesOxyz,sothat x=-at, y=T)-bt,z=-ct, wherea,b,caretheconstant componentsofvelocityoftheoriginofthemoving axes. Then theresultalready provedisthat d.4m(e+^+C2 )=Xd+Yd?,+ +(aX+bY+cZ)dt. 64 Principlesavailable fortheintegration [OH.m Nowwehave d.m( =m(at;+bij+c)dt =(aX+bY+cZ)dt, andtherefore d.im(x*+y2+zi }=Xdx+ Ydy+Zdz, which establishes thetheorem. Itmaybenoted thatfrom this result thethree equationsofmotion oftheparticle canbederived, bytaking x= atetc.,andsubtracting theequationofenergyinthe coordinates (#,y,2)from theequationofenergyinthecoordinates (, 77, ). 42.Reductionofadynamical problemtoaproblemwithfewer degrees of freedom, bymeans oftheenergy-equation. When aconservative dynamical systemhasonlyonedegreeoffreedom, theintegralofenergyisalone sufficient togivethesolutionbyquadratures. For ifqbethecoordinate, theintegralofenergy .dL q-TT-.L=h dq isarelation betweenqandq;.ifthereforeqbefoundexplicitlyinterms ofq from thisequation,sothat ittakes theform 9=/(?), wecanintegrate againandobtain theequation t=-.^r-r+constant. which constitutes thesolution oftheproblem. When thesystemhasmore than onedegreeoffreedom, theintegralof energyisnotinitself sufficient forthesolution;butweshallnowshew that itcanbeused forthesamepurposeastheintegrals correspondingtoignor- ablecoordinates were used,namelytoreduce thesystemtoanotherdynamical systemwithasmaller number ofdegreesoffreedom*. Inthefunction L,replacethequantities q.2,q3,...,qnbyftq 2,ftq^, ..., ftq^,respectively,whereqrdenotes :anddenote theresultingfunction by n(&lt;j,, &lt;//,qs,...,qn, &lt;?i, q-2,...,qn)-Thendifferentiatingtheequation wehaveq.2,...,qn,ft,q*,,qn)=&(ft, q2,q3 dLan raii (-1^,3,...,,).........(3). *Whittaker, Mess, ofMath. xxx.(1900). 41,42] Principlesavailable fortheintegration 65 Equations (1)and(2)give **.* 80_*A+Iir^ ~.o. I^-1~i~^TT~ ........................... \^) oqi0$ ,-=2 ^ioqr Now intheintegralofenergy ndL2qr~]r-L=h, r=l Off replace gvby^gvforallvalues ofrfrom 2towinclusive, andthenfrom this equation obtain^asafunction ofthequantities (q2,q3,...,q n &gt;qi, &lt;?2&gt;&gt;qn), andbyusingthisexpressionforql}expressthefunction |dLg, r=l9^r ?i interms of(^2,^3,...,qn,q1}q2,...,qn).Letthefunction thus obtained r)O bedenoted byL;then from(4)weseethatListhesame as^-r,but 5^i differently expressed. Differentiatingtheequationofenergy,whichby(4)maybewritten in theform .an , (ft-r-li=h, 1 fyi andregardingitasarelation whichimplicitly determines ^asafunction of thevariables (&lt;?/, q.,,...,qn,qltq2,...,qn),wehave Butdifferentiatingtheequation r,an= 8^ regardedasanidentityinthevariables(g/,q3,...,q n,q-i,qz, -,qn),wehave 9X_j^ a^na^ " Comparing equations (5)and(7),wehave az/ iao 8= ^8" andcomparing equations (6)and(8),wehave dL 130 Wr=^Wr W.D. 66 Principlesavailable fortheintegration [CH.in Combiningthese withequations (2)and(3),wehave dL^_dLdL_1dL *\ / -i &gt;anQ _. . dqrdqr dqr?idqr Substitutingfrom theseequationsintheLagrangian equationsofmotion, weobtain thesystem d(dL\.dL -T.(5,-ql~=0, (r=2,3,...,TO), dt\dq rj* dqr orfinally dL\dL (r=2,3,...,n). Now #zesemayberegarded astheequations ofmotionofanewdynamical systeminwhichListhekineticpotential, (qz,q3,...,qn)arethecoordinates, andqlplaysthepartofthetimeastheindependentvariable. Thenewsystem will, likethesystems obtained byignorationofcoordinates, beingeneral non-natural, i.e.Lwillnotconsistsolelyofterms ofdegrees2and inthe velocities(q2,q3f ,...,qn),butonaccount ofitspossessionoftheLagrangian form, most ofthetheoremsrelatingtodynamical systemswillbeapplicable toit.Theintegral ofenergythus enables ustoreduce agiven dynamical systemwithndegrees offreedomtoanotherdynamical systemwithonly(n 1) degrees offreedom. Thenewdynamical systemwillnotingeneral possess anintegralof energy,since theindependentvariableqloccursexplicitlyinthenewkinetic potential L .But ifqlisanignorablecoordinate intheoriginal system, thenqlwillnotoccurexplicitlyinanystageoftheaboveprocess, andthere fore willnotoccurexplicitlyinL .From this itfollows thatthenewsystem willalsopossess anintegralofenergy, namely 2qr3,L=constant, r=2 Oqr andthiscaninitsturnbeused toreduce further thenumber ofdegreesof freedom ofthesystem. Theprecedingtheorems shew thatanyconservativedynamical systemwith ndegreesoffreedom and(n 1)ignorablecoordinates canbecompletely integrated byquadratures ;wecanproceedeither(a)byfirstperformingthe processofignorationofthecoordinates, soarrivingatasystemwithonlyone degreeoffreedom, whichpossesses anintegralofenergyandcantherefore be solved inthemanner indicated atthebeginningofthepresentarticle;or (@)wecan firstusetheintegralofenergytolower thenumber ofdegreesof freedom byunity,then usetheintegralofenergyofthenewsystemtolower thenumber ofdegreesoffreedomagain byunity, andsoon,obtaining finally asystemwithonedegreeoffreedom whichagaincanbesolved inthemanner indicated. 42,43] Principles available fortheintegration 67 Example. Thekineticpotentialofadynamical systemis Shew that therelation between thevariablesqlandqzisgiven bythedifferential equation ____ 5ft\3$iY 9?2~ whereJj=^,andwhereZisdenned bytheequation Shew that thenon-naturaldynamical system represented bythe last differential equation possesses anintegralofenergy, andhence solve thesystem byquadratures. 43.Separation ofthevariables;dynamical systems ofLiouville stype.Aclass ofdynamical equations which areobviouslysolublebyquadratures fsconstituted bytheequationsofthosesystemsforwhich thekineticenergy isoftheform andthepotential energyisoftheform V= u&gt;i(?i)+W2(q2)+ ...+wn(qn), where vltv2,...,vn,wltw2,...,wnarearbitrary functions oftheir-respective arguments;sothatthekineticpotential breaksupintoasum ofparts,each ofwhich involvesonlyoneofthevariables. ForinthiscasetheLagrangian equationsofmotion are ^K(qr)qr]-\vr(qr)q,?=-iff[(qr), (r=1,2,...,n), or vr(qr)qr+%vr(qr)qr2=-w/(qr\ (r=1,2,...,n). Theseequations canbeimmediately integrated, andgive %vr(qr).qr2+wr(qr)=cr,^/(r=1,2,...,n), where d,c2,...,cnareconstants ofintegration; theseequations canbe furtherintegrated,since thevariablesqrand tareseparable, andwethus obtain ^, (r-!, 2,...,), where yltya,...,y narenewconstants ofintegration. These lastequations constitute thesolution oftheproblem. Animportant extension ofthis class ofdynamical systems wasmadeby Liouville*, whoshewed that alldynamical problemsforwhich thekinetic andpotential energies canrespectivelybeputintheforms T=K(?,)+Ui(q2)+...+* (qn)} \Vl(ft)^+v2(q2)qs*+...+Vn(qn)jn*}} V=Wl^+w*^+---+wn(qn) canbesolvedbyquadratures. *Journal deMath. xiv.(1849), p.257. 52 68 Principlesavailable fortheintegration [CH.in Forbytaking J&gt;Jvr(qr)dqr=qr, (r=1,2,... ,w), whereft,ft,...,qnarenew variables, wecanreplaceallthefunctions vi(ft)&gt;V2(ft),&gt;vn(qn)byunity;weshallsupposethis done, sothat the kinetic andpotential energiestake theform T=^(ft2+ft2 +...+ft&gt;2 ), V=-K(ft)+w*(ft)++wn(qn}}, where ustands fortheexpression i(ft)+u2(ft)+...+un(qn). TheLagrangian equationforthecoordinateq1is dfdr\_ar=_dv dt\9ft/3^ dql Multiplyingthisequation throughout by2uq ltwehave ~ (%&gt;-*gw+^+..:+.)=-2*g. Butfrom theintegralofenergyofthesystem, wehave %u(q12+q-&gt;+...+qn*)=h-V, where hisaconstant. Theequationforthecoordinateq^cantherefore bewritten intheform =2ft-{h^ (?j)-w,(q,)} Integrating,wehave ^u2q^=AM,(ft)-Wj(ft)+7j, where71isaconstant ofintegration. Weobtain similarequationsforeach ofthecoordinates(ft, ft,...,qn),thecorrespondingconstants(7!, j.2,...,yn) mustsatisfytherelation 7i+72++7=0, invirtue oftheintegralofenergyofthesystem. 43] Principlesavailable fortheintegration69 Theseequations give {hu-t(qj-Wt(&lt;?i)+71}*dq l=[hu 2(g-2)w2(q2)+ *)%}""*dq2=... ={hun(qn)-wn(qn)+yn]~%dqn, and this setofequations,which canbeimmediately integratedsince the variables areseparated,furnishes thesolution ofthesystem. Forfurther investigations onthissubjectcf.Hadamard, Bull, desSc.Math. xxxv.(1911), p.106,andBurgatti, Rom. Ace. L.Rend.(5)xx.(1911), p.108. MISCELLANEOUS EXAMPLES. 1.Ifthecomponents (X,Y}oftheforce acting onaparticle ofunitmass atthe point (#,y)inaplanedonotinvolve thetimet,shew thatbyelimination oftfrom the differential equationsthesolution oftheproblemismade todepend onthedifferential equationofthethird order dx&y-- Idx* ) 2.Asystemoffreeparticlesisinmotion, andtheirpotential energy, which depends onlyontheir coordinates,isunaltered when thesysteminanyconfigurationistranslated asifrigid through anydistance inanydirection. Whatintegralsofthemotion can atonce bewritten down ? 3.Inadynamical systemwithtwodegreesoffreedom thekineticenergyis andthepotential energyis wherea,b,c,dareconstants. Shew thatthevalue ofq2interms ofthetime isgiven by anequationoftheform whereh,k,and tareconstants. 4.Thekineticpotentialofadynamical systemis wherea,b,caregiven constants :shew thatq^isgiveninterms oftbytheequation where eisanarbitraryconstant and $&gt;denotes aWeierstrassianelliptic function. 5.Prove that inasystem withignorable coordinates thekineticenergyisthesum ofaquadraticfunction Tofthevelocities ofthenon-ignored coordinates andaquadratic functionKofthecyclic momenta. Inthecasewhere there arethree coordinatesx,y,&lt;andonecoordinate $isignored investigatetheequationsofmotion ofthetype 8/30\) oy\dxjl~ 70Principles available fortheintegration [CH.in where Fisthepotential energy,Jcisthecyclicmomentum, andthedifferential coefficients of &lt;/&gt;withrespecttox,and$arecalculated from theJinear equation bywhich kis expressedinterms ofx, i/,0. (Camb. Math.Tripos, 1904.) 6.Thekinetic potential ofadynamical system withtwodegreesoffreedom is Byusing theintegralofenergy, shew that thesolution depends onthesolution ofthe problemforwhich thekineticpotentialis andbyusingtheintegralofenergyofthis lattersystem, shew thattherelation between qiandq2isoftheform where cand eareconstants ofintegration, and|Pdenotes theWeierstrassianelliptic- function. 7.Thekineticenergyofadynamical systemis T= ri(qi2+ &lt;722 )(&lt;?12+?22 )&gt; andthepotential energyis F=-J Shew(byuseofLiouville stheorem, orotherwise) that the relation betweenql andq2is a2 qi2+b2q%2+2abq lq2cosy=sin2 y, wherea,b,yareconstants ofintegration. 8.Thekineticenergyofaparticle whoserectangular coordinates are(x,y)is^(x2+^2 ), and itspotential energyis where(A,A ,5,B,C~)areconstants andwhere(r, r")arethedistances oftheparticle from thepoints whose coordinates are(c,0)and(c, 0),where cisaconstant. Shew thatwhen thequantities J(V+/)and\(r-rf }aretaken asnewvariables, thesystemis ofLiouville stype, andhence obtain itssolution. 9.The observation that"acatalwaysfallsonitsfeet" suggested theproblem: Asystem, whose state atanyinstant iscompletely specified bytheposition andvelocity ofeach element,isinitially withoutvelocityinfreespace invacua. Can itatasubsequent instant resume itsinitialconfiguration butwithadifferentorientation inspace? Shew that ifthesystemisnotconservative, oriftheforces arederived from apotential which isnotone-valued, thereplyisaffirmative :but ifthesystemisconservative witha one-valued potential, thereplyisnegative. (Of.Painleve, Comptes Rendus, cxxxix.(1904), p.1170.) CHAPTER IV THESOLUBLE PROBLEMS OFPARTICLE DYNAMICS 44.Theparticlewith onedegree offreedom:thependulum. Asexamplesofthemethods described intheforegoing chapters, weshall nowdiscuss those cases ofthemotion ofasingle particlewhich canbesolved byquadratures. We shall consider firstthemotion ofaparticleofmass m,which isfree tomove intheinterior ofagivenfixed smooth tube ofsmall bore, under theaction offorces whichdepend onlyonthepositionoftheparticleinthe tube. Thetubecaninthemostgeneralcasebesupposedtohave theform ofatwisted curve inspace. Let sbethedistance oftheparticleattime tfromsome fixedpointof thetube,measuredalongthearcofthecurve formedbythetube: and let f(s)bethecomponentoftheexternal forcesactingontheparticle,inthe direction ofthetangenttothetube. Thekineticenergyoftheparticleis and itspotential energyisevidently -ff(s)d, where sisaconstant. Theequationofenergyistherefore rs |-ms2=f(s)ds+c, JSo where cisaconstant. Integratingthisequation, wehave where Iisanother constant ofintegration.This equation representsthe solution oftheproblem,since itisanintegralrelation between sandt, involvingtwoconstants ofintegration. 72 TheSoluble Problems ofParticle Dynamics [OH.iv Thetwoconstants cand Icanbephysically interpretedinterms ofthe initial circumstances oftheparticlesmotion;thus iftheparticlestarts at time t=tfrom thepoints=s,withvelocity u,then onsubstitutingthese values intheequationofenergy, wehave andonsubstitutingthesame values inthe finalequation connectingsandt, wehave lt^. Themost famousproblemofthistypeisthat ofthesimple pendulum ; inthiscase thetube issupposedtobeintheform ofacircle ofradius a whoseplaneisvertical, andtheonlyexternal forceactingontheparticleis gravity*. Using9todenote theanglemade with thedownward verticalby theradius vector from thecentre ofthecircle totheparticle, wehave s=adandf(s)=mgsin9; sotheequationofenergyis a$2=2$rcos9+constant =4#sin2\Q+constant. Supposethatwhen theparticleisatthelowestpointofthecircle, the a2 /92 quantity-=hasthevalue h.Then this lastequation canbewritten Takingsin^9=y,thisbecomes Now inthependulum-problemthere aretwo distincttypesofmotion, namelythe" oscillatory,"inwhich theparticle swingstoand froabout the lowestpointofthecircle, aridthe" circulating,"inwhich thevelocityofthe particleislarge enoughtocarryitover thehighest pointofthecircle, so that itmoves round andround the circle, alwaysinthesame sense.We shall consider these casesseparately., (i)Intheoscillatory typeofmotion, since theparticle comes torest beforeattainingthehighest pointofthecircle, ymust bezero forsome value ofylessthanunity, andtherefore h/2amust belessthanunity. Writing h=2ak2 , where kisanewpositiveconstant lessthanunity,theequation becomes *Inactual pendulums, thetube isreplaced byarigid barconnecting theparticle tothe centre ofthe circle, which serves thesame purpose ofconstraining theparticle todescribe the circle. Theisochronism ofsmall oscillations ofthependulum wasdiscovered byGalileo in1632, andtheformula fortheperiod wasgiven byHuygensin1673. Oscillations offinite amplitude were firststudied byEuler in1736. 44] TheSoluble Problems ofParticle Dynamics73 thesolution ofthis is* where tisanarbitraryconstant. Thisequation representsthesolution ofthependulum-probleminthe oscillatorycase :thetwoarbitraryconstants ofthesolution are tand k,and these must bedetermined from the initial conditions. From theknown propertiesoftheellipticfunction sn,weseethatthemotion isperiodic,its period (i.e.theinterval oftimebetween twoconsecutive occasions onwhich thependulumisinthesameconfigurationwith thesamevelocity) being .(O^jr,4- }K,where 9 K=\ (1-t2)~?(1-kHzy%dt. (ii)Next, supposethat themotion isofthecirculating type;inthis casehisgreaterthan 2a,soifwewrite 2a=A&2 ,thequantitykwillbeless thanunity. The differentialequation nowbecomes thesolution ofwhich is and inthis tandkarethetwoconstants which must bedetermined in accordance with the initial conditions. (iii) Lastly,lethbeequalto2a,sothat theparticle justreaches the vertex ofthe circle. Theequation nowbecomes thesolution ofwhich is Itwasremarked byAppellf thataninsightintothemeaningoftheimaginary period oftheelliptic functions which occur inthesolution ofthependulum-problemisafforded bythetheorem of 34.Forwehave seenthat iftheparticleissetfreewithnoinitial velocityatapoint ofthecircle which isataverticalheight habove thelowestpoint, themotion isgiven by where &2=: *Cf.Whittaker andWatson, ModernAnalysis,22-11. tComptes Rendiis, LXXXVII.(1878). 74 TheSoluble Problems ofParticle Dynamics [OH.iv andtherefore by 34, if,with thesame initial conditions, gravity were supposedtoact upwards, themotion would begiven by |(r-r,),A. Buttheperiod ofthismotion isthesame asifthe initial position were ataheight (2a-A),gravity acting downwards :andthesolution ofthis is y=Vsn {A/1(r-TO),V\ , where2=1-k2 . Thelatter motion hasarealperiod4(- jK;andtherefore thefunction \t7/ must have aperiod4^-j/r,sothefunctionsn(w, )must have aperiod4*/f. The \t// doubleperiodicityoftheelliptic function snisthus inferred fromdynamical considerations. Example. Aparticle ofunitmassmoves onanepicycloid,traced byapoint onthe circumference ofacircle ofradius bwhich rollsonafixed circle ofradius a.Theparticle isacted onbyarepulsive forceprdirected from thecentre ofthefixedcircle, where ris thedistance from thiscentre. Shew thatthemotion isperiodic,itsperiod being [This result ismosteasily obtained when theequation oftheepicycloidistaken in theform sbeing thearcmeasured from thevertex oftheepicycloid.] 45.Motion inamovingtube. Weshallnowdiscuss some cases ofthemotion ofaparticlewhich isfree tomove inagiven smooth tube,when thetube isitself constrained tomove inagiven manner. (i)Tuberotating uniformly. Supposefirstthatthetube isconstrained torotate withuniformvelocity atabout afixed axisinspace. Weshallsupposethattheparticleisofunit mass, asthisinvolves noreal lossofgenerality. Weshallmoreoversupposethat thefield ofexternal forceactingonthe particleisderivable from apotential- energyfunction which issymmetrical withrespecttothefixed axis,and socanbeexpressedinterms ofthe cylindrical coordinates zandr,where zismeasuredparalleltothefixed axisandristheperpendiculardistance from thefixed axis;foraparticle inthetube, thispotential energycantherefore beexpressedinterms of thearcs :weshall denote itbyF(s), andtheequationofthetube willbe written intheform r=g(s). 44,45]TheSoluble Problems ofParticle Dynamics 75 By 29,themotion oftheparticleisthesame asiftheprescribed angular velocityo&gt;were zero,andthepotential energywere tocontain anadditional term^r2 &&gt;2 .Hence wecanatonce writedown theequationofenergyin theform ^2 -ia&gt;2 {#(s)}2+F(&gt;)=c, where cisaconstant. Integrating again, wehave t=^[2c+ &&gt;2[g(s)}2-2V(s)Y^ds +constant, andthisrelation between tand srepresentsthesolution oftheproblem. Example1.Iftherotating tube isplane, andtheparticle candescribe itwith constantvelocity when thefixed axis isvertical and intheplaneofthetube,andthe field offorce isthatduetogravity, shew thatthetubemust beintheform ofaparabola with itsaxis vertical andvertex downwards. Example2.Aparticle moves under gravityinacircular tube ofradius awhich rotatesuniformly about afixed vertical axis inclined atanangle atoitsplane;if6be theangular distance oftheparticle from thelowestpointofthecircle, shew that facosV 2a where thefunction |jf&gt;isformed withtheroots aco2cosa aw2cosaa2cosa ty&gt; Off* ~3g~&gt; andt(]isaconstant. (ii)Tubemovingwith constant accelerationparalleltoafixeddirection. Consider nowthemotion ofaparticleinastraight tube, inclined atan angleatothehorizontal, which isconstrained tomove initsown vertical plane with constant horizontalacceleration/. Takingtheaxisofxhorizontal andthat ofyvertically upwards,with the originattheinitialpositionoftheparticle, wehave forthekineticenergy r-i(?+A where x=ycota+^ft2 , so T=i(ycota+/02+^2 =-ly2cosec2a+ycota.ft+%f*t2 , andthepotential energyis r=w. Theequationofmotion d_(dT_\=_dV_ dt(dy)~ dy 76 TheSoluble Problems ofParticle Dynamics [CH.iv givestherefore d,. "T.(ycosec-a+jtcota)=g, ory=(gfcota)sin2a. Integrating, wehave, supposingtheparticletobeinitiallyatrest, andtherefore x=\V (gcosa+/sin a)sina. Theseequations constitute thesolution oftheproblem:itwillbeobserved that inthissystemthekineticenergyinvolves thetimeexplicitly,sono integralofenergyexists. 46.Motionoftwointeracting free particles. Weshallnext consider themotion oftwoparticles,ofmasses m^and ra2 respectively,which arefreetomove inspaceunder theinfluence ofmutual forces ofattraction orrepulsion, actinginthelinejoiningtheparticlesand dependent ontheir distance fromeach other. Thesystemhassixdegreesoffreedom, since thethreerectangularcoordi nates ofeitherparticle canhaveanyvalues whatever. Weshall take, asthe sixcoordinatesdefiningthepositionofthesystem,thecoordinates(X,Y,Z} ofthecentre ofgravityoftheparticles,referred toanyfixed axes,andthe coordinates (x,y,z)oftheparticlem.2referred tomovingaxeswhoseoriginis attheparticlemlandwhich areparallel.tothefixed axes. Thecoordinates ofm1}referred tothefixed axes, are Z^ TWj+TO2 andthose ofw2,referred tothefixed axes, are Z+-2V(x+m1+m2 Thekineticenergyofthesystemistherefore /T7 1 /TT m,+m.2/ m,+m,J *\m1+m2 \ml+mj raj+mj2 \m^m2 or T=|(wj+w2)(X-+Y2+Z"-)+J^L771L^+^+^ Thepotential energyofthesystem depends onlyontheposition,ofthe particles relative toeach other, socanbeexpressedinterms of(x, //,z) ;let itbeV(x, y,z}. 45-47] TheSoluble Problems ofParticle Dynamics 77 TheLagrangian equationsofmotion ofthesystemare l=0, r=0, Z=Q, mlm.i ..__8F m^m^.._dV m-jn^ .._dV ml+mz dx m-L4-mzJ "byml+m2 dz The firstthree oftheseequations shew that thecentreofgravity moves in astraightlinewithuniform velocity, andtheother threeequations shew that themotionofmrelative tom-^isthesame asifmlluerefixed andm2were attracted tom^with theforce derived fromthepotential energy--2V*. lit/] Example.Iftwo freeparticles move inspace under anylawofmutualattraction, shew that thetangentstotheir paths meet anarbitraryfixed planeintwopoints, the linejoining whichpasses through afixedpoint. (Mehmke.) 47. Centralforcesingeneral:Hamilton stheorem. The lastarticle shews thattheproblemoftwointeractingfreeparticles isreducible totheproblemofthemotion ofasinglefreeparticle acted onby aforce directed towards orfromafixed centre. This isknown astheproblem ofcentralforces.There isclearlynoloss ofgeneralityifwesuppose the mass oftheparticletobeunity. Iftheparticlebeprojectedinanyway,itwillalways remain intheplane whichpasses throughthecentre offorceandthe initial direction ofprojec tion :foratnotime doesanyforce acttoremove itfrom thisplane.Wecan therefore define thepositionoftheparticle bypolar coordinates(r,6)inthis plane,thecentre offorcebeingtheorigin. LetPdenote theacceleration directed tothecentre offorce.Weshall notsupposeforthepresent thatP isnecessarilyafunction ofralone. Thekineticenergyoftheparticleis T=^(r2+r*62 ), andthework donebytheforce inanarbitraryinfinitesimaldisplacement (o&gt;,BO)is -PSr. TheLagrangian equationsofmotion oftheparticlearetherefore (r-rfc=-P, The latterequation gives onintegration r26=h, where hisaconstant; this istheintegral correspondingtotheignorable coordinate 6,andcanbe physically interpretedastheintegralofangular momentum oftheparticle about thecentre offorce. *Newton, Principia, Book i.Sect. 11. 78 TheSoluble Problems ofParticle Dynamics [OH.iv Tofindthedifferentialequationofthepathdescribed (whichisgenerally called theorbit ortrajectory}, weeliminate dtfrom the firstequation byusing therelation d_hd ~dt~~r2d0 wethusobtain theequation hdfhdr\ h2_ r2dd(r2d~6)~ r*~" or,writingufor1/r, dh* P h*u2 This isthedifferentialequationoftheorbit *,inpolarcoordinates;its integrationwillintroduce twonewarbitraryconstants inaddition tothe constant h,andafourtharbitraryconstant willoccur inthedetermination of tbytheequation t=TIr2d0+constant. fiJ The differentialequationoftheorbit in(r,p)coordinates, (where p denotes theperpendicular from thecentre offorce onthetangenttothe orbit),isoften ofuse :itmaybeobtaineddirectlyfrom Siacci stheorem (18),which(since hisnowconstant) givesatonce h?r PY Dft2dpor P=-.-f,p3dr which isthedifferentialequationoftheorbit. Since h=vp,where visthevelocityintheorbit,wehavefrom thisequation v*=Pp-P,r whichmaybewritten intheform "-iPfr whereqisthechord ofcurvature oftheorbit throughthecentre offorce. Wefrequently requiretoknow thelawofforcewhich must acttowards a given pointinorder thatagivencurvemaybedescribed;this isgivenatonce bytheequation - iftheequationofthecurve isgiveninpolarcoordinates;while iftheequa tion isgivenin(r,p)coordinates, theforce isgiven bytheequation ph2dp p3dr *This issubstantially giveninNewton sPrincipia, Book i. 2and3,and inClairaut s Theorie delaLune(1765);andintheabove form inWhewell sDynamics (1823). 47]TheSoluble Problems ofParticle Dynamics 79 Iftheequationofthecurve isgiveninrectangular coordinates, wepro ceed asfollows : Take thecentre offorce asorigin,and letf(x, y)=betheequationof thegivencurve. Theequationofangular momentum is xyyx=h. Differentiatingtheequationofthecurve, wehave fx+fyy=0, wherefxstands for~- . ooo From these twoequations weobtain _~ Differentiating again, wehave =j.&lt;M&gt; .dx=hfy d_(_hfy\ ___hfx_8f__hfy_ dxydyxfx+yfy"dx\xf x~+yfy) xfx+yfydy\xfx+yfy Performingthedifferentiations, thisgives _"?(fy2fxx+tfxfyfxy~fxfyy) ocButtherequired force isP,where x=P- :andtherefore wehaver _n?T{jiff xx^J^Jy thisequation givestherequiredcentral force. Themostimportantcase ofthis result isthat inwhich thecurve f(x, 2/)= isaconic, 2/O, y)=ax-+Wixy+by-+2gx+2fy+c=0. Inthiscasewefindatonce thattheexpression Jxxjy~tfxyjxjy +fyyjx has, forpointsontheconic, theconstant value -(abc+2fgh-of2-bg*-ch2 ), while thequantity _,9/x^-+yf-cxady hasthevalue -(gv+fy +c}, andsoisaconstantmultipleoftheperpendicular from thepoint (x,y}onthe polaroftheoriginwithrespecttotheconic.Wethus obtain, fortheforce under which agivenconic canbedescribed, anelegant expression due to Hamilton*, namelythat theforce actingontheparticleintheposition (x,y) *Proc. Roy.Irish Acad. 1846. 80 TheSoluble Problems ofParticle Dynamics [CH.iv variesdirectlyastheradius fromthecentreofforcetothepoint (x,y\and inverselyasthecubeoftheperpendicular from (a;,y}onthepolar ofthecentre offorce. Thetwofollowing theorems, theproof ofwhich islefttothestudent, maytogether be regarded astheconverse ofHamilton stheorem. (i)Ifaparticle moves under theaction ofaforce directed toafixedpoint, varying directlyasthedistance from thefixedpoint andinverselyasthecube ofthedistance from agiven straight line, theorbit isalways aconic. (ii)Ifaparticle moves under theaction ofaforce directed totheorigin, of magnitude where(x,y}arerectangular coordinates andp,a,/3,yareconstants, theorbits areconies which touch thelines ax2+Zftxy+yy~=0. Darboux(Comptes Rendus, LXXXIV.p.936)hasshewn that these twolaws offorce are theonlylaws forwhich theorbits arealways conies,iftheforcedepends onlyonthe positionoftheparticle. Suchar(Nouv. Ann.1vi.p.532)hasfound other laws offorce, which involve thecomponentsofvelocityoftheparticle. Example1.Ifaconic bedescribed under theforce^given byHamilton stheorem, shew that theperiodic time is-~pj?,wherepistheperpendicular from thecentre of &lt;V&gt; theconic onthepolar ofthecentre offorce.(Glaisher.) Example2.Shew that iftheforce be aparticlewilldescribe aconichavingitsasymptotes parallel tothelines Ax2+2Hxy+By*=0, ifproperly projected.(Glaisher.) 48.Theintegrable casesofcentralforces ;problems soluble intermsof circular andelliptic functions. Themostimportantcase ofmotion under central forces isthat inwhich themagnitudeoftheforcedepends onlyonthedistance r.Denotingthe forcebyf(r),thedifferentialequationoftheorbit is &lt;$&gt;.(,_f(r) d&+ ~hW Integrating, wehave du where cisaconstant :integratingthisequation again, wehave ~ JV~ h2]fy*r~r*\ r* 47,48]TheSoluble Problems ofParticle Dynamics81 andthis istheequationoftheorbit inpolarcoordinates. When rhasbeen found from thisequationinterms of8,thetime isgiven bytheintegral t=Tr2dO+constant. hj Theproblem ofmotion under centralforcesistherefore alwayssolubleby quadratures when theforceisafunction ofthedistanceonly. Example. Shew that thedifferential equationsofmotion ofapointParealways integrable byasimple quadrature when thecentral forceFisoftheform F= where isafunction of6only,while aandbarearbitraryconstants.(Armellini.) Weshallnowdiscuss thecases inwhich thequadraturecanbeeffected interms ofknown functions, thecentral forcebeing supposedtovaryassome positiveornegative integral power, saythenth, ofthedistance. Letusfirst findthoseproblemsforwhich theintegrationcanbeeffected interms ofcircular functions. Theaboveintegralforthedetermination of canbewritten intheform 8= I(a+bu*+0M-?-1 )"*d where a,b,careconstants;except when n=1,when alogarithm replaces theterm inu~n~1 .Iftheprofolemistobesoluble interms ofcircular func tions, thepolynomial under theradical intheintegrandmustbeatmost ofthe seconddegree;thisgives -w-l=0, I,or2, andconsequently n=-l,-2,or-3. Thecasen=1ishowever excluded bywhat hasalready been said,and thecasen=1istobeadded, since inthis case theirrationality becomes quadratic when w2istaken asanew variable. Next, letusfindthecases inwhich theintegrationcanbeeffectedbythe aidofellipticfunctions*. Forthis itisnecessarythattheirrationalitytobe integratedshould beofthethird orfourthdegree -f-inthevariable with respecttowhich theintegrationistaken. But thiscondition isfulfilled if n=0, 4,or5,when uistaken astheindependentvariable; n=3, 5,or7,when w2istaken astheindependentvariable. Itfollows that theproblem ofmotion under acentralforce which varies as thenthpower ofthedistance issolublebycircular orelliptic functionsinthe cases n=o,3,1,0,-2,-3,-4,-5,-7. *These cases were first investigated byLegendre, Theorie desFonctionsElliptiques (1825) andafterwards byJ.F.Stader, Crelle sJournal, XLVI.(1853), p.262. tWhittaker andWatson, ModernAnalysis, 22-7. W.D. 6 82 TheSoluble Problems ofParticle Dynamics [CH.iv Example. Shew that theproblemissoluble byelliptic functions when nhasthe followingfractional values :=_n _A_i_s_T2 2 3 S) Thegeneral case offractional values ofnisdiscussed byNobile, Giornale diMat. XLVI. (1908), p.313. The cases inwhich motion under acentral forcevaryingasapowerof thedistance issolublebymeans ofcircular functions areofspecialinterest. They correspond,asshewn above, tothevalues 1,-2,-3ofn ;thecase n=2willbeconsidered inthenext article :thecases n=1andn=3 canbetreated inthefollowing way. (i)tt-1. Inthis case theattractive force is sotheequationoftheorbit becomes f*Y M \~^ I (cv~-tf Jdv, where u?=v, J\ fl / so _c or 2(6-j)=arccos,where 7isaconstant ofintegration, /2 \2 = This istheequationofanellipse (when fi&gt;0)orhyperbola (when p&lt;0) referred toitscentre. The orbits arethereforeconies whose centre isat thecentreofforce*. (ii)n=-3. Inthis case theattractive force is sotheequationoftheorbit becomes *Newton found that ifabodymove inanellipse under theaction ofaforce directed tothe centre oftheellipse, theforce isdirectly proportional tothedistance :Principia, Book i. 2, Prop.x. 48] TheSoluble Problems ofParticle Dynamics 83 Integrating,wehave (u=Acos(kd+e),where k2=1-^,when//,&lt;h2 ,h |u=J.cosh(kd+e),where &2=^1,whenp&gt;h2 , \u=A6+e, when/A=h2 , where ineach caseAand eareconstants ofintegration. These curves aresometimes known asCotesspirals;the last isthe reciprocal spiral*. Inconnexion with forces varying astheinverse cube ofthedistance,itmaybeobserved that if beanorbit described under acentral forceP(r)totheorigin, thentheorbit where kisanyconstant, canbedescribed under acentral force P(r)+-=,where cis T3 aconstant :theintervals oftimebetweencorresponding points,i.e.points forwhich the radius vector hasthesamevalue,inthetwoorbitsbeing thesame. For,ifaccented letters refer tothesecondorbit,wehave Iftherefore wechoose thenewconstant ofmomentum hsothat K=hk thisequation implies thattheintervals oftimebetweencorresponding pointsinthetwo orbits arethesame, since itcanbewritten-=),wehaveriI*/ which establishes theresult. This issometimes known asNewton stheoremofrevolving orbits. kThetypesofcentral motioncorrespondingto n=5, 3,0,-4,-5,-7 lead, ashasbeen shewn, toelliptic integrals:oninvertingtheintegrals, we obtain thesolution interms ofellipticfunctions. Asanexample weshall take thecase ofn=5. *Newton, Principia, Book i.2,Prop.ix.;R.Cotes, Harmonia Mensurarum, pp.31,98. 62 84 TheSoluble Problems ofParticle Dynamics [OH.iv Letfj,u6betheforce towards thecentre ofattraction;weshallsuppose theparticle initially projectedwith avelocitylessthan thatwhich would be acquired byafallfrom restataninfinite distance tothepointofprojection, sothatthetotalenergy isnegative:callthisquantity ^7.Then theequationofenergy togetherwith theequation sdr\2 &lt;y , it Introducinginplaceofranewvariablepdefinedbytheequation v* 1 r=(^ thedifferentialequation becomes {^-^ The roots ofthequadratic arerealwhen 7ispositive;theirsum is,andthesmaller[ofthem isless than i.Hence ifthegreater and lessoftheroots bedenotedbyetand e3 respectively,and ife2denotes,wehave therelations &lt;?i+ez+es=0, 61&gt;62&gt;63, sop where eisaconstant ofintegration, andthefunctiongjisformed with the roots BI,ez,e$.Thuswehave Now risrealandpositive, and, asweseefrom theequation ofenergy,VAtcannot begreaterthan ./&lt;-. So&gt;(6 e)+ isrealandpositive and hasafinite lower limit;butwhen e1&gt;e2&gt;es,thefunction p(# e)isreal 48J TheSoluble Problems ofParticle Dynamics 85 andhasafinite lower limit forallrealvalues ofonlywhen eisreal; soeispurely real,andbymeasuringdfrom asuitable initial linewecan take etobezero.Wehave therefore and this istheequation oftheorbit inpolarcoordinates*. Thetime cannowbedetermined from theequation orh. u,fdd Performingtheintegration, wehave where(6)istheWeierstrassianzeta-functionf. Thisequationdetermines t. Example1.Shew thattheequation oftheorbit ofaparticle which moves under the influence ofacentral attractive force^./r5canbewritten intheform r=asn (K = orelse intheform a7 Ivl-=snfK =zr provided A4 &gt;4p,E&gt;0,where histheangular momentum round theorigin andEisthe excess ofthetotalenergyover thepotential energyatinfinity. (Cambridge Math.Tripos, PartI,1894.) Example2.Aparticleisattracted totheorigin with constant accelerationp. ;shew thattheradiusvector, vectorialangle, andtime, aregiveninterms ofarealauxiliary angle ubyequationsofthetype -^- _ (Schoute .) o)2-a)o- (&lt;D!+a&gt;2+a) Amongthepointsofspecialinterest onanorbit arethepointsatwhich theradius vector, afterhavingincreased forsome time, beginstodecrease : orafterhaving decreased forsome time, beginstoincrease. Apoint belongingtotheformer ofthese classes iscalled anapocentre,whilepoints ofthelatter class arecalledpericentres ;both classes areincluded under the *The orbits arediscussed and classified byW.D.MacMillan, Amer. Journ. Math. xxx. (1908), p.282. tCf.Whittaker andWatson, ModernAnalysis, 20-4. 86 TheSoluble Problems ofParticle Dynamics [CH.iv generaltermapse. Atanapse,iftheapseisnotasingularityoftheorbit (e.g.acusp), wehave whichimpliesthat thetangenttotheorbit isperpendiculartotheradius vector. Thewordsaphelion andperihelionaregenerallyused instead ofapocentre andpericentrewhen thecentre offorce issupposedtobetheSun. Example. Aparticle moves under anattraction r+* r2r3 toafixed centre;shew thattheangle subtended atthecentre offorcebytwoconsecutive apsesis where histheconstant ofangular momentum. 49.Motion under theNewtonian law*. Theremainingcase inwhich motion under acentral forcevaryingasan integral powerofthedistance canbesolved interms ofcircular functions is that inwhich theforce varies astheinversesquareofthedistance. This case isofgreat importanceinCelestial Mechanics, since themutual attractions oftheheavenlybodiesvaryastheinversesquaresoftheir distancesapart,in accordance with theNewtonian lawofuniversalgravitation. (i)The orbits. Consider then themotion ofaparticlewhich isacted onbyaforce directed toafixedpoint (which wecantake astheoriginofcoordinates), of magnitude pu?,where uisthereciprocalofthedistance from thefixedpoint. Lettheparticlebeprojectedfrom thepointwhosepolarcoordinates are (c,a)withvelocityvinadirection makinganangle 7with c;sothatthe angular momentum is h=CVQsin7. The differential equationoftheorbit is dhi P a this isalinear differential equationwith constant coefficients, and its integralis ,.v2c2sin27 *Newton, Principia, Book i.3,Props. XL,xn.,xm. 48,49]TheSoluble Problems ofParticle Dynamics 87 where eand CTareconstants ofintegration. This istheequation,inpolar coordinates, ofaconic whose focus isattheorigin, whoseeccentricityise, andwhose semi-latus rectum Iisgiven bytheequation ,_v2c2sin27I ; theconstant CTdetermines thepositionoftheapse-line, and iscalled the perihelion-constant. Thecircumstance thatthefocus oftheconic isatthecentre offorce isinaccord with Hamilton stheorem;for ifthecentre offorce isatthefocus oftheconic theperpen dicular onthepolar ofthecentre offorce istheperpendicular onthedirectrix, which is proportionaltor,asbyHamilton stheorem theforcemust beproportionalto1/r2 . Todetermine theconstants eand orinterms oftheinitial data c,a,7,v, weobserve thatinitially 1du 1#=a, w=- -775= cot7 ;cdO c substitutingthese values intheequationoftheorbit andtheequation obtainedbydifferentiatingitwithrespectto6,we.have v2csin27= /ti+/j.ecos(a or), \v2csin7cos7=pesin(a- or). SolvingtheseequationsforeandCT,weobtain 2_,v4c2sin272v2csin276 ~~u? cot(a CT)=-- -.\-tan7. I cvsin7cos7 Thesemi-major axis,when theconic isanellipse,isgenerallycalled the mean distance oftheparticle;denotingitbya,wehave 1-e2 andsubstitutingthevalues ofIand e2already found, wehave v.c a/ thisequation determines ainterms oftheinitial data. Thetimeoccupiedindescribingthewhole circumference oftheellipse, which isgenerallycalled theperiodic time, is Txareaofellipse, .3~88 TheSoluble Problems ofParticle Dynamics [CH.iv since hrepresentstwice therateatwhich thearea issweptoutbytheradius vector;theperiodictime istherefore7 ,where bisthesemi-minor axis. Butwehave h=vcsin7=Vpib\/ , V Cl IfiS1 .3 sotheperiodic time is2?r*/.Itisusual todenote thequantity n?a~2Vp byn;theperiodic timecanthenbewritten niscalled themean motion, beingthemean value of8foracomplete period. Ithasbeenshewn byBertrand andKoenigs that ofalllaws offorcewhich giveazero force ataninfinitedistance, theNewtonian law istheonlyoneforwhich alltheorbits are algebraic curves, andalsotheonlyoneforwhich alltheorbits areclosed curves. Example. Shew that ifacentre offorcerepels aparticle withaforce varyingasthe inversesquare ofthedistance, theorbit isabranch ofahyperbola, described about its outer focus. (ii)Thevelocity. Consider nowthecase inwhich theorbit isanellipse;theequation establishes aconnexion between themean distance aandthevelocityvand radius vector cattheinitialpointofthepath.Since anypointoftheorbit canbetaken asinitialpoint, wecanwrite thisequation 2r where visthevelocityoftheparticleatthepointwhose radius vector isr. Similarlyiftheorbit isahyperbola,whosesemi-majoraxis isa,wefind and iftheorbit isaparabola,therelation becomes Itisclear from thisthat theorbit isanellipse, parabola,orhyperbola, 2a accordingasv2= ,i.e.accordingastheinitialvelocity oftheparticleis less than, equal to,orgreater than, thevelocity which theparticle would acquireinfalling fromaposition ofrest ataninfinitedistance fromthe centreofforcetotheinitialposition. 49] TheSoluble Problems ofParticle Dynamics89 Itcanfurther beshewn that thevelocityatanypointcanberesolved into acomponent jperpendiculartotheradius vector andacomponent ^rperpen dicular totheaxisoftheconic;each ofthese components beingconstant. For ifSbethecentre offorce,Pthepositionofthemoving particle, Gtheintersection ofthenormal atPtotheconic with themajor axis,GL theperpendicularonSPfrom G,andSYtheperpendicularonthetangentat Pfrom S,itisknown that thesides ofthetriangle SPG arerespectively perpendiculartothevelocityand tothecomponentsofthevelocityinthe twospecified directions; andtherefore wehave v.SPh.SP h Component perpendiculartotheradius vector=p~=~yp~=^j h_fji= l= h SfitandComponent perpendiculartotheaxis=-~pxComponent perpendicular totheradius vector _Bfl=T which establishes theresult stated. Example1.Shew that inelliptic motion under Newton slaw,theprojections,onthe external bisector oftworadii, ofthevelocities correspondingtothese radii, areequal. Shew alsothatthesum oftheprojections ontheinner bisector isequaltotheprojection ofalineconstant inmagnitude and direction.(Cailler.) Example2.Shew that inelliptic motion under Newton slaw,thequantityITdt, whereTdenotes thekineticenergy, integrated overacomplete period, depends onlyon themean distance andnotontheeccentricity. (Grinwis.) Example3.Atacertainpointinanellipticorbit described under aforce/i/r2 ,the constant pissuddenly changed byasmall amount. Iftheeccentricities oftheformer and neworbits areequal, shew thatthepointisanextremityoftheminor axis. (iii) Theanomalies inellipticmotion. Ifaparticleisdescribing anellipse under acentre offorce inthefocus S, thevectorialangleASP ofthepointPatwhich theparticleissituated on theellipse, measured from theapseAwhich isnearer tothefocus, iscalled thetrueanomalyoftheparticle and willbedenotedbyQ\theeccentric angle correspondingtothepointPiscalled theeccentricanomalyofthe particle, and willbedenotedbyu :andthequantity nt,where nisthe mean motion and tisthetime ofdescribingthearcAP, iscalled themean anomalyoftheparticle. Weshallnow findtheconnexion between thethree anomalies. 90 TheSoluble Problems ofParticle Dynamics [CH.iv Therelation between anduisfound thus : Wehave -=1+ecos6,r and r=aex,where xistherectangular coordinate ofPreferred tothecentre oftheellipseasorigin, or r=a(l-ecosu). Hence(1-ecosu) (1+ecos6)=1-e\ anequation which canalsobewritten intheforms M/l-e\4 and sin-*. 1+ecos Therelation between uandntcanbeobtained inthefollowing way: Wehave ft aXrea^^ whereQisthePointontneauxiliary circlecorrespondingtothepointPontheellipse 2=-[AreaACQ-AreaSGQ}, where Cisthecentre ofthe I\J\A/ ellipse 2a2a2e . f-jrw x-smwV,wa2(22j 80 nt=uesinw. This isknown asKeplersequation. Aiiomogram forthesolution ofthisequationisdescribed byH.Chretien, Assoc.Franf. Congres, Keims(1907), p.83.Thesolution byanalytical expansion hasbeen discussed by many writers, animportant recent memoirbeing thatbyLevi-Civita, Atti delta R.Ace. delLincei, Rendiconti, (5)xin.(1904), p.260. Lastly,therelation between andntcanbefound asfollows : Wehave nt=u-e sinu. Replacing ubyitsvalue interms of0,thisbecomes . nt=arcsm(1-e^sin0}e(l-e^sin 1+ecosj1+ecos which istherequiredrelation;thisequation givesthetime interms ofthe vectorialangleofthemoving particle. Asolution oftheproblemofcalculating theTrueAnomaly from theMean Anomaly, based onageometrical deduction, wasfoundamong theunpublished papers ofNewton. 49] TheSoluble Problems ofParticle Dynamics91 Example1.Shew that 7"* r=lT where thesymbols Jdenote Besselcoefficients*. Forwehave 1du 1 ndt\ecosu &lt;d(nt} 1-ecosucosrntf27rcosrnt .d(nt),^., .,2-I--,byFourier stheorem t ,.=1TTjo1ecosu 1f2jr .j ,~cosrnt[2*. .Y, T= du+2- /cosr(w-esinu)}aw &lt;S7r_/o r=iTTjo =1+22Jr(re)cosr?z^J. r=l Integrating, wehave therequiredresult. Example2.Shew that ^e2sin2n+.... Example3.Inhyperbolic motion under theNewtonian law,shew that andinparabolic motion, shew that wherepisthedistance from thefocus tothevertex. Example4.Inelliptic motion under Newton slaw,shew thatthesum ofthefour times(counted fromperihelion)totheintersections ofacircle with theellipseisthesame forallconcentriccircles, andremains constant when thecentre ofthecirclemovesparallel tothemajoraxis.(Oekinghaus.) (iv) Lambert stheorem. Lambert in1731shewed that inellipticmotion under theNewtonian law,thetimeoccupiedindescribing anyarcdepends onlyonthemajor axis, thesum ofthedistances from thecentre offorce tothe initial and final points, andthelengthofthechordjoiningthesepoints:sothat ifthese three elements aregiven,thetime isdeterminate, whatever betheform oftheellipse. *Thename ofBessel iscommonly connected with thisexpansion:but itisreally dueto Lagrange, Oeuvres, in.p.130. tOf.Whittaker andWatson, ModernAnalysis, Chapterix. Ibid. Chapter xvn. Lambert soriginal demonstration wasgeometrical andsynthetic:thetheorem wasproved analytically andgeneralised byLagrangein1778 (OeuvresdeLagrange,iv.p.559). 92 TheSoluble Problems ofParticle Dynamics [CH.iv Letuand iifbetheeccentric anomalies ofthepoints ;thenwehave nxtherequired time=u esinu(u esinu) / , ,.uuu+u=(u-u) Zesm cos - .2"2 Now ifcbethelengthofthechord, andrandrbetheradii vectores, wehave r+r, u+uu-u =1ecosu+1ecosu=22ecos-cos, and c2=a2(cosucosu)2+b2(sin%-sinu}2 4a2sin2 ;=(1 e2cos2 2 so i=2sinU^ (l-e2cos2^-Y.a2V 2J Hence wehave r+r+c (uf-u _ f u+ r+r+c . . fw w, / --=22cos j^--Harccos (ecos i r+r-c fu-u f u+u\] and =22cos{-----harccos ecos-U-aI2 V 2yj andtherefore* r+r+c\u-u2arcsin5[ =1-arccos ecos2\ a J 2 V v-,^ o.\(r+rc\% uu( u+u\and 2arcsm51- I=^~+arccos[ecos-- .Z\ fl / 2 \ 2J Thus ifquantities aand/3aredennedbytheequations .a lfr+r+c\$. 1/r+r-sm= Of 0.+ap=uu,and cos-=ecosthelastequations give Oap Thusfinally wehave nxtherequired time=a-j3-2cos =--sin-~ 2 =(a-sina) (/3sin/3). This isLambert stheorem. Example1.Examine thelimiting casewhen theminor axisoftheellipse vanishes, so thattheorbit isrectilinear. *Itwillbenoticed thatowingtothepresence oftheradicals, Lambert stheorem isnotfree fromambiguityofsign. Thereader willbeable todetermine withoutdifficulty theinterpretation ofsigncorresponding toanygiven position oftheinitial and finalpoints. 49,50]TheSoluble Problems ofParticle Dynamics 93 Example2.Toobtain theform ofLambert stheoremapplicabletoparabolic motion. Ifwesuppose themean distance atobecomelarge, theangles aand/3becomevery small, soLambert stheorem canbewritten intheapproximate form a3-83 Required tirne=- 1 3 3=r{(?+?+c)*-(?+?-c)* }, 6M* andthis istherequired form*. Example3.Establish Lambert stheorem forparabolic motiondirectly from theformulae ofparabolic motion. 50.Themutualtransformation offields ofcentralforce andfields of parallel force. Ifinthegeneral problemofcentral forces wesupposethecentre offorce tobeatavery great distance from thepartofthe fieldconsidered, thelines ofaction oftheforce indifferentpositionsoftheparticlewillbealmost paralleltoeach other; andonpassingtothelimitingcase inwhich the centre offorce isregardedasbeingataninfinite distance, wearrive atthe problem ofthemotion ofaparticle under theinfluence ofaforcewhich is always paralleltoagivenfixed direction. Forthediscussion ofthisproblem, takerectangular axes Ox,Oyinthe planeofthemotion, Oxbeing paralleltothedirection oftheforce;and letX(x)bethemagnitudeoftheforce,which willbesupposedtobeindependent ofthecoordinatey.Theequationsofmotion are ,x=X(x\ y=0, andthemotion isthereforerepresented bytheequations x)dx+c}~4dx+I, where a,b,c,Iaretheconstants ofintegration; thevalues ofthese are determinedbythecircumstances ofprojection,i.e.bythe initial values of x,y,x,y. While theproblem ofmotion inaparallel field offorce isalimitingcase oftheproblem ofmotion under centralforces, itisnotdifficult toreduce the latter moregeneral problem totheformer morespecialone. For ifaparticleisinmotion under aforce ofmagnitude Pdirected to *This result wasgiven byEuler inhisDeterminate Orbitae Cometae Anni 1742 (1743), beforeLambert published thegeneral theorem. 94 TheSoluble Problems ofParticle Dynamics [CH.iv afixed centre (which wemaytake asoriginofcoordinates), theequations ofmotion are y- Theangular momentum oftheparticle round theorigin (whichiscon stant)isxyyx:letthisbedenotedbyA.Introduce newcoordinates X,Y, defined bythehomographictransformation X-?, F-i, .y yVand letTbeanewvariable defined bytheequation *-/? Thenwehave W/^A. W I**s \VUV IU/ U*JU\ _ 7-_ |1^/_7 I^.2f&gt; d2T dt\y) dT\yf)y dF_^/l\ dt^__y_2.^^**jxf,/T^i 7^y~^rfT tx tx C}/-\ _ || _J/-7/-^_/-"*feU 1/Tfc Vi 3/T^ VV -* Theseequationsshew thataparticlewhose coordinates are(X,Y)would, ifTwereinterpretedasthetime,move asifacted onbyaforceparallelto P* theaxis ofFandofmagnitude ^-.Asthesolution ofthistransformed problemwillyieldthesolution oftheoriginal problem,itfollows that the general problem ofmotion under centralforcesisreducible totheproblem of motion inaparallel fieldofforce. Example1.Shew thatthepathofafreeparticle moving under theinfluence ofgravity alone isaparabola with itsaxis vertical andvertex upwards. Example2.Shew thatthemagnitudeoftheforceparalleltotheaxisofxunder which thecurve/(a?, y)=canbedescribed isaconstant multipleof /a/\-3 r_ay/a/yj2/yv_ay/8/yi \dxj (dx*\dy)" 84%dxty dy*\dx} J Example3.Ifaparallelfield offorce issuchthatthepathdescribed byafreeparticle isaconic whatever betheinitial conditions, shew thattheforce varies astheinverse cube ofthedistance fromsome lineperpendiculartothedirection oftheforce. 51.Bonnet stheorem. Wenowproceedtodiscuss themotion ofaparticlewhich issimultaneously attracted bymore thanonecentre offorce.Anindefinite number ofparticular cases ofmotion ofthiskindcanbeobtained bymeans ofatheorem due to Bonnet*, whichmaybestated thus : *Liouville sJournal, ix.(1844), p.113andNote iv.oft.n.ofthelastedition ofLagranges Mec. Anal.(OeuvresdeLagrange, xn.p.353). 50-52]TheSoluble Problems ofParticle Dynamics 95 Ifagiven orbit canbedescribed ineachofngiven fields offorce,taken separately,thevelocities atanypointPoftheorbitbeingv1}v2,...,vn, respectively,then thesame orbit canbedescribed inthefield offorcewhich isobtained bysuperposingallthesefields,thevelocityatthepointPbeing Or+v22+...+n2 )4 Forsupposethat inthe field offorcewhich isobtainedbysuperposing theoriginal fields, anadditional normal forceRisrequiredinorder tomake theparticle move onthecurve inquestion;and let itbeprojectedfrom apointAsothatthesquareofitsvelocityatAisequaltothesum ofthe squaresofitsvelocities atAintheoriginalfields offorce. Then onadding theequationsofenergy correspondingtotheoriginal motions, andcomparing with theequationofenergyforthemotion inquestion, weseethat the kinetic energyofthemotion inquestionisthesum ofthekineticenergies oftheoriginal motions, i.e.thatthevelocityatanypointPis Hence, resolving alongthenormal totheorbit,wehave . . P wheremisthemass oftheparticle, ptheradius ofcurvature oftheorbit, andF1,F?, ...,Fnarethenormalcomponentsoftheoriginalfields offorce atP. P P P andtherefore Riszero;thegivenorbit istherefore afreepathinthefield offorcewhich isobtainedbysuperposingtheoriginalfields. Example. Shew thatanellipsecanbedescribedifforces respectivelyactinthedirectionsofthefoci. This result follows atoncefrom Bonnet stheorem when itisobserved thatthegiven forces areequivalenttoforces~and~actinginthedirections ofthefoci,together with aforce-^xdistanceactinginthedirection ofthecentre oftheellipse. 52.Determinationofthemostgeneral field offorce under which agiven curve orfamily ofcurves canbedescribed. Let (/&gt;(x,y)=c betheequationofacurve;onvaryingtheconstantc,thisequationwill representafamilyofcurves. We shallnow findanexpressionforthe mostgeneralfield offorce(the forcebeing supposedtodepend onlyonthe 96 TheSoluble Problems ofParticle Dynamics [CH.iv positionoftheparticleonwhich itacts)forwhich thisfamilyofcurves is afamilyoforbits ofaparticle. Letvdenote thevelocityoftheparticle,and(X,Y}thecomponentsof forceperunitmassparalleltothecoordinate axes. Thetangentialand 1dv^ v^ normal componentsofacceleration being ^-y-andrespectively,wehave CtS O Substitutingfor-itsvalue, namely wehave 4&gt;y*&lt;l&gt;xx~~ Y, .~ Writingv-=-u($x*+&lt;/), andreplacingj-by ((}&gt;X2+(/y)~^ (&lt;f&gt;x^-- &lt;j&gt;y^},thisequation becomes X=U (4&gt;X^&gt;yy~ &lt;j&gt;y &lt;f)Xy)+\$yjg(&lt;j&gt;X+ &lt;/&gt;/) Nowuisarbitrary,since itdependsonthevelocitywithwhich thegiven orbits aredescribed; andasXandFaretobefunctions oftheposition,of theparticle, wecantakeutobeanarbitraryfunction ofxandy ;wehave thereforeX=U(QxQw~ 4&gt;y$xy) + ^(f&gt;y (&lt;f&gt;xUy~^y^x), andsimilarlyY=U (4&gt;y&lt;j&gt;xx- &lt;t&gt;x&lt;l&gt;xy)+%$x (&lt;&/U&gt;x- &lt;t&gt;xUy), where uisanarbitraryfunction ofxandy.Theseexpressionsforthefield offorceunder which thecurves ofthegiven familyareorbits were firstgiven byDainelli*. Example1.Shew thataparticle candescribe agivencurve uy\der anyarbitrary forces PI,P2,...directed togiven fixed points, providedtheseforces satisfytherelations 97\ kPk2as\rk vohere rkistheradius andpktheperpendicular onthetangent, fromthekthofthegiven fixed points,andwherepistheradiusofcurvatureofthegivencurve. Forthetangential andnormal componentsofforceontheparticleare T=-?P k~fcandN=SPk&, kels krk *Giornale diMat.xvm. (1880), p.271. 52,53]TheSoluble Problems ofParticle Dynamics97 sofrom theequation wehave Example2.Aparticlecandescribe agiven curve under thesingleaction ofanyone oftheforces &lt;i, 2,...,actingingiven (variable)directions. Shew thatthecondition to besatisfied inorder thatthesame curvemaybedescribed under thejointaction offorces F\,F2,...,actinginthedirections of$1,$2,&gt;respectively,is ^]-0j,u &gt; where ckisthechord ofcurvature ofthecurve inthedirection of$t. (Curtis.) Example3.Apointmoves inafield offorce intwodimensions ofwhich thework function isV;shew thatanequipotential curve isapossible path, provided Vsatisfy the equation rwn 53.Theproblem oftwocentresofgravitation. Theequationsofmotion ofaparticle movinginaplaneunderarbitrary forces cannot beintegrated byquadraturesinthegeneralcase. Themost famous oftheknown solubleproblemsofthis class, other thanproblemsof central motion, istheproblem oftwocentres ofgravitation,i.e.theproblem ofdeterminingthemotion ofafreeparticleinaplane,attracted bytwofixed Newtonian centres offorce intheplane;itsintegrabilitywasdiscovered by Euler*. Let2cdenote thedistance between thetwocentres offorce;andtake thepointmidway between them asorigin,andthelinejoiningthem asaxis ofx,sothat their coordinates canbetaken tobe(c,0)and(c,0).The potential energyoftheparticle (whose mass istaken tobeunity)istherefore F--p(0-cY+2/2}~*- ft{(x+cr+f\~* whereyu,andp,areconstantsdependingonthestrengthofthecentres of attraction. Nowanyellipseorhyperbolawith thetwocentres offorce asfoci isa possibleorbitwhen either centre offorce actsalone, andtherefore byBonnet s theorem itisapossibleorbitwhen both centres offorce areacting.It istherefore natural, indefiningthepositionoftheparticle,toreplacethe rectangular coordinates(x,y)byellipticcoordinates (, ?;),defined bythe equations x=ccosh cos?;,y=csinh sin77. *Euler, Mem. deBerlin, 1760, p.228; Nov. Cornm. Pctrop.x.(1764), p.207; xi.(1765), p.152 :Lagrange, M4m. deTurin, iv.(1706-9), pp.118,215, orOeuvres, n.p.67. w.D. 7 98 TheSoluble Problems ofParticle Dynamics [CH.iv Theequations=Constant and77=Constant thenrepresent respectively ellipses andhyperbolas whose fociareatthecentres offorce; andthese are aparticular familyoforbits. Thepotential energy, whenexpressedinterms ofand77,becomes c(coshcos77) c(cosh f+cos77) andthekineticenergyTisgiven bytheequations ThisproblemisevidentlyofLiouville stype (43),andcantherefore beintegrated bythemethodapplicabletothis class ofquestions. The Lagrangian equationforthecoordinate is J^TT- c2 -j-{(cosh2f-cos2 77) }-c2cosh sinh(2+i}2 )=-~ , Cut (jc or c2 -j-{(cosh21-cos2 r;)22 }-2c2cosh sinh(cosh2f-cos2 77) (2+r)2 ) or,usingtheequationofenergy T+V=h, c2 ^-{(cosh2-cos2 77)22 } =-2(cosh2-cos2 77)-+2(A-F)^(cosh2-cos2 17) =2^ j(A-F)(cosh2-cos2 77)} Og&gt; = "2^jA(cosh21cos2 77)+-(cosh +00377)+(cosh cos77)[Og (_C C =2-j-(h cosh2+ cosh j. Integrating, wehave c2tt+u! (cosh2-cos2 77)2 12=Acosh2+"-coshf-7, where 7isaconstant ofintegration. Subtractingthisfrom theequationofenergy, which canbewritten |-(cosh2-cos2 77)2(I2+f) =h(cosh2fcos2 77)H(cosh +cos77)+(cosh cos77), 53,54]TheSoluble Problems ofParticle Dynamics 99 wehave p2 _ (cosh2cos2 77)2 ?7a=Acos2 T;---cos77+7.^ c Eliminatingcfabetween theseequations, wehave_ ^__ ,. it+/U/ ,,. , fJUU, Aicosh-ff-1- cosh 7 hcos2 77-- cos77+7c c Introducinganauxiliaryvariable u,wehave therefore u=I\hcosh2+---- cosh 7[c, JIc] f( /i~At )~^ 7w=N A,cos2 ?;-cos77+7&gt;dtj. ./(cj These areelliptic integrals,andwecanthereforeexpressand77aselliptic functions oftheparameter u,say =%(M), 17= &lt;t&gt;O)- Theseequations determine theorbit oftheparticle,theellipticcoordinates (&gt;7?)being expressedinterms oftheparameteru*. 54.Motion onasurface f. Weshallnextproceedtoconsider themotion ofaparticlewhich isfree tomove onasmooth surface, and isacted onbyanyforces. Let(X,Y,Z)bethecomponents, paralleltofixedrectangular axes, of theexternal forceontheparticle,notincludingthepressureofthesurface : let(x,y,z)bethecoordinates and vthevelocityoftheparticle,sthearcand ptheradius ofcurvature ofitspath,^theangle between theprincipal normal tothepathandthenormal tothesurface, and(X,p,v)thedirection- cosines ofthelinewhich liesinthetangent-planetothesurface and is perpendiculartothepathattime t ;themass oftheparticleistaken asunity. The acceleration oftheparticleconsists ofcomponents vdv/ds alongthe tangenttothepathandv2 /palongtheprincipalnormal;thelatter component canberesolved into(v2/p)sin^ alongthe linewhose direction-cosines are (X, p.,v)and(v2 /p)cos^alongthenormal tothesurface. Wehave therefore theequationsofmotion dv dx ,,dy dz-r=X-r+Yf-+z ..................(A),ds ds ds ds YfM+Zv...........................(B), *Some generalisations oftheproblem oftwo fixed centres will befound inapaper by Hiltebeitel, Amer. Journ. Math, xxxni.(1911), p.337. tThe earliest investigation ofmotion onasurface wasGalileo sstudyofthemotion ofa heavy particle onaninclined plane (Discourses, Third Dialogue, 1638). Themotion ofaheavy particle moving inahorizontal circle onasphere wasexamined byHuygens (Horolog. oscill., 1673). 72 100 TheSoluble Problems ofParticle Dynamics [CH.iv andthese, togetherwith theequationofthesurface, aresufficient todetermine themotion;fortheequationofthesurface mayberegardedasgivingzin terms ofxandy,andbyusingthisvalue forzwecanexpressallthe quantities occurringinequations (A)and(B)interms ofx,y,x,y,x,y: equations (A)and(B)thusbecome asystemofdifferentialequationsofthe fourth order forthedetermination ofxandyinterms of t. Iftheforces areconservative, theexpression -Xdx-Ydy-Zdz willbethedifferential ofapotential-energyfunction V(x, y,z);equation (A) cantherefore beintegrated, andgivesonintegrationtheequationofenergy $iP+V(ae ty,z}=c, where cisaconstant.Substitutingthevalue ofv2given bythisequation in(B),wehave This is(oneliminatingzbymeans oftheequationtothesurface) a differentialequationofthesecond order between xandy,and isinfactthe differentialequationoftheorbits onthesurface. The differentialequationsofmotion onasurface arenotintegrable by quadratures inthegeneralcase :there arehowever twocases inwhich the problemcanbeformulated insuch awayastoutilise results obtained in other connexions. (i)Motion under noforces. iWhen noexternal forces actontheparticle, equation (B)gives^=0,so theorbit isageodesic onthesurface* ;theintegralofenergy shews that this geodesicisdescribed with constantvelocity. Example. Aparticlemoves under noforcesonthefixed smooth ruledsurface whose line ofstriction istheaxisofz,thedirection-cosinesofthegeneratoratthepointzbeing . sinacos,sinasin,cosa,m m respectively.Todetermine themotion. Letvdenote thedistance ofthepoint onthesurface whose coordinates are(x,y,z) from thelineofstriction, measured along thegenerator,arid let(0,0,f)bethecoordinates ofthepointinwhich thisgenerator meets thelineofstriction. Thenwehave x=vsinacos,y=vsinasin,2=+vcos a. Thistheorem isduetoEuler, Mechanica(1736),n.cap.4. 54]TheSoluble Problems ofParticle Dynamics 101 Thekinetic energyoftheparticleis Wecantake vandfasthetwocoordinates which define thepositionoftheparticle ;itis evident thatthecoordinate isignorable, andthecorresponding integralis dT r=k, where kisaconstant, +vcosa=k. Theintegralofenergyis T=/i, where hisaconstant. Eliminating fbetween these twointegrals, wehave v2(v2+m2 )=2hv2+(2h-F)m2cosec2a. Ifvisinitially sufficiently largecompared withf,thequantity (2AF)ispositive ; shallsupposethistobethecase,andshall write (2h2 )m2cosec2a=2AX2 , where Xisanewconstant; theequation thusbecomes Theintegrationofthisequation canbeeffected byintroducing arealauxiliaryvariable M, defined bytheequation Writingv=\m.v~^, thisbecomes andthis isequivalenttotheequation *-f&gt;()-ij where theroots ex,32,e3ofthefunction|p(M)arerealandaredefined bytheequations !e2=X2 ,exes=m2 ,e1+e.,+e3=0. Theconnexion between thevariables vanduisthereforeexpressed bytheequation v=\m{^&gt;(u)-el}~^. Substitutingthisvalue ofvintheequation which connects vandt,wehave =e%u f(M+ !)+Constant. Cf.Whittaker andWatson, ModernAnalysis,20-33. 102 TheSoluble Problems ofParticle Dynamics [CH.iv This equation expresses thetime tinterms oftheauxiliary variableu,andthus in conjunction with theequation gives theconnexion between vand t. (ii)Motion onadevelopable surface. Ifthesurface onwhich theparticle moves isdevelopable, wecanutilise theknown theorems thatthearcsandthequantity-%areunalteredby developingthesurface onaplane:these results, appliedtotheequationsof motiongiven above, shew thatifinthemotionofaparticle onadevelopable surface under anyforcesthesurfaceisdevelopedonaplane,theparticlewill describe theplanecurve thus derived fromitsorbit with thesamevelocity as before, providedtheforce actingintheplane-motionisthesame inamount and direction relative tothecurve asthecomponent offorceinthetangent- planetothesurface inthesurface-motion. Example1.Asmoothparticleisprojected alongthesurface ofaright circularcone, whose axis isvertical and vertex upwards,icith thevelocitydue tothedepth below thevertex. Prove that thepathtraced outonthecone,whendevelopedintoaplane,icillbeoftheform r%sin|6=a*.(Coll. Exam.) Forondevelopingthecone, theproblem becomes thesame asthat ofmotion inaplane under aconstantrepulsiveforce from theorigin, andwith thevelocity compatible with rest attheorigin. Wetherefore have theintegrals r*+r22=Cr, whereCisaconstant, r2=h, where hisaconstant. These equations give 1dr\* Cr3 =-gsay, where aisanewconstant, soifu=-,wehave r du\21a3w3 andtherefore 0=1 (l*-(* J(l-a where v=u-a?, V2)2 =sin~1 v, which isequivalenttotheequation 3. 3 r*sin%6=a?. Example2.Ifinthemotion ofapointPonadevelopable surface thetangent IPto theedgeofregressiondescribes areasproportionaltothetimes, shew that thecomponent offorce perpendiculartoIPand inthetangent-planeisproportionalto^,wherepis theradius ofcurvature oftheedgeofregression. (Hazzidakis.) 54,55]TheSoluble Problems ofParticle Dynamics103 55.Motion onasurface ofrevolution ;cases soluble intermsofcircular andelliptic functions*. Themost importantcase ofsurface-motion which issoluble byquadra tures isthemotion ofaparticleonasmooth surface ofrevolution, under forces derivable from apotential-energyfunction which issymmetricalwith respecttotheaxisofrevolution ofthesurface. Letthepositionofapointinspacebedenned bycylindricalcoordinates (z,r, &lt;),where zisacoordinate measuredparalleltotheaxis ofthesurface, ristheperpendiculardistance ofthepointfrom this axis,and &lt;f&gt;isthe azimuthalanglemadebyrwith afixedplane throughthe axis. The equationofthesurface willbearelation between zandr,say r-/(*), andthepotential energywillbeafunction ofzandr(itcannot involve &lt;f&gt;, since itissymmetricalwithrespecttotheaxis), which forpointsonthe surface can,onreplacingrbyitsvaluef(z\beexpressedasafunction ofz only, sayV(z);themass oftheparticlecanbetaken asunity. Thekineticenergyis.by 18, Thecoordinate &lt;f&gt;isevidently ignorable;thecorresponding integralis =k. where kisaconstant, or thisequationcanbeinterpretedastheintegralofangular momentum about theaxisofthesurface. Theequationofenergyis T+V=h, where Aisaconstant, andsubstitutingfor (j&gt;inthisequationfrom thepreceding, wehave integratingthisequation, wehave i Constant. The relation between tandzisthusgiven byaquadrature ;thevalues ofrand &lt;/&gt;arethen obtained from theequationofthesurface andthe equation (/(*)}*-*, respectively. hThemotion ofaparticle onasurface ofrevolution wasinvestigated byNewton, Principia, Jook i.Section 10. 104 TheSoluble Problems ofParticle Dynamics [OH.iv Weshallnow discuss themotion onthose surfaces forwhich thisquad rature canbeeffectedbymeans ofknown functions, when theaxis ofthe surface isvertical (zbeing measuredpositively upwards) andgravityisthe onlyexternal force, sothat (i)Thecircularcylinder. When thesurface isthe circularcylinderr=a,theaboveintegral becomes t= and iftheoriginofcoordinates issochosen that2ha-=k2 ,wehave *== or z=^g(ttoy&gt;, where tisaconstant. Theequation thengives &lt;/&gt;- &lt;o= 2(tt), where &lt;isaconstant. (ii)Thesphere. The case inwhich thesurface isthesphere iscalled theproblem ofthespherical pendulum*, andcanberealisedby supposingaheavy particleattached toafixedpoint byalight rigidwire capableofmoving freely about thepoint. Inthis case thequadraturefor tbecomes ~2 or t=l Theintegralontheright-handside ofthisequationisanelliptic integral,which weshallnowreduce toWeierstrass canonical form. Denote by*i,#2.Z3theroots ofthecubic since theexpression 2(/i-gz) (I*-z*)-k* *Lagrange, Mecavique Analytiqve. Thecomplete solution interms of(Jacobian) elliptic functions wasobtained byA.Tissot, Liouville sJournal, (1)xvn.(1852), p.88 :Jacobi sown solution oftheproblemofarotating rigidbody interms ofelliptic functions hadbeen published previously, in1839. Theanalysis connected with thespherical pendulumisessentially that for Lame sequation oforder 2. 55] TheSoluble Problems ofParticle Dynamics 105 isnegativeforthevalues Iand Iofz,andpositiveforvery large positive values ofzandalso forthevalues ofzwhich occur intheproblemconsidered (which mustnecessarilyliebetween Iand+1,since theparticleisonthe sphere) weseethatoneoftheroots(sayz^isgreater than Iandtheother two(sayz2andz3,where z2&gt;zs)arebetween Iand I.Thevalues ofzin theactual motion will liebetween2andz3,since forthem thecubic must bepositive. Write =Hr where tisanew variable,6gg -&gt;d*-4+f"(-1.2.3) sothat e1}e2,esarenew constants, whichsatisfytherelation &lt; 6j4-e.&gt;+es=-- andalsosatisfytheinequalitiesel&gt;e?&gt;es. Therelation between tandznowbecomes or where eisaconstant ofintegration andthefunction g&gt;isformed with the rootse-i,e2,e3. Nowwhen elte2,e3arerealand indescendingorder ofmagnitude, $(u)and@(u)areboth realwhen uisreal, inwhich case$(u)isgreater than eltand alsowhen uisoftheform &&gt;..,+arealquantity, where &&gt;3is thehalf-period correspondingtotheroot es;inthis latter case,#&gt;(t)lies between e2and e3.Since intheactual motion zliesbetween2andz3,it follows that fliesbetween &lt;?2and e3,andtherefore theconstant emust con sistofanimaginary part&&gt;3andarealpartdependingontheinstant from which time ismeasured :byasuitable choice oftheoriginoftime,wecan take this realpartofetobezero,andwethenhave hW *=^+7?&lt; +&lt;** Thisequation givestheconnexion between zand t.Wehavenow to determine theazimuth &lt;j&gt;.Forthiswehave theequation -,."7, Icclt dt where &lt;/&gt;isaconstant ofintegration. 106 TheSoluble Problems ofParticle Dynamics [CH.iv Toeffect theintegration,wetakeXand/*tobethe(imaginary)values of t4- &&gt;3correspondingtothevalues Iand Iofzrespectively;sothatXand/* arenewconstants defined bytheequations hW,k 21*-l~- theseequations give 8&gt;(x)=f,W=. Theintegralnowbecomes . ._%2 [_dt___ 4J*J {$&gt;(t+w3)- &gt;(X)} {&gt;(t+0,3)-900} kg( dt dt~ a(j(X)dt %&gt;00dt $\t-\ But*wehave (X) andtherefore -t(\)\t a(t+o)3+/JL)a-(t+o&gt;3X) thisequation expressestheangle&lt;/&gt;asafunction oft,and socompletesthe solution oftheproblem. Weseethatwhen tincreases by2a&gt;1(faincreases by -2^(X-/*). Example. When thebobofthespherical pendulumisexecuting periodicoscillations between twoparallels onthesphere, shew thatoneofthepointsreached onthehigher parallel, andthepoint onthelowerparallelatwhich thebobarrives after ahalf-period, have adifference ofazimuth which alwaysliesbetween oneandtworight angles. (Puiseux andHalphen.) Theproblemofthespherical pendulumhasbeen discussed from thestandpointof periodic solutions byF.R.Moulton, Palermo Rend. xxxn. (1911), p..338. (iii) Theparaboloid. Consider next theproblemofmotion ontheparaboloid,whoseequationis r=2a?z%. Inthiscasethequadraturefor tbecomes r i/ A*2\~ - t=(a+z)*(2hz-2gz*-^-\dz. *Cf.Whittaker andWatson, Modern Analysis, %20-53, Ex. 2. 55] TheSoluble Problems ofParticle Dynamics 107 Toobtain thesolution oftheprobleminterms ofelliptic functions, we introduce anauxiliary quantity v,definedbytheequation v=(\a+z)~a(2hz-2gz*-^-}* dz. Ifaand$(wherea^ft)denote theroots ofthequadratic 2hz-2gz*-k~=0,4a wecanwrite thisintegralintheform v=(~f) /*{4(*+)(* Define anewvariablebytheequation and letelfe2,e3bethevalues ofcorrespondingtothevalues of-a, ft,a respectivelyofz ;then theintegrals become and itiseasily proved thatthequantitieselye2,e3satisfytherelations e1+e2+es=0, el&gt;e.2&gt;e3. Theauxiliary quantityvcannowbereplaced byanauxiliary quantity u, definedbytheequation j2)*V=4-- }u, {ff(&lt;*+*)) andthen theinversion oftheintegral gives =p(w+e), where eisaconstant ofintegration andthefunction g&gt;isformed with the roots e1}e2,e3,which aregiven bytheequations =2a+a+ft _-ajf_a -2ft _-a-2a +ft 3~(a+o) 3(a+a) 3(a+a) Asintheactual motion zevidentlyliesbetween a.andft,itfollows that p(u+e)liesbetween esand e2,andtherefore (aswewishutobereal) the imaginary partoftheconstant emust bethehalf-period&&gt;3;therealpartcan betaken tobezero, since itdepends merelyonthelower limit oftheintegral foru. Wehave therefore -~a -, since a+ft=- 69 9 108 TheSoluble Problems ofParticle Dynamics [CH.TV Theequationtodetermine tis t t= I(a+z)dv andthisequation givesinterms oftheauxiliaryvariable u. Lastly,\vehave todetermine theazimuth(j):forthiswehave ,kdt kdt k 2a fr(M+a,,)-e, du, / v4ala(a+o)J-a+a+ andtherefore 4a 3(a+a) k where &lt;/&gt;isaconstant ofintegration, and Iisanauxiliaryconstant denned by theequation j ft&gt;(I)=7-- ; ,so 3(+) Theequation cannowbewritten ku,,_" a a theintegralofwhich isfound (asintheproblemofthespherical pendulum) tobe i(0- )=e &lt;TM+0)3 thisequation expresses^&gt;interms oftheauxiliaryvariable u,andsocompletes thesolution. (iv) Thecone. Consider next thecone,whoseequationis rztan o, where aisthesemi-verticalangle. 55,56]TheSoluble Problems ofParticle Dynamics 109 Since this isadevelopable surface, wecanapplythetheorem of54,and weseethattheorbit ofaparticleontheconeundergravity becomes, when thecone isdevelopedonaplane,thesame astheorbit ofaparticleofunit mass intheplaneunder aforce ofconstantmagnitude gcosaacting towards afixed centre offorce(namelythepointontheplanewhichcorrespondsto thevertex ofthecone). This(48)isoneoftheknown cases inwhich the problemofcentral motion canbesolved interms ofelliptic functions, and thissolution furnishes atoncethesolution oftheproblemofmotion onthe cone. Example1.Shew thatthemotion ofaparticle undergravityonasurface ofrevolution whose axis isvertical canalsobesolved interms ofelliptic functions when thesurface is given byanyoneofthefollowing equations z(z 3a)2 , (r2az a2 )2=a3z. (Kobb andStackel.) Example2.Shew thatthesame problem canbesolved interms ofelliptic functions when thesurface is (a*+3/2 )3+2a6=8a%(x*-fy2 ). (Salkowski. ) Example3.Shew that ifanalgebraicsurface ofrevolution issuch thattheequations ofitsgeodesies canbeexpressedinterms ofellipticfunctions ofaparameter, thesurface must besuch that r2and zcanbeexpressedasrational functions ofaparameter,i.e.the equation ofthesurfaceregardedasanequation between r2and zistheequation of aunicursal curve;wherez,r, arethecylindrical coordinates ofapointoilthe surface.(Kobb.) Example4.Shew that inthefollowingcases ofthemotion ofaparticle onasurface ofrevolution, thetrajectoriesareallclosed curves : 1.When thesurface isasphere, andtheforce isdirectedalong thetangent tothe meridian andproportionaltocosec2 #,where 6istheangular distance from theparticle to thepole. (The trajectoriesareinthiscasesphero-conics having onefocus inthepole.) 2.When thesurface isasphere, andtheforce isdirectedalong thetangent tothe meridian andproportionaltotan6sec26.(The trajectoriesinthiscasearesphero-conics havingthepoleascentre*.) 56.Joukovskystheorem. Weshallnowshewhow todetermine thepotential-energyfunction under which agiven familyofcurves onasurface canbedescribed astheorbits of aparticleconstrained tomove onthesurface. Thethreerectangularcoordinates ofapointonthesurface canbeexpressed interms oftwoparameters, sayuandv,sothatanelement ofarcdsonthe surface isgiveninterms oftheincrements ofuandvtowhich itcorresponds byanequationoftheform ds2=Edu2+ZFdudv +Gdv*, where E,F,Gareknown functions ofuand v. *Darboux hasexamined thepossibilityofother cases, inBull, delaSoc.Math, deFrance, v. (1877). 110 TheSoluble Problems ofParticle Dynamics [OH.iv Letthefamilyofcurves which aretobetheorbits under therequired system offorces bedefined byanequation q(u,v)=Constant, and let p(u,v)=Constant denote thefamilyofcurves which isorthogonaltothese. Then instead ofuand vwecantakepandqasthetwoparameters which define thepositionofapointonthesurface;lettheline-element inthissystemofparametersbeexpressed bytheequation ds*=Edf+G dpz , theterm indqdp being absent, because thecurves p=Constant and q=Constant cutatright angles:EandGbeing known functions of pandq. The kineticenergyofaparticlewhich moves onthesurface is theLagrangian equationsofmotion aretherefore d, i-T-(Gp} [-5 &lt;f+^p2 )=^, ^cto V8p op / op whereVdenotes theunknownpotential-energy function, which itisrequired todetermine. Theseequationsaretobesatisfied bythevalueq= ;theythenbecome 2~dq~P= 85" 19//v^ Eliminating _p2 ,wehave dq Integratingthisequation, wehave ,,;+V=f(q\wherefisanarbitrary function, dq TheSoluble Problems ofParticle Dynamics 111 or^- andtherefore V9 G^G where gdenotes anarbitraryfunction. Now -~~ igAI(P\thedifferentialparameter*ofthe first order ofthe function p;andthuswehave atheorem enunciated byJoukovskyin1890, thatifq=Constant istheequation ofafamily ofcurves onasurface, and p=Constant denotes thefamily ofcurves orthogonaltothese, then thecurves q=Constant canbefreelydescribed byaparticleunder theinfluence offorces derived fromthepotential-energy function V=*l(p)g(p) +&l(p)ff(q)^\^--ldq, wherefandgarearbitrary functions,andA!denotes thefirst differential parameter. Theaboveequations give _*~ dqdq~ G G andhence theequation ofenergyinthemotion is MISCELLANEOUS EXAMPLES. / 1.Aparticle moves undergravityonthesmoothcycloid whoseequationis s=4asin$, where sdenotes thearcand &lt;f&gt;theanglemade bythetangenttothecurve with the horizontal :shew that themotion isperiodic,theperiod being 4v*/- . t/ 2.Aparticle moves inasmooth circular tubeunder theinfluence ofaforce directed toafixedpoint andproportionaltothedistance from thepoint. Shew thatthemotion is ofthesame character asinthependulum-problem. *Iftheline-element onasurface isgiven bytheequation d2=Edu2+2Fdu dv+Gdv*, the first differential parameter ofafunction &lt;f&gt;(u,v)isgiven bytheformula The differential parameterisadeformation-covariant ofthesurface,i.e.when achange of variables ismade from(u,v)to(u,v),thedifferential parameter transforms intotheexpression formed inthesamewaywith thenew variables(u ,v)andthecorresponding new coefficients ( ,F,G). 112 TheSoluble Problems ofParticle Dynamics [CH. 3.Aparticle moves inastraightlineunder theaction oftwocentres ofrepulsive force ofequal strength p.,eachvaryingastheinverse squareofthedistance. Shew that, ifthecentres offorce areatadistance 2capart andtheparticle starts from rest at adistance kc,where k&lt;1,from themiddlepointofthelinejoining them,itwillperform oscillations ofperiod IT [2 (1-Fsin26^dQ. Jo (Camb. Math.Tripos, PartI,1899.) 4.Aparticleunder theaction ofgravitytravels inasmooth curvedtube, starting from restatagiven point ofthetube. Iftheparticle describeseveryarcOPin thesame time thatwould betaken toslidedown thecorresponding chord OP,shew that thetubehastheform ofalemniscate. 5.Aparticleisprojected downwards along theconcave sideofthecurvey3+ax2= with avelocity f(2#)2 from theorigin, theaxis ofxbeing horizontal ;shew that the vertical componentofthevelocityisconstant.(Nicomedi.) 6.Aparticle moves inasmooth tube intheform ofthecurve r2=2a2cos2$,under theaction oftwoattractive forces, varying inverselyasthecube ofthedistance, towards thetwopoints ontheinitial linewhich areatadistance afrom thepole. Prove that if theabsolute force isp.,andthevelocityatthenode 2/^/a,thetime ofdescribing oneloop ofthecurveis 7ra2 /2/ii. (Camb. Math. Tripos, PartI,1898.) 7.Aparticle describes aspace-curve under theinfluence ofaforcewhose direction alwaysintersects agiven straightline. Shew that itsvelocityisinversely proportional tothedistance oftheparticle from thelineand tothecosine oftheangle which the plane throughtheparticle andthelinemakes with thenormalplanetotheorbit. (Dainelli.) 8.Aheavy particleisconstrained tomove onastraight line,which ismade to rotate with constant angular velocitycoround afixed vertical axisatgiven distance from it.Shew thatthemotion isgiven bytheequation where risthedistance oftheparticle from afixedpoint ontheline, aistheanglemade bythelinewiththehorizontal, andA,Bareconstants.(H.amEnde.) 9.Aheavy particleisconstrained tomove onastraight line,which ismade to rotate withgiven variable angular velocity round afixed horizontal axis. Shew thatthe equationofmotion is r=+gsinasinBr82sin2a+adsina, where aistheangle between thelineandtheaxis ofrotation, 6theangle made with thevertical bytheshortest distance abetween thelines, and rthedistance ofthe particle from theintersection ofthisshortest distance with themovingline. (Vollhering.) 10.Aparticleslides inasmoothstraight tubewhich ismade torotate withuniform angular velocityo&gt;about avertical axis :shewthat,iftheparticle starts from relative restfrom thepoint where theshortest distance between theaxisandthetubemeets the tube, thedistance through which theparticle movesalongthetube intime tis 2(7 -^cotacosec asinh2 (^u&gt;tsina), where aistheinclination ofthetube tothe vertical. (Camb. Math.Tripos, PartI,1899.) iv] TheSoluble Problems ofParticle Dynamics 113 11.Aparticleisconstrained tomove under noexternal forces inaplanecircular tube which isconstrained torotate uniformly about anypointinitsplane. Shew that the motion oftheparticleinthetube issimilar tothat inthependulum-problem. 12.Asmall bead isstrung upon asmooth circular wire ofradiusa,which iscon strained torotate withuniform angular velocityo&gt;about apoint onitself. Thebead is initiallyattheextremityofthediameter through thecentre ofrotation, and isprojected withvelocity 2& relative tothewire :shew that thepositionofthebead attime t isgiven bytheequation sin &lt;f)=snbu&gt;t\a (modulus a/b) or isin(f)=(bja)sncot, (modulus b/a) accordingasa&lt;or&gt;b,$beingtheangle which theradius vector tothebeadmakes with thediameter ofthecirclethrough thecentre ofrotation. (Camb. Math.Tripos, PartI,1900.) 13.Shew that theforceperpendicular totheasymptote under which thecurve .r3+^3=a3 canbedescribed isproportional to xy(x2+y2}~3 . 14.Aparticleisacted onbyaforcewhosecomponents (X,Y}parallel tofixed axes areconjugate functions ofthecoordinates(x,y).Shew thattheproblem ofitsmotion is always soluble byquadratures. 15. If((7)beaclosed orbit described byaparticle under theaction ofacentralforce,Sthecentre offorce, thecentre ofgravityofthecurve (},Gthecentre ofgravityof thecurve(C)onthesupposition that thedensityateachpoint variesinversely as thevelocity, shew that thepoints S,0,Garecollinear andthat 2&lt;7=3SO. (Laisant.) 16.Shew that themotion ofaparticle which isconstrained tomove inaplane,under aconstant force directed toapoint outoftheplane, canbeexpressed bymeans ofelliptic functions. 17.Shew that thecurves wherea,b,carearbitrary constants and/isagiven function, canbedescribed under the same lawofcentral force totheorigin. 18.Shew thatwhen acircle isdescribed under acentral attraction directed to apointinitscircumference, thelawofforce istheinverse fifthpowerofthedistance. 19.Aparticle describes thepedal ofacircle, taken withrespecttoanypoint in itsplane, under theinfluence ofacentre offorce atthispoint. Shew that thelaw offorce isoftheform whereAandBareconstants. Shew that thelawofforce isalso ofthisformwhen theinverse ofanellipse with respect toafocus isdescribed under acentre offorce inthefocus.(Curtis.) w.D.. 114 TheSoluble Problems ofParticle Dynamics [en. 20.Provethat,ifwhenprojected fromr=R, 6=with avelocity Vinadirection making anangleawith theradius vector thepathofaparticle be/(r, 0,R,F,sina)=0, thepath with thesame initial conditions butunder theaction ofanadditional central force ^-.is f(r,nd,R, where (Coll- Exam.) 21.Aparticleofunitmass describes anorbit under anattractive forcePtothe origin andatransverse forceTperpendiculartotheradius vector. Prove that, the differential equationoftheorbit isgiven by &lt;Pu PTdu &lt;M?_ _3+~~ dd&gt;d6~ Iftheattractive force isalways zero,andtheparticle moves inanequiangular spiral ofanglea,provethat ^,.2860*0-3 andA=(^sincosa)^rsec2a . (Camb. Math. Tripos, Part I,1901.) 22.Aparticle,acted onbyacentral force towards apoint varyingasthedistance, isprojectedfrom apointPsoastopassthrough apointQsuch thatOP isequaltoOQ ; shew that theleast possible velocityofprojectionisOP(^s,mPOQ}^ ,wherep.OP isthe force perunitmass atP. (Camb. Math. Tripos,PartI,1901.) 23.Find aplane curve such that thecurve and itspedal,withregardtosomepoint intheplane, canbesimultaneouslydescribed byparticles under central forces tothat point,insuch amanner that themoving particles arealwaysatcorresponding points ofthecurve andthepedal:and findthelawofforce forthepedalcurve. (Camb. Math.Tripos, PartI,1897.) 24. If/(#, y)beahomogeneousfunction ofonedimension, then thenecessary and sufficient condition thatthecurvef(x,y)=lbecapableofdescriptionunder accele ration tendingtotheorigin andvarying with thedistance alone,isthatfbesubject toacondition oftheform Hence shew thattheonlycurves ofthis class arenecessarilyincluded intheequation r(A+Bsin6+Ccosff)=1. Proceed tothediscussion ofthecase wherein f(x, y)ishomogeneous and ofn dimensions. (Coll. Exam.) 25.Anellipseofcentre Cisdescribed under theinfluence ofacentre offorce atapointonthemajoraxis oftheellipse ;shew that ntuesinu, where %Tr/nistheperiodic time,eistheratio ofCOtothesemi-major axis,anduisthe eccentric angleofthepointreached bytheparticleintime tfrom thevertex. 26.Two freeparticles p.andMmove inaplane under theinfluence ofacentral force toafixed point0.Shew that theratio ofthevelocityoftheparticle patanarbitrary pointTOofitspath,tothevelocitywhich ispossessedatTObythecentral projectionofM ontheorbit ofp.,isequaltotheconstant ratio oftheareas described inunittimebythe radii0/x,OM,multiplied bythesquareofacertain function /ofthecoordinates ofTO, which expressestheratio ofOM,Om. (Dainelli.) iv]TheSoluble Problems ofParticle Dynamics115 27.Aparticleismoving freelyinaparabolaunder anattraction tothefocus. Shew that,ifateveryinstant apointbetaken onthetangent throughtheparticle,atdistance 4acos0/(&lt;9 +sin&lt;9) from theparticle,this pointwill describe acentral orbit about thefocus, andtherateofdescriptionofareas willbethesame asintheparabola;where 4aisthelatus rectum, and6thevertical angleoftheparticle measured from theapse line. (Camb.Math. Tripos,PartI,1896.) 28.When aperiodiccomet isatitsgreatestdistance from thesun,itsvelocity receives asmall increment 8v.Shew that thecomet sleast distance from thesun willbeincreased bythequantity 48v.{3(l-e)//x(l +&lt;?)}-. (Coll. Exam.) 29. IfPOP isafocal chord ofanelliptic pathdescribed round thesun,shew that thetime fromPtoPthrough perihelionisequaltothetime offalling towards the sunfrom adistance 2atoadistance a(1+cosa),where a=Zir-(uf-u\andu-uis the difference oftheeccentric anomalies ofthepoints P,P . (Cayley.) 30.Aparticle moves inaplaneunder attractive forces ^/rV2 ,/z/rV3along the radiir,rdrawn totwofixedpointsatdistance 2dapart. Shew that,ifitisprojected with thevelocity duetoafallfrom restatinfinity,apossible pathisacircle withregard towhich thetwofixed centres areinversepoints,andthat,iftheradius ofthis circle isa, theperiodic time is - (Coll. Exam.) 31.Aheavy particleisprojected horizontally with avelocity vinside asmooth sphereatanangular distance afrom thevertical diameter drawn downwards :shew that itwillnever fallbelow ornever riseabove itsinitial level according as v2 &gt;or&lt;agsinatan a. (Coll. Exam.) 32.Aparticleisprojected horizontallywithvelocity Valongtheinterior ofasmooth sphereofradius afrom apoint whose angulardistance from thelowestpointisa.Shew that thehighest pointofthesphericalsurface attained isatanangular distance/3from thelowestpoint, where/3isthesmaller ofthevalues of^,xgiven respectively by theequations (3cos-^-2cosa)#+F2=0j ,,9nfivXMi. Hixam.j (cosx+cosa)V*2agsin*1^=OJ 33. Ifthemotion ofaspherical pendulumoflength abewholly between thelevels fa,iabelow thepointofsupport, shew thatatatime tafterpassing apointofgreatest depth, thedepthofthebob is fca{4-sn2V(130/14a)} (mod. V(?/65)) andthatahorizontal coordinate referred tothepoint ofsupportasoriginisdetermined bytheequation which isacase ofLame sequation. (Coll. Exam.) 34.Shew that ifaconical pendulumisexecutingsmalloscillations, thehorizontal projectionofthebobdescribes anellipse whose axes turn inthesense ofthemotion with theangular velocityH where $ istheangleofgreatest deviation from thevertical, lthat ofleastdeviation, Ithelengthofthependulum, andggravity. (Resal.) 32 116 TheSoluble Problems ofParticle Dynamics [CH.iv 35.Aparticleisconstrained tomove onthesurface ofasphere, and isattracted toa fixedpointMonthesurface ofthespherewithaforce that varies asr~2 (o?2r2 )%where disthediameter ofthesphere and ristherectilinear distance from theparticletoM. Ifthepositionoftheparticle onthespherebedefined byitscolatitude 6andlongitude $, withMaspole,shew thattheequationsofmotion furnish thedifferential equation where aand bareconstants;andintegratethisequation, shewing that theorbit is asphero-conic. 36.Aparticleofmassmmoves ontheinner surface ofacone ofrevolution whose semi-vertical angleisa,under theaction ofarepulsiveforce mp/r3from theaxis;the angular momentum oftheparticleabout theaxisbeingm-Jptana,shew thatthepath isanarcofahyperbola whoseeccentricityisseca. (Camb. Math.Tripos, PartI,1897.) 37.Shew thatthenecessarycentral force tothevertex ofacircular cone inorder thatthepathontheconemaybeaplane section is 4-^. (Coll. Exam.) ..1ifO* 38.Aparticleofunitmassmoves ontheinner surface ofaparaboloidofrevolution, latus rectum 4a,under theaction ofarepulsiveforceprfrom theaxis, where risthe distance from theaxis;shewthat,iftheparticleisprojected along thesurface ina directionperpendiculartotheaxiswithvelocity 2a/*2 ?itwilldescribe aparabola. (Coll. Exam.) 39.Asmooth surface ofrevolution isformed byrotating thecatenarys= cta,n&lt;p about itsaxis ofsymmetry, andaparticleisprojected alongitssurface from apoint distant bfrom that axiswithvelocityh(a2+62 )^/fe2 .Thedirection ofprojectionissuch that thecomponent velocity perpendiculartotheaxis ish/bandtheparticle moves in contact with thesurface, under theinfluence ofaforce ofattraction A2 (r2+2a2 )/r5inthe direction oftheperpendicularrtothe axis. Shewthat,ifgravity beneglected, the projectionofthepathonaplaneatright anglestotheaxis willhave apolar equation rcsmh- =a#.(Coll. Exam.) 40.Aparticle moves onasmoothhelicoid, z=a(f&gt;,under theaction ofaforcepr perunitmass directed ateachpoint along thegenerator inwards,rbeing thedistance from theaxis ofz.Theparticleisprojected alongthesurfaceperpendicularlytothe generatoratapoint where thetangent plane makes anangle awith theplane ofxy,its velocityofprojection being p*a. Shew thattheequationoftheprojection ofitspathon theplaneofxyis 2 /r2=sec2acosh2 ((f&gt;cosa)1. (Camb. Math.Tripos, PartI,1896.) 41.Shew that theproblemofthemotion ofaparticle under noforces onaruled surface, whose generatorscuttheline ofstriction ataconstantangle, andforwhich the ratio ofthelengthofthecommonperpendiculartotwoadjacent generators totheangle between these generatorsisconstant, canbesolved byquadratures. (Astor.) 42.Aparticle (x,y,z}whose potential energyis(ax2+by2+cz2 )isconstrained to move onthesphere x2+y2+z2l.Determine themotion. (C.Neumann, Journal furMath. LVI.(1859), p.46.) CHAPTER V THEDYNAMICAL SPECIFICATION OFBODIES 57.Definitions. Beforeproceedingtodiscuss thoseproblemsinthedynamicsofrigid bodies which canbesolved byquadratures,itisconvenient tointroduce and calculate anumber ofconstants which canbeassignedtoarigid body,and which dependonitsconstitution :itwillbefound that these constants determine thedynamicalbehaviour ofthebody. Letanyrigid bodybeconsidered;and lettheparticlesofwhich (from thedynamical pointofview)itisconstituted betypified byaparticleof massmsituated atapointwhose coordinates referred tofixedrectangular axes are(x,y,z). Thequantity 2m(y2+z2 ), where thesymbol 2denotes asummation extended over alltheparticlesof thesystem,iscalled themomentofinertia ofthebodyabout theaxisOx*. Similarlythemoment ofinertia aboutanyother line isdefined tobethesum ofthemasses oftheparticlesofthebody,eachmultiplied bythesquareof itsperpendiculardistance from theline. These summations areevidentlyin thecaseofordinary rigidbodiesequivalenttointegrations ;thus2m(y2+z2 ) fff isequivalentto I I(y2+z2 )pdxdydz, wherepisthedensity,ormassper unitvolume, ofthebodyatthepoint (x,y,z). Thequantity ^mxy iscalled theproduct ofinertia ofthebodyabout theaxes Ox,Oy;and similarlythequantities myand^nizx aretheproductsofinertia about theotherpairsofaxes. Forthemoments andproductsofinertia with reference tothecoordinate axes, thenotation A=2m(y2+z2 ),B=Zm(z* +x2 ),G=2m(x2+y2 ), F=Zmyz, G=ILmzx, H=ILmxy willbegenerallyused. Twobodies whose moments ofinertia abouteverylineinspaceareequal toeach other aresaid tobeequimomental.Itwillbeseen later that this involves alsotheequalityoftheproductsofinertia ofthebodies withrespect toanypairoforthogonallines. *Moments ofinertia were firstintroduced byHuygensinhisresearches onthependulum (Horolog. oscill., 1673). Thename isduetoEuler. 118 Thedynamical specification ofbodies[CH.v IfMdenotes themass ofabodyand ifkisaquantitysuch thatMk2is equaltothemoment ofinertia ofthebodyabout agiven line,thequantity kiscalled theradius ofgyrationofthebody about the line. Inthecase ofaplane body,themoment ofinertia about alineperpen dicular toitsplaneisoftenspokenofasthemoment ofinertia about the pointinwhich this linemeets theplane. 58.Themomentsofinertiaofsomesimplebodies*. (i)Therectangle. Let itberequiredtofindthemoment ofinertia ofauniformrectangular plate, whose sides areoflengths2aand26respectively, about alinethrough itscentreparalleltothesides oflength2a.Takingthis line asaxisOx, andalinethrough paralleltotheother sides asaxisOy,therequired moment ofinertia is or where cristhemassperunitarea oftheplate,orthe"surface-densityasitis frequentlycalled;evaluatingtheintegral, wehave fortherequired moment ofinertia |&lt;ra63 ,orMass ofrectanglex^62 . Themoment ofinertia ofauniform rod,about alinethroughitsmiddle point perpendiculartotherod,canbededuced from this resultbyregarding therodasthelimitingform ofarectangleinwhich thelengthofonepair ofsides isindefinitelysmall. Itfollows that themoment ofinertia in questionis Mass ofrodx|62 , where 26isthelengthoftherod. (ii) Therectangularblock. Consider next auniformrectangularblock whoseedgesareoflengths 2a, 26,2c;letitberequiredtofindthemoment ofinertia about anaxisOx passing throughthecentre andparalleltotheedgesoflength2a.This moment ofinertia is rarbre Zm(y2+^2 ),or Ip(yn+z2}dzdydx, J-aJ-b--c wherepisthedensity. Evaluatingtheintegral,wehave forthemoment of inertia ^Lc (fca+c^}orMass ofblock x1(62+c2 ). *Forpractical purposes themoments ofinertia ofabody aredetermined experimentally ; convenient apparatusisdescribed byW.H.Derriman, Phil. Mag.v.(1903), p.648,andby W.R.Cassie, Phys.Soc. Proc. xxi.(1909), p.497. 57,58]Thedynamical specification ofbodies 119 (iii) Theellipse and thecircle. Let itnowberequiredtofindthemoment ofinertia ofauniformelliptic platewhoseequationis about theaxisofx.Itis. rara I Ia-y^dydx, where cristhesurface-density. Evaluatingtheintegral, wehave fortherequired moment ofinertia \7rabz ar,orMass ofellipsex62 . Themoment ofinertia ofacircle ofradius babout adiameter istherefore Mass ofcircle x\62 . (iv) Theellipsoid andthesphere. Themoment ofinertia ofauniform solidellipsoidofdensity p,whose equationis T2ift ?%71 /T^ fc r2= tv L/ O about theaxisofxissimilarly \p(y*+z^dxdydz, integrated throughouttheellipsoid. Toevaluate thisintegral,write where,77,arenew variables :theintegral becomes pab3cIII ?2 where theintegrationisnowtakenthroughoutasphere whoseequationis Since theintegrals ^,and areevidently equal,therequired moment ofinertia canbewritten inthe form orTrpabc (fe2+c2 )[|2 (1-f2 )J-i or-^Trpabc (b2+c2 ), or Mass ofellipsoidxi(62+c2 ). 120 Tliedynamical specification ofbodies[CH.v Themoment ofinertia ofauniformsphereofradius aabout adiameter istherefore Mass ofspherex^a2 . (v)Thetriangle. Let itnowberequiredtofindthemoment ofinertia ofauniform triangular plateofsurface-density a,withrespecttoanyline initsplane; thepositionofthelinecanbespecified bythelengths a,ft,yoftheper pendicularsdrawn toitfrom thevertices ofthetriangle. Taking (x,y,z}tobetheareal coordinates ofapointoftheplate,the perpendiculardistance from thispointtothegivenline is(ax+/3y+yz),and therequired moment ofinertia istherefore ax4-fiy4-yz)2dS, wheredSdenotes anelement ofarea oftheplate. Now ifYdenotes thelengthoftheperpendicularfrom thepoint (x,y,z) ontheside cofthetriangle,and ifXdenotes thelength interceptedonthe side cbetween thevertexAandthefootofthisperpendicular, wehave Y=zbsinA andXsinAFcosA=perpendicularfrom(x,y,z)ontheside b =ycsinA. Wehave therefore dydz=38 7(|-4- xdXdY =,-.dXdY=^dS,d(X, Y) besinA 2A whereAdenotes thearea ofthetriangle.Hence theintegral \\y-dS, where theintegrationisextended over thearea ofthetriangle,canbewritten in theform2Ally-dydz, where theintegrationisextended over allpositive values ofyand zwhose sum islessthanunity:this isequalto rr rr orA.Bysymmetry,theintegralsIla^dS and IIz-dShave thesame value, andasimilar calculation shews thattheintegrals ffyzdS, ffxedS, \\xydS eachhavethevalueTVA. 58,59] Thedynamical specification ofbodies 121 Substitutingthese values intheintegralcr 11(ax+/3y+yz^dS, the moment ofinertia ofthetriangleabout thegivenlinebecomes &lt;rA(a2+@-+f+7+7+a/3), ixMass oftrianglexJ(*?Y +(?-?Y +f*+*Yl. (\2/ \2/ \2 /j But thisexpression evidently representsthemoment ofinertia about the givenlineofthreeparticlessituatedrespectivelyatthemiddlepointsofthe sides ofthetriangle,themass ofeachparticle beingone-third themass of thetriangle; thetriangleistherefore equimomentaltothis setofthree particles. Example. Shew thatauniform solid tetrahedron ofmassMisequimomentaltoaset offiveparticles, four ofwhich areeach ofmass-^Mandaresituated atthevertices ofthetetrahedron, while the fifthparticleisatthecentre ofgravityofthetetrahedron and isofmass -iM. 59. Derivationofthemomentofinertia about anyaxiswhen themoment ofinertia about aparallelaxisthroughthecentreofgravityisknown. Themoments ofinertia found intheprecedingarticle were forthemost parttaken withrespecttolinesspeciallyrelated tothebodies concerned : these results canhowever beappliedtodetermine themoments ofinertia of thesame bodies withrespecttootherlines,bymeans ofatheorem which will nowbegiven. Letf(x., y,z,x,y,z,x,y,z)beanypolynomial (not necessarily homo geneous)oftheseconddegreeinthecoordinates andthecomponentsof velocity andacceleration ofaparticleofmass m.Let (x,y,I)denote the coordinates ofthecentre ofgravityofabodywhich isformed ofsuchparticles, aridwrite Ifnowwesubstitute these values forx,y,z,respectively,inthefunction/, weobtain thefollowingclasses ofterms : (1)Terms which donotinvolve xl,y1,z1:these termstogether evidently give f(x, y,z,x,y,z,x,y,z). (2)Terms which donotinvolvex,y,z: these termsgive /(!, 2/j,z1}xltyl}zltxltyl}z\). (3)Terms which arelinear inxltyltzltxltylyzltxltyltzl;when the expression ^mf(x, y,z,x,y,z,x,y,z)isformed, thesummationbeing taken over alltheparticlesofthebody, these terms willvanish inconsequenceof therelations ^mx l=0, Sray!=0, Stmzl=0. 122 Thedynamical specification ofbodies[CH.v Wehave therefore theequation ^mf(x, y,z,x,y,z,x,y,z)=(xlty,,zltx,,y,,zltx,,y,,z,} +/(&gt; V&gt;z&gt; x&gt;y&gt;z,5,y,z).Sw, andconsequentlythevalue oftheexpression 2m/, taken withrespectto anysystemofcoordinate axes, isequaltoitsvalue taken withrespecttoa parallelsetofaxesthroughthecentre ofgravityofthebody, togetherwith themass ofthebody multiplied bythevalue ofthefunction fatthecentre ofgravity,taken withrespecttotheoriginal systemofaxes. From this itimmediatelyfollows that themoments andproducts ofinertia, ofabodywithrespecttoanyaxesareequaltothecorresponding moments and products ofinertia, withrespecttoasetofparallelaxesthroughthecentreof gravity ofthebody, together with thecorresponding moments andproducts ofinertia, withrespecttotheoriginal axes, ofaparticle ofmassequaltothat ofthebodyandplacedatthecentre ofgravity. Asanexample ofthisresult,let itberequiredtodetermine themoment ofinertia ofastraight uniform rodofmassMandlengthIabout alinethrough oneextremity perpendicular totherod. Itfollows from thelast article that themoment ofinertia 2 ;andhence, applying the aboveresult, weseethattherequired moment ofinertia is 60. Connexion between momentsofinertia withrespecttodifferentsetsof axesthroughthesameorigin. Theresult ofthelastarticle enables ustofindthemoments ofinertia of agiven body withrespecttoanysetofaxes,when themoments ofinertia arealready known withrespecttoasetofaxesparalleltothese. Weshall nowshewhowthemoments ofinertia ofabodywithrespecttoanysetof rectangularaxescanbefound when themoments ofinertia areknown with respecttoanother setofrectangular axeshavingthesameorigin. LetA,B,C,F,G,Hbethemoments andproductsofinertia withrespect toasetofaxesOxyz, and letOxyzbeanother setofrectangularaxes havingthesameorigin;thedirection-cosines ofeither setofaxes with respecttotheother willbesupposedtobegiven bythescheme ^/ * li ml 59,60] Thedynamical specification ofbodies 123 Ifthemoments andproductsofinertia withrespecttotheaxesOxyz aredenoted byA,B,C",F,G,H,wehave A=2m(y2+z2 ),where thesummation isextended over alltheparticlesof thebody, =2m{(I2x+m2y+n2z)2+(I3x+m3y+nsz)2 } =2m[x2 (l2+I2 }+y2(m2+w32 )+z2 (n22+n/)+2yz(m2n.2+m3n3) =2m[x2(m,2+n,2 )+if(n2+I,2 )+z2 (I,2+m2 )-Zm^yz-ZnJ.zx-Zl^ =2m{I,2 (y2+z2 )+m*(z2+x2 )+n2(x2+r/2 )-Zmj^yz-Zn^zx-Zljn.xy andsimilarly B=A122+Bm.?+On?-2Fm.2n2-2Gn2l.2-2Hl.2m,, C=Al,2+Bms*+Cn.?-2Fm3n3-2Gn,l 3-2Hl3ms. Wehave also F=2myz =2m(l^+m2y+n2z](I3x+m3y+nsz) =1213.2mx2+m2ms.2my2+n2n3.2mz2+(m. 2ns+m3n2). +(n2l3+n3l2).2mzx+(I2m3+I3m2).2mxy =^IJ,(B+G-A) +^&gt;n2m3(C+A-B)+in.2ns(A+B-C) +(w2i3+msn,)F+(n.2l3+nzl2}G+(I2m3+ or F=Al.2l3+Bm2m3+Cn.2n3-F(m. 2n3+m3n2)-G(I3n.2+-l.M3)-H(l. 2m3 andsimilarly -G=Al,l,+Bm^n,+Cn-.n,-F(m. An,+m,7i 3)-G(I}n3+lsn^-H(l^ni,+I^m 3),-H=Al,L+Bm.in,+Cn,n 2-F(m,n 2+m^)-G(Ln,+l^)-H(l^t,,+l^n,). Thequantities A ,B,C,F ,G,Harethusdetermined; these results, combined with those ofthe last article, are sufficient todetermine the moments andproductsofinertia ofagiven bodywithrespecttoanysetof rectangular axeswhen themoments andproductsofinertia withrespectto anyother setofrectangular axes areknown. Example.Iftheoriginofcoordinates isatthecentre ofgravityofthebody, prove that themoments andproductsofinertia withrespect tothreemutually orthogonalandintersecting lineswhose coordinates are (l\ir "h, i&gt;Xj,Mi, "i), (12, &lt;i,n2,X2,^, 2), (?3, 3&gt;%,X3,/x3,i/3) are A+Jf^S+rf+vft etc.andF-M(X2X3+M:,/X3+I,2J,3)etc., where A\B,C,F, 0",Hhave thesarue values asabove andMisthemass ofthe (Coll. Exam.) 124 Thedynamical specification ofbodies[OH.v 61.Theprincipalaxesofinertia;Cauchys momentalellipsoid. Ifnowweconsider thequadricsurface whoseequationis Ax-+By2+Cz*-&lt;LFyz- 2Gzx-2Hxy=1, where A,B,0,F,G,Harethemoments andproductsofinertia ofagiven bodywithrespecttotheaxes ofreferenceOxyz,itfollows from theequation thatthereciprocalofthesquareofanyradius vector ofthequadricisequal tothemoment ofinertia ofthebodyabout this radius. Thequadricis therefore thesame whatever betheaxes ofreferenceprovidedtheorigin isunchanged,andconsequentlyitsequationreferred toanyotherrectangular axesOxyzhavingthesameoriginis Aa?+Bf-+Cz*-ZFyz-ZGzx-IRxy=1; whereA,B,C,F,G,Harethemoments andproductsofinertia with respecttothese axes. Thisquadriciscalled themomentalellipsoidofthebodyatthepoint\ itsprincipalaxes arecalled theprincipalaxesofinertia ofthebodyat ; theequationofthequadricreferred tothese axescontains noproduct-terms, andtherefore theproductsofinertia withrespecttothem arezero :and themoments ofinertia withrespecttothese axes arecalled theprincipal momentsofinertia ofthebodyatthepoint 0*. Themomentalellipsoidisalsocalled theellipsoid ofinertia;itspolar reciprocal with regardtoitscentre isanotherellipsoid, which issometimes called theellipsoid ofgyration. Example. Theheightofasolidhomogeneous rightcircular cone ishalftheradius ofitsbase. Shew that itsmomentalellipsoidatthevertex isasphere. 62. Calculation oftheangular momentumofamoving rigid body. Weshallnowshewhowtheangular momentum ofamoving rigidbody about any line, atanyinstant ofitsmotion, canbedetermined. LetMbethemass ofthebody, (x,y,z)thecoordinates ofitscentre of gravity G,and(u,v,w)thecomponentsofvelocityofthepoint G,atthe instantt,resolvedalong any (fixedormoving) rectangularaxesOxyzwhose originisfixed; and let(oj1;&lt;u2,&&gt;3)bethecomponentsoftheangular velocityofthebodyabout G,resolvedalongaxesGxlylzl,paralleltotheaxes Oxyzandpassing throughG.Letmdenote atypical particleofthebody, and let(x,y,z)beitscoordinates and(u,v,w)itscomponentsofvelocityat theinstant t;andwrite x=x+xi, y=y+yi,z=z+z1, UU+11^ , V=V+Vl,W=W+Wl, *Theexistence ofprincipalaxeswasdiscovered byEuler, Mem. deBerl., 1750, 1758,andby J.A.Segner, Specimen Th.Turbinum, 1755. Themomentalellipsoid wasintroduced byCauchy in1827, Exerc. demath. i.p.93. 61,62] Thedynamical s2)ecification ofbodies 125 soinvirtue ofthepropertiesofthecentre ofgravity wehave 2m^j=0, ^myj=0, Sm^j= ; moreover since(17)wehave Ul=Zlwzy^w^,Vl=Xlo&gt;32l&)j ,Wl=y1(0l itfollows that Sm^!=0, Sravjs=0,mwl=0. IfA3denotes theangular momentum ofthebodyabout theaxis Oz,we have therefore h3=Sm(xv yu) =2m{(x+#,)(v+Wj)-(y+yj(u+u,)} =2m(xv yu)+2w(x^ y^iii) =M(xv yu)+2m(X^W Axlzla)1y^z^w*+y^w^) =M(xv yu)GwlFwz+C(D3, where A,B,C,F,G,Harethemoments andproductsofinertia ofthebody withrespecttotheaxesGx-^y^. Similarlytheangular momenta about theaxesOxandOyrespectively are h1=M(yw zv}+AwlHw n_Gco3, h?=M(zuxw} Ha)1+Bwo Fo)3. Theangular momentum aboutanyother linethroughtheorigin canbe found(39)byresolvingtheseangular momentaalongthelineinquestion. Corollary.Ifthebodyisconstrained toturnround oneofitspoints, which isfixed inspace,itisunnecessarytointroduce thecentre ofgravity. For let (&&gt;!,o)2,&)3)bethecomponentsoftheangular velocityofthebody about thefixedpointwithrespecttoanyrectangularaxes(fixed ormoving) which have thefixedpointasorigin,and letA,B,C,F,G,Hdenote the moments andproductsofinertia withrespecttothese axes. Thecom ponentsofvelocity (u,v,w)withrespecttothese axes ofaparticleinwhose coordinates are(x,y,z)are(17) v= andtheangular momentum about theaxis ofz,which is2m(xv yu},can therefore bewritten intheform Sm or Similarlytheangular momenta ofthebody about theaxes ofxandy respectivelyare A(i)lHwzGu&gt;s and 126 Thedynamical specification ofbodies[on.v 63. Calculationofthekineticenergy ofamoving rigid body. Thekineticenergyofarigidbodywhich isinmotion canbecalculated inthesamewayastheangular momenta. Ifthe-general theorem obtained in59isappliedtothecase inwhich thepolynomial f(x, y,z,x,y,z,x,y,z) hastheform(x2+y2+z2 ),weimmediatelyobtain theresult that thekinetic energy ofamoving rigid bodyofmassMisequaltothekineticenergy ofa particle ofmassMwhich moves with thecentreofgravity ofthebody, together with thekineticenergy ofthemotionofthebody relative toitscentreof gravity. Todetermine thekineticenergyofthemotion ofthebodyrelative toits centre ofgravity G,takeanyrectangularaxes(whose directions maybefixed ormoving) havingtheiroriginatG;let (a&gt;1;&&gt;2,o&gt;3)bethecomponentsof theangular velocityofthebody about 0,relative tothese axes,and let (x,y,z)denote thecoordinates ofatypical particlemofthebody referred tothese axes. Thecomponentsofvelocityoftheparticle paralleltothese axes, inthemotion relative toG,are(17) andtherefore thekineticenergyofthemotion relative tothecentre of gravityis \2m\(za&gt; z-yo s}2+(.v&lt;o 3-zco^2+(yWl- xca^-}, or (Aw,-+Bw.?+Co)./-2Fco2(,)3-2&&gt;3&&gt;1-2#&&gt; 1&&gt;,), where A,B,C,F,G,Harethemoments andproductsofinertia relative to theaxes. Thisexpression may(byuseof60)beinterpretedashalfthesquare oftheresultantangular velocityofthebodyinthemotion relative tothe centre ofgravity, multiplied bythemoment ofinertia ofthebody about theinstantaneous axis ofrotation inthismotion. Corollary.Ifoneofthepointsofthebodyisfixed inspace,itisnot necessarytointroduce thecentre ofgravity. For let(co1}a&gt;2,&&gt;3)denote the componentsofangular velocityofthebodyabout thefixedpoint resolved along anyrectangularaxes(fixed ormoving) Oxyz which have thepoint asorigin,and let(x,y,z}bethecoordinates ofatypical particlemofthe bodyreferred tothese axes. Thecomponentsofvelocityoftheparticle are(17) zo)2yo)3,xais zw^yw lxo)z, andsoasbefore weseethatthekineticenergyofthemotion is where A,B,C,F,G,Hdenote themoments andproductsofinertia ofthe bodywithrespecttotheaxesOxyz. 63,64] Thedynamical specification ofbodies 127 From this itfollows that ifoneofthecoordinate axessaytheaxis ofx istheinstantaneous axis ofrotation ofthebody,thekineticenergyis ^Aa)^; andhence, since thedirections oftheaxescanbearbitrarily chosen, thekineticenergyofanybodymoving about oneofitspoints,which isfixed, is^/&)2 ,where /isthemoment ofinertia ofthebodyabout theinstantaneous axisofrotation, and &&gt;istheangular velocityofthebodyabout this axis. Example. Alamina canturnfreely about ahorizontal axisinitsownplane, andthe axisturns about afixed vertical line,which itintersects. If (f&gt;betheazimuth ofthe horizontalaxis,and^theinclination oftheplane ofthelamina tothevertical, shew that thekineticenergyis where A,B,Zfarethemoments andproductofinertia ofthelamina about thehorizontal axisandaperpendiculartoitatthepointofintersection with thevertical.(Coll. Exam.) 64.Independence ofthemotionofthecentreofgravity and themotion relative toit. The result ofthe last article shews thatthekineticenergyofamoving bodycanberegardedasconsistingoftwoparts,ofwhich onedepends onthe motion ofthecentre ofgravityandtheother isthekineticenergyofthemotion relative tothecentre ofgravity. Weshallnowshew thatthese twopartsof themotion ofthebodycanbetreatedquite independentlyofeach other*. Letarigid bodyofmassMbeinmotion under theinfluence ofany forces Ascoordinatesdefiningitsposition wecantakethethreerectangular coordinates(x,y,z)ofitscentre ofgravity G,relative toaxes fixed inspace, andthethree Eulerianangles (6,0,i/r)whichspecifytheposition, relative toaxes fixed indirection, ofanythreeorthogonal lines, intersectinginG, which arefixed inthebodyandmove with it.Thekineticenergyistherefore T=iM(x*+p+*)+/(B, 0,+,e,0,^), wheref(0, 0,&,0,0,^)denotes thekineticenergyofthemotion relative toG. Let denote theworkdoneonthebodybytheexternal forces inanarbitrarydis placement (Bx, By,Bz,B0,80,8^r)ofthebody. TheLagrangian equations of motion are MX=X,My=Y,Mz=Z, &lt;*L dt di d dtdf80 *Euler, Scientianavalis,i.(1749), 128. 128 Thedynamical specification ofbodies[CH.v The first three oftheseequations shew that themotionofthecentreof gravity ofthebodyisthesame asthatofaparticle ofmassequaltothewhole massofthebody, under theinfluence offorces equivalenttothetotal external forces acting onthebody, appliedtotheparticle paralleltotheir actual directions;since thework done onsuch aparticleinanarbitrary displace ment wouldevidentlybeXBx+YBy+Z8z. Thesecond threeequations shew that themotionofthebodyabout its centreofgravityisthesame asifthecentreofgravitywerefixedand thebody subjectedtotheaction ofthesameforces ;forinthemotion relative tothe centre ofgravity,thekineticenergyofthebodyisf(6, &lt;f&gt;,-^r,6,$,i^),and theworkdonebytheforces inanarbitrary displacementis + &lt;J&gt;S+ These results areevidentlytrue alsoforimpulsive motion. Corollary.Ifaplane rigidbody (e.g.adiscofanyshape)isinmotion in itsplane, and if(x,y)arethecoordinates ofitscentre ofgravity,Mitsmass, 6theanglemadebyalinefixed inthebodywith aline fixed intheplane,Mk2themoment ofinertia ofthebody about itscentre ofgravity,and if (A,F)arethetotalcomponents paralleltotheaxes oftheexternal forces actingonthebody,andLthemoment oftheexternal forces about thecentre ofgravity,then thekineticenergyis \M(x2+y-+k*fr\ andthework donebytheexternal forces inadisplacement (Bx, By,86)is andtherefore theequationsofmotion ofthebodyare MX=X,My=Y,M&0=L. Example. Obtain oneoftheequations ofmotion ofarigidbodyintwodimensions in theform whereMisthemass ofthebody,/istheacceleration ofitscentre ofgravity, pisthe perpendicular from theorigin uponthisvector, J/F isthemoment ofinertia about the origin,6istheanglemade byaline fixed inthebody withalinefixed initsplane, andL isthemoment about theorigin oftheexternal forces.(Coll. Exam.) 64]Thedynamical specification ofbodies 129 MISCELLANEOUS EXAMPLES. 1.Ahomogeneous rightcircular cone isofmassM;itssemi-vertical angleis/3,and thelengthofaslant side isI.Shew that itsmoment ofinertia about itsaxis is andthat itsmoment ofinertia about alinethroughitsvertex perpendiculartoitsaxis is fM2(l-fsin2 /3), atid itsmoment ofinertia about ageneratoris 2.Shew that themoment ofinertia ofthearea enclosed bythetwoloopsofthe lemniscate r2=2cos20 about theaxisofthecurve is (37r-8)a2-xmass ofarea. 3.Anynumber ofparticlesareinoneplane;ifthemasses areWj,m2,...,their distancesaparto?12,...,therelative descriptionsofareaA12,...,andtherelative velocities- *i2,...,prove that arerespectively themoment ofinertia about thecentre ofinertia, theangular momentum about thecentre ofinertia, andthekinetic energyrelative tothecentre ofinertia. (Coll. Exam.) 4.Prove that themoment ofinertia ofahollow cubical boxabout anaxisthrough thecentre ofgravityoftheboxandperpendiculartooneofthefaces is whereMisthemass oftheboxand2athelengthofanedge. The sides oftheboxare supposedtobethin. (Coll. Exam.) 5.Shew thatthemoment ofinertia ofananchor-ring about itsaxis is where aistheradius ofthegenerating circle, cisthedistance ofitscentre from theaxis oftheanchor-ring, andpisthedensity. 6.Shewhow tofindatwhatpoint,ifany,agiven straightline isaprincipalaxisofa- body, and ifthere issuch apointfindtheother twoprincipalaxesthroughit. Auniform square lamina isbounded bytheaxes ofxandyandthelinesx=2c, y=2c, andacorner iscutoffitbythelinexja+y/b=2.Shew thatthetwoprincipalaxesat thecentre ofthesquare which areinitsownplaneareinclined totheaxisofxatangle? given by ab2ctan20=--j--.(Coll. Exam.)v (a- 6)(a+b2c) 7.Shew thattheenvelopeoflines intheplaneofanareaabout which thatareahasa constant moment ofinertia isasetofconfocalellipses andhyperbolas. Hence findthe direction oftheprincipalaxes atanypoint. (Coll. Exam.) w.D. 9 130 Thedynamical specification ofbodies[CH.v 8.Find theprincipalmoments ofinertia atthevertex ofaparabolic lamina, latus rectum 4a,bounded byalineperpendiculartotheaxisatadistance hfrom thevertex. Provethat,if15A&gt;28, twoprincipal axes atthepoint ontheparabola whose abscissa is-a+(a2-4aA/5+3A2/7)iarethetangent andnormal.(Coll. Exam.) 9.Findhowtheprincipal axes ofinertia arearrangedinaplane body. Write down theconditions thatparticles m^at(x{,?/,-),where z=l, 2,...,maybeequimomentaltoa given plate. Shew thatthesixquantities m1,m2,&\,x2,y^y2canbeeliminated from these conditions. Ifthreeequal particles areequimomental toagiven plate, thearea ofthetriangle formed bythem is3^/3/2 times theproduct oftheprincipalradii ofgyrationatthe centre ofgravity. (Coll. Exam.) 10.Auniform lamina bounded bytheellipseb2x2+a2y2=aWhasanelliptic hole (semi-axes c,d)initwhosemajor axis lies inthe linex=y,thecentrebeingata distance rfrom theorigin ;prove that ifoneoftheprincipal axes atthepoint (x,y) makes anangle6with theaxis ofx,then tan26=-Sabxy~d^(x^~^("^~r)~(c*~^} ab[4(x*-y*)+a2-&*]-cd[2(x^-r)2-2(y^2-r)2 ] (Coll. Exam.) 11. Ifasystemofbodies orparticlesismoved ordeformed inanyway,shew that thesumoftheproductsofthemass ofeachparticle intothesquare ofitsdisplacement isequaltotheproductofthemass ofthesystem intothesquare oftheprojectioninany given direction ofthedisplacementofthecentre ofgravity, together with thesumofthe products ofthemasses oftheparticles intothesquares ofthedistancesthrough which theymust bemoved inorder tobring them totheir finalpositions aftercommunicating tothem adisplacement equaltotheprojectioninthegiven direction ofthedisplacement ofthecentre ofgravity. (Fouret.) 12.Theprincipal moments ofinertia ofabodyatitscentre ofgravityare(A,B,G); ifasmall mass, whose moments ofinertia referred tothese axesare(A ,B,C),beadded tothebody, shew that themoments ofinertia ofthecompound body about itsnew principal axes atitsnewcentre ofgravityare A+A,B+B ,C+C, accuratelytothe firstorder ofsmallquantities. (Hoppe.) 13.Shew that theprincipal axes ofagiven materialsystematanypoint arethe normals tothethreequadrics whichpassthrough thepoint andbelongtoacertain confocalsystem. If(I,m, ,,X, /*, i/)bethe sixcoordinates ofaprincipalaxisandtheassociated Cartesian system betheprincipal axes atthecentre ofgravity, thenshew that Al\+Bmp.+Cnv=0, andtherefore allprincipal axes ofagiven system belongtoaquadratic complex. (Coll. Exam.) 14.Asmoothly jointed framework isintheform ofaparallelogram formed bv attaching theends ofapairofrods ofmassmandlength 2atothose ofapairofrods of massmandlength26.MassesMareattached toeach ofthefour corners. Express the angular momentum ofthesystem about theoriginofcoordinates, interms ofthe coordinates(A;y)ofthecentre ofgravity andtheangles 6and&lt;between thetwopairsof sidesandtheaxisofx.(Coll. Exam.) CHAPTER VI THESOLUBLE PROBLEMS OFRIGID DYNAMICS 65.Themotionofsystemswith onedegree offreedom:motion round afixed axis, etc. Wenowproceedtoapplytheprincipleswhich havebeendevelopedin theforegoing chaptersinorder todetermine themotion ofholonomicsystems ofrigidbodies inthose caseswhich admit ofsolution byquadratures. Itisnatural toconsider firstthosesystemswhich haveonlyonedegreeof freedom. Wehave seen(42)thatsuch asystemisimmediatelysoluble by quadratures when itpossessesanintegralofenergy:and thisprincipleis sufficient fortheintegrationinmost cases. Sometimes, however(e.g.when wearedealingwith systemsinwhich oneofthesurfaces orcurves ofcon straint isforced tomove inagiven manner), theproblemasoriginallyformu lated does notpossessanintegralofenergy,butcanbereduced(e.g.bythe theorem of29)toanotherproblemforwhich theintegralofenergyholds; when thisreduction hasbeenperformed,theproblemcanbeintegratedas before. Thefollowing exampleswill illustrate theapplicationoftheseprinciples. (i)Motionofarigid bodyround afixedaxis. Consider themotion ofasingle rigidbodywhich isfreetoturnabout anaxis, fixed in thebodyandinspace.Let/bethemoment ofinertia ofthebody about theaxis, sothat itskinetic energyis/02 ,where 6istheanglemadebyamoveableplane, passing through theaxisandfixed inthebody, with aplane passing through theaxisandfixed inspace. Let6bethemoment round theaxis ofalltheexternal forcesacting onthebody,sothat $istheworkdonebythese forces intheinfinitesimal displacement which changes 6to 6-\-fid.TheLagrangian equationofmotion d_SVT\_dT_ dt\ti) dd~ thengives IB=0, which isadifferential equationofthesecond order forthedetermination ofG. 92 132 TheSoluble Problems ofRigid Dynamics [CH.vi Iftheforces areconservative, andV(6}denotes thepotential energy,thisequation becomes 10=-^ 80 which onintegration gives theequationofenergy $I62+V(ff)=c, where cisaconstant. Integrating again, wehave t=I\f{2(c-V}}"%dd+constant, andthisrelation between 9and tdetermines themotion, thetwoconstants ofintegration being determined bytheinitial conditions. Themost important case isthat inwhich gravityistheonlyexternal force, andthe axis ishorizontal. Inthiscase letGbethecentre ofgravityofthebody,Cthefootof theperpendicular drawn fromGtotheaxis,and letCG=h.The potential energyis Mghcos0,whereMisthemass ofthebodyand6istheanglemade byCGwith the downward vertical :andtheequationofmotion is This isthesame astheequationofmotion ofasimple pendulumoflength J/Mh, and themotion cantherefore beexpressedinterms ofellipticfunctions asin44,thesolution beingoftheform .6,sin-=Asn-- intheoscillatory case,andoftheform inthecirculatingcase. Thequantity IjMhiscalled thelength oftheequivalent simple pendulum. Ifbeapoint onthelineCGsuch thatOC=IjMh,thepoints andCarecalled respectively thecentreofoscillation andthecentreofsuspension. Acurious result inthis connexion isthat thecentreofoscillation and thecentre ofsuspensionareconvertible, i.e. if isthecentre ofoscillation whenCisthecentre ofsuspension,thenCwillbe thecentre ofoscillation when isthecentre ofsuspension.Toprovethisresult, we haveby59 Moment ofinertia ofthebody about Moment ofinertia aboutG+M.GO2 andtherefore wehave Moment ofinertia ofbodyabout _I-Mh2+M(I/Mh-A)2 Distance ofcentre ofgravity from IjMh-h Iftherefore thebodyweresuspendedfrom 0,theequationofmotion would stillbe vMqh. 6+J-sin=0, which establishes the result. Itisevident that theperiodofoscillation would bethe same about either ofthepointsCand 0. 65] TheSoluble Problems ofRigid Dynamics 133 (ii)Motionofarodonwhich aninsect iscrawling. Weshall nextstudythemotion ofastraight uniform rod, ofmassm.andlength 2a, whose extremities can slide onthecircumference ofasmooth fixed horizontal circle of radius c;aninsect ofmass equaltothat oftherod issupposedtocrawlalongtherodat aconstant ratevrelative totherod. Let6betheanglemadebytherodattime twithsome fixed direction, and letxbe thedistance traversedbytheinsect from themiddlepointofthe rod. The kinetic (2a2\ c25-1#2 ,andthekineticenergyoftheinsect isdue to3/ acomponentofvelocity {x-(c2-a2 )20}alongtherodandacomponentofvelocity xB perpendiculartotherod, sothetotal kinetic energyofthesystemis there isnopotential energy. Since x=vt, (tbeing measured from theepochwhenxiszero),wehave T=\m(c2-2a2 /3)62+\m{v-(c2-a2 )^0}2+mv*t*0\ Thecoordinate6,which isnowtheonly coordinate,isigriorable, andwehave therefore =constant, or m(c2- -|-)0-m(c2-a2 )^{v-(c2-a2 )*0}+mv2t2=constant, or 0(2c232+t;2 i!2 )=constant. Integratingthisequation, wehave #-0 arctan{vt(2c2 -|a2 )~ }, where andkareconstants. Thisformula determines thepositionoftherodatanytime. (iii) Motionofaconeonaperfectly rough inclinedplane. Consider nowthemotion ofahomogeneous solidrightcircular cone, ofmassMand semi-vertical angle /3,which moves onaperfectly rough plane (i.e.aplane onwhichonly rolling withoutsliding cantakeplace) inclined atanangle atothehorizon. Let Ibethe lengthofaslant sideofthecone, and let betheangle between thegenerator which is incontact withtheplane attime tandthelineofgreatest slopedownwards intheplane. Then if^betheanglemade bytheaxisoftheconewith theupward vertical, ^isone side ofaspherical triangle whose verticesrepresent respectivelythenormal totheplane, theupward vertical, andtheaxis ofthecone;theother twosides areaand(?), the angle included bythese sides being (TT-0}.Wehave therefore. cosx=cosasinftsinacosftcos; buttheverticalheightofthecentre ofgravity oftheconeabove itsvertex is Icos ftcos^, andthepotential energyofthecone isMgxthisheight ;iftherefore wedenotebyVthe potential energyofthecone,wehave(disregardingaconstant term) V=-^Mglsinacos2 ftcos0. 134 TheSoluble Problems ofRigid Dynamics [CH.vi Wehave next tocalculate thekinetic energyofthecone;forthisthemoments of inertia oftheconeabout itsaxisandabout alinethroughthevertexperpendiculartothe axisarerequired:these areeasily found(bydirectintegration, regardingthecone as composedofdiscs perpendiculartoitsaxis)tobe^MPsin2 /3and J/72(cos2 /3+Jsin2 /3) respectively, andsothemoment ofinertia about agenerator is,bythetheorem of60 (since thedirection-cosines ofthegenerator canbetaken tobesinft,0,cos$withrespect torectangular axes atthevertex, ofwhich theaxisofzistheaxisofthecone), MV(cos2 ft+1sin2 ft)sin2 ft+^MPsin2 ftcos2 ft, or Jft2sin20(cos2+i). Now allpointsofthatgenerator which isincontact withtheplane areinstantaneously atrest, since themotion isoneofpure rolling, and therefore thisgeneratoristhe instantaneous axis ofrotation ofthecone. Ifwdenotes theangular velocityofthe cone about this generator, thekinetic energyofthecone istherefore(63,Corollary) jj-MPsin2 ft(cos2 ft+J)w2 . But(15)wehave a)=6cotft, andsubstitutingthisvalue for o&gt;,wehavefinallyforthekinetic energy Tofthecone thevalue T=%MPcos2 ft(cos2 ( TheLagrangian equationofmotion becomes therefore inthiscase |Ml*cos2 ft(cos2 ft+ )6+Mglsinacos2 ftsin6=0, This isthesame astheequationofmotion ofasimple pendulumoflength Icosec a(cos2 ft+ ); theintegration cantherefore beeffected interms ofelliptic functions, asin 44. (iv)Motionofarodonarotating frame. Consider nextthemotion ofaheavy uniform rod,whose ends areconstrained tomove inhorizontal andverticalgrooves respectively, when theframeworkcontaining thegrooves ismade torotate withconstantangular velocitycoabout thelineoftheverticalgroove. Let2abethelengthoftherod,Mitsmass, and6itsinclination tothe vertical. By 29,theeffect oftherotation maybeallowed forbyaddingtothepotential energy aterm where pisthedensityoftherodandxdenotes distance measured from theendoftherod which isinthevertical groove ;integrating,thisterm canbewritten Theterm inthepotential energy duetogravityis Mgacos6, andthetotalpotential energy Vistherefore given bytheequation V=-Mgacosd-%Jfo*a* sin26. 65] TheSoluble Problems ofRigid Dynamics135 Thehorizontal andvertical componentsofvelocityofthecentre ofgravityoftherod areasin6 .6andacos6 .$,sothepartofthekinetic energyduetothemotion ofthe centre ofgravityis\Ma?6z ;andsince themoment ofinertia oftherodabout itscentre isItMa?, thepartofthekineticenergyduetotherotation oftherodabout itscentre isJMa262 ;wehave therefore forthetotal kinetic energy Ttheequation Theintegralofenergyistherefore Ma262Mgacos6fM&lt;o2a2sin26=constant, or,writing where edenotes aconstant :thisconstant mustevidentlybepositive,sincex2and(1x2 ) arepositive. Weshall supposefordefiniteness that isnotvery largeandthat3&lt;7/4or islessthanunity,sothatxoscillates between thevalues3^/4aco2 e/&&gt;. Tointegratethisequation, wewrite* #=!+. +^8a 18" 642 o&gt;2^12 where isanewdependentvariable. Substitutingthisvalue forxinthedifferential equation, wehave where thevalues correspond respectivelytothevalues itiseasilyseenthatei+e 2+e3iszeroandthat el&gt;e2&gt;e3. Wehave therefore = |jf&gt;(&lt;+y),where thefunction $&gt;isformed with theroots e},e2,es, andwhereydenotes aconstant. Since e^~&gt;e&lt;{&gt; e^,and(P(&lt;+y)liesbetween e2and e3for realvalues oft(sincexliesbetween 3#/4co2- e/o&gt;and3^/4aa)2+e/a)),theimaginary partof theconstantymust bethehalf-periodo&gt;;!;therealpartofycanthen betaken aszero, since itdepends onlyonthechoice oftheoriginoftime.Wehave therefore andhence thisequation determines ^interms of &lt;. (v)Motion ofadisc,oneofwhosepointsisforcedtomove inagivenmanner. Consider nextthemotion ofadiscofmassM.resting onaperfectly smooth horizontal plane, when oneofthepointsAofthedisc isconstrained todescribe acircle ofradius c inthehorizontalplane, withuniform angular velocityo&gt;. *Cf.Whittaker andWatson, ACourse ofModern Analysis, %206. 136 TheSoluble Problems ofRigid Dynamics [OH.vi LetGbethecentre ofgravityofthedisc,and letAGbeoflength a.Theacceleration ofthepointAisofmagnitude cw2 ,and isdirected alongtheinward normal tothecircle : iftherefore weimpressanacceleration cw2 ,directedalong theoutward normal tothe circle, onalltheparticlesofthebodyandsuppose thatAisatrest,weshall obtain the motion relative toA.Theresultant forceacting onthebodyinthismotion relative toA istherefore J/co&gt;2 ,actingatGinadirectionparalleltotheoutward normal tothecircle. Let6and betheangles made withafixed direction intheplane bythelineAGand theoutward normal tothecirclerespectively;then theworkdonebythisforce inasmall displacement 86is Mca&gt;2asin(0- &lt;9) 8&lt;9, andthekinetic energyofthebodyisiJ/2#2 ,whereMk2isthemoment ofinertia ofthe body about thepoint A.TheLagrangian equationofmotion istherefore Mm=Macrf sin(0-6\ Butsince= a&gt;,wehave=0;soif^bewritten for(60),wehave acco2 . ^+-p-sm^=0. This isthesame astheequationofmotion ofasimple pendulumoflength k2 gjaca&gt;2 ; theintegration cantherefore beperformed bymeans ofellipticfunctions asin 44. (vi) Motionofadiscrollingonaconstrained discandlinked toit. Consider themotion oftwoequalcirculardiscs, ofradius aandmassJ/,withedges perfectly rough, which arekeptincontact inavertical plane bymeans ofalink(inthe form ofauniform barofmassm)whichjoinstheir centres :thecentre ofonedisc isfixed, andthisdiscAisconstrained torotate withuniform angular acceleration a;itisrequired todetermine themotion oftheother discBandthelink. Let betheangle which thelinkmakes with thedownward vertical attimet,and let6betheangleturned throughattime tbythediscA.Theangular velocityofdiscA is0,andthevelocities ofthepointsofthediscswhich areinstantaneouslyincontact are therefore each [email protected] thevelocityofthecentre ofthediscBis20,itfollows thatthe angular velocityofthediscBabout itscentre is20-0. Since themoment ofinertia of each discabout itscentre isiMa2 ,thekinetic energyofthesystemis T=M.^V +lM.a *-(2$-eT- +lM.(2a?&lt;tf +m.~&-m iu o and=at+f,where fisaconstant. Thepotential energyofthesystemis F=-(2M+m) agcos0, andtheLagrangian equationofmotion is d(o_T\_dT__dV &lt;ti\d&lt;p) 80~ 80 orjt{(6Af+f m)a20-Ma*0}=-(2J/+wi)agsin0. Since d=a,thisequation gives (6M+1m)a2-J/a2a+(2M+m)agsin=0. Integrating, wehave (3J/+ %m)a2 (fi-J/u2n0(2J/+ m)agcos=c, 65,66]TheSoluble Problems ofRigid Dynamics137 where cisaconstant dependingontheinitial conditions :andasthevariables tand &lt;/&gt;are separable,thisequation canagain beintegrated byaquadrature:this final integral determines themotion. Example.Ifthesystemisinitiallyatrestwith thebarvertically downwards, shew thatthebarwillreach thehorizontal positionif 66.Themotionofsystemswith twodegrees offreedom. Inthedynamicsofrigid bodies, asinthedynamicsofaparticle,the possibilityofsolving byquadraturesaproblemwithtwodegreesoffreedom generally dependsonthepresenceofanignorablecoordinate. Theintegral correspondingtotheignorablecoordinate canoften beinterpreted physically asanintegralofmomentum orangularmomentum. Theformation and solution ofthe differentialequationsiseffected byapplicationofthe principles developedinthepreceding chapters:this willbeshewn bythe followingillustrativeexamples. (i)Hodpassing through ring. Consider, asafirstexample,themotion ofauniform straightrodwhichpasses through asmall fixedringonahorizontalplane, beingable toslidethroughtheringorturn inany wayabout itintheplane. Letthedistance from theringtothemiddlepointoftherodattime tber,and letthe rodmake anangle 6withafixed lineintheplane ;let21bethelengthoftherod,andM itsmass. Themoment ofinertia oftherodabout itsmiddle pointis\MP,andthekinetic energy istherefore there isnopotential energy. Thecoordinate Bisignorable, andthecorresponding integralis dT r=constant, vB or(?&gt;2+%l-)6=constant. Theintegralofenergyis ;,2_|_,.202 _|_^Ilfri_constant. Dividingthesecond oftheseintegrals bythesquareofthefirst,wehave dry dft) 1where C1Saconstant , or B+constant = j{(r2+}I2 )(cr*+Jcl2-1)}~4dr. Writingcr1=s,thisbecomes 6+constant = I{4s(s+\cP) (+JcP-1)}~*^s. 138 TheSoluble Problems ofRigid Dynamics [CH.vi Iftherefore g&gt;denotes theWeierstrassianelliptic function with theroots drwhichsatisfytherelation e}&gt;e2&gt;e3if-^issufficiently great initially, wehave s=ft&gt;(0- )gj ,where isaconstant ofintegration; since sispositive, wehave0&gt;(0-0)&gt;eiforrealvalues of0,andconsequently the constant $isreal. The solution oftheproblemistherefore contained intheequation (ii)0^ecylinder rolling onanother undergravity. Let itnowberequiredtodetermine themotion ofaperfectly rough heavysolid homogeneous cylinderofmassmandradiusr,which rolls inside ahollowcylinderofmassMandradiusR,which inturn isfreetoturnabout itsaxis(supposed horizontal). Let &lt;j)denote theangle which theplane through theaxes ofthecylindersattime t makes withthedownward vertical, and let6betheangle through which thecylinderof massMhasturned sincesome fixedepoch. Theangular velocities ofthecylinders about their axes areeasilyseen tobe6and{(R r}-R6}jr respectively ;andthemoments of inertia ofthecylinders about their axes areMB?and\mr2respectively ;sothekinetic energyTofthesystemisgiven bytheequation im(R-r?A* while thepotential energyisgiven bytheequation F=mg(R r)cos$. Thecoordinate 6isclearly ignorable ;theintegral correspondingtoitis dT r=constant, 30 or MR^-\mR{(R-r)4&gt;-RQ}=k, where kisaconstant. Theintegralofenergyis T+V=/i, where hisaconstant, or \MRW +}m{(R-r)-BflY+m (R-r)2 tf&gt;2-mg(R-r}cos=h. Eliminating6between thetwointegrals, weobtain theequation m(3J/+m). F This isthesame astheequation ofenergyofasimple pendulum oflength (l thesolution canbeeffected bymeans ofelliptic functions asin 44. (iii)Rodmovinginafreecircularframe. Weshall next consider themotion ofarod,whose ends can slidefreely onasmooth vertical circularring, theringbeingfreetoturnabout itsverticaldiameter, which isfixed. 66] TheSoluble Problems ofRigid Dynamics139 Letmbethemass oftherodand2aitslength ;letMbethemass oftheringandr itsradius;let6betheinclination oftherodtothehorizontal, and$theazimuth ofthe ringreferred tosome fixed verticalplane,atanytime t. Themoment ofinertia oftherodabout anaxisthroughthecentre ofthering perpendicular toitsplaneism(r2fa2 ),andthemoment ofinertia oftherodabout the vertical diameter oftheringisr{(r2-a2 )sin25+Ja2cos2 0}.The kinetic energyofthe systemistherefore T=\m (r2-a2 )2+}Mr^+lmty (r2sin26-a2sin26+Ja2cos2 0). Thepotential energyis Vmg(r2 2 )icos6. Thecoordinate &lt;pisevidently ignorable ;thecorresponding integralis dT=constant, d&lt;j&gt; or IMr2 (j&gt;+ m&lt;j)(r2sin26-a2sin26+\a2cos26)=k, where kisaconstant. Substituting thevalue of$found from thisequationinthe integralofenergy T+V=h, wehave x. k2 lm(r2-fa2 )G2h+mg(r2a2 )acos-1--....- .9, ,n9-2~m -\Mi&+m(r2sin2^-a2sin20+^a2cos25) Inthisequationthevariables 6and tareseparable;afurther integrationwill thereforegive 6interms oft,andsofurnish thesolution oftheproblem. (iv)Hoop andring. Weshall next discuss themotion ofasystem consistingofauniform smooth circular hoopofradiusa,which liesinasmooth horizontalplane, and issoconstrained that itcan onlymove byrolling onafixedstraightlineinthatplane, while asmall ringwhose mass is1/Xthatofthehoopslides on it.Thehoopisinitiallyatrest,andtheringisprojected from thepoint furthest from thefixed linewithvelocityv. Let &lt;/&gt;denote theangle turnedthrough bythehoopafter atime tfrom thecommence ment ofthemotion, andsuppose thatthediameter ofthehoopwhichpasses through the ringhasthenturned through anangle \^.Taking theringtobeofunitmass,sothatthe mass ofthehoopisX,themoment ofinertia ofthehoopabout itscentre isXa2 ,andthis centre moves withvelocity 0,while thevelocityoftheringiscompoundedofcomponents a$anda\^,whose directions areinclined toeachother atanangle \^.Thekinetic energy ofthesystemistherefore T= andthepotential energyiszero. Thecoordinate &lt;j)isevidently ignorable, andthecorresponding integralis cT r=constant, 80 or(2X+1)a-^+a2^cos\/^=theinitial value ofthisexpression =av. 140 TheSoluble Problems ofRigid Dynamics [CH.vi Integratingthisequation, wehave vt (2X+1)(f&gt;+sin &gt;//--=the initial value ofthisexpression =0, 1 vt Thisequation determines interms of\^. Theequationofenergyis Titsinitial value=4*2 &gt; andsubstitutingfor$itsvalue (v/a-cos-^-.4-)/(2X+l)inthisequation, wehave -j= vv2X Jo Writing sin\^=.r,thisbecomes --%= P(2X+ vv^2X Jo Inorder toevaluate thisintegral, weintroduce anauxiliary variable,defined bythe equation u= IX (2X+.r2 )-4(1-.r2 )~4dx. Jo Write x2=2X/,where |isanewvariable;thelastintegral becomes which isequivalentto where thefunction $&gt;()isformed with theroots these roots arerealandsatisfytheinequality 6i&gt;e 2&gt;e3,so $&gt;(w)isrealandgreater than eiforrealvalues ofu. Nowwehave dt=%=.(2X+#2)4(1-a;2 )~4dx. vv/2X JVKvdt f 2X ,7or -=^2X+^^-, \du.a Integrating, wehave where(u)denotes theWeierstrassian Zeta-function. Thusfinallythecoordinate ^and thetime tareexpressedintermsofanauxiliary variable ubytheequations 2X 66,67] TheSoluble Problems ofRigid Dynamics 141 67. Initial motions. Wehavealready explainedin32thegeneral principlesused infinding the initial character ofthemotion ofasystemwhich starts from rest at agiventime. Thefollowing exampleswillserve toillustrate theprocedure forsystemsofrigidbodies. (i)Aparticle hangs byastring oflengthbfromapointinthecircumference ofadisc oftwice itsmassandofradius a.Thedisccanturnabout itsaxis,which ishorizontal, and thediameterthroughthepoint ofattachment ofthestringisinitiallyhorizontal. Tofindthe initial pathoftheparticle. Let6denote theangle through which thedischasturned, and theinclination ofthe stringtothe vertical,, attime tfrom thebeginningofthemotion :letmbethemass ofthe particle. Thehorizontal and(downward)vertical coordinates oftheparticle withrespect tothecentre ofthediscare acos6+bsin and asin6+bcos0, sothesquareoftheparticlesvelocityis a22+&22-2absin(6+0)00, andthekinetic energyofthesystemis T=ma-6z+-|m&22-mobsin(6+0)#0, while thepotential energyis V-mg(asin6+bcos0). TheLagrangian equations ofmotion are dt\d0 d_/dT\_dT_ _9F dt\d(p/ c0 30 J22&lt;9-a&cos (0+0)02-#acos$ a&sin(0+0)=0, =0. 620-a6cos(0 +0)02+#6sin0-a6sin(0 +0)6=0. Initiallythequantities 6,0,6,are allzero :theseequations therefore give initially &lt;9=gr/2a and=0,sotheexpansionof6begins withatermgt2/4aandthat ofwitha termhiher than thesuare of t.Assumin, termhigher than thesquareof t.Assuming =Ct--+Dt*+Et+Ff&lt; +..., substitutingintheabove differentialequations, andequating powersoft,wecanevaluate thecoefficients A,B,C,...;wethus find 0-^+0. +...4a *=ff2 *^ 32a6 1920a62 Now ifxandyarethecoordinates oftheparticle referred tohorizontal and(downward) vertical axesthroughitsinitialposition, wehave x=a(1-cos6}-bsin=\ad2-60=-E- ,approximately, andy=asin6+b(cos0-l)=a0=~, approximately. 142 TheSoluble Problems ofRigid Dynamics [CH.vi Eliminatingtbetween these equations, wehave and this istherequired approximate equationofthepathoftheparticleinthe neighbourhood ofitsinitialposition. (ii)AringofmassmcanslidefreelyonauniformrodofmassMandlength 2a,which canturnabout oneend.Initiallytherod ishorizontal, with theringatadistance rafrom thefixedend. Tofind theinitial curvatureofthepathoftheringinspace. Let(r,6)denote thepolarcoordinates oftheringattimet,referred tothefixedendof therodandahorizontal initialline,6being measured downwards from the initial line. Forthekinetic andpotential energies wehave 4/72- T=%m(r* +rW}+%M.~6\ V=mrgsin&Magsin6. TheLagrangian equationsofmotion are --._. ~dtdfW~ dr rr82gsin6=0, [fMa*d+mrzd+2mrf0-Mgacos6-mgrcos &lt;9=0. Sincer,0,and6areinitially zero,wecanassume expansionsoftheform substitutingthese expansionsinthedifferential equations, andequatingcoefficients of powersoft,wefind _ 2~ Thecoordinates oftheparticle,referred tohorizontal and vertical axes atitsinitial position,are x=rcos6randy=rsin6, orapproximatelyx= (4-rb^} t*,y=r6212 . Thecurvature ofthepathisgiven bytheequation - p yr andonsubstituting theabove values of62and45wehave 1Ma(4a3r()) p~~ 9r2(Ma+mr) This istherequiredinitial curvature ofthepathofthering. Example. Twouniform rodsAB,SC,ofmasses miandm.2,andlengths aand b respectively,arefreely hingedatB,andcanturnround thepoint A,which isfixed. Initially, AB ishorizontal andECvertical. Shew that,ifCbereleased, theequationof theinitial pathofthepointoftrisection ofBCnearer toGcanbeputintheform f=60(1+2?n2/m1)abx. (Camb.Math. Tripos, PartI,1896.) 67,68] TheSoluble Problems ofRigid Dynamics 143 68.Themotionofsystemswith threedegrees offreedom. Thepossibilityofsolving byquadraturesthemotion ofasystemofrigid bodies which hasthreedegreesoffreedomdepends generally (asinthecase ofsystemswithtwodegreesoffreedom)either ontheoccurrence ofignorable coordinates, givingrise tointegralsofmomentum andangular momentum, or onadisjunctionofthekineticpotentialintoparts whichdependonthe coordinatesseparately.Thefollowing examplesillustrate theprocedure. (i)Motion ofarodinagiven field offorce. Consider themotion ofauniform rod,ofmassmandlength 2a,which isfreetomove onasmooth table,when eachelement oftherod isattracted toafixed lineofthetable withaforceproportionaltoitsmassand itsdistance from theline. Let(#,y)bethecoordinates ofthemiddlepoint oftherod,and6itsinclination tothe fixed line. Thekineticenergyis andthepotential energyis 11m I^r=~I(y+rsm6}zdr, where pisaconstant, 4&lt;2J-a orV=pm(iy2+%a2$in2 d). TheLagrangian equationsofmotion aretherefore |#=0, y w&gt; ((20)+fisin20=0. The firsttwoequations give xct+d, wherec,d,/,eareconstants ofintegration;thecentre oftherodtherefore describes asinecurve intheplane. Theequationfor6isofthependulum type, andcanbe integrated asin 44. (ii)Motionofarodandcylinderonaplane. Weshall nextdiscuss themotion ofasystem consistingofasmooth solidhomogeneous circularcylinder,ofmassMandradiusc,which ismoveable onasmooth horizontalplane, andaheavy straightrailofmassmandlength 2a,placed with itslength incontact with thecylinder,inavertical plane perpendiculartotheaxis ofthecylinder andpassing through thecentre ofgravityofthecylinder, andwith oneextremity ontheplane. Let6betheinclination ofthe railtothevertical, andxthedistance traversed onthe plane bythelineofcontact ofthecylinder andplane, atanytime t.Thecoordinates of thecentre oftherodreferred tohorizontal andverticalaxes, theorigin being theinitial pointofcontact ofthecylinder andplane, areeasily seen tobe xccot(--)+asin$ and acosd. \4 zj Let$betheangle through which thecylinder hasturned attime t.The kinetic energyofthesystemis +Ama2sin26 .d2+MX 144 TheSoluble Problems ofRigid Dynamics [CH.vi Thepotential energyisgiven bytheequation V=-mgacos6. Thecoordinates xand &lt;areevidently ignorable ;thecorresponding integralsare ar=constantox (which maybeinterpretedastheintegralofmomentum ofthesystem paralleltotheaxis ofx)and tiT ^=constant (which maybeinterpretedastheintegralofangular momentum ofthecylinder about its axis). Theseintegrals canbewritten xiccosec2 [-- j.B+acos6. f-+Mx= constant,\42/ J c;2 &lt;&gt;=constant. Substitutingforxand &lt;/&gt;thevahies obtained from these equationsintheintegralof energy T+F=constant, wehave theequation If( ,/TT 6-kccosec2--- where disaconstant. Thisequationisagain integrable, since thevariables tand6are separable ;initsintegrated form itgives theexpressionof6interms of t:thetwo integrals found above thengivexand$interms of t. 69.Motionofabodyabout afixed point under noforces. One ofthemostimportant problemsinthedynamicsofsystemswith threedegreesoffreedom isthat ofdeterminingthemotion ofarigid body, oneofwhosepointsisfixed,when noexternal forces aresupposedtoact*. Thisproblemisrealised(64)inthemotion ofarigidbodyrelative toits centre ofgravity,under theaction ofanyforces whose resultantpasses throughthecentre ofgravity. Inthissystemtheangular momentum ofthebodyabouteverylinewhich passes throughthefixedpointand isfixed inspaceisconstant(40),and consequentlythelinethroughthefixedpointforwhich thisangular momen tum-has itsgreatestvalue isfixed inspace. Letthis line,which iscalled the invariable line,betaken asaxisOZ,and letOXandYbetwoother axes throughthefixedpointwhich areperpendiculartoOZand toeach other. Theangularmomenta about theaxesOXandOFarezero, for ifthiswere notthecasetheresultant oftheangular momenta about OX,OY,OZwould givealineabout which theangular momentum would begreaterthan the *Euler, Memoires deBerlin, Annee 1758. Elliptic functions were appliedtotheproblem firstbyEueb, Specimen inaugurale... (Utrecht, 1834):andthesolution wascompleted byJacobi, Journal furMath, xxxix.(1849), p.293. 68,69JTheSoluble Problems ofRigid Dynamics145 angular momentum about OZ,which iscontrarytohypothesis.Itfollows (39)that theangular momentum about anylinethrough makingan angle6withOZisdcos6,where ddenotes theangular momentum about OZ. Thepositionofthebodyatanytime tissufficiently specified bythe knowledgeofthepositionsatthattime ofitsthreeprincipalaxes ofinertia atthefixedpoint:letthese linesbetaken asmovingaxesOxyz;let(6,&lt;f&gt;,ty) denote thethree EulerianangleswhichspecifythepositionoftheaxesOxyz with reference totheaxesOXYZ, let(A,B,C)betheprincipal moments of inertia ofthebodyat0,supposed arrangedindescendingorder ofmagnitude, and let (&&gt;1(to2,&&gt;3)bethethreecomponentsofangular velocityofthesystem about theaxes Ox,Oy,Ozrespectively,sothat(10,62) A &&gt;!=dsin6costy, Bw&lt;,=dsin9sinty, Ca)s=dcos0, or(16) /sin-v/rcj&gt;sin6cos^=-jsin6costy, &lt;6cos-fy+(f)sin6sin-v/r=^sin6sinty, ; i d \ Y+9cos"=TYcos" These arereallythreeintegralsofthedifferentialequationsofmotion of thesystem (onlyonearbitraryconstant however occurs, namely d,ourspecial setofaxesbeingsuch astomake theother twoconstants ofintegration zero); wecantherefore take these instead oftheusualLagrangiandiffer entialequationsofmotion inorder todetermine6, &lt;f&gt;,ty. Solvingfor6,0, -\jr,wehave ,*(A-B)d. rj-- ClT\ifPrid I//"1GlTl lf/&gt; vj.-j-jO-L1-1 t/\_v*Jo \lfolll \Lf A.tj ,d d . q&gt;=-T-cos2Y+^DsinTJA &gt; (dd d .,\Y=\n JcosrDsmrcos" VOA B ] Theintegralofenergy (whichisaconsequenceofthese threeequations) maybewritten down atoncebyuseof63;itis where cisaconstant :replacinga&gt;i,o&gt;2,wsbytheir values interms of6and thisequationcanbewritten ineither oftheforms w.D. 10 146 TheSoluble Problems ofRigid Dynamics [OH.vi A-B . Bc-d*B-Csm"cs"=~-+-cs &gt; - .ZQ. 2^- - or .Msm26sin2 -v/r= --- cos20.AB Ad2A-B . . ^lc-d2A-G AC SinceA &gt;B &gt;C,thequantity (cA-d?)orB(A-B)&&gt;22+C(A-C)o&gt;32is positive,and(cCd1 }isnegative:thequantity (.Be d2 )maybeeither positiveornegative:fordefiniteness weshallsupposeittobepositive. The first ofthethree differentialequations may,byuseofthe last equations,bewritten d Bc-dr-B-C Ac-d2A-G* AdT Thisequationshews that cos6isaJacobianellipticfunction* ofalinear function oft;andthetwopreceding equations shew that sin6costyand sin6sintyaretheother twoJacobian functions. Wetherefore write sin6cosi|r=Penu, sin6sin-fyQsnu, cos6=Rdnu, where P,Q,Rareconstants anduisalinear function oft,say\t-fe ;the quantities P,Q,R,A,andthemodulus koftheelliptic functions, arethen to bechosen soastomake theaboveequationscoincide with theequations &cn2u=-k2+dn2u, A?sn2u= 1dn2u, ~rdnu=k2snuenu.du Thecomparison gives A(d*-cC} B(d*-cC) C(cA-ffi) d*(A-C) d*(B-C) d*(A-C} (A-B)(d2-cC) (B-C)(cA-d*) (B-C) (Ac-d*) ABC Theequationfork2shews thatkisreal,andtheequation (A-Q(Bc-.ffi) (B-C)(Ac-d2 ) shews that (1k2 )ispositive,i.e.that k&lt; 1.Thequantities P,Q,R,X,are alsoevidentlyreal. Now arealquantityamaybedefined bythemutuallyconsistent equations snla=i-T-TT:^t ,cma= *Thetheoryofelliptic functions required inthisandthesucceeding problemswillbefound inWhittaker andWatson sModern Analysis,Chs. xx.xxn. 69]TheSoluble Problems ofRigid Dynamics147 _i Since (A)I where thetheta-functions aredefined bytheexpansions $(y)=1+2qcos2iri/+2g4cos4?+2g9cos ^01(y)=1-2gcos27Ti/+2q-4cos47TZ/-2g9cosGTTV+..., %0)=2icosTTV+2q*cosSirv+ 2^"*"cos5?+..., Sn(v) %&lt;fsin^TV~2?4sinSTTV+2g"sin andq=e~7&gt; ,wehave 1+2gcosh27+2g4cosh 4-y+...=,,_i 1-2gcosh 27+2^4cosh47-...~ where 7stands forTra/2K:from thisequation 7(and consequently a)may readilybedetermined bysuccessive approximation. TheEulerianangles6and-^attime tarenowgiven bytheequations sin6cos^r= sin6sin\|r=cnza dniasn(\t+e) cnm Asnladn(\i+e)cos6=- :^-- ienla or(omittingthee) Themodulus koftheellipticfunctions isknown; wecantherefore determine theparameter qofthetheta-functions bytheequation orbythemorerapidly convergentseries q=$tan2 /3+TVtan10+-$fatan18 /3+ ..., where cos/9=(ky.Kmaythenbecalculated from theseries andthus theperiod 4&lt;K/\oftheinclinations oftheaxesOxyztotheline OZ isdetermined. 102 148 TheSoluble Problems ofRigid Dynamics [CH.vi Ifnowwewrite(&gt;jra/2K)=7and(-TrX/2K}=/A,wehave (1-2acosh27-f2o4cosh47- ...)(cosu,t+ ... sin i/cos\f (cosh7+(fcosh 87+ ...)(1-2qcos2yu,+2#4cos4/4+...) (1+2acosh27+2o4cosh 4&gt;y+...)(sin /4o2sin3/4+...)sinusin "vj/" (cosh7+ &lt;?2cosh87+...)(1 2&lt;?cos2/4+2(?4cos4^+...) (sinh7y2sinh87+...)(!+ 2gcos2__~ (cosh74-92cosh87+...)(1-2gcos2/j.t+2q*cos4/u^+...) Thequantities q,p,7mayberegardedastheconstants whichspecifythe motion. Example. Supposethatthebodyisahomogeneous ellipsoidofunitdensity, whose three semi-axes are a=l, 6=2, c=3. Thethreeprincipal moments ofinertia are ^=^7ra6c(62+c2 )=20-87T, =1677-, =&*. Supposethat the initial velocities ofrotation round theprincipalaxes are ft)!=J,C02=i, W3=l. Theconstant ofenergyis c=Aa)!2+Ha&gt;22+f&lt;032=133r, andtheconstant ofangular momentum isgiven bytheequation so Themodulus oftheellipticfunctions isgiven bytheequation (4-5X^-c)_ -(.g-cw-rf2 )" whence wehave 2=i_2 =0.760, ....=1-0342, #=1-68013, K TT Wehave also so A=0-6045 and M=|^=0-5651. Theperiodoftheangles 6and^is-r-or,which hasthevalue 11118. 69] TheSoluble Problems ofRigid Dynamics 149 Inorder toexpress and^astrigonometricseries interms oft,wemust determiney. Forthiswehave andtherefore if"*beneglected wehave 1+2?cosh2y_1-1094~ 0-9337 giving cosh2y=2-503, andhence2y=1-568 and y=0-784. Thequantity aisthengiven bytheequation 2A"a=y=0-8385. 7T Alimiting case ofthegeneral problemisthat inwhichA=B,sothatkreduces to zeroandtheelliptic functions become circular functions. Inthiscasethesolution may bewritten n &lt;9^=cosX&lt; \ {\-\(A-C)(Ac-d^ &gt;b^~cosha\A2Cj sinAn (C(Ac-d2)}%sm(9sm^= r&gt;. where&lt;smha=\.\.,^J-coshof (A(d2-cC) cos0=tanha cosha=~^ (A(d^ cC)} sothemotion isasteady precession about theinvariable lineOZ,thebody rotatingalso about itsownaxisofsymmetryOz. Another limitingcase isthat inwhich d2=cB,sothatF=landtheelliptic functions degenerateintohyperbolic functions;this isillustratedbythefollowing examples. Example1.Arigid bodyismoving about afixed pointunder noforces: shew thatij (inthenotation used above) d2=Bc,and if&lt;a 2iszerowhen tiszero, a&gt;iand a&gt;3being initially positive,then thedirection-cosinesoftheB-axis attimet,referredtotheinitial directionsof theprincipal axes, are atanhxysin/*sech^,cos /j.sech^,ytanhx+asinp-sech^, where _dt _dt ((A-B)(B-C)\% (A(B-C)\ (C(A-B)}^f-BX-B\~-AC~- )= \B(A-C}\y=\B(A-C]} (Camb. Math.Tripos,PartI,1899.) Toobtain thisresult, weobserve thatwhenBc=d2 ,thedifferential equationforthe coordinate Qbecomes dt\BC)\Ad2 AC) theintegralofwhich is cos6=ysech^, 150 TheSoluble Problems ofRigid Dynamics [CH.vi whereyandxarethequantities above defined. Theequation then gives andtheequationA-B...... Ac-d2A-Cj^-sm20sm2 \//-= ---^cos2 J.D u4d2J(7 sin6sin-^=tanh^, d)=-7cos2-= JD gives sin (&lt; /*)=ysin\//-. Theseequations shew that thedirection-cosines ofthe.5-axis referred totheaxes OXYZ, which(10)are cos &lt;cos6sin\|^sin&lt;cos\^,sin &lt;pcos$sin^+cos &lt;cos\|f,sin$sin\^, canbewritten sinpsech^,cosp.sech^,tanh^. But ifo)10,W2o, 0)30denote theinitial directions oftheprincipal axes, since AW+CW=d2=Be=B(Aon2+Ccos2 ), sothat^w1=ao?and C&lt;o3=yd,weseethatthedirection-cosines of &lt;BIO,0)20, 0)30,referred to ,aregiven bythescheme X YZ 0)20 0)30 andhence thedirection-cosines oftheZ?-axis, referred to o&gt;10,0)20, 0)30,are -ysin /j.sechx+atanh^,cos/*sech^,asin^sechx+7tanh^. Example2.AVhen d2=cB,shew thattheaxis6tydescribes, onasphere with thefixed point ascentre, arhumb linewith respecttothemeridianspassing through theinvariable line.(Coll. Exam.) Returning now tothegeneral case,wehave toexpressthethird Eulerian angle&lt;/&gt;interms ofthetime.Wehave d,/l 1 Now whenceCOt T= sn2-vr=dniasn\t dn2iasn2\t This function oftvanishes witht,andhaspoleswhen thedenominator vanishes,i.e.when sn\t=+ -,=+sn(ia+iK}; ~ksnla soinoneperiod-parallelogram (2K,2iK)ithaspolesatthepoints \t=ia+iKand\t=ia+iK . 69] TheSoluble Problems ofRigid Dynamics 151 Near theformer ofthesepoints, writing\t=ia+iK+e,andretaining onlythelowestpowersofe,wehave . 2dn2ia/&2sn2m l2"= dn2ia k2sn-ia+e&2 .2sniaeniadnm A;2sn2ia sotheresidue atthispoleofsin2 ty,considered asafunction of\t,is dnia I ((B-G)(Ac- &)AB}* 2A;2sniaenia 2id(A)\ Cj Therefore theresidue ofdl-f.- )sin2 -\lratthispoint (consideredasa VDA] function of\t)is ABC andtheresidue when\t/2Kisregardedasthevariable isconsequently i\/4&lt;K. Aswenowknow thezeros, polesandresidues ofthisfunction, wecanwrite down itsexpressionasasum oflogarithmicderivates oftheta-functions :in fact, since^01(i/)hasasimplezero atv^&&gt;=iK/2K,wehave ,r:z= Mp^^f**!2i &lt;v J01 I &gt;/^- MOl I^Tr -J01 +2A 4A" /\t-ia andtherefore /Xm\(2id I~9~V~ I~\~A e constant.: .e AT /\t^ia\ /./\t-\-ia\ . iNowS01( I/S-011 jispurely periodicwithrespecttothereal \ZtJ^L //V -f\ / period 2K/X of^,sotheexponentialontheright-handsidegivesthemean motion of &lt;,i.e.theprecessionalmotion ofthesystem round theinvariable line.Wehave ^01(*)=125-COS 27TI&gt;+2(?4COS4t7TV-.., ^o/(^)=4?r^sinZTTVS-rrq*sinkirv+..., sothecoefficient of in &lt;/&gt;,i.e.theconstantpartof &lt;j),ortheprecession, which is maybewritten oIqsinh2y2^4sinh 4&lt;y+... A **,1-2qcosh27+2^4cosh47- ... inwhich form itmaybecalculatedreadily. 152 TheSoluble Problems ofRigid Dynamics [CH.vi Example1.Inthecasepreviously discussed, ofanellipsoid \vhose semi-axes area=1, 6=2,c=3,wehave 2y=l 568, sinh2y=2-294, cosh2-y=2503, , 7r,^= ,= sothemean motion of$,which whenj4isneglected maybewritten dqsinh2-yIM1-2qcosh2y is 0-5986+0-0970, or 0-6956. Example2.Auniform circular dischas itscentrefixed, andmoves under the action ofnoexternal forces. Thedisc isgiveninitialangular velocities Qabout adiameter coinciding with inspace, andnabout itsaxiscoinciding withOfinspace. Shew that atanysubsequent time X=2arcsinf- -sin{(Q2+42)^LQ2 4?l2i =arccot--tan{(Q2+4?i2)i.It] \,LQ2+4%2* where^istheangle between andtheaxisofthediscOzand o&gt;istheangle between the planes 0%and(Oz. (Coll. Exam.) For letOZdenote asusual theinvariableline,andconsider thespherical triangle Z&, whose vertices aretheintersections ofthelines OZ,0,Ozrespectively with asphere of centre 0.Inthisspherical triangle wehaveZz=d,Zz= (fr.Moreover wehave forthe discC=&lt;2B=2A, so and Theequationsofmotion for6and &lt;f&gt;therefore become 0=0, &lt;j&gt;=d/A Inthespherical triangle Zz,wehave therefore v Qandhence sin^=sinZsin\Zz= and cot a&gt;=cosZftan1^= tan (Q2+4n2)* which aretherequired equations. 70. Poinsot skinematicalrepresentation ofthemotion; thepolhode and herpolhode. Anelegantmethod ofrepresenting kinematicallythemotion ofabody about afixedpointunder noforces isthefollowing, which isduetoPoinsot*. *Poinsot, Theorie nouvelle dehirotation descorps, Paris, 1834. 69,70] TheSoluble Problems ofRigid Dynamics 153 Theequationofthemomentalellipsoidofthebodyatthefixedpoint, referred tothemovingaxesOxyz,is Consider thetangent-planetotheellipsoidwhich isperpendiculartothe invariable line. Ifpdenotes theperpendicularonthistangent-planefrom theorigin, wehave(since thedirection-cosines ofpareAwjd, Bw^d,Ca)3/d) =-r,which isconstant. d- Since theperpendicular ontheplaneisconstant inmagnitudeand direction, theplaneisfixed inspace:sothemomentalellipsoid always touches afixedplane. Moreover, if(x ,y,z)arethecoordinates ofthepointofcontact ofthe ellipsoid andtheplane, wehaveonidentifyingtheequations Axx+Byy+Czz=1andA(olx+Bw2y+Cwsz=pd ,11 , &&gt;l l /Mo G&gt;2 I &&gt;) U&gt;3rhpvnUPC; m_-_ ?/ ?_1_- uiicvdiiico i// ,.,u-- -. T .z i T~ . pdyc pd \/c pdyc andhence theradius vector tothepoint (x,y,z)istheinstantaneous axis ofrotation ofthebody.Itfollows that thebodymoves asifitwererigidly connected toitsmomentalellipsoid, and thelatter bodywere torollabout the fixed pointonafixed plane perpendiculartotheinvariable line, without sliding ;theangular velocity being proportionaltotheradius tothepoint of contact, sothatthecomponent ofangular velocityabout theinvariable line is constant. Example1.Ifabodywhich isinoveable about afixed pointisinitiallyatrestand then isacted oncontinually byacoupleofconstant magnitude andorientation, shew that Poinsot sconstruction stillholds good, butthatthecomponent angular velocity about the invariable line isnolonger constant butvariesdirectlyasthetime.(Coll. Exam.) Forinanyinterval oftime dttheaddition ofangular momentum tothebodyis+Ydt about thefixed axisOZofthecouple;sothat theresultant angular momentum ofthe system attime tisNtabout OZ.Now thecomponentsofangular momentum about the principal axes ofinertiaOxyzare A&lt;al,Bu&gt;z,C(o3,where A,B,Caretheprincipal moments ofinertia and (&lt;!,o&gt;2,MS)arethecomponentsofangular velocity:hencewehave Aa&gt;i= JVtsin 6cos!//,Bu&gt;.2=Ntsin6sin\^,C(o3=Ntcosd, where6,0,-^aretheEulerianangles which fixthepositionoftheaxesOxyz with reference tofixed axesOXYZ. Butthese equationsdiffer from those which occur inthe motion ofabodyunder noforcesonlyinthesubstitution of tdtfordt;sothemotion willbethesame asintheproblemofmotion under noforces, exceptthatthevelocities are multiplied byt;whence theresult follows. Example2.Inthemotion ofabody, oneofwhosepointsisfixed, under noforces, letahyperboloid berigidly connected with thebody,soastohave theprincipal axes of 154 TheSoluble Problems ofRigid Dynamics [CH.vi inertia ofthebodyatthepointasaxes, and tohave thesquares ofitsaxesrespectively proportionaltod2Ac,d2Bc,cP Cc,where A,B,Carethemoments ofinertia ofthe bodyatthefixedpoint,cistwice itskineticenergy, anddistheresultant angular momentum. Shew thatthemotion ofthishyperboloid canberepresented bycausingit torollwithoutsliding onacircularcylinder, whose axispasses throughthefixedpointand isparalleltotheaxisofresultant angular momentum.(Siacci.) Thecurve which inPoinsot sconstruction istraced onthemomental ellipsoid bythepointofcontact with the fixedplaneiscalled thepolkode. Itsequations,referred totheprincipal moments ofinertia, areclearlythe equationoftheellipsoid togetherwith theequation p=constant, i.e.they are Ax*+By-+Cz2=1, ExampleI.Shew thatwhenA=B,thepolhodeisacircle. Example2.Taking A^B^C,shew thatthere aretwokinds ofpolhodes, onekind consistingofcurves which surround theaxisOzofthemomentalellipsoid, andcorrespond to cB&gt;d"&gt; cC,while theother kind consists ofcurves which surround theaxisOx,and correspondtocA &gt;dz &gt;cB;andthatthelimitingcasebetween thesetwokinds ofpolhodes isasingular polhode which correspondstocBd2=0,andconsists oftwoellipses which passthroughtheextremities ofthemean axis. Thecurve which istraced onthefixedplane bythepointofcontact with themoving ellipsoidiscalled theherpolhode. Tofindtheequationoftheherpolhode,letp,%bethepolarcoordinates ofthepointofcontact, when thefoot oftheperpendicularfrom thefixed pointonthefixedplaneistaken aspole.If(x,y,z}denote thecoordinates ofthesame pointreferred tothemovingaxesOxyz,wehave 3/2 _|_y?.+z-isquareofradius frompointofsuspensiontopointofcontact 2c ""+# Substitutingforx,y,ztheir values asgiven bytheequations x=eoj/Vc=dsin6costy/A^/c, y=a).2/\/c=dsinBsinty/B^c, {z= &&gt;3/\/c=dcos6/C\/c, wehave cd2d2d2 p-=--=-+-.sin2cos2 T/T+Tsin2sin2-\lr+T~cos2#. d?A-c B-c G2c ReplacingBand^rbytheir values interms oft,thisbecomes (B-C)(A- _ _ cd?A*B*G*( _(cA-d*}(c?2-cC) cd*AC 70,71] TheSoluble Problems ofRigid Dynamics 155 where wdenotes thehalf-period correspondingtotheroote^,thisequation expressestheradius vector oftheherpolhodeinterms ofthetime. Wehave next tofindthevectorialangle%interms of t.For thiswe observe thatVcp2 ^/c?issixtimes thevolume ofthetetrahedron whose vertices arethefixedpoint,thefoot oftheperpendicular from thefixed pointonthefixedplane, andtwoconsecutivepositions ofthepointofcontact, dividedbytheinterval oftimeelapsed between thesepositions, andthat this quantitycanalsobeexpressedintheform x, y,z Acx\d\BeyId2 ,Ccz/d2 A, y, zA, B, C t/x &gt;y/y,tl Allthequantities involved, except %,areknown functions of t:on substitutingtheir values interms oft,andreducing, wehave A-C which canbewritten intheform ._d i$(I+&))%~ ~B+2 Thisequationcanbeintegratedinthesamewayastheequationforthe Eulerianangle c/&gt;,andgives a(Ita) where^oisaconstant ofintegration. Thecurrent coordinates(p,^)ofthe herpolhodearethusexpressedasfunctions of t. Example1.Aparticle moves insuch awaythat itsangular momentum round the originisalinear function ofthesquare oftheradiusvector, while thesquare ofitsvelocity isaquadratic function ofthesquareoftheradiusvector, thecoefficient ofthehighest power being negative ;shew thatthepathistheherpolhodeofaPoinsot motion, inwhich however J,JB,Carenotrestricted tobepositive. Jfxample2.Discuss thecases inwhich thepolhode consists of(a)twoellipsesinter secting onthemean axisofthemomenta!ellipsoid, (/3)twoparallel circles, (y)twopoints; shewing that inthese cases theherpolhode becomesrespectively aspiral curve (whose equation canbeexpressedinterms ofelementary functions), acircle, orapoint. 71.Motionofatoponaperfectly rough plane ;determinationofthe Eulerianangle6. Atopisdefined tobeamaterialbodywhich issymmetrical about anaxis andterminates inasharp point (called theapexorvertex) atoneendof the axis. We shallnowstudythemotion ofatopwhenspinningwith itsapex placed onaperfectly rough plane,sothat ispracticallyafixedpoint. The 156 TheSoluble Problems ofRigid Dynamics [OH.vi problemisessentiallythatofdeterminingthemotion ofasolid ofrevolution under theinfluence ofgravity, when apointonitsaxis isfixed inspace*. Let(A,A,G)denote themoments ofinertia ofthetopaboutrectangular axesOxyz,fixed relative tothetopandmovingwithit,theorigin beingthe apexandtheaxisOzbeingtheaxisofsymmetryofthetop;let(6,&lt;,\Jr)be theEulerianangles definingthepositionofthese axeswith reference tofixed rectangularaxesOXYZ, ofwhichOZ isdirectedvertically upwards. Thekineticenergyis(63) where wl,w2,&&gt;3denote thecomponentsrelative tothemovingaxes ofthe angular velocityofthetop,sothat(16)wehave &&gt;j=6sin^r$sin6costy, co2=6costy+sin6sinty, &&gt;s=^-t- 4&gt;cos6 ; thekineticenergyistherefore T=\AB-+^Aftsin26+^C(jr+ &lt;j&gt;cos0?, andthepotential energyisV=Mghcos6,whereMisthemass ofthetop andhisthedistance ofitscentre ofgravityfrom theapex. Thekineticpotentialistherefore L=T- V=$A6* +4ssin2+4C(t+&lt;cos0^-MghcoaB. The coordinates &lt;f&gt;and^areevidently ignorable; thecorresponding integralsare dT .dT ^-r=constant, and --.-=constant, d(f&gt; d-fr or A&lt;j&gt;sin2+C(^+cos0)cos-a, Cty+fj)cos &lt;9) =b, where aand bareconstants :thesemaybeinterpretedasintegralsofangular momentum about theaxesOZand Oz,and soareobvious apriorifrom general dynamical principles. Themodified kineticpotential (38)is R=L a(&gt; b\s -cos Theterm-b2/2Ccanbeneglected,as itismerelyaconstant; the equationofmotion is d/dR\ dR_ *Lagrange, Mec. Anal.(Oeuvres, xn.p.251). 71] TheSoluble Problems ofRigid Dynamics 157 sothevariation of6isthesame asinadynamical systemwith onedegree offreedom forwhich thekineticenergyis^A62andthepotential energyis (a bcos0)2 The connexion between dand tisthereforegiven bytheintegralof energyofthisreducedsystem, namely Aa-2 (a-bcos0)2 \AQ2=-^--Mghcos6+c,2Asin2 where cisaconstant. Writingcos=x,thisequation becomes A*a?=-(a-bx)2-2AMgh (x-a?)+2Ac(1-x9 ). Theright-handside ofthisequationisacubicpolynomialinx;now whenx=1,thecubic isnegative;forsome realvalues of6,i.e.forsome values ofxbetween 1and1,thecubic must bepositive,since theleft-hand sideoftheequationispositive; when x=I,thecubic isagain negative; and when x=+oo,thecubic ispositive. Thecubic hastherefore tworealroots which liebetween 1and1,andtheremainingroot isalso realand is greaterthanunity. Letthese roots bedenotedby cos a,cosft,cosh7, where cos/3&gt;cosa,sothat a&gt;/3. The differentialequation nowbecomes \MghftA$dt={4(x-cosa)(x-cos/3)(x-cosh7}}~*dx. Ifwewrite wehave therefore t+constant ={4(zej(ze.2)(z where theconstants el}e?,e3aregiven bytheequations 2Ac+b2 _Mgh 2Ac+b2 =~~ sothate-i,e,e3areallrealandsatisfytherelations el+e2+e3=0, el&gt;e, &gt;e3 158 TheSoluble Problems ofRigid Dynamics [CH.vi Theconnexion between zand tistherefore where eisaconstant ofintegration,andthefunction@isformed with the roots e1}e2,fy ,andhence wehave 2A Now inorder thatxmaybereal forrealvalues oft,itisevident thatxmust liebetween cosaand cosfi,i.e.$(t+e)must liebetween e2and e3forreal values oft:andtherefore theimaginary partoftheconstant emust bethe half-period&lt;w3correspondingtotheroot es.The realpartofedependson theepochfromwhich thetime ismeasured, and socanbetaken tobezero bysuitably choosingthisepoch. Wehave thereforefinally ^= and this istheequation whichexpressestheEulerianangle6interms of thetime*. Example1.Ifthecircumstancesofprojection ofthetoparesuch thatinitially sheiv that thevalueofQatanytime tisgiven bytheequation Mgk sec0=l+sech(^p/), sothat theaxisofthetopcontinually approachesthevertical. Forinthiscasewereadilyfindfortheconstantsa,b,cthevalues sothedifferential equationtodetermine xis whence theresult follows. Example2.Asolidofrevolution canturnfreelyabout afixed pointinitsaxisof symmetry, and isacted onbyforces derived fromapotential-energy function p.cot2 6,where 6istheanglebetween thisaxisandafixedline ;shew that theequations ofmotion canbe integratedintermsofelementary functions. Forproceeding asintheproblem ofthetopontheperfectly rough plane, wefindfor theintegralofenergyofthereducedproblemtheequation (a-ft. Writing cos6=x,thisbecomes Thequadratic ontheright-handside isnegative when#=1andx=1,but ispositive forsome values ofxbetween 1and+1,since theleft-hand side ispositiveforsome real *Itmayberemarked that thepresent problem reduces tothat ofthespherical pendulum (55)when thequantities M,C,A,h,a,b,c,cos0, &lt;f&gt;, I,karereplaced respectively by 1,0,P,I,k,0,h,z/l,&lt;f&gt;,\, fjL. 71,72] TheSoluble Problems ofRigid Dynamics159 values of6 :thequadratichastherefore two real roots between -1and+1.Calling these cosaandcos/3,theequationisoftheform \-xz=(cosa-x](xcos/3), thesolution ofwhich is x=cosasin2 (&lt;/2X)+cos/3cos2 (Z/2X). 72.Determination oftheremaining Eulerianangles, andoftheCayley- Klein parameters;thespherical top. When theEulerianangle6hasbeen obtained interms ofthetime, asin thelast article, itremains todetermine theother Eulerianangles&lt;f&gt;and -\|/-. For thispurposeweusethetwointegrals correspondingtotheignorable coordinates :these, when solved for &lt;f)and -\jr,give (._abcos6 1^= ~Tsin2 I.b(abcos0)cos6 \^=C~ Asin*0 Ifweregardthemotion asspecified bytheconstants ofthebody (M,A,C,h)andtheconstants ofintegration (a, b,c),itisevident from these equations andtheequationfor thatCdoes notoccurexceptin theconstant term oftheexpressionforty;andtherefore anauxiliary top whose moments ofinertia are(A,A,A)canbeprojectedinsuchawaythat itsaxisofsymmetry always occupiesthesamepositionastheaxisofsymmetry ofthetopconsidered, theonlydifference inthemotion ofthetwotopsbeing that theauxiliary tophasthroughoutthemotion aconstant extraspin b(CA)/AG about itsaxis ofsymmetry. Atopsuch asthisauxiliary top, whose moments ofinertia are allequal,iscalledaspherical top. Itfollows therefore thatthemotion ofanytopcanbesimply expressedinterms ofthe motion ofaspherical top,andthat there isnoreal lossofgeneralityin supposing anytopunder consideration tobespherical. IfthenwetakeG=A,theequationstodetermine &lt;f&gt;andi/rbecome _abcos_a+b ab = :Asin2~24(cos0+1)~ 2A(cos-f) bacos a+b ab (*=Asin2= 2X(cos&lt;9+1)+2A(coslT^Tj Substitutingforcos itsvalue from theequation 2A andwriting Om-M9h 2A 12A* Mgh^~ 2A 160 TheSoluble Problems ofRigid Dynamics [CH.vi sothat Iandkareknownimaginaryconstants(beinginfactthevalues of t+o&gt;3correspondingtothevalues and TTof6},thedifferentialequations become =Mgh(a +b)1 Mgh(a-b)1 2 6)1^%/i(a-6)1 Now theconnexion between thefunction %&gt;and itsderivate $canbeat oncewritten downbysubstitutingforxfrom theequation 2AX=Wh intheequation 2 ^}+2Ac(1-a?) ; iftheargumentofthe^-functionisk,itfollows from thedefinition ofkthat thecorrespondingvalue ofxis 1;andsothelastequation gives A2 .{2Ap (k)/Mgh\*=-(a+6)2 , or &&gt;(k)=iMgh (a+b)/2A-. Similarlywehave &gt;(/)=iMgh (a-b)/2A2 , andtherefore theequationsfor &lt;/&gt;and-fycanbewritten intheform 2^=*-,. 7\;...T.+ Now thefunction isanelliptic function, whosepolesinanyperiod-parallelogramarecongruent with t+w3=kand t+ o&gt;3=k,thecorrespondingresiduesbeing1and 1; andthefunction iszerowhen t+&&gt;3=0.Hence wehave $(t+&&gt;3)- andtherefore (k}dt, cr(-|-&lt;w3 A;) ^&lt;,/7x , v =logV- ~y^+2^(A;)+constant.& - bv y 72] TheSoluble Problems ofRigid Dynamics 161 Theintegralsoftheequationsfor &lt;/&gt;and-fycantherefore bewritten in theform .. tr(t+&&gt;3+k)a-(t+a)3 I) a-(t+o)3+k)a-(t+o&gt;3+/) where &lt;andi/rareconstants ofintegration. Theseequationslead tosimple expressionsfortheCayley-Klein parameters a,fi,7,5(12),which define thepositionofthemovingaxesOxyzwith reference tothefixed axesOXYZ: forbydefinition wehave a=cos16 .e$*&++\ {3=ism$6. e*l {&lt;t&gt;~ *\ y=ism^e.e^1^-^, 8=cos^.e Butwehave 2cos20=1+cos0 or cos\= Similarly wefind sin6=2A &lt;r(t+(03+k)a-(t+ 3k) Mgli~ a2 (k)a2(t~+w3) -A\l{a-(t+a&gt;3+k)a-(t+a)3-k)}$ (ft , i7\ /4 i 7\)i I(T\Tf~T~^3v)O~\v~T~(J&v vIf* andoncombining these with theexpressionsfore21*and e2i*already found, wehave _ =7"o-(0o-(+ &lt;o3) i(^"0o) &lt;?(&lt;+ft&gt;- \M~gh~ &lt;r(l)~tr(t+os).et) Theseequations expresstheparameters a,^7,8asfunctions ofthe time. W.D. 11 162 TheSoluble Problems ofRigid Dynamics [OH.vi Example1.Agyrostat ofmassMmoves about afaced pointinitsaxisofsymmetry: themoments ofinertia about theaxisoffigure andaperpendiculartoitthroughthefixed pointareCandArespectively, and thecentreofgravityisatadistance hfromthefixed point.Thegyrostatisheld sothat itsaxismakes ananglearccosl/v/3with thedownward vertical, and isgivenanangular velocity \jAJMgh */3/C about itsaxis.Iftheaxis benow leftfreetomove about thefixed point,shew that itwilldescribe thecone sin26sin20= (-cos6-l/v/3)* (-cos6+v/3)*, or sin26cos20= 3(^3/2+cos0)^, where istheazimuthalangleand6theinclinationoftheaxis totheupwardvertical. (Camb. Math.Tripos, PartI,1894.) Forinthisproblem wehaveinitially cos0= -l/v/3,=0,6=0,=0,^= andthese initial values give a=-jMAghltlZ,6=4/3jKAgh, c=- Substitutinginthegeneraldifferential equationfor6,namely wehave A62sin26--Mgh (cos6+l/v/3) (N/3+2cos6)(-cos6+v/3), while theequation abcos6 Dividingthisequation bythesquarerootofthepreceding equation, wehave =si I(_cos6-l/v/3)* (v/3+2cos6)"* (-cos6+^3)~icosec 6dd, =3*l(x- 1/^/3)* (v/3-2a;)"*(^+v/3)~*(1-^)-i ote, where .r=-cos Now ifwewrite u=(x- l/N/3)i(x+N/3)* (v3/2-x)~^ wehavebydifferentiation Wehave therefore * or tan20=32" (-cos6-l/v/3) (-cos which isequivalenttotheresult given above.&lt;9)~ *, 72,73] TheSoluble Problems ofRigid Dynamics163 Example2.Shew thatthelogarithmsoftheCayley-Klein parameters,considered as functions ofcos6,areelliptic integralsofthethird kind. Example3.Obtain theexpressionsfound above fortheCayley-Klein parametersas functions ofthetime tbyshewing thatthey satisfydifferential equations typified by where I7"denotes adoubly-periodicfunctionoff,these equations beingoftheHermite-Lame typewhich issoluble bydoubly-periodicfunctions ofthesecond kind. Asimple typeofmotion ofthetopisthat inwhich theaxisofsymmetry maintains aconstant inclination tothe vertical; inthis case, which is generallyknown asthesteadymotion ofthetop,6and6arepermanently zero;sincewehave (a bcos0)2 - /iCOSitfollows that d((a bcosQY=-Ja\ n ,,add{2Asm2 Performingthedifferentiation, andsubstitutingfor(abcos6)itsvalue A&lt;f&gt;sin-0, wehave =- bcj&gt;+Aftcos6+Mgh. Thisequation givestherelation between theconstants(j),0,and b(which dependsontherateofspinningofthetoponitsaxis)insteadymotion. 73.Motionofatoponaperfectly smoothplane. Weshallnowconsider themotion ofatopwhich isspinningwith itsapex incontact with asmooth horizontalplane*.Thereaction oftheplaneisnow vertical, sothehorizontal componentofthevelocityofthecentre ofgravity, G,ofthetopisconstant;wecantherefore without lossofgenerality suppose that thiscomponentiszero, sothatthepointGmovesverticallyinafixed line,which weshall take asaxisofZ;twohorizontal lines fixed inspaceand perpendiculartoeach other willbetaken asaxes ofXandY. LetGxyz betheprincipalaxes ofinertia ofthetopatG,and(A,A,C) themoments ofinertia about them, Gzbeingtheaxis ofsymmetry:and let (0,&lt;/&gt;, -\/r)betheEulerianangles definingtheirpositionwith reference tothe axes ofX,Y,Z. TheheightofGabove theplaneishcos6,where hdenotes thedistance ofGfrom theapexofthetop;thepartofthekineticenergy duetothe motion ofGistherefore ^J/A2sin26 .62 ,whereMisthemass ofthetop;and so,asin71,thetotal kineticenergyis cos2 , 112andthepotential energyis =Mgh cos6. *Poisson, TraitS deMtcanique (1811),n.p.198. 164 TheSoluble Problems ofRigid Dynamics [OH.vi Proceedingnowexactlyasin 71,wehavetwointegrals corresponding totheignorablecoordinates &lt;/&gt;andty,namely \A&lt;j&gt;sin26+G(^+&lt;cos6)cos6=a, C(^+(j)cos0)=b, where aand bareconstants; andonperformingtheprocessofignorationof coordinates weobtain forthemodified kineticpotentialtheexpression i(A+Mh* sin*0)P-(^~~?-Mghcos0, sothevariation of6isthesame asinthesystemwithonedegreeoffreedom forwhich thekineticenergyis (A+Mh*sin20)fr, andthepotential energyis (a bcosBY, n^- =~+Mqh cos0. 2J.sin2 Theconnexion between and tisgiven bytheintegralofenergyofthis lattersystem, namely $(A+Mh*sin20)ft=-(a"6C ;?)2-Jfyfecos+c, -/Isin c/ where cisaconstant. Writingcos=x,thisbecomes A(A+Mh2-Mtta?) x2=-(a- bx}2-ZAMgh (x-a?)+2Ac (1-x2 ). Thevariables xand tareseparatedinthisequation,sothesolution can beexpressedasaquadrature;buttheevaluation oftheintegralinvolved willrequireingeneral hyperelliptic functions, orautomorphicfunctions of genustwo. 74.Kowalevski stop. Theproblemofthemotion undergravityofabodyoneofwhosepointsis fixed isnotingeneralsoluble byquadratures:andthecases considered in 69(inwhich thefixedpointisthecentre ofgravityofthebody,sothat gravitydoes notinfluence themotion), andin71(inwhich thefixedpoint andthecentre ofgravitylieonanaxis ofsymmetryofthebody),were for longtheonlyonesknown tobeintegrable.In1888 however Mme. S. Kowalevski* shewed that theproblemisalso soluble when two ofthe principalmoments ofinertia atthefixedpointareequalanddouble the third, sothatA=B=2C,andwhen further thecentre ofgravityissituated intheplaneoftheequalmoments ofinertia. Letthelinethroughthefixedpointandthecentre ofgravitybetaken astheaxisOx,and letthecentre ofgravitybeatadistance afrom thefixed *ActaMath. xn.(1888), p.177. 73,74] TheSoluble Problems ofRigid Dynamics 165 point;let(6, &lt;f&gt;,i/r)betheEulerianangleswhich define thepositionofthe principalaxes ofinertia Oxyzwith reference tofixedrectangularaxesOXYZ, ofwhich theaxisOZ isvertical; let (&lt;uj,&&gt;2,co3)bethecomponents alongthe axesOxyzoftheangular velocityofthebody,and letMbeitsmass. The kinetic andpotential energiesaregiven bytheequations T=i(Aw*+Aw?+Ca&gt;32 ) =C{fc+ &lt;j&gt;2sin26+(^+ &lt;/&gt;cos0)2 }, V=Mgasin costy. Thecoordinate &lt;j&gt;isevidently ignorable, givinganintegral dT ;r-r=constant, Off) or 20sin29+(-^+ &lt;j&gt;cos0)cos=k, where kisaconstant :andtheintegralofenergyis T+V=constant, or2+&lt;2sin2+(^+ &lt;j&gt;cos0)2-^sin6cos-f=A. Mme. Kowalevski shewed thatanotheralgebraic integral exists, which can befound inthefollowing way. Thekineticpotentialis L=C0* + C&lt;j&gt;2sin2+^C(^r+ &lt;j&gt;cos0)~+Mgasin cos -i/r, andtheequationsofmotion are ^(?^_ = dt\M)W _ dt d dt the firstofthese is 20= (rf&gt;cosd- -dr)6sin6+^cos costy, (j andoneliminating -fybetween thesecond andthird,weobtain 2^ (4&gt;sin0)=-(Acos0-&lt;dr)0+^a cos sin^. rit Addingthe firstoftheseequations multiplied byitothese.cond, wehave 2-= ((j&gt;sin0+t0)=i (&lt;cos- -^)(&lt;f&gt;sin0+i0)+i.-^cos0e~^, 166 TheSoluble Problems ofRigid Dynamics [CH.vi anequation which canbewritten intheform sin6+i6)*+ sin1 (.]=i (&lt;cos- ijr)j(4sin+i6)-+-sn or ~=lcos^~" where 7=(isin+i0)n~+ sin Similarly,if F=(0sin(9-i8?+^sin wehave "FT"=~*(0cos^~ 1 Itfollows that J.d7 1.,. tfdtVdt= or UV=constant. Wehave therefore theequation (&lt;j&gt;sin+id)2H7^-sin 0ar~*Kj(0sin i6)2+L "^~sin$6** [=constant, or ($2+ &lt;/&gt;2sin2 #)2+(~^r )sin2#-|^-sin0{el*l ((j&gt;sm0 +i0)2+e~i*l((j)sm0i0)2]- &gt; =constant, and this istherequiredthirdalgebraic integral ofthesystem. The firstintegralswhich havebeen found constitute asystemofthree differentialequations,each ofthe first order, forthedetermination of0,(j),ty, andtheycanberegardedasreplacingtheoriginaldifferentialequationsof motion. Thevariable &lt;f&gt;doesnotoccurexplicitlyinthem andwecanthere foreuseoneofthethreeequationsinorder toeliminate&lt;from theother two: weshall then have asystemoftwodifferentialequations,each ofthe first order, todetermine and-fy.IthasbeenshewnbyMme. Kowalevski that theseequationscanbesolvedbymeans ofhyperelliptic functions :forthis solution reference maybemade tothememoiralreadyreferred to*. *Cf.also Kotter, ActaMath. xvn.(1893), p.209;Stekloff, Gorjatscheff, andTcbapligine, Trav. Soc.Imp. Nat. Moscou, x.(1899) andxn.(1904) ;G.Dumas, Nonv. Ann.(4)iv.(1904), p.355;Husson, Toulouse Ann.(2)vm.(1906), p.73;Husson, ActaMath. xxxi.(1907), p.71; N.Kowalevski, Math. Ann. LXV.(1908), p.528; P.StackeJ, Math. Ann. LXV.(1908), p.538; 0.Olsson, Arkiv forMat. iv.Nr.7(1908);E.Marcolongo, Rom. Ace.Rend.(5)xvn.(1908), p.698;F.deBrun, Arkiv forMat. vi.Nr.9(1910) ;P.Burgatti, Palermo Rend. xxix.(1910), p.396; 0.Lazzarino, Rend. d.Soc. reale diNapoli, (3a )xvn.(1911), p.68. 74,75] TheSoluble Problems ofRigid Dynamics 167 Example.Lety1,y2,y3denote thedirection-cosines ofOx,Oy,Qzreferred toOZ,and letvariablesx,y,rbedenned bytheequations Mgay-&gt; 2o&gt;1o&gt;2+- -,I0)30)!+ - -2a&gt;3w2 =0,3(0!+ +a,32a,2^/ ShewbyuseofKowalevski sintegral (without usingtheintegralsofenergyorangular momentum) thattheequationsofmotion canbewritten intheform whereVisafunction ofxandyonly,sothattheproblemistransformed intothat ofthe motion ofaparticleinaplane conservative field offorce.(Kolosoff.) E.Liouville* hasstated that theonlyother generalcase inwhich themotion under gravityofarigidbody withonepointfixed hasathird algebraic integralisthat inwhich 1.Themomentalellipsoidofthepointofsuspensionisanellipsoidofrevolution. 2.Thecentre ofgravityofthebodyisintheequatorial planeofthemomental ellipsoid. 3. If(J,A,C)aretheprincipal moments ofinertia atthepointofsuspension, the ratio2C/Aisaninteger:thisinteger canbearbitrarilychosen. Onthis,cf.thememoirs cited inthefootnote onthepreceding page. Example. Aheavy bodyrotates about afixedpoint 0,theprincipal moments of inertia atwhichsatisfytherelation A=B=4C :andthecentre ofgravityofthebodyliesin theequatorial plane ofthemomentalellipsoid,atadistance hfrom 0.Shew that ifthe constant ofangular momentum about thevertical through vanishes, there exists anintegral w3 (o&gt;!2+co22 )+gha&gt;icos6=constant, wherecoj,&lt;u2,o&gt;3arethecomponentsofangular velocityabout theprincipal axesOxyz, Oxbeingthelinefrom tothecentre ofgravity ;andhence thattheproblem canbe solved byquadratures, leadingtohyperelliptic integrals. (Tchapligine.) 75.Impulsivemotion. Ashasbeen observed in 36,thesolution ofproblemsinimpulsive motion doesnotdependontheintegrationofdifferentialequations, andcan generallybeeffectedbysimple algebraic methods. Thefollowing examples illustrate varioustypesofimpulsive systems. Example1.TwouniformrodsAB,BC,eachoflength 2a,aresmoothly jointedatB and restonahorizontal table with their directions atright angles. Animpulseisappliedto themiddle point ofAB,and therods startmovingasarigid body:determine thedirection oftheimpulsethat thismaybethecase,andprovethat thevelocities ofA,Cwillbeinthe ratioV13:1.(Coll. Exam.) Wecanwithout lossofgenerality suppose themass ofeachrodtobeunity. Let(x,y} bethecomponentvelocities ofBreferred tofixed axes Ox,Oyparalleltotheundisturbed *ActaMath. xx.(1897), p.239. 168 TheSoluble Problems ofRigid Dynamics [CH.vi position BA,BCoftherods,and let6,&lt;betheangularvelocities ofBAandBC.The componentsofvelocityofthemiddlepointofABare(x,y+ad),andthecomponentsof velocityofthemiddle pointofBCare(x- a&lt;,y),sothekinetic energyofthesystemis given bytheequation Letthecomponents paralleltotheaxes oftheimpulse be/,J.Thecomponentsof thedisplacementofthepointofapplicationoftheimpulseinasmall displacementofthe systemare(bx,By+add) ;andhence theequationsof36become 0=-ax+^a2 ), while thecondition thatthesystem moves asifrigidis6= &lt;p.These equations give \x=y=lae= lai&gt;=\I=lJ. Hence /=/, which shews thatthedirection oftheimpulse makes anangle of45withBA; andasthecomponentsofvelocityofAare(x,y+2a0),andthecomponentsofvelocityof Care(x2a&lt;, y),wehave forthevelocities ofAandofCthevalues \/65y and*Jby respectively,sothevelocityofAis^/13xthevelocityofC:which istherequiredresult. Example2.Aframeworkintheform ofaparallelogramismadebysmoothly jointing theendsoftivopairs ofuniformbarsoflengths 2a,26,masses m,m ,andradii ofgyration k,k.Theparallelogramismovingwithout anyrotationofitssides,andwithvelocity V,in thedirection ofoneofitsdiagonals;itimpinges onasmoothfixedwallwithwhich thesides makeangles 6,$and thedirection ofthevelocity Varight angle,thevertex which impinges being broughttorestbytheimpact.Shew that theimpulse onthewall is (Coll. Exam.) Letxandybethecoordinates ofthecentre oftheparallelogram, xbeing measured at right anglestothewallandtowards it.Thekineticenergyis T=(m+m)(x2+y2 )+(mk2+ma2 }2+(mb2+m k"2 )2 . The^-coordinate ofthepointofcontact is#+asin#+&sin0,sothedisplacementofthe pointofcontact paralleltotheaxis ofxcorrespondingtoanarbitrary displacement (&r, By,B6,80)isS.r+acos#50 +&cos080. Theequations ofmotion, denotingthe impulse by/,aretherefore fdT_ (dT\__ dx /^T\=lacos0, ,=-Ibcos 80/o 2(mkz+ma2 )6=lacos6, 2(mb2+mk2 )&lt;p=-Ibcos0. 75] TheSoluble Problems ofRigid Dynamics 169 Moreover since thefinalvelocityofthepointofcontact iszero,wehave x+acos6.6+bcos &lt;j&gt;. &lt;f&gt;=0. Eliminating x,6,&lt;from these equations, wehave b2cos2 __ [2(m+mf) which istheresult stated. Thenextexamplerelates toacaseofsuddenfixture;ifonepoint (orline) ofafreely-moving rigidbodyissuddenlyseized andcompelledtomove ina given manner, there willbeanimpulsive changeinthemotion ofthebody, which canbedetermined from thecondition thattheangular momentum of thebodyaboutanylinethroughthepointseized(orabout the line seized) isunchanged bytheseizure;this follows from thefactthattheimpulseof seizure hasnomoment about thepoint (orline). Example3.Auniformcircular disc isspinningwithanangular velocity Qabout a diameter when apointPonitsrim issuddenly fixed. Prove that thesubsequent velocity of thecentre isequaltoJofthevelocity ofthepointPimmediately beforetheimpact. (Coll. Exam.) Letmbethemass ofthedisc,and letabetheangle between theradius toPandthe diameter about which thediscwasoriginally spinning. Theoriginal velocity ofPis Q.Csina,where cistheradius ofthedisc. Theoriginal angular momentum aboutPis about anaxisthroughPparalleltotheoriginalaxisofrotation, andofmagnitude ^mc2Q.; andthis isunchanged bythefixingofP,sowhenPhasbeenfixed, theangular momentum about thetangent atPisrac2I2sina.Butthemoment ofinertia ofthediscabout its tangentatPisfme2 ,andsotheangular velocity about thetangent atPis^12sina.The velocityofthecentre ofthedisc isthereforeJflcsina,which isoftheoriginal velocity ofP. Example4.Alamina intheform ofaparallelogram whose mass ismhasasmooth pivotateach ofthemiddle points oftwoparallel sides. Itisstruck atanangular point byaparticle ofmassmwhich adheres toitafter theblow. Shew that theimpulsive reaction atoneofthepivotsiszero.(Coll. Exam.) MISCELLANEOUS EXAMPLES. 1.Prove that foradisc freetoturnabout ahorizontal axisperpendicular toitsplane thelocus onthedisc ofthecentres ofsuspensionforwhich thesimple equivalent pendulum hasagiven lengthLconsists oftwo circles;andthat,ifAandBaretwo points, oneoneachcircle, andListhelength ofthesimple equivalent pendulum when thecentre ofsuspensionisthemiddle pointofAB,theradius ofgyration kofthedisc about itscentre ofinertia isgiven bytheequation where 2cisthelength ofAB.(Coll. Exam.) 2.Aheavy rigidbody canturnabout afixed horizontal axis. Ifonepoint inthe bodyisgiven through which thehorizontal axishastopass, discuss theproblem of choosing thedirection oftheaxis inthebodyinsuch awaythat thesimple equivalent pendulumshallhaveagiven length ;shewing thattheaxeswhichsatisfythiscondition are thegeneratorsofaquartic cone.(Coll. Exam.) 170 TheSoluble Problems ofRigid Dynamics [CH. 3.Asphere ofradius brollswithoutslipping down thecycloid x=a(0 +sin6), y=a(l costf). Itstarts from restwith itscentre onthehorizontal liney=2a.Prove thatthevelocity V ofitscentre when atthelowestpointisgiven by F2=-W (2o- 6). (Coll. Exam.) 4.Auniform smooth cube ofedge2aandmassMrests symmetrically ontwoshelves each ofbreadth bandmassmandattached towalls atadistance 2capart. Shewthat,if oneoftheshelvesgivesw&yandbeginstoturnabout theedgewhere itisattached tothe wall, theinitial angular acceleration ofthecube willbe :-&+a)2 whereMb2and/arerespectivelythemoments ofinertia ofthecubeabout itscentre and oftheshelfabout itsedge. (Camb. Math. Tripos, PartI,1899.) 5.Ahomogeneous rodofmassMandlength2amoves onahorizontalplane, oneend beingconstrained toslide without friction inafixed straightline. Therod isinitially perpendiculartotheline,and isstruck atthefreeendbyablow/paralleltothe line. Shew that aftertime ttheperpendiculardistance yofthemiddle pointoftherodfrom theline isgivenbytheequation (l-$xtf(l-x2)-%dx=3lt/2Ma. (Coll. Exam.) t 6.Four equal uniform rods,oflength 2a,aresmoothly jointedsoastoform a rhombus ABCD. ThejointAisfixed, whilstCisfree tomove onasmooth vertical rod through A.Initially Ccoincides withAandthewhole systemisrotating about the vertical withangular velocityo&gt;.Prove that,ifinthesubsequent motion 2aistheleast angle between theupper rods, w2cosa=3gsin2a. (Camb. Math. Tripos, PartI,1900.) 7.AdiscofmassMrestsonasmooth horizontal table, andasmooth circular groove ofradius aiscutinit,passing throughthecentre ofgravityofthe disc.Aparticleof mass^Misstarted inthegroove from thecentre ofgravityofthedisc.Investigate the motion. Prove that ifa$isthearctraversed bytheparticle and6theangle turned round bythedisc,then Mk2beingthemoment ofinertia ofthediscabout avertical linethroughitscentre ofgravity. (Coll. Exam.) 8.Arigidbodyismoving freely under theaction ofgravity androtating withangular velocityG&gt;about anaxisthroughitscentre ofgravity perpendiculartotheplaneofits motion. Shew that theaxis ofinstantaneous rotation describes aparabolic cylinderof latusrectum (^/4a+ *j2g/ca}2 ,whose vertex isatadistance vS^a/wabove that ofthepath ofthecentre ofgravityofthebody;where 4aisthelatus rectum oftheparabola described bythecentre ofgravity. (Coll. Exam.) 9.Aparticleofmassmisplacedinasmooth uniform tubewhich canrotate ina vertical plane about itsmiddlepoint. Thesystemstarts from restwhen thetube is horizontal. If6istheangle thetubemakes with thevertical when itsangular velocityis amaximum andequalto o&gt;,provethat 4(mr*+J/F)&&gt;4-Smgrco2cos6+mg2sin26=0, vi] TheSoluble Problems ofRigid Dynamics 171 whereMkzisthemoment ofinertia ofthetubeabout itscentre andrthedistance ofthe particle from thecentre ofthetube.(Coll. Exam.) 10.Four uniformrods,smoothly jointedattheir ends, form aparallelogram which canmove smoothlyonahorizontalsurface, oneoftheangular points beingfixed. Initially theconfigurationisrectangular andtheframework issetinmotion insuch a manner thattheangular velocityofonepairofoppositesides isQ,that oftheotherpair beingzero. Shew thatwhen theangle between therods isamaximum orminimum, the angular velocityofthesystemisQ.(Coll. Exam.) 11.Twohomogeneous rough spheresofequal radii aandofmasses m,mrestona smooth horizontal plane withmatthehighest point ofm. Ifthesystemisdisturbed, shew that theinclination oftheircommon normal 6tothevertical isgiven bythe equation (l-cos&lt;9). (Coll. Exam.) 12.Auniform rodAB isoflength 2aand isattached atoneendtoalightinexten- siblestringoflengthc.Theother endofthisstringisfixed at toapointinasmooth horizontal plane onwhich therodmoves.Initially GAB isastraightlineandtherod is projected without rotation withvelocity Vinthedirection perpendiculartoitslength. Prove that thecosine ofthegreatest subsequent angle between therodandstringis l-/6c. (Coll. Exam.) 13.Toafixedpoint arcsmoothly jointed twouniform rods oflength 2a,andupon themslides, bymeans ofasmoothringateach end,athird rodsimilar inallrespects. Initiallythethree rods areinahorizontal linewith theends ofthethird rodatthe middlepointsoftheother twoand,ontheapplicationofanimpulse, therods beginto rotate with angular velocity Qinahorizontalplane. Shew that thethird rodwill slide rightofftheother twounless J3. (Coll. Exam.) 14.Ahollow thincylinderofradius aandmassMismaintained atrest ina horizontalposition onarough plane whose inclination isa,andcontains aninsect ofmass matrestonthelineofcontact with theplane. Thecylinderisreleased astheinsect starts offwithvelocity V:ifthisrelativevelocity bemaintained andthecylinderrollup hill,shew that itwillcome toinstantaneous restwhen theradius through theinsect makes anangle dwith theverticalgiven by F2 {1-cos(0-a)}+ag (cosa-cos &lt;9)=(1+M/m) ag(d-a)sin a. (Coll. Exam.) 15.Auniform smoothplane tube canturnsmoothly about afixed axis ofrotation lyinginitsplane andintersectingit :themoment ofinertia ofthetubeabout theaxis is1.Initiallythetube isrotating withangular velocity Qabout theaxis,andaparticle ofmassmisprojectedwithvelocity Vwithin thetubefrom thepointofintersection of thetubewith theaxis. Thesystem thenmoves under noexternal forces. Provethat, when theparticleisatadistance rfrom theaxis, thesquare ofitsvelocityrelative tothe tube is 1 Q2 .(Coll. Exam.) 16.Auniform straightrodofmassMislaidacross twosmooth horizontal pegs so thateach ofitsendsprojects beyondthecorresponding peg.Asecond uniform rodof massmandlength21isfastened tothe first atsomepoint between thepegsbya 172 TheSoluble Problems ofRigid Dynamics [CH. universaljoint. Thisrod isinitiallyheld horizontal andincontact with thefirstrod :and then letgo,soastooscillate inthevertical plane through the first rod. Prove that if6 betheangle which thesecond rodmakes with thevertical atany instant, andxthe distance through which the firstrodhasmoved from rest, (M+m) x+mlsin6=ml, and (t-- ^&gt;cos26\ifr=2gcos6. (Coll. Exam.) \dm+M / 17.Aplane bodyisfreetorotate initsplane about afixedpoint, andasecondplane bodyisfreetoslidealong asmoothstraight grooveinthe firstbody,itsmotion beingin thesame plane ;shew that therelation between therelative advance xalong thegroove andtheangle ofrotation &(noexternal forces being supposedtoactonthesystem) isoftheform wherePandQarerespectivelylinear andquadraticfunctions ofx1 -.(Coll. Exam.) 18.Apendulumisformed ofastraight rodandahollow circular bob,andfitting inside thebob isasmooth vertical lamina intheshapeofasegmentofacircle, the distances ofthecentre(C)ofthebobfrom thepointofsuspension (0)andfrom the centre ofgravity (G}ofthelamina beingIand crespectively.Prove that ifM,marethe masses ofthependulum andlamina, kand 1Jtheir respectiveradii ofgyrationabout andG,6and$theangles whichOCandCGmake with thevertical, then twice the workdonebygravityonthesystem duringitsmotion from rest isequalto (2+c2 )&lt;j&gt;2+2mcZcos (6-$) 6$. (Coll. Exam.) 19.Aparticleofmassmisattached totheendofafinestringwhichpassesround thecircumference ofawheel ofmassM,theother endofthestring being attached toa pointinthatcircumference, alengthIofthestring being straight initially, andthewheel (radius aandradius ofgyration k)beingfreetomove about afixed vertical axisthrough itscentre; theparticle, which liesonasmooth horizontalplane,isprojectedatright anglestothestring,sothatthestring beginstowrapround thewheel;prove that,ifthe string eventually unwinds from thewheel, theshortest lengthofthestraight portionis (P-a2-J/F/m)*. (Coll. Exam.) 20.Acarriageisplacedonaninclinedplanemaking anangleawith thehorizon and rollsstraight down without anyslippingbetween thewheels andtheplane. The floor of thecarriageisparalleltotheplane andaperfectly roughball isplaced freelyon it.Shew thattheacceleration ofthecarriage down theplaneis whereMisthemass ofthecarriage excludingthewheels,mthesumofthemasses ofthe wheels, which areuniform discs, andMthat oftheball.The friction between thewheels andtheaxes isneglected. (Coll. Exam.) 21.Auniform rodofmassmtandlength2aiscapableofrotating freely about its fixedupper extremity and isinitiallyinclined atanangleofTr/6tothevertical. Asecond rod,ofmassm2andlength 2a,issmoothly attached tothelower endofthe firstandrests initiallyatanangleof2-/3with itand inahorizontalposition. Shew that ifthecentre ofthelower rodcommence tomove inadirection making anangle -rr/Gwith thevertical, then3wH=14m2. (Coll. Exam.) vi]TheSoluble Problems ofRigid Dynamics 173 22.Auniform circular disc issymmetrically suspended bytwo elasticstrings of natural lengthcinclined atanangle atothevertical, andattached tothehighest point of thedisc. Ifoneofthestringsiscut,provethattheinitial curvature ofthepathofthe centre ofthedisc is (csin4a bsin2a)/6 (b c), where bistheequilibrium length ofeachstring. (Coll. Exam.) 23.Two rodsAC,CBofequal length 2aarefreely jointedat(7,therodACbeing freely moveable about afixedpoint A,andtheendBoftherodCBisattached toAby aninextensiblestringoflength 4a/x/3.Thesystem beinginequilibrium, thestringiscut; shew thattheradius ofcurvature oftheinitial pathofBatBis _!_f4 181V" (Camb. Math.Tripos, PartI,1897.) 24.Arodoflength 2aissupportedinahorizontal position bytwolight strings which passovertwosmooth pegsinahorizontal lineatadistance 2aapart andhave attheir other extremities weights each equaltoonehalfthat oftherod.Oneofthestringsiscut; prove thattheinitial curvature ofthepathofthatendoftherodtowhich thecutstring wasattached is27/25. (Coll. Exam.) 25.Aheavy plank, straight andveryrough,isfreetoturn inaverticalplane about ahorizontal axisfromwhich thedistance ofitscentre ofgravityisc.Arough heavy sphereisplaced onthisplankatadistance bfrom theaxis, onthesideremote from the centre ofgravity ;theplank being held horizontal. Thesystemisnow leftfreetomove. Prove that the initial radius ofcurvature ofthepathofthecentre ofthesphereis 21&#/(5 110),where d=(tnbMc)j(mb+Ma)imandMarethemasses ofthesphere and theplank, andMob isthemoment ofinertia oftheplank about theaxis. (Coll. Exam.) 26.Alightstiffrodoflength2ccarries twoequal particlesofmassmatdistances k from thecentre oneach sideofit;toeachendoftherod istiedanendofaninextensible stringoflength 2aonwhich isaringofmassm .Initiallythestring androdareinone straight lineonasmooth horizontal table with thestring tautandtheringattheloop ; theringisthenprojected atright anglestotherod,shew thattherelative motion willbe oscillatoryif c2/P&gt;1+2i/m.(Coll. Exam.) 27.Threeequal uniformrods,each oflength c,arefirmly joinedtoformanequilateral triangle ABC ofweight W;auniform baroflength26andweightWisfreely jointed to thetriangleatC.Thissystemrests inequilibriumincontact withthesurface ofafixed smooth sphere ofradiusa,ABbeing horizontal and incontact with thesphere, andthe barbeingintheverticalplane through thecentre ofthetriangle ;thebar,andthecentre of thetriangle, areonoppositesides ofthevertical linethroughC.Prove thattheinclination oftheplaneofthetriangletothehorizon istheangle whose tangentis [abfj.+2c\2 ]-T-[np(a2+1c2 )+AV-2a6c] , where \2=a2-rc2-^c, M2=12a2-c2 ,andn=W/W. (Camb. Math.Tripos, PartI,1896.) 28.Abody, under theaction ofnoforces, moves sothattheresolvedpartofitsangular velocity about oneoftheprincipal axesatthecentre ofgravityisconstant;shew thatthe angular velocityofthebodymust beconstant, andfind itsresolvedparts about theother twoprincipal axeswhen themoments ofinertia about these axesareequal. (Coll Exam.) 174 TheSoluble Problems ofRigid Dynamics [CH. 29.Shew thataherpolhode cannot have apointofinflexion.(Hess.) (Asimple proofofthis result isgiven byLecornu, Bull, delaSoc.Math, deFrance, xxxiv.(1906), p.40.) 30.Inthemotion under noforces ofabodyoneofwhosepointsisfixed,shew thatthe motion ofevery quadric homocyclic withthemomentalellipsoidrelative tothefixedpoint, andrigidly connected with thebody,isthesame asifitweremade torollwithoutsliding onafixed quadricofrevolution, which has itscentre atthefixedpoint, andwhose axis is theinvariable line.(Gebbia.) 31.Inthemotion ofabodyunder noforces round afixedpoint, shew thatthethree diameters ofthemomentalellipsoidatthefixedpoint andthediameter oftheellipsoid reciprocaltothemomentalellipsoid, determined respectively bytheintersection ofthe invariable plane with thethreeprincipal planes andwiththeplane perpendiculartothe instantaneous axis, describe areasproportionaltothetimes, sothat theaccelerations of their extremities aredirected tothecentre.(Siacci.) 32.When abodymoveable about afixedpointisacted onbyforces whosemoment round theinstantaneous axis isalways zero, shew that thevelocityofrotation is proportionaltothat radius vector ofthemomentalellipsoid which isinthedirection ofthis axis. Shew that thistheorem isstilltrue ifthebodyismoveable about afixed pointand alsoconstrained toslideonafixed surface.(Flye StMarie.) 33.Aplanelamina isinitially moving withequal angularvelocities Qabout the principalaxes ofgreatest and leastmoment ofinertia atitscentre ofmass, andhas noangular velocity about thethirdprincipalaxis;expresstheangularvelocities about these axes asellipticfunctions ofthetime, supposing noforces toactonthelamina. If6betheangle between theplaneofthelamina andanyfixedplane, shew that (Camb. Math.Tripos, PartI,1896.) 34.Arigidbodyiskinetically symmetrical about anaxiswhichpasses through afixed point above itscentre ofgravity and issetinmotion inanymanner;shew that inthe subsequent motion, exceptinone case, thecentre ofgravity cannever bevertically over thefixed point ;andfindthegreatest heightitattains.(Coll. Exam.) 35.Inthemotion ofthetopontherough plane, shew thatthere exists anauxiliary setofaxesOTJwhose motion withrespecttothefixed axesOXYZ &n&lt;\alsowithrespect tothemovingaxesOxyzisaPoinsot motion;theinvariable planes being thehorizontal planeintheformer case,andtheplane perpendiculartotheaxisofthebodyinthesecond case.(Jacobi.) 36.Auniform solid ofrevolution moves about apoint, sothat itsmotion maybe represented bytheuniformrollingofacone ofsemi-verticalangleafixed inthebodyupon anequalcone fixed inspace,theaxisoftheformer being theaxisofrevolution. Shew that thecouple necessarytomaintain themotion isofmagnitude Q2tana{C+(C-A]cos2a}, whereQistheresultant angular velocity andAandCtheprincipal moments ofinertia at thepoint, andthatthecoupleliesintheplaneoftheaxes ofthecones.(Coll. Exam.) vi]TheSoluble Problems ofRigid Dynamics 175 37.Avertical planeismade torotate withuniform angular velocity about avertical axis initself, andaperfectly rough cone ofrevolution has itsvertex fixed atapointof that axis.Shew that,ifthelineofcontact make anangle 6with thevertical, and/3andy betheextreme values of#,andabethesemi-vertical angle ofthecone, ,2A?#\2_,sin2a(cos&-cos/3)(cosy-cosK i -,- ]&Qn, \dt/ cosa cos@+cosy where histhedistance ofthecentre ofgravityoftheconefrom itsvertex, andkitsradius ofgyrationabout agenerator. (Camb. Math.Tripos, PartI,1896.) 38.Abodycanrotatefreely about afixed vertical axis forwhich itsmoment of inertia is/ :thebodycarries asecond bodyintheform ofadiscwhich canrotate about ahorizontal axis, fixed inthefirstbodyandintersecting thevertical axis. Intheposition ofequilibrium themoments andproduct ofinertia ofthediscwithregardtothevertical andhorizontal axesrespectively areA,B,F.Prove that ifthesystem startfrom rest with theplaneofthediscinclined atanangle atothevertical, the firstbodywilloscillate through anangle IFf^sinalarctan I- }-.(Coll.Exam.) r 39.Agyrostat consists ofaheavy symmetrical flywheel freely mounted inaheavy sphericalcaseand issuspended from afixedpoint byastringoflengthIfixed toapoint inthecase. Thecentres ofgravityoftheflywheel andcasearecoincident. Shewthat, ifthewhole revolve insteadymotion round thevertical withangular velocity i2,thestrin- andtheaxisofthegyrostatinclined atangles a,?tothevertical, then Q2 (Isina+asin/3+bcos)=gtana, and /Qsinj3-Afl2sin/3cos/3=Mgseca(asin(/3 a)+bcos(/3 a)}, whoreMisthemass ofthegyrostat,aandbthecoordinates ofthepoint ofattachment ofthestring with reference toaxescoinciding with,andatright angles to,theaxisofthe flywheel, /theangular momentum oftheflywheel about itsaxisandAthemoment of inertia about alineperpendiculartoitsaxis. (Camb. Math.Tripos, PartI,1900.) 40.Asystem consisting ofanynumber ofequal uniform rodsloosely jointed and initiallyinthesame straightline isstruck atanypointbyablowperpendicular totherods. Shew that ifu,v,wbetheinitial velocities ofthemiddlepoints ofanythree consecutive rods,u+4;V+w=Q.(Coll. Exam.) 41.Anynumber ofuniform rods ofmasses A,B,C,...,Zaresmoothly jointed to eachother insuccession and laidinastraightlineonasmooth table. IftheendZbe freeandtheendAmoved withvelocity Finadirectionperpendicular totheline ofthe rods, thentheinitial velocities ofthejoints (AB\ (BC),...andtheendZarea,b,..., z, where 0=A(F+2a)+(2a+&),=B(a andy+22=0.(Coll. Exam.) 42. Sixequal uniform rodsform aregular hexagon loosely jointedattheangular points:ablow isgiven atright angles tooneofthem atitsmiddlepoint, shew that the opposite rodbeginstomove withT]oofthevelocityoftherodstruck. (Camb. Math.Tripos, 1882.) 176 TheSoluble Problems ofRigid Dynamics [OH.vi 43.Abodyatrest,withonepoint fixed,isstruck :shew that theinitial axis of rotation ofthebodyisthediametralline,with respecttothemomentalellipsoidat0,of theplaneoftheimpulsive couple acting onthebody. 44.The positiveoctant oftheellipsoid^72/a2+3/2/62+22 /62=lhastheoriginfixed. Shew that ifanimpulsive coupleintheplane .- ba2\ba/c actupontheoctant,itwillbegintorevolve about theaxis ofz.(Coll. Exam.) 45.Anellipsoidisrotating about itscentre withangular velocity (&lt;BI,&&gt;2,ws)referred toitsprincipal axes;thecentre isfreeandapoint (#,y,z)onthesurface issuddenly broughttorest. Find theimpulsivereaction atthatpoint. (Coll. Exam.) 46.TwoequalrodsAB,BCinclined atanangle aaresmoothly jointedatB ;Ais made tomoveparalleltotheexternal bisector oftheangleABC :prove thattheinitial angularvelocities ofAB,BCareintheratio 2+3sin25:2-15sin|.(Coll. Exam.) 2t 2* 47.Auniform cone isrotating withangular velocityo&gt;about agenerator whensuddenly thisgeneratorisloosed andthediameter ofthebasewhich intersects thegeneratorisfixed. Prove thatthenewangular velocityis where histhealtitude, athesemi-vertical angle, andktheradius ofgyration about a diameter ofthebase. (Coll. Exam.) 48.Aroughdisccanturnabout anaxisperpendiculartoitsplane, anda1ough circular cone rests onthediscwith itsvertex justatthe axis. Ifthediscbemade toturnwith angular velocity Q,shew that thecone takes anamount ofkineticenergy equalto |Q2/{cos2a/J+sin2 /{7}. (Coll. Exam.) 49.Oneendofaninelasticstringisattached toafixedpointandtheother toapoint inthesurface ofabodyofmass M.Thebodyisallowed tofallfreely undergravity without rotation. Shew thatjustafter thestring becomestight thelossofkineticenergy dvietotheimpactis where Vistheresolvedvelocityofthebodyinthedirection ofthestring just before impact, thestring onlytouching thebodyatthepointofattachment, (I,m,n,A,p,v)are thecoordinates ofthestringattheinstant itbecomestight, andA,B,Caretheprincipal moments ofinertia ofthebody with respecttoitsprincipalaxes atitscentre ofinertia. (Coll. Exam.) CHAPTER VII THEORY OFVIBRATIONS 76. Vibrations aboutequilibrium. InDynamics wefrequentlyhave todealwithsystemsforwhich there exists anequilibrium-configuration,i.e.aconfigurationinwhich thesystem canremainpermanentlyatrest :thus inthecaseofthespherical pendulum, theconfigurationsinwhich thebob isverticallyover orverticallyunder the pointofsupportareofthis character. If(q1}q2,...,qn)arethecoordinates ofasystem andLitskineticpotential,and if(1? 2,..., n)arethevalues of thecoordinates inanequilibrium-configuration,theequationsofmotion dfdL\ dLA "7-,Up- -3-=(r=l, 2,...,w)at\oqrjoqr must besatisfiedbythesetofvalues ji=0,#2=0, ...,#n=0,^=0,#2=0,...,gn=0,ql=al, &lt;/2=or2,...,qn=an. The values ofthecoordinates inthevariouspossible equilibrium-con figurationsofasystemaretherefore obtained bysolvingforqltqz,...,qnthe equations |=(,.1,2,. ..,), inwhichql}q2,...,qnaretobereplaced byzero. Inmany cases, ifthesystemisinitially placednearanequilibrium-con figuration,itsparticles having verysmall initial velocities, thedivergence from theequilibrium-configurationwillnever becomevery marked, the particles always remaininginthevicinityoftheiroriginal positions and neveracquiring largevelocities. Weshallnowstudy motions ofthistype*; theyarecalled vibrations about anequilibrium-configuration^. *Morestrictly speaking, westudyinthischapter thelimiting form towhich thistype of motion approximates when theinitial divergence from astate ofrestintheequilibrium-configu ration tends tozero;thestudyofthemotions which differ byafinite, though notlarge, amount from astate ofrestintheequilibrium-configurationisgiven later inChapter XVI :thediscussion ofthepresent chapter mayberegarded asafirstapproximation tothat ofChapter XVI. tThetheoryofvibrations hasdeveloped from Galileo sstudy ofthesmall oscillations ofa pendulum. Inthe firsthalf oftheeighteenth century thevibrations ofastretched cordwere investigated byBrook Taylor, DAlembert, Euler, andDaniel Bernoulli, thelast-named ofwhom in1753enunciated theprincipleoftheresolution ofallcompound typesofvibration intoinde pendent simple modes. Thegeneral theoryofthevibrations ofadynamical system with afinite number ofdegrees offreedom wasgiven byLagrange in1762-5 (Oenvres,i.p.520). w.D. 12 178 Theory ofVibrations[CH.vn Inthepresent workweareofcourse concernedonlywith thevibrations ofsystems which have afinitenumber ofdegreesoffreedom;thestudyofthevibrations ofsystems which haveaninfinite number ofdegreesoffreedom, which ishere excluded,willbefound intreatises ontheAnalytical TheoryofSound. Weshallsupposethatthesystemisdefinedbyitskineticenergy Tand itspotential energy F,andthatthepositionofthesystemisspecified bythe coordinates(qltq2,,..,q n)independentlyofthetime, sothatTdoes not involve texplicitly:weshall alsosupposethatnocoordinates have been ignored;thekineticenergy Tistherefore ahomogeneous quadraticfunction of &lt;?i,^2. t^n,with coefficientsinvolving qltq2,...,qninanyway.There isevidentlynoloss ofgeneralityinassumingthat theequilibrium-con figuration correspondstozerovalues ofthecoordinatesqltq.2&gt;...,q n;sothat &lt;?i* &lt;?*&gt; 9 9i&gt; &lt;?t&gt; &gt;9nareverysmallthroughoutthemotion considered. The coefficients ofthesquares andproductsofq1}q2,...,qninTare functions ofqltq.2,...,qn:ashowever allthecoordinates and velocities are small, wecaninapproximatingtothemotion retainonlytheterms oflowest order inT,and socanreplaceallthese coefficientsbytheconstant values whichtheyassume whenql,q.2&gt;...,qnarereplaced byzero. The kinetic energyistherefore forourpurposesahomogeneous quadraticfunction of &lt;jn i2,&gt;??iwith constant coefficients. Moreover, ifweexpandthefunction VbyTaylorstheorem inascending powersofql,q2,...,qnthetermindependentofqlyq.2,...,qncanbeomitted, since itexercises noinfluence ontheequationsofmotion;andthere areno dVterms linear inq1}qz,...,qn,since ifsuch terms existed thequantities- dqr would notbezero intheequilibrium position,astheymust be.Theterms oflowest order inVaretherefore thetermsquadraticinql}q2,...,q n. Neglectingthehigherterms oftheexpansionincomparison with these, wehave therefore Vexpressedasahomogeneous quadratic form inthe variablesqltq.2,...,qnwith constant coefficients. Thus theproblem ofvibratorymotions about aconfiguration ofequilibrium dependsonthesolutionofLagrangian equations ofmotion inwhich thekinetic andpotential energiesarehomogeneous quadratic formsinthevelocities and coordinatesrespectively,with constantcoefficients. 77.Normal coordinates. Inorder tosolve theequationsofmotion ofavibrating system, wewrite theexpressionsforthekinetic andpotential energiesintheform T=|(anqf+a2.2q./+...+annqn2+2al2q,q2+2alsqlq3+...+ 2an_ 1&gt;qn-iqn), 76,77] Theory ofVibrations 179 oftheseTis(26)apositivedefinite form;andthedeterminant formed ofthequantitiesarsisnotzero (sinceifthis condition isnot satisfied, Twilldependonlessthannindependent velocities). Theequationsof motion are d/dT\ 8F/ -io \-o-i, 2,....), ifachangeofvariables ismade, such thatthenewvariables (&lt;?/,q2,...,q n) arelinear functions of(qltqa,...,q n),thenewequationsofmotion willbe (]b IUJ-\"Of/ ~t Ck \ -J-.brri)=-;, (r=l, 2,...,), c\oqrj oqr andtheseequationsareclearlylinear combinations oftheoriginal equations. Supposethen* that theoriginal equationsofmotion aremultiplied respectively byundetermined constants m,,m.2,...,mn,andaddedtogether. Theresulting equationwillbeoftheform where Q=h1q1+h^q2+...+hnqn, providedtheconstants mltra2,...,mn,hl}ha,...,hn,Xsatisfytheequations 7 "\/ II \ "\/ ~i~tinn^^ii^~^\^m ^^i ~i^/i2^^2 *~ *^nH***fn,)^=A/t7j,. TheseequationscancoexistonlyifXisarootofthedeterminantalequation ftjiA/^ji&gt; Gfi2X Oj2 , &gt;flinA/ Omi a.21X62i,a22X622)..., 2riX b.2r bnl ,,b Moreover, ifX:isanyrootofthisequation, wecandetermine from the preceding equationsapossiblesetofratios form^m2,...,mn,hl,h2,...,h n; these ratiosmay,incertain cases, bepartly indeterminate, butinallcases at leastonefunction Qcanbeobtained inthisway, satisfyingtheequation +U&gt;=o. Now letalinearchangeofvariables beeffected sothatthequantity Q sodetermined isoneofthenew variables :there willbenoambiguityin Thismethod ofproofisduetoJordan, Comptes Rendus, LXXIV.(1872) p.1395. 122 180 Theory ofVibrations[OH.vn denotingthenewvariables byq1}q2,...,qn\weshall takeqtobeidentical with Q,sothattheaboveequationsaresatisfied bythevalues h-L=\,A2=0, ...,hn=0.Since theformTisapositivedefinite form, the coefficients "22, 33&gt; ,annofthesquaresofqz,q3)...,qnwillnotbezero: soinstead ofq2,q3, ,qnwecanagaintakenew variablesrespectively equalto .^12 ^is,**in q*+ft.&+?i,..,2+?i- 1*22 1*33"Tin Bythischangeofvariables theterms inq^, q^q z,...,q^qnareremoved from T :sowecanassume thata21,a31,...,amarezero. Nowintroducingtheconditions hl=\,h2=0,/i3=0,...,hn=0,a21=0, ...,anl=intheequationswhich determine mltm2,...,mn,hlth2,...,hn,X, weobtain thevalues m1=l/an,ra.2=0,m3=0,...,mn=0, 611=Xiu, fea=0, 631=0,...,6M1=0. Itfollows thattheequation cfia hastheform -+\qi=0,. dt\dqj +\qi d/dT\ dV ji(o^~ )"~*5~~ (**"*, o &gt; while theequations ji(o^~ )"~*5~~dt\dqrJ oqr d/dT \dVhave theform3-=T-=^ (r=2,3,...,), CV9^r/ 9^r where T=T-a nq*,V=V-^a^q*, sothatTandFdonotinvolveq^andqlf This lastsystemofequations mayberegardedasthesystemofequations correspondingtoavibrational problemwith (n 1)degreesoffreedom. Treating them inthesame manner, wecan isolate another coordinateq2 such that if T"=T-lavtf,V"-F-faoKqf (where X2and a%,arecertainconstants), then T"and V"donotinvolveq2or q2,andthecoordinatesq3,q4,...,qnaredetermined bytheequationsof avibrational problemwith(n 2)degreesoffreedom, inwhich thekinetic andpotential energiesarerespectivelyT"and V". Proceedinginthisway,weshallfinallyhave thevariables chosen sothat thekinetic andpotential energiesoftheoriginal systemcanbewritten in terms ofthenewvariables intheform T=%(angi2+22922+...+annqn*), V=i(&i2i2+/3*q z*+...+&m?2 ), where an,a^,...,, Ai. ^22, ,Pnnareconstants. 77] Theory ofVibrations 181 Iffinally wetake asvariables thequantities^anqi,Va^g,,....^a instead ofql}q2,...,qn,thekinetic andpotential energiestaketheform V=|(/ij^2+fJ,2q&lt;?+. where pkstands forfikklakk- Inthisreduction itisimmaterial whether thedeterminantal equationhas itsroots alldistinct orhasgroupsofrepeatedroots. The final result canbe expressed bythestatement thatifthekinetic andpotential energies ofa vibrating systemaregivenintheform V=...+annqn2+2a12^q2+...+2a, l_1)nqn-^qn\ ...+bnnqn2+2612 &lt;7,q2+...4-2bn-i,nqn-iqn), itisalways possibletofindalineartransformation ofthecoordinates such that thekinetic andpotential energies, whenexpressedintermsofthenew coordinates, have theform =\/Mi2+^29V++Mns , where thequantities ^, /i.2)..., fjtnareconstants. These newcoordinates are called thenormal coordinates orprincipalcoordinates ofthevibrating system. Now itisawell-knownalgebraicaltheorem thattheroots ofthedetermi nantalequation bln 2A,=0 aniA,oni.................. annA,onn arethevalues ofXforwhich theexpression (an\-6n)q*+(a, 2X-622) q&lt;?+...+(ann\-bnn)qn2+2(au\- canbemade todependonlessthannindependentvariables (whichwillbe linear functions ofq1}q.2,...,qn).Since this isapropertywhichpersists through anylinearchangeofvariables, weseethatthedeterminantalequation isinvariantive,i.e.ifg/,q2f ,...,qnareanynindependentlinear functions of &lt;?i,92,,qn,and ifTandVwhenexpressedinterms ofq^,q2,...,qntake theform T=\ qi2+a*q^++2a12?/q,+...), 182 Theory ofVibrations[OH.vii then theroots ofthenewdeterminantalequation a,./X &,./=0arethe same astheroots oftheoriginal determinantalequation arg\br^=0. Butwhen thekinetic andpotential energieshavebeenbrought bythe introduction ofnormal coordinates totheform thedeterminantalequationis =0, soitsroots are^ ,//,2,...,//,.Itfollows that theconstants ^,/JLZ,...,pn,which occur asthecoefficients ofthesquares ofthenormal coordinates inthepotential energy,arethenroots (distinct orrepeated) ofthedeterminantalequation |ars\brs ]|=0,where an,a12,...,bn,612,...arethecoefficientsintheoriginal expressions forthekinetic andpotential energies. Ltwillbeseenthattheproblemofreducing thekinetic andpotential energiestotheir expressionsinterms ofnormal coordinates isessentiallytheproblemofsimultaneously reducing each oftwogiven homogeneous quadratic expressionsinnvariables toasumof squaresofnnew variables :thefactthatTisafunction ofthevelocities whileVisa function ofthecoordinates doesnotaffect thequestion,since theformulae oftransforma tion forthevelocitiesjl5j2, &gt;^narethesame astheformulae oftransformation forthe coordinatesqi,q2,...,qn. Itmight besupposed from theforegoing that itisalways possible totransform simultaneously each oftwogiven homogeneous quadratic expressionsinnvariables toa sum ofsquares ofnnewvariables; butthis isnotthecase;forexample,itisnotpossible totransform thetwoquadratic expressions ax2+bxy+az2and 2and ototheforms where ,77,farelinear functions ofx,y,z. Theconditions which must besatisfied inorder thattwogiven quadratic expressions maybesimultaneously reducible totheform 77,78] Theory ofVibrations 183 are,infact,thattheelementarydivisors (Elementartheiler) ofthedeterminant j|ars\brs \\ shall belinear*. Ifhowever oneofthetwogiven forms isadefiniteform(aswesawwas thecasewith thekinetic energyinthedynamical problem),theelementarydivisors are always linear, andthesimultaneous reduction tosums ofsquaresistherefore possible; thisexplains thecircumstance thatthereduction canalways beeffected inthedynamical problem ofvibrations. Theuniversalpossibilityofthereduction tonormal coordinates fordynamical systems was established byWeierstrass in18581;previouswriters (following Lagrange)had supposed that incases where thedeterminantal equation hadrepeatedroots asetof normal coordinates would notexist, andthatterms involvingthetime otherwise than in trigonometric andexponential functions would occur inthefinal solution oftheequations ofmotion. 78.Sylvesters theorem onthereality oftheroots ofthedeterminantal equation. Wehave seen intheprecedingarticle thatbyintroducing new variables which arelinear functions oftheoriginal variables, itisalways possibleto reduce thekinetic andpotential energiesofavibrating systemtotheform iq+Ma+ Thequestionarises astowhether thistransformation isreal, i.e.whether the coefficients m1}m2,...,mn,hlyh2,...,hnwhich occur inthetrans formation arerealorcomplex. Sijice these coefficients aregiven bylinear equations whose coefficients, withthepossible exceptionoftheroots Xj ,A,2,...,\n ofthedeterminantalequation,arecertainly real,thequestion reduces toan investigationoftherealityorotherwise oftheroots oftheequation au\b ua12X-&i2 ...... amX bln=0; ttojA,b.2l ooX^ ...... a.2n\b2n \ am^~bnlan2\bn.2 unn\bnn itbeingknown thatthequantities arsand brsareallreal,andthat Oii^i2+a^q 22+...+annqn2+2awg1ga+...+2a, l_ ]&gt;ngn_1gn isapositivedefinite form. LetAdenote| thedeterminant\\arg\brs ,and let Aj.denote the determinant obtained from itbystrikingoutthe firstrowand firstcolumn; letA.,denote thedeterminant obtained fromAbystrikingoutthe firsttwo *Cf.Muth streatise onElementartheiler (Leipzig, 1899);orBocher sIntroduction toHigher Algebra (New York, 1907). tCf.Weierstrass Collected Works, Vol. i.p.233. Thefollowing proofisduetoNanson, Mess, ofMath. xxvi.(1896), p.59. 184Theory ofVibrations[CH.vii rowsand firsttwocolumns, and soon.Then inanysymmetrical deter minant, say ,where arg=agr, itisknown that 9a12/ andhence if^vanishes thequantities Dand5-must have opposite0*u OOufa* signs;thuswehave theresult that intheseries ofquantities A,AlfA2,...,An (whereAn=1), ifanyonemember oftheseries vanishes foragivenvalue ofX,thetwo adjacent members must haveopposite signsforthat value ofX. LetArdenote thedeterminant formed fromArbyreplacingXbyunity andeach ofthequantitiesbrsbyzero, sothatAristhecoefficient ofthe highest powerofXinAr.Since isapositivedefinite form,Arispositiveforallvalues ofrfrom ton. Thus thecoefficients ofthehighest powersofXinthefunctions A,A1;...,An are allofthesamesign ;andtherefore asXincreases from ooto+oc, these functions losenchangesofsign. Now sinceAnisnotzeroandAr_a,Ar+1haveopposite signswhen A,. vanishes, itfollows thatthefunctions A,A1;A2,...,Ancannot lose orgaina changeofsignexcept whenXpasses througharoot ofA.But asXpasses from ooto -foo ,thefunctions losenchangesofsign;andhence the nrootsofthedeterminant Aare allreal. Thetransformationtonormal coordinates istherefore always arealtransformation*. Moreover, since achangeofsignislostinthepairA,Aj ,every time thatX passes througharoot ofA,itisevident that Ajmustchange signwhenX increases from oneroot ofAtotheconsecutive root,andhence that the nroots ofAareseparated bythe(nl) roots ofA^similarly theroots of each ofthefunctions A,,areseparated bytheroots ofthefunction A,.+1. NowAnhasnoroots :and ifAn_jhasthesamesignatX=asatX= oo, theroot ofthefunction An_!willnotbenegative.Ifmoreover An_2has thesamesignatX=asatX= oo,neither oftheroots ofAn_2willbe negative:for ifthis condition issatisfied, An_2must have either two negativeroots ornonegative roots, and there cannot betwonegative roots since there isnonegativerootofAn_jtoseparatethem.Similarlyin * Sylvester, Phil.Mag. (4)iv.(1852), p.138: Coll. Papers,i.p.378. 78,79] Theory ofVibrations generalthecondition thatnone ofthefunctions intheseries A,AlsA2,...,An shall have anegativeroot isthat each ofthefunctions must have the samesignatX=asatA,=-oo .Hence thecondition tobesatisfied inorder that alltheroots ofAmaybepositiveisthat each quantityA,, shall have atX= thesamesignas(-l)n~r ,i.e.that each ofthe determinants shall bepositive. Butthese arethewell-known conditions thatthequadratic form Ai &lt;?i2+b^q?++bnnqn2+Zb^q* +...+2&n_lingn_1gn shall beapositivedefinite form. Hencefinallythecondition that thedeter- minantalequationIars\brs\\= shall have all itsrootspositiveisthat thequadratic form bnnq n_ 1&gt;nqn_lq shall beapositive definite form,i.e.that thepotential energyinthevibratory motion shall beessentially positive. 79. Solutionofthedifferential equations ;theperiods; stability. Inorder toexpresstheconfigurationofanyvibrating systeminterms of thetime, we firstdetermine thenormal coordinates ofthesystem,and expressthekinetic andpotential energiesinterms ofthem, sothat these take theform where(ql}q2,...,q,i)arethenormal coordinates, and(Xj,A, 2,...,\ n)arethe roots ofthedeterminantalequation ||ars\brs\\= ;thesequantities (A,!,X&gt;,...,\n)havebeenshewn inthe last article tobeallreal. TheLagrangian equationofmotion foranycoordinateqr,namely isthereforedt\dq=__ dqr dqr r=0. The solution ofthisequationis .rt+Br) ,ifX,.ispositive, */ j. ,ifX,.iszero, +Bre~~rt ,ifX,.isnegative, where ineach caseArandBrdenote constants ofintegration. 186 Theory ofVibrations[CH.vu Itappears from theseequationsthat ifallthenormal coordinatesexcept one,sayqr,areinitially zero,and iftheconstant \.correspondingtothe non-zero coordinate ispositive,thenthecoordinates(qi,q 2, ,qr-i,q r+i,&gt;&lt;Jn) willbepermanently zero,andthesystemwillperformvibrations inwhich thecoordinateqrisalone affected. Moreover theconfigurationofthe systemwillrepeatitself after aninterval oftime2?r/Vx,.. This isusually expressed bysayingthat eachofthenormal coordinates correspondstoan independent modeofvibrationofthesystem, providedthecorresponding constant \rispositive;and theperiod ofthisvibration is2?r/Vx,.. Moreover,ifthesystembereferred toanyother setofcoordinates which arenotnormal coordinates, these coordinates arelinear functions ofthe normal coordinates;anjithenormal coordinatesperformtheir vibrations quite independentlyofeach other;.thuseveryconceivable vibration ofthe system mayberegardedasthesuperpositionofnindependentnormal vibrations. This isgenerallyknown asDaniel Bernoulli sprinciple ofthe superposition ofvibrations*. Ifthequantities (Xj,\2,...,X n)arenot allpositive,itappearsfrom the above solution that those normal coordinatesqrwhichcorrespondtothe non-positiveroots X,.willnotoscillate about azerovaluewhen thesystemis slightlydisturbed from astate ofrest initsequilibrium position,but will increase soastoinvalidate theassumptionmade attheoutset ofthework, namelythat thehigher powersofthecoordinates canbeneglected.In thiscase therefore, there willnotbeavibration atall,andtheequilibrium configuration.issaid tobeunstable. Ifhowever the initial disturbance is such that these normal coordinates whichcorrespondtonon-positiveroots Xrarenotaffected, thesystemwillperformvibrations inwhich therest ofthenormal coordinates oscillate about zero values. The normal modes ofvibration, whichcorrespondtothose normal coordinates forwhich thecorrespondingroot X,.ispositive,aresaid to bestable. Iftheconstants Xrareallpositive,theequilibrium-configuration asawhole issaid tobestable. The condition for stabilityoftheequi librium-configurationistherefore, bythetheorem ofthe last article, that thepotential energy ofthevibrating systemshall beapositive definite form. This result might havebeen expectedfrom aconsideration oftheintegralofenergy ; forthisintegralis T+V=h, whereTandFarethequadraticforms which represent thekinetic andpotential energies, andwhere hisaconstant. Thisconstant hwillbesmall iftheinitial divergence from the equilibriumstate issmall. ButT7isapositivedefinite form;and ifVisalsoapositive definite form,wemust haveTandVeach lessthanh,soTandVwillremain small throughoutthemotion :themotion willtherefore never differ greatlyfrom theequilibrium- configuration,i.e. itwillbestable. *Histoire deVAcademic deBerlin, annee 1753, p.147. 79,80] Theory ofVibrations 187 80.Examples ofvibrations aboutequilibrium. We shallnow discuss anumber ofillustrative cases ofvibration about equilibrium. (i)Tofindthevibration-period ofacylinder ofanycross-section which canrollonthe outsideofaperfectly rough fixed cylinder. Let sbethearcdescribed onthefixedcylinder bythepointofcontact,sbeing measured from theequilibrium position ;letpandpbetheradii ofcurvature ofthe cross-sections ofthefixedandmoving cylinders respectivelyatthepoints which arein contact intheequilibrium position ;pandpbeing supposed positive when thecylinders areconvex toeach other :letMbethemass ofthemoving cylinder, Mk* itsmoment of inertia about itscentre ofgravity, and cthedistance ofthecentre ofgravity from the initialposition ofthepointofcontact inthemoving cylinder. Ifadenotes the initial angle between thecommon normal tothecylinders andthe vertical, then a+s/pistheangle between thecommon normal attime taridthevertical, a+s/p+s/pistheanglemade with theverticalbythelinejoiningthecentre ofcurvature ofthemoving cylinder with theoriginal pointofcontact inthemoving cylinder, and s/p+s/pistheanglemade with thevertical bythelinejoining thelast-namedpointto thecentre ofgravityofthemoving cylinder. Theangular velocityofthemoving cylinder istherefore I 1 ..,.,.. \P P soitskineticenergyis ppThepotential energyis VMgxheightofthecentre ofgravityofthemoving cylinder above some fixedposition =Mg\(p Neglectings3thisgives(p+p)cos(o+-] -pcos(a+-+J+ccos [-+ PP/ \Pp PP \PP TheLagrangian equation ofmotion, dfdT\dT dt\di gives Jf(#+c2)(1+-- .- - PP/ (PP \PP andthevibrations arethereforegiven bytheequation whereAand eareconstants ofintegration tobedetermined bytheinitialconditions, and Xisgiven bytheequation V9 ff (PP }X-r~5{ ,COSa C[.K+C2 (p+p }Thevibration-periodis27T/X. (ii)Tofindtheperiods ofthenormal modesofvibration aboutanequilibrium-configura tionofaparticle moving onafixed smoothsurface undergravity. Thetangent-plane tothesurface atthepoint occupied bytheparticleinthe equilibrium-configuration isevidently horizontal :take asaxes ofxandythetangents to 188 Theory ofVibrations[OH.vn thelines ofcurvature ofthesurface atthispoint, andasaxisofzalinedrawnvertically upwards:sothattheequationtothesurface isapproximately whereplandp2denote theprincipalradii ofcurvature, measuredpositively upwards. The kinetic energy andpotential energyareapproximately T=\m(xi -\-i/2 } (whereinisthemass), and V=mgz &y* Itisevident from these expressionsthatxandyarethenormal coordinates :the equations ofmotion are Pi P2 andtheperiodsofthenormal modes ofvibration aretherefore 9^-fl 1anH 9ZiTT\IclIlLl -. (iii) ^oymc?zAenormal modesofvibration ofarigid body,oneofwhosepointsisfixed, andwhich isvibratingabout aposition ofstableequilibriumunder theaction ofanysystem ofconservativeforces. Take asfixed axes ofreference OXYZ theequilibrium positionsoftheprincipal axes of inertia ofthebodyatthefixedpoint;themovingaxes willbetaken asusual tobethese principalaxes ofinertia. We shall supposethepositionofthebodyatanyinstant defined bythesymmetrical parameters (, ij,x)of9;weshall regard ,77,asthe independentcoordinates ofthesystem, xbeingdefined interms ofthem bytheequation Thecomponentsofangular velocityofthebody about themoving axes are(16) Onaccount ofthesmallness ofthevibration, weregard ,?;,fassmallquantitiesof thefirstorder;xtherefore differs from unity byasmall quantityofthesecond order, and sowehave, correctlytothe firstorder ofsmallquantities, andthekinetic energyofthebody, which isgiven bytheequation where A,B,Caretheprincipal moments ofinertia atthepointofsuspension, canbe written Thepotential energyissome function ofthepositionofthebody, andtherefore ofthe parameters (, r;, );letitbedenoted byV(, 77,f). 80] Theory ofVibrations 189 Since zerovalues of(|, 77, )correspondtotheequilibrium position, there willbe noterms linear in(, 77,f)whenFisexpandedinascending powersof(, 77,f):thelowest terms aretherefore ofthesecond order;neglectingterms ofhigher order, wecantherefore write V=a?+b^+o?+2fyC+2g&+Zhfa where a,b,c,f,g,hareconstants. Theproblemofdeterminingthenormal coordinates istherefore thesame asthat ofreducing thetwoquadratic expressions totheform where(#,y,z)arelinear functions of(, 77, ). Now theequation,referred tothefixed axes, ofthemomentalellipsoidinitsequi libriumpositionis consider inconnexion with thisthequadric whose equationis which weshall callthe" ellipsoidofequal potential energy" ;anddetermine thecommon setofconjugate diameters ofthesequadrics.Let(X\Y,Z)bethecoordinates, referred totheseconjugate diameters,ofapoint whose coordinates referred tothefixed axes are(X,Y,Z\and lettheequations connecting (X ,Y,Z}and(X,J7 ,Z)be Bythistransformation theequationsofthequadricsarereduced totheform X^+b^ +CjZ2=1, andtherefore thetransformation whichgives thenormal coordinates inthedynamical problemis Itfollows that inanormal mode ofvibration, saythat inwhich xalonevaries, the quantities (, 77,f)willbepermanentlyintheratio Butfrom thedefinitions of 9itisevident that,r;, are,tothe firstorder ofsmall quantities, proportionaltothedirection-cosines ofthelineabout which therotation oftherigidbody takesplace, andconsequentlythenormal mode ofvibration ofthe rigidbodyconsists ofasmall oscillation about alinewhose equationis X :Y :Z=li:12:/,, i.e.about theline r=o,zr=o, which isoneofthecommonconjugate diameters ofthetwoquadrics. 190 Theory ofVibrations[CH.vn Hencefinally wehave theresult that thenormal vibrationsofthebodyaresmall oscillations about thecommonconjugatediametersofthemomentalellipsoid and theellipsoid ofequal potential energy. (iv) Tofindthenormal coordinates and theperiods ofnormal vibration inthesystem ofthreedegrees offreedom forwhich F=i{ where aissmall incomparisonwithpandq;and toshew thatifsuchasystembelet gofromrestwithyand zinitially zero, thevibration inxwillhavetemporarilyceased afteratimeirp(qz-jo2 )/a2 ,andthat there willthen beavibrationofthesameamplitudein yastheoriginal onewasinx.(Coll. Exam.) Theform ofthekinetic andpotential energies suggests thetransformation whichgives (f- \7= . Thevariable77istherefore anormal coordinate :toreduce theremaining terms inthe kinetic andpotential energiestosums ofsquares, wewrite andthenwehave The variables?/, &lt;^&gt;,^aretherefore thenormal coordinates. Supposethatinitially wehave x= lc,y=0,z=0, z=0,^=0,2=0, andsupposethatkissosmall that itsproduct with other smallquantities canbe neglected. Then tothisdegreeofapproximation wehaveinitially r,=U,&lt;f&gt;=U,f=0. Thevibrations ofthenormal coordinatesTJand&lt;aretherefore given bytheequations rj=kcospt, I=1COS The lastequationcanbewritten 4.-Ucos[pt(l----jL-Jl,L IjrW-tfU or a2t a-t (f&gt;=Mcosptcosr-7.r+ksinptsin p(qi-p-)~ 80,81] Theory ofVibrations 191 Themotion cantherefore beapproximately represented initially by T)=kcospt, 4&gt;=^kcospt, or x=kcospt,y=0. After aninterval oftimeTrp(q2p2 )/a?,themotion isapproximately represented by ?7=\kcospt, &lt;f)=l;kcospt, or #0,y~ kcospt; which establishes theresult stated. 81.Effect ofanew constraint ontheperiods ofavibrating system. We shallnow consider the effectproducedontheperiodsofnormal vibration ofadynamical systemabout aconfigurationofstableequilibrium when thenumber ofdegreesoffreedom ofthesystemisdiminishedbythe introduction ofanadditional constraint Supposethat theoriginal systemisspecifiedinterms ofitsnormal coordinates(q1,q,...,qn),sothat thekinetic andpotential energies have theform and lettheadditional constraint beexpressed bytheequation Sinceql, &lt;?2 &gt; &gt;&lt;?naresmall, wecanexpandthefunction /inascending powersofql}q.2,...,qn,andretainonlythe firstterms oftheexpansion: we canthusexpresstheconstraintbytheequation where A,, ...,Anareconstants. Astheequilibrium-configurationissupposed tobecompatiblewith theconstraint, there willbenoconstant term.By means ofthisequation wecaneliminateqn:wethushave V=\X.V++^n-i /n-!+ (A ,9l+ ...+A_,qn^y. TheLagrangian equationsofmotion oftheconstrainedsystemarethere forethe(n 1)equations Ar. 2 I* (r=l,2,...,n-1), qr+\r*qr+f*Ar=(r=I,2,...,n-1) 192Theory ofVibrations[CH.vn where 1 X- P= -T--. i(A&++-4_!&lt;/_,) +-j-iCAtfi +...+^-i^n-i)^ln-d.n~ sotheequationsofmotion oftheconstrainedsystemcanbewritten inthe form ofthenequations qr+Vgv+pA r=(r=1,2,...,n), where//,isundetermined. Now consider anormal mode ofvibration ofthemodifiedsystem,defined byequations (/!= !cos\t, qo=2cos\t,...,qn=a.ncos\t,//,=vcos\t. Substitutingintheequationsofmotion, wehave ar(X,.2-X2 )+vAr=(r=l, 2,...,n). Substitutingthevalues ofa1,a. 2&gt;...,angiven bytheseequationsinthe equationA^+A2a.2+...+An*n=0, wehave A*_AJ_ AJ_. \i&gt;-VV-V^n"-^ Thisequationin\-has(w 1)roots, which from theform oftheequa tion areevidently interspaced between thequantities Xf,A22 ,...,XH2 :the quantities 2?r/X correspondingtothese roots aretheperiodsofthenormal modes ofvibration oftheconstrainedsystem, and ittherefore follows that the(n l)periods ofnormal vibrationoftheconstrainedsystemarespaced between thenperiods oftheoriginal system. 82.Thestationary characterofnormal vibrations. We shall next consider the effect ofaddingconstraints toadynamical systemtosuchanextent thatonlyonedegreeoffreedom isleft tothe system. Let(q1}q2,....qn}bethenormal coordinates oftheoriginal system;theconstraintsmay,asinthe last article, berepresented bylinear equations between these coordinates, andcantherefore beexpressedinthe form gi=/*i?i&=Wl,,q,i=pnq, where/^1,/i2,..., fj.nareconstants andqisanew variable which maybe taken asdefiningtheconfigurationoftheconstrainedsystemattime t. Letthekinetic andpotential energiesoftheoriginal system be V= 81-83] Theory ofVibrations 193 so27T/XJ, 27T/X 2,...,2TT/\ nare itsperiodsofnormal vibration: thekinetic andpotential energiesoftheconstrainedsystemarethen V=i(XV+V^2+.+xn&gt;n2 )f. Theperiodofavibration oftheconstrainedsystemistherefore2-Tr/X, where Xisgiven bytheequation _ Iftheconstraints arevaried, thisexpressionhasastationaryvaluewhen (n 1)ofthequantities /i1}/u2, , f*&gt;narezero :thisstationaryvalue isone ofthequantitiesXa2 ,X22 ,...,Xn2 :andthuswehave thetheorem thatwhen constraints areputonthesystemsoastoreduce itsnumberofdegrees of freedomtounity,theperiod oftheconstrainedsystemhasastationaryvalue forthose constraints which make thevibration tobeanormal vibrationofthe unconstrainedsystem. 83. Vibrations aboutsteadymotion. Atypeofmotion whichpresents many analogieswith theequilibrium- configurationisthatknown asthesteady motion ofsystems whichpossess ignorablecoordinates: this isdefined tobeamotion inwhich thenon- ignorable coordinates ofthesystem have constant values, while thevelocities correspondingtotheignorable coordinates have alsoconstant values. Oneexampleofasteady motion isthat ofthetop,discussed in 72;asanother example wemaytake thecase ofaparticle which isfree tomove inaplane and is attracted byafixed centre offorce, thepotential energy depending onlyonthedistance from thecentre offorce; forsuch aparticle, acircular orbit described with constant velocityisalwaysapossible orbit, andthis isaform ofsteady motion, since theradius vector isconstant andtheangular velocity correspondingtotheignorable coordinate 6 isalsoconstant. Inmany cases, ifasystemisinitiallyinastate ofmotiondiffering only slightlyfromagivenform ofsteady motion, thedivergence from thisform of motion willneversubsequently becomeverymarked;weshallnowconsider motions ofthiskind,which arecalled vibrations aboutsteadymotion. Thesteady motion issaid tobestable* ifthevibratory motion tends to acertainlimiting form,namelythesteady motion, when the initial disturb ancefromsteady motion tends tozero. Let (&gt;!, j&gt;2,...,pk)betheignorable and(qlt&,..., qn)bethenon-ignorable coordinates ofthesystem. Correspondingtotheignorable coordinates, there willbekintegrals dL *This definition isduetoKlein andSommerfeld. W.D. 194 Theory ofVibrations [OH.vn where &,/32, &gt;ftkareconstants. We shall supposethat these constants have thesame value inthevibratorymotion asintheundisturbed steady motion ofwhich itisregardedasthedisturbed form; this ofcourse only amounts tocoordinatingeachvibratorymotion tosomeparticular steady motion. Wesupposethesystem conservative, with constraints independentofthe time;letitskineticenergybe nn nk kk T=\2,2ayfay+22byqipj+b22 where thecoefficientsa#,by,cv-arefunctions ofqltqz....,qn. Theintegrals correspondingtotheignorablecoordinates are pi+2b{jqi=fy (j=1,2,...,k). LetCijbetheminor ofcy-inthedeterminant formed ofthecoefficients Cy,divided bythisdeterminant; thensolvingthe lastequationsforthe quantities pr,wehave ]jr=2Cm(Ps2bigqi). s I Substitutingforpi,pz,...,ptintheaboveexpressionforT,andutilising thepropertiesofminors ofdeterminants, wehave T=i2(a*,-2Clsbubjs)qiqj+^2Cufrfr. i,j I,s I,s Nowperformtheprocessofignorationofcoordinates. LetRbethe modified kineticpotential,so R=T-V- 2pr/3r =\2(ay-2Cubabjg) qiqj+2Crs@rblsqi-$2C&&&-V. i,j I,s l,f,s l,s Wecanwithout lossofgenerality supposethatthevalues ofqltq2,...,qn inthesteadymotion areallzero. Ifthen thecoefficients inRareexpanded inascending powersofq-i, q.2,...,qnbyTaylorstheorem, and allterms inthe expressionofRthus obtained which areabove theseconddegreeinthe variablesqi,qz,...,qn,qi,qz,...,qnareneglectedincomparisonwith the terms oftheseconddegree,weobtain forRanexpression consistingofterms linear andquadraticinql}qz,...,qn,ql}q2,...,qn.Now theterms which arelinear inq^,qz,...,qnandindependentofq1;q2,...,qndisappearauto maticallyfrom theequationsofmotion dfdR\ dR A N -r.(^)-a=(r-1, 2,...,n),dt\dqj dqr andthese terms cantherefore beomitted. Moreover, since theequationsare satisfied bypermanentzerovalues ofq^q2,...,qn,itisevident thatnoterms 83,84] Theory ofVibrations 195 linear inq1}q2,...,qnandindependentofql,q2,...,qncanbepresentinR. Itfollows that theproblem ofvibrations aboutsteadymotion dependsonthe solutionofLagrangian equations ofmotion inwhich thekineticpotentialisa homogeneous quadratic function ofthevelocities andcoordinates, with constant coefficients. Thedifference between vibrations aboutequilibriumandvibrations about steadymotion consists inthepossible presenceinthelatter case ofterms of thetype qrqs(i.e.productsofacoordinate andavelocity)inthekinetic potential.These arecalledgyroscopicterms. The vibrations aboutsteady motion ofasystemareinfactthesamethingasthevibrations about equilibriumofthereduced ornon-natural(38)systemtowhich theproblem isbrought byignorationofcoordinates. Theequationsofmotion forthevibrating systemaretherefore dfiR\dR -nlo-r- -5=(r=l, 2,...,w),dt\dqrjdqr whereRcanbewritten intheform R=i2arf!qrqs+2&.,gr &lt;/+2jrsqrqs (r,s=1,2,...,ri), r,s r,s r,s andwhere =a,,.,$n=/3sr , butwhere yrsisnotingeneral equaltojsr.Theequationsofmotion inthe expandedform are (11ql-&1qi+12#a+(72!-Tia)J2-&292+13&+(731- 7l3&gt;&~$^3 j2i&lt;/i+(712~721)qi~0*qi+&&-j3vq z+a,3q3+(y3,-yzi)q,-/3&q3 |etc. These arelinearequations withconstant coefficients, which areofthesame generalcharacter asthecorresponding equationsinthecaseofvibrations about equilibrium ;theydifferonlyinthepresenceofthegyroscopic terms, which involve the coefficients(7^. 7^).Thepresenceofthese terms makes it impossibletotransform thesystemtonormalcoordinates*; butasweshall nextsee,themain characteristic ofvibrations aboutequilibriumisretained, namelythatanyvibration canberegardedasasuperpositionofnpurely periodic vibrations, which weshall call(asbefore) thenormal modes of vibration ofthesystem. 84.Theintegration oftheequations. Weshallnowshewhowthenature ofthevibrations canbedetermined, byintegrationoftheequationsofmotion. That istosay,impossible totransform thesystem tonormal coordinates byapoint-trans formation:itispossible toeffect thetransformation tonormal coordinates byacontact-trans formation, andthis isactually done inChapter XVI. 132 196 Theory ofVibrations [OH.vn Itwillbeconvenient first totransform them intoasystemofequations each ofthe first order. LetRdenote themodified kineticpotentialofthe system,sothat inthevibratory problem Risahomogeneous quadratic function ofq1}q2,...,qn,ql,qz,...,qn-Write ^R-&lt;V-1 2^ &lt;5~T~==Qn+T v" &gt;*/&gt; sothatqn+i,qn+2, &narelinear functions ofq1}q2,...,qnand viceversa: theequationsofmotion canbewritten =(r=l2n) ^n+r dqr Now ifBdenote anincrement ofafunction ofthevariablesqltqz,...,qn, qn+i,---,qzn,duetosmall changesinthese variables, wehave 8R= ] (^Bqr+ -. &&lt;]&gt;, r=i\oqr- oqr n2/*^ S^"\ \qii-\-roqr~Tqn.^-roqr) /n \ n=SSqn+rqr}+S(qn+r&qrqr$qn+r)- \r=1 r=l n LetthequantitySqn+rqrR, whenexpressedasafunction ofql}q2,...,qzn ,bedenoted byH,sothatHis aknown homogeneous quadraticfunction ofthevariablesq1}qz,..-,q 2n ,the lastequationcanbewritten andtherefore* theequations ofmotion, which consistedoriginally ofnequations eachofthesecond order, canbereplaced byasystem of2nequations,eachof thefirstorder, namely dH dH/no \ 9=^0n+r=-g-(r-1,2,...,), oqn+r oqr theindependentvariablesbeing ql,q2,...,q2n. Weshallnowshew that thefunction H,which hasreplaced Rasthe determiningfunction oftheequations, representsthesum ofthekinetic and potential energiesofthedynamical systemconsidered. ForRcontains terms ofdegrees 2,1,and inthevelocities, and *This transformation isreally acase oftheHamiltonian transformation given later in Chapter X. Theory ofVibrations 197 isequivalenttotwice theterms ofdegreetwotogetherwith theterms of degree one,byEuler stheorem;itfollows thatH,beingdefined as vdRT? 2qr^-R,r=\ oqr willbeequaltotheterms ofdegreetwointhevelocities inR,togetherwith theterms ofzerodegreeinRwith theirsigns changed:oncomparingthe expressionsforTandRgivenonpage 194, itfollows that H=T+V, soHisthetotalenergy ofthedynamical system, expressedintermsofthe variablesqltqz,...,q.M. Inthecase ofvibrations about anequilibrium-configuration,wehave seen that thecondition forstabilityisthat thepotentialaswell asthe kineticenergyshall beapositivedefinite form;weshallnowmake asimilar assumptionforthecase ofvibrations aboutsteady motion, namelythat the totalenergyHisapositivedefinite forminthevariablesq1}q2,...,gwj n thisassumption weshallshew that thesteady motion isstable, andinfact thattheequationsofmotion dqrdffdqn+r dH -rr=5- ,-^f^=^(atoqn+r dtdqr canbeintegratedinthefollowing way*. Consider thesetoflinearequationsinthevariablesqltqz,...,q^n, dH( qi,q2,...,q 2n)\ Sqnj-r+ ~ ~-yr (r-1,2,...,); ifwedenote thedeterminant ofthesystem by/(s), andtheminor ofthe element intheXthrowand/ithcolumnby f(s) (X, /*=!,2,...,2n), theexpressionofqltq2,...,q^interms ofylty9,...,ymisgiven bythe equations andthedegreeoff(s)insis2n,while thedegreeoff(s)^isnotgreater than(2n-l). Inorder tosolve theequationsofmotion, considerexpressionsfor q\,q?, -,q.2noftheform *Themethod ofintegration which follows isduetoWeierstrass, Berlin.Monatsberichte, 1879. 198 Theory ofVibrations[CH.vn where theintegrationistaken round alargecircleCwhich encloses allthe roots oftheequation f(s)=0.These values ofq1}q2,...,q 2nwillsatisfy theequationsofmotion, providedtheequations ,, (r=l2...n) dpn+r aresatisfied. Iftherefore pl9p2,...,p2narepolynomialsin5sochosen that theexpressionsinbrackets under theintegral signvanish when sisequalto oneoftheroots oftheequation f(s)=0,theseequationswillbesatisfied, since theintegrandswillthenhavenosingularitieswithin thecontour C*. Itfollows thatPi,j}.2,",pzn must beasetofsolutions oftheequations lie ,....gn _~r o Pr when sisaroot oftheequation f(s)= ;thiscondition issatisfied bythe expressions P*(s)=I/(*)I F+o2/(5) 2M+...+a2nf(s}. 2n&gt;&gt;l (^=1,2,...,2n), where a:,a2,...,a^arearbitraryconstants. Theequationsofmotion aretherefore satisfied bythevalues &lt;?M=coefficient of1/5intheLaurentexpansion-|-inpositiveandnega tivepowersofsoftheexpression es(t-t ) {Oi/()m +Ot/(*). ++t&gt;mf(*)m,t] -77-7- (p=1,2,..., 2?i).7v5^ Now oninspectionofthedeterminant/(s)weseethatminors ofthe types (X&lt;H* (#*!, 2,...,) areofdegree (2?i 1)ins,andtheother minors areofdegree (2n 2)ins; sothecoefficient of1/5intheLaurentexpansionoff(s)\n/f(s)iszero unless \=n+p,orfj,=n+\;intheformer case itis 1,andinthelatter case it is1.Hence ontakingt=t,weseethatthequantities ttj ,OF-2 ,...,dzn arerespectivelythevalues of atthetime t . *Whittaker andWatson, ACourse ofModernAnalysis,5-2. t76id. 5-6. Je*-84] Theory ofVibrations 199 Iftherefore wewrite (f)()A/x=coefficient ofl/sintheLaurent expansionof .. and ifft,&gt;,...,ftarethevalues offt,%...,22respectively corresponding toanydefinite value toft,wehave H ^=2{ft,+a&lt;MO,M- Inorder toevaluate thequantities &lt;f&gt;(t)^,itisnecessarytodiscuss the nature oftheroots ofthedeterminantal equation f(s)= ;letki+l, where kand Iarerealand idenotes \/-l, beanyroot ofthisequation;then the2nequations n ,7\ .9j^(?i&gt; ?,&gt;&n) A) (ki+I)qn+a+- (-1.2,....) canbesatisfied byvalues ofq1}q^, ...,q^which arenot allzero. Leta systemofsuch values be 1+7i, 2+M?2&gt; ,&n+^2i , where %1,gz,...,%zn ,rji,T), ,^2iarerealquantities.Then ifwewrite dH(q l&gt;q2,...,q 2n)_o7-" Wi&gt; &&gt; &lt;12n)n&gt; wehave, onseparatingthelastequationsintotheir realandimaginary parts, + +,-^n+.= 0| +-^a+Atya= i (a=1;2,. .,W). +a= a= ButsinceHishomogeneousandofdegreetwoinitsarguments, wehave 2tf(&, |a,-.,)=2fxfld!, -..,f)A, A=l andusingthe firsttwoofthepreceding equationsthisgives ,&,.... )=AS (A) I Similarly 2H(77,,77., ,...,77^)=A;2(.??+- i;an+a ). a=l Moreover onmultiplyingthe first ofthepreceding equations by t)aandthe second by r)n+a,adding, andsummingforvalues ofafrom 1ton,wehave 2 n X=l a=l andsimilarly 2/1 71 200 Theory ofVibrations[CH.vn Since theleft-hand sides oftheseequationsareequal, wemust have 2( =0. Butfromequations (A)weseethat, asHisapositivedefinite form, neither n knor2(fa^n+a yaZn+a) canbezero;wemust therefore have Izero; and sotheequation f(s)=haseachofitsrootsoftheform ik,where kisareal quantity different fromzero. Weshall nextshew that inthecase inwhich theequation f(s)=has aJ-tupleroot s,each ofthefunctionsf(s)\nisdivisibleby(ss)i~l . For let Cj,c2,...,c2nbeasetofdefinite realquantities;definequanti tiesq1}q2,...,q2nbytheequations Sqn+a+H((fr, q.2,...,&lt;?2Tl)a=Ca -sq a sothatwehave(a=l,2, ...,n)... (/=!, 2,..., 2n). Lets-iibeanyroot oftheequation f(s)=Q,and letmbethesmallest positive integerforwhich allthefunctions (s Sji)mf. arefinite forthevalue s^iofs.When sistakensufficientlynear sxi,wecan expand q^inaseries oftheform (&lt;7M+h^i) (s s^i)""1+((//+hpi)(ssii)~m+l+..., where &lt;jrM,h^,g^,h^,...denote realconstants; andwecansupposethe quantitiesc:,c2,-.,c.2nsochosen thatthequantities g^andAMarenotzero. Substitutingthisvalue ofq^inequations (B),andequatingthecoefficients of(s Sii)~m ,wehave H(gi,g2,...,gzn)a Sihn+a= H(gl,g2&gt;...,gm)n+a+sji*= TT/I T T\ /\ andonequatingthecoefficients of(ss1t)~m+1 ,wehave (0whenm &gt;1 [0whenm &gt;1 [cn+awhenm= w*=o la=0(C), (a=1,2,...,)(D). 84] Theory ofVibrations 201 NowbyEuler stheorem onhomogeneousfunctions wehave Zn 2H(g 1}g2,...,gm)=2yJHfr, gz,...,g^)\, A=l orby(C),n%H(gl;gs, ...,g2n)=s,2(gahn+a-hagn+a ), a=l andsimilarlyn2H(/*!, 7*3,...,hm)=Si2(gjin+*-hagn+a ), a=l n fromwhich itisevident that2(gahn+a hagn+a)isnotzero. o.=l Moreover, the firsttwoofequations (C)give 2n w 2hiH(g l)gi,...,gm)i+8i2(hjin+a-hahn+a)=Q(E), A=l a=l andthelasttwoofequations (C)give 2n n 2g^H(h, ,A.,...,Aan)A- !S(gagn+-ggn+)=(F). A.=l a=l Butfrom the firsttwoofequations (D),whenm &gt;1,\vehave 2 n n 2hi,H(g1,g2,...,g 2n)\-s l2(hahn+aL-hahn+ai)-2(gahn+a-hagn+a)= \=1 a=l a=l (G), andfrom thelasttwoofequations (D)wehave Zn n n 2gjl(&,k{&gt; &gt;..,h^}),+sl2(gagn+a-gagn+a)+2(gahn+a-hagn+a)= X=l a=l a=l (H). Also since .STishomogeneousoftheseconddegreeinitsarguments, we have theidentities 2n Zn 2h^H(g,,g 2,...,gzn)\=2#*# (A/,V, ,^//)A (K)A=l X=l I 2w 2n and 2gJH Qh,h2,...,A2n)A=2h^H (g,,g2f ,...,gm\ (L).A=l A=l Fromequations (E),(H),(K)wehave n n n 2(gjin+a-h agn+a)=sl2(ha.hn+a-haihn+a)-s l2(gagn+a-gagn+a ), o=l&lt;x=l a=l andfromequations (F),(G),(L)wehave n n n 2(gahn+a-hagn+a)=-sl2(hahn+a-hahn+a)+sl2(gagf n+*~gagn+*).a=l a=l a=l Comparingtheseequations, wehave n 2(gahn+ai-hagn+&lt;1)=0, 202Theory ofVibrations[CH.vn which iscontrarytowhat hasalready beenproved. Theassumptionthatm &gt;1,which wasused inobtaining equation (G),must therefore befalse;mmust therefore beunity, andconsequently when f(s)isdivisibleby(s-s^if, eachofthefunctions f(s)^isdivisibleby(s-s^i)k-\ Now letSj ,s2,...,srbethemoduli ofthedistinct roots oftheequation/(s)=0, sothatthefunctionsf(s)^/f(s)areinfiniteonlyfors= s^, s2i,..., sri; thendenotingthe coefficient of(s Spi)1intheLaurentexpansionof /()W/(*)inpowersof(s-spi)by (X,fi)p+i(X, fj,)p, where(X,/i)pand(X,/*)/arereal,andobservingthat theonly polesofthe functionf(s)^/f(s)arethepointss= spi,andthat these aresimple poles, wehave (X,n)p+i(\M(\,f*)-i\,n s-si andtherefore &lt;f&gt;(t)^isthecoefficient of1/5intheLaurentexpansionof e&lt;-)f [(X?/*)P+i(\rip ,(\rip~i(\AQpl p=i ( sSpi s+Spi j inpowers ofs. Butthecoefficient ofI/sintheLaurentexpansionofe8(t~^] f(sspi)is -&lt;o)i ?an(jt^ecoefficient ofl/sintheLaurentexpansionofes{t-t) /(s+spi) ise~Sf&gt;(t~t &lt;&gt;)l ;wehave therefore =2S{(X, /i)pcossp(t-t)-(X,fi)pfsinsp(- )},P=I and sofinally nr qn=222[&+.{(,AO PcosSp (*-*&lt;&gt;) -(a,M)Psin5P(-^)}a=lp=l -^a{(n+a,/i)pcossp(i- )-(n+a,/x)psinsp(t-1)}]O=1,2,...,2w). T/itsformula constitutes thegeneralsolutionofthedifferential equations of motion. Hencefinally weseethatwhen thetotalenergy ofasystem vibrating about astateofsteady motion isapositive definite form,thevibratorymotion canbeexpressedintermsofcircularfunctions of t,and thesteadymotion is stable; theperiods ofthenormal vibrations areZTT/S!, 27r/s 2,...,where is1} is.2,...aretheroots ofthedeterminantalequation f(s)=0,whose order ins2 isequaltothenumberofnon-ignorable coordinatesofthesystem. Theaboveinvestigationisvalidwhether thedeterminantalequationhas repeatedroots ornot. Between thecoefficients(X, p.}p,(X, /x)p,there exist therelations (^M)P=- (p-,x)P,(x,/i)p =(M&gt; ^)p t andsoinparticular (X,X)p=0. 84,85J Theory ofVibrations 203 These relations follow from equationswhich Cinvirtue oftheirdefinitions)aretrue for /()&gt;/(*)*! namely /(*)-/(-), Example.Ifthenumber ofdegreesoffreedom ofthesystem,afterignorationofthe ignorable coordinates,iseven, shew thatwhen theignorablevelocities arelarge (e.g.if theignorablecoordinates aretheangles through which certainfly-wheelshave rotated, thiswouldimplythat thefly-wheelsarerotating very rapidly),half theperiodsof vibration areverylongandtheother halfarevery short, theonesetbeing proportional totheignorablevelocities andtheother setbeing inversely proportionaltothese velocities. ItwaspointedoutbyPoincare* that thediscussion ofstability bythe method ofsmall oscillations doesnottakeaccount ofsome features which are likelytobepresentinactualproblems. Thusf, consider aparticlemoveable ontheinner surface ofasphericalbowlwhich rotates with constantangular velocityabout itsvertical diameter. Ifthebowl beperfectly smooth, the equilibriumoftheparticleinthelowestpositioniscertainly stable, the rotation ofthebowl havingnoeffect on it.But ifthere betheslightest friction between theparticleandthebowl, and iftheangular velocityofthe bowl exceeds acertain value, theparticlewillwork itswayoutwards in aspiral pathtowards thepositioninwhich itrotates with thebowl likethe bobofaconicalpendulum. 85.Examples ofvibrations aboutsteadymotion. Anumber ofillustrative cases ofvibration about astate ofsteadymotion willnowbeconsidered. (i)Aparticleisdescribingthecircle r=a,z=b,inthecylindrical field offorcein which thepotential energyisF=((r, 2),where r2=#24-y2 ,itbeing giventhatclV/dzis zerowhenr=a,z=b.Tofindtheconditions forstability ofthemotion. Ifwewrite x=rcos8, y=rsir\8, wehave forthekinetic andpotential energiesoftheparticle, whose mass willbedenoted bym, Theintegral correspondingtotheignorablecoordinate 6ismr26=k,where kisa constant. Themodified kinetic potentialafterignorationof6istherefore R=T-V-W *ActaMath. vn.(1885), p.259. tThis illustration isduetoLamb, Proc. Roy. Soc.LXXX.(1908), p.168. 204Theory ofVibrations[CH.vn Forthesteady motion wemust have thelatter condition issatisfied byhypothesis, andtheformergives k2=ma3 c(f&gt;/da. We have therefore is2- &lt;r,z-^-. Writingr andneglecting terms above theseconddegreeinpandf,wehave 3 \ 0aa+- (Pa- Asnoterms linear inporfoccur, this isessentially thesame asaproblemofvibrations about equilibrium, andthecondition forstabilityis(79)that shall beapositivedefinite form,i.e.that 3\ a+-(f)a)&lt;f&gt;bb~ &lt;t&gt;2 aband(pbba/ shall both bepositive. These aretherequiredconditions forstabilityofthesteady motion. Corollary.Ifaparticleofunitmass isdescribingacircular orbit ofradius aina plane about acentre offorce atthecentre ofthecircle, thepotential energy being &lt;p(r) where risthedistance from thecentre, themodified kineticpotentialis (3\ &lt;aa+-&lt;M,a/ wherer=a+p,sothecondition forstabilityis 3 &lt;Paa+a&lt;P*&gt; andtheperiodofavibration about thecircular motion is 3 (ii)Tofind theperiod ofthevibrations aboutsteadycircular motion ofaparticle moving undergravity onasurface ofrevolution whose axis isvertical. Letzf(r)betheequationofthesurface, where(z,r,ff)arecylindricalcoordinates with theaxisofthesurface asaxis ofz.Iftheparticleisprojected alongthehorizontal tangenttothesurface atanypoint withasuitable velocity,itwilldescribe ahorizontal circle onthesurface with constantvelocity. Letabetheradius ofthecircle;weshall takethemass oftheparticletobeunity,asthisinvolves nolossofgenerality. The kinetic potentialis The integral correspondingtotheignorablecoordinate 6isr2d=k,andthemodified kinetic potentialofthesystemafter ignorationofQistherefore r}}-gf(r)-F/2r2 . 85] Theory ofVibrations 205 Theproblemisthusreduced tothat offindingthevibrations about equilibriumofthe systemwithonedegreeoffreedom forwhichRisthekineticpotential. Thecondition for equilibriumis (d/)=0,orV=ga\f (a\ \or lr=a , and this gives R=ir*{1+/2(r)}-gf(r) -ga?f (a)/2r2 . Writing r=a+p,where pissmall, andexpandinginpowersofp,wehave Theequationofmotion &lt;^/ _ dt\df&gt; dp istherefore p{l+/2 (a)}+0p{/"()+|/()]=0, andthecondition forstabilityis r&lt; theperiodofavibration being 27T Tg Example.Ifthesurface isaparaboloidofrevolution whose axis isvertical and vertex downwards, shew that thevibration-periodis where Iisthesemi-latus rectum oftheparaboloid. (iii)Todetermine thevibrations about steadymotion ofatoponaperfectly rough plane. LetAdenote themoment ofinertia ofthetopabout alinethroughitsapexperpen dicular toitsaxisofsymmetry,and let6denote theanglemade bytheaxiswith the vertical,Mthemass ofthetop,andhthedistance ofitscentre ofgravity from itsapex: thenwehave seen(71)that after ignoringtheEulerian angles (pand^,theangleis determined bysolvingthedynamical systemdefined bythekineticpotential where aandbareconstants dependingontheinitial circumstances ofthemotion. Let a,nbethevalues of6and &lt;j&gt;respectivelyinthesteady motion,so(72) wehave Anzcosa+Mgk=bn, Ansin2a=ab cos a. Todiscuss thevibratorymotion ofthetopabout thisform ofsteady motion, wewrite e=a+xwhere xisasmall quantity, andexpandRinascending powersofx,neglecting powersofxabove thesecond andeliminatingaand bbyuseofthelasttwoequations ; wethus obtain forRthevalue R^AJP-lAx* {n2sin2a+(ncosa-MghjA n)2 }. 206 Theory ofVibrations[OH.vn Theequationofmotion forxistherefore x+{n2sin2a+(ncosa-Mgh/Anf]x=0. Asthecoefficient ofxispositive,thestateofsteady motion isstable;and theperiod ofavibration is 27T{*-2Mghcosa/A+J*gih*IA*n*}~t. (iv) Thesleeping top. Ifweconsider thatform ofsteadymotion ofthetopinwhich aiszero, sothat theaxisofthetopispermanentlydirectedvertically upwards, thetoprotating about this axiswithagiven angular velocity,themethod ofthepreceding example must bemodified, sincenowtheform ofsteady motion inwhich aisasmall constant istoberegardedasa vibration about thetypeofmotion inwhich aiszero :sothatwemaynowexpecttohave twoindependent periodsofnormal vibration, theanaloguesofwhich intheprevious examplearetheperiodofthesteady motion andtheperiodofvibration about it. Asin71,thekinetic andpotential energiesofthetopare T=\4&lt;92+M(/&gt;2sin2 &lt;?+|tf(^+ &lt;j&gt;cos &lt;9)2 , V=Mghcos6. Theintegral correspondingtotheignorablecoordinate ^is 6=^(^+cos &lt;9), andhence afterignorationof^weobtain forthekineticpotentialofthesystemthevalue R=\Afr+%A&lt;&sin26+b$cos6-Mghcos6. Inthetwolastterms wecanreplacecos6by(cos61),since theterms b&lt;j)andMgh thusaddeddisappearfrom theequationsofmotion. As &lt;isnotasmall quantity throughoutthemotion, wetake ascoordinates inplace of 6and &lt;f)thequantities and77,where From these equations, neglectingterms above theseconddegreein,77, ,rj,wehave andsowehave &-lAt* Theequations ofmotion are _ _ " dt bf,-Mg (Arj-b-Mghr,=0. IfSTT/Xistheperiodofa.normal vibration, onsubstituting =7 ,r)KeKinthese differential equations andeliminating JandKweobtain theequation -A2/I-Mghib\ =0, ib\ -\2AMg, or(X2A+Mgh)2-62X2=0. Thetworoots ofthisquadraticinX2givethevalues ofXcorrespondingtothetwo normal vibrations :wehave therefore todetermine thenature ofthese roots. 85,86] Theory ofVibrations 207 Thesolution ofthequadraticis Thevalues ofXaretherefore realornotaccordingasb2isgreater orlessthan4AMgh. Intheformer casethesteady spinning motion round thevertical isstable :inthelatter case, unstable. Itmust notbesupposed, however, that intheunstable case theaxis ofthetop necessarily departs veryfarfrom thevertical: allthat ismeant bytheterm "unstable" isthatwhen b2 &lt;.4AMghthedisturbed motion does not,asthedisturbance isindefinitely diminished, tend toalimiting form coincident withtheundisturbed motion. Asamatter offact,ifb2-4AMgk, though negative,isvery small,itispossible forthe axisofthetopinits "unstable" motion toremainpermanentlyclose tothevertical :but inthiscasethemaximumdivergence from thevertical cannot bemadeindefinitely small (foragiven value of6)bymakingtheinitial disturbanceindefinitely small*. 86. Vibrationsofsystems involving movingconstraints. Ifadynamical systeminvolves aconstraint which varies with thetime (e.g.ifoneoftheparticlesofthesystemismoveable onasmooth wire or surface which ismade torotateuniformlyabout agiven axis), thekinetic potentialofthesystemisnolonger necessarily composedofterms ofdegrees 2and inthevelocities;terms which arelinear inthevelocities mayalso occur. Theequationswhich determine thevibrations ofsuch asystemwill therefore ingeneralincludegyroscopic terms, evenwhen thevibration is about relativeequilibrium:thesolution canbeeffected bythemethods above developedfortheproblemofvibrations aboutsteadymotion. Thefollowing examplewillillustrate this. Example.Tofindtheperiods ofthenormal vibrationsofaheavy particleabout its position ofequilibriumatthelowestpoint ofasurface which isrotatingwith constant angular velocityu&gt;about avertical axisthroughthepoint. Let(.v, ?/,z)bethecoordinates oftheparticle,referred toaxeswhich revolve with the surface, theaxes ofxandybeingthetangentstothelines ofcurvature atthelowest point, andtheaxis ofzbeingvertical. Lettheequation ofthesurface be i2 2 z=~+~-+terms ofhigher order. %&gt;i 2p2 Thekinetic andpotential energiesoftheparticle are V=mgz. Thekinetic potentialofthevibration-problemistherefore (y&y2\ 5T-+j- )/Pi ^Pz/ *Adiscussion ofthestability ofthesleeping topisgiven byKlein, Ball.Amer. Math. Soc.in. (1897), pp.129132, 292. 208 Theory ofVibrations [CH. Theequationsofmotion are _._ = ~ cy Pi Iftheperiodofanormal vibration is27T/X,wehave (substituting XAe, y=BelXin thedifferential equations, andeliminating AandB) -2wiX =0, or (X2+2-g/pi) (X2+o&gt;2-glpz}-4X22=0. Theroots ofthisquadraticinX2determine theperiodsofthenormal vibrations. MISCELLANEOUS EXAMPLES. 1.Aparticlemoves onacurve which rotates uniformlyabout afixed axis, the potential energy V()oftheparticle depending onlyonitspositionasdefined bythe arc s.Shew that theperiodofavibration about apositionofrelative restonthe curve is dVd, where risthedistance oftheparticlefrom theaxis. 2.Determine thevibrations ofasolid horizontal circular cylinder rollinginside a hollow horizontal circular cylinderwhose axis isfixed, shewingthat thelengthofthe simple equivalent pendulumis(b-a)(3M+m)l(2M+m);where bistheradius andJ/the mass, oftheoutercylinder,andaistheradius andmthemass, oftheinner cylinder. (Coll. Exam.) 3.Athin hemisphericalbowl ofmassMandradius aisonaperfectly rough horizontalplane,andaparticleofmassmisincontact with theinner surface ofthebowl, supposedsmooth. Shew thatwhen thesystem performssmall oscillations, themotion of theparticleandthecentre ofgravityofthebowl beinginoneplane,theperiodsofthe normal vibrations are2^/^X7 and2^/^X3,where AtandX2aretheroots oftheequation ma\g-(g-aA)(\g-%aA)M= 0. (Coll.Exam.) 4.Astringoflength4aisloaded atequalintervals with three weights m,Mandm respectively, and issuspendedfromtwopointsAandBsymmetrically.Shew that ifM performsmall vertical vibrations, thelengthofthesimple equivalent pendulumis acosacosftsin(a- ft}cos(a-0) sinacos2a+sinftcos2 ft where aandftaretheinclinations ofthepartsofthestringtothevertical. (Coll. Exam.) 5.Auniform barwhose lengthis2aissuspended byashort string whose lengthisI; provethat thetime ofvibration isgreater than ifthebarwere swingingabout one extremityintheratiol+9/32a:1nearly.(Coll. Exam.) vn] Theory ofVibrations 209 6.Anelliptic cylinderwithplane ends atright anglestoitsaxisrestsupon twofixed smoothperpendicular planes which areeach inclined at45tothehorizon. Shew that there aretwostable configurations andoneunstable, andthat intheformer casethe length oftheequivalent pendulumis ab(a2+62)/2V2(a-6)2(a+6), aandbbeing thelengthsofthesemi-axes.(Coll. Exam.) 7.Arough circularcylinderofradius aandmassmisloaded sothat itscentre of gravityisatadistance hfrom theaxis,and isplaced onaboard ofequal masswhich canmove onasmooth horizontalplaneiIfthesystemisdisturbedslightly when ina position ofstableequilibrium, shew thatthelengthofthesimple equivalent pendulumis k2lh+\(a-h}2 /k,wheremk1isthemoment ofinertia ofthecylinder about ahorizontal axisthroughitscentre ofgravity. (Coll. Exam.) 8.Oneendofauniform rodoflengthbandmassmisfreely jointedtoapointina smooth vertical wall;theother end isfreely jointedtoapointinthesurface ofauniform sphere ofmassMandradius awhich restsagainstthewall. Shew thattheperiod ofthe vibrations about thepositionofequilibriumisZTT/JO,where p2(sin sin2 (a- /3)+|cosasin(a-/3)+fsin/3cos2 /3}=--?--(asinacos2a+bsin/3cos2 /3), aand/3being given bytheequations asina+bsin/3-a=0, (m+M)tanft-Mtana=0.(Coll.Exam.) 9.Athin circularcylinder ofmassMand radius brests onaperfectly rough horizontalplane, andinside itisplacedaperfectly rough sphereofmassmandradius a. Ifthesystem bedisturbed inaplane perpendiculartothegeneratorsofthecylinder,find theequations offinite motion, anddeduce two firstintegrals ofthem;and ifthemotion besmall, shew thatthelengthofthesimple equivalent pendulumis l4M(b-a)/(WM+7m.). (Camb. Math.Tripos, PartI,1899.) 10.Asphereofradius cisplaced upon ahorizontalperfectly rough wire inthe form ofanellipse ofaxes 2a,2b.Prove thatthetime ofavibration undergravity about theposition ofstableequilibriumisthat ofasimple pendulumoflengthIgiven by whereF=2c2 /5andd2=c2-62 .(Coll. Exam.) 11.Arhombus offourequal uniform rods oflength afreely jointed togetherislaid onasmooth horizontalplane with oneangle equal to2a.Theopposite corners are connected bysimilar elasticstringsofnaturallengths 2acosa,2asin a.Prove that if onestringbeslightly extended andtherhombus leftfree, theperiods during which thestrings areextended inthesubsequent motion areintheratio (cosa)$:(sina)^. (Coll. Exam.) 12.Aparticleofmassmisattachedbynequalelasticstringsofnaturallength ato thefixed angular pointsofaregular polygonofnsides, theradius ofwhosecircumscribing circle isc.Shew that iftheparticle beslightly displaced from itsequilibrium positionin theplaneofthepolygon,itwillexecute harmonic vibrations inastraight line,thelength ofthesimple equivalent pendulum being 2mgac/n\ (2c-a),and that forvibrations perpendiculartotheplaneofthepolygon, thecorresponding lengthwillbemgacjriX (c-a), Xbeing themodulus ofeachstring. (Camb. Math.Tripos, PartI,1900.) w.D. 14 210 Theory ofVibrations[en. 13.Theenergy -equationofaparticleis f(x)x2= 2&lt;f)(x)+constant, andaisavalue ofxforwhich &lt;$&gt;(x~)iszero. If &lt;&lt;2p)(x)isthe first derivative of &lt;j&gt;(x) which doesnotvanish forx=a,shew thattheperiodofavibration about theposition ais 4r (i/2/&gt;) fr (2/&gt;)/ra hv1F(1/2/3+^) ( 4/30(2p)(a) where histhevalue of(x-a)correspondingtotheextremedisplacement. (Elliott.) ]4.Aconehas itscentre ofgravityatadistance cfrom itsaxis,there beinginother respectstheusual kinetic symmetryatthevertex. Ifthecone oscillates onahorizontal plane andtheplane beperfectly rough, shew that thelength ofthesimple equivalent pendulumis (cosa/Me)(Asin2a+Ccos2 a), whereas ifthisplane beperfectly smooth, thelengthis (cos a/Me)(sin2a/A+cos2a/C). (Coll. Exam.) 15.Anumber ofequal uniform rodseach oflength 2aarefreely jointedatacommon extremity andarrangedatequal angularintervals liketheribsofanumbrella. Thiscone ofrods isputoverasmooth fixed sphereofradius6,each rodbeingincontact withthe sphere, andrests inequilibrium. Shew that,ifthesystem beslightlydisturbed sothat thehinge performsvertical vibrations about thepositionofequilibrium, theirperiodis l+3sin2a\4 /wsm a.1+2 sin2a, where sec3asina=a/6. (Camb. Math. Tripos, PartI,1896.) 16.Aheavy rectangular board issymmetrically suspendedinahorizontalposition byfourlightelasticstrings attached tothecorners oftheboard andtoafixedpoint vertically above itscentre. Shew thattheperiodofthevertical vibrations is ^WM) where cistheequilibriumdistance oftheboard below thefixedpoint,aisthelengthof asemi-diagonal, =(2+c2 )2,andXisthemodulus.(Coll. Exam.) 17..Aheavy lamina hangsinequilibriuminahorizontalposition suspended bythree vertical inextensiblestringsofunequal lengths. Shew that thenormal vibrations are (1)arotation about either oftwo vertical lines inaplane through thecentroid, and (2)ahorizontal swing paralleltothisplane. (Coll. Exam.) 18.Auniform rodoflength 2aisfreely hingedatoneend, attheother endastring oflengthbisattached which isfastened atitsfurther endtoapoint onthesurface ofa homogeneous sphereofradius c.Ifthemasses oftherodandsphereareequal,findthe motion ofthesystem whenslightlydisturbed from thevertical, andshew that the equationtodetermine theperiodsis 2a6c/x3-gp?(66c+19ca+5a6)+g2n(35a+156+2k1 )-g3=0. (Coll. Exam.) 19.Auniform wire,intheshapeofanellipseofsemi-axesa,6,restsuponarough horizontal planewith itsminor axis vertical andaparticleofequal mass issuspended by afinestringoflengthIattached tothehighest point.Ifvibrations inaverticalplane beperformed, provethat their periodswillbethose ofpendulums whose lengthsarethe value ofxgiven bytheequation {x(36-2a2/6)+562+i?}(x-l)+462/=0, where kistheradius ofgyrationabout thecentre ofgravity. (Coll. Exarn.) vn] Theory ofVibrations 211 20.Afineinextensiblestring has itsends tied totwo fixedpegsinahorizontal linewhose distanceapartisthree-quarters ofthelength ofthestring. Thestring alsopasses through twosmall smoothrings which arefixed totheends ofauniform straightrodwhoselengthishalf that ofthestring. The rodhangsinequilibrium inahorizontalposition and receives asmall disturbance intheverticalplaneofthe string. Shew thatinitiallyitsnormal coordinates interms ofthetime areLcos(pt+a) andMcosh(qt+fi),wherep2andq2aretheroots oftheequation yA_^9X2_3ff*=o.(ColLExam .)4a 4a-* 21.Aheavyuniform rodoflength 2a,suspended from afixedpoint byastring oflength b,isslightly disturbed from itsverticalposition. Shew thattheperiodsofthe normal vibrations areI2nlp land27r/p 2,wherepi2and p&gt;22aretheroots oftheequation abp*-(4a 22.Acirculardisc,mass M,isattached byastring from itscentre Ctoafixed point0.Aparticle ofmassmisfixed tothedisc atapointPontherim. Find the equations ofmotion onaverticalplaneinterms oftheangles 6and whichOCandCP make with thevertical, andprove that ifthesystem vibrates about theposition of equilibrium theperiodsinthese coordinates aregiven bytheequation (M+m)(p~a-g]{(M+2i)cp2-2mg]=2m2cap\ where aisthelength ofthestringOCand ctheradius ofthedisc.(Coll. Exam.) 23.Ahemispherical bowl ofradius 2brestsonasmooth table with theplane ofits rimhorizontal;within itandinequilibriumliesaperfectly rough sphereofradiusb,and massone-quarterofthat ofthebowl.Aslight displacementinaverticalplane con taining thecentres ofthesphere andthebowl isgiven:prove that theperiods ofthe consequent vibrations are2ir/pi and2ir/p z,wherep^andp%2aretheroots ofthe equation 1566%2-2606a^+75#2=0.(Coll. Exam.) 24.Auniform circular disc ofmassmandradius aisheld inequilibrium ona smooth horizontalplane bythree equalelasticstringsofmodulusX,naturallength1and stretchedlengthI.Thestrings areattached tothediscattheextremities ofthree radii equally inclined tooneanother andtheir other ends areattached topoints oftheplane lying ontheradiiproduced. Shew thattheperiods ofvibration ofthediscare J)}and where /i=2m^ /3X. (Camb. Math.Tripos, PartI,1898.) 25.Aparticleisdescribing acircle under theinfluence ofaforce tothecentre varyingasthenthpower ofthedistance. Shew that thisstate ofmotion isunstable ifn belessthan-3. r Shewthat,iftheforcevaryase"/r2 ,themotion isstable orunstableaccordingas theradius ofthecircle islessorgreater than a.(Coll. Exam.) 26.Aparticle moves infreespace under theaction ofacentre offorcewhich varies astheinverse square ofthedistance andafield ofconstant force :shew thatacircle describeduniformlyisapossible state ofsteady motion, butthis willbestableonly provided thecircle asviewed from thecentre offorceappearstolieonaright circular conewhose semi-verticalangleisgreater thanarccosj. (Coll. Exam.) 142 212 Theory ofVibrations [CH. 27.Aparticledescribes acircle uniformlyunder theinfluence oftwocentres offorce which attract inverselyasthesquareofthedistance. Prove thatthemotion isstable if 3cosecos&lt;&lt;1,where 6and &lt;f&gt;aretheangleswhich aradius ofthe circle subtends atthecentres offorce. (Camb.Math. Tripos,PartI,1889.) 28.Aheavy particleisprojected horizontallyontheinterior ofasmooth conewith itsaxis vertical andapexdownwards; the initial distance from theapexiscandthe semi-vertical angleofthecone isa.Find thecondition thatahorizontal circle should be described; andshew that thetime ofavibration about this steadymotion isthat ofasimple pendulumoflength ^cseca. (Coll. Exam.) 29.Acircular dischasathinrodpushed throughitscentre perpendiculartoits plane,thelengthoftherodbeing equaltotheradius ofthedisc;provethatthesystem cannot spinwith therodvertical unless thevelocityofapointonthecircumference ofthedisc isgreaterthan thevelocity acquired byabodyafter falling from rest vertically throughtentimes theradius ofthe disc. (Coll. Exam.) 30.Prove that forasymmetrical topspinning uprightwith sufficient angular velocityforstability,thetwotypesofmotion, differing slightlyfrom thesteadymotion intheupright position,which aredetermined bysimpleharmonic functions ofthetime, arethelimits ofsteadymotions with theaxisslightlyinclined tothevertical, andthat theperiodofthevibrations isthelimitingvalue ofthatwhich correspondstosteady motion inaninclined positionwhen theinclination isindefinitelydiminished. (Coll. Exam.) 31.Oneendofauniform rodoflength2awhose radius ofgyration about one end iskiscompelledtodescribe ahorizontal circle ofradius cwithuniform angular velocityw.Prove thatwhen themotion issteadytherod lies intheverticalplane throughthecentre ofthecircle andmakes anangleawith thevertical given by w2(k2+accoseca)=ayseca. Shew that theperiodsofthenormal vibrations are27r/\i, 27r/X 2,where Xl5X2arethe (FX2sina- a&gt;2ac) (2X2sina- a&gt;2ac- a&gt;2&2sin3a)=4a&gt;24X2sin2acos2a. (Camb.Math. Tripos,PartI,1889.) 32. Investigatethemotion ofaconical pendulumwhen disturbed from itsstate of steadymotion byasmall vertical harmonic oscillation ofthepointofsupport. Canthe steadymotion berendered unstable bysuchadisturbance? (Coll. Exam.) 33.Themiddle pointofonesideofauniform rectangleisfixedandthelinejoining ittothemiddle pointoftheoppositeside isconstrained todescribe acircular cone ofsemi -angleawith uniform angular velocity. The rectangle being otherwise free, find thepositionsofsteadymotion andprovethat thetime ofavibration about the positionofstable steadymotion isequaltotheperiodofrevolution divided bysina. (Coll. Exam.) 34.Asolid ofrevolution, symmetricalabout aplane throughitscentre ofgravity perpendiculartoitsaxis,issuspendedfrom afixed pointbyastringoflengthbwhich is attached tooneend oftheaxis ofthe solid,this axisbeingoflength2a.Themass ofthesolid isM,and itsprincipalmoments ofinertia atitscentre ofgravityare (A,A,C}.Ifthesolid isslightlydisturbed from thestate ofsteadymotion inwhich the string and axis arevertical, andthebodyisspinningonitsaxiswithangular velocity , shew thattheperiodsofthenormal vibrations are^irlp land 27r//&gt; 2,wherep^andp22are theroots oftheequation HaPgp*-=(g-fy2 )(Mag+Cnp-Ap*}. vn] Theory ofVibrations 213 35.Asymmetrical topspins with itsaxisvertical, thetipofthepegrestingin afixed socket. Asecondtop,alsospinning,isplaced onthesummit ofthefirst, thetip ofthepegrestinginasmall socket. Shew thatthearrangementisstableprovidedthe equation (Mcgx*+CQx+A){(Mc1+MK)gx*+CQx+(A+Mh?}}=J/2AV) has allitsroots real;fl,iibeingthespinsoftheupper andlowertops respectively, M,Mtheir masses, C,Ctheirmoments ofinertia about theaxis offigure, A,Aabout perpendiculars throughthepegs, c,cthedistances ofthecentroids from thepegs,andh thedistance between thepegs. (Camb. Math. Tripos, PartI,1898.) 36.Ahomogeneous body spins onasmooth horizontalplaneinstablesteady motion, withangular velocity wabout thevertical through thepointofcontact andthecentre of gravity. Thebodyissymmetrical about each oftwoperpendicular planes through the vertical. Theprincipalradii ofcurvature atthevertex onwhich itrests arepi ,p2;the moments ofinertia about theprincipalaxesthrough thecentre ofgravity (parallel tothe lines ofcurvature) arerespectively Aand,and thatabout the vertical isC.The heightofthecentre ofgravity about thevertex isa=al+pl=a.2+p2;andXw2isthe weightofthebody. Shew thatthefollowing conditions must besatisfied : (i) (ii) (iii)Thevalue ofXmust not liebetween thetwovalues (A+B-C]\JB{M+ 2(A-Cf)YJA[a.2B4-av(E- ifthetworadicals intheexpression areboth real. (Camb. Math.Tripos, PartI,1897.) CHAPTER VIII NON-HOLONOMIC SYSTEMS. DISSIPATIVE SYSTEMS 87.Lagranges equationswithundeterminedmultipliers. Wenowproceedtotheconsideration ofnon-holonomic dynamical systems. Inthesesystems,aswasseen in 25,thenumber ofindependentcoordinates (&lt;?i&gt; &lt;?2,&gt;In)requiredinorder tospecifytheconfigurationofthesystemat anytime isgreaterthan thenumber ofdegreesoffreedom ofthesystem, owingtothefactthat thesystemissubjecttoconstraints which willbe supposedtodonowork, andwhich areexpressed byanumber ofnon- integrable*kinematical relations oftheform Alkdq1+A2kdq2+...+Ankdqn+Tkdt=(k=l,2, ...,m), where An,A12,...,Anm ,7\,T2,...,Tmaregivenfunctions ofq1} q%, iQn&gt;t. Themost familiar exampleofsuchasystemisthat ofabodywhich isconstrained to rollwithoutsliding onagivenfixed surface :thecondition thatnosliding takesplaceis expressed bytworelations ofthetype givenabove. Astillsimpler exampleisthatof,a vertical wheel with asharp edgewhich rolls onahorizontal sheet ofpaper,asinthe integraphofAbdank-Abakanowicz andtheintegratorofPascal :thewheel movesonlyin itsowninstantaneous plane,thefriction atthesharp edge preventingitfromslipping sideways.If(x,y}aretherectangularcoordinates ofitspointofcontact with the paper, and$theazimuth ofitsplane, wehave inthiscasethenon-holonomicequation ofcondition dytan &lt;p.dx=0. Thenumber ofkinematical relationsbeing m,thesystemwillhave (nm)degreesoffreedom;itisnotpossibletoapply Lagrangesequations directlytosuchasystem,butanextension oftheLagrangian equationswill nowbegivenwhich willenable ustodiscuss themotion ofnon-holonomic systemsinawayanalogoustothatpreviously developedforholonomic systems. Consider then anon-holonomicsystem,whoseconfigurationatany instant iscompletely specified byncoordinatesql}q2,...,qn;letthe kinetic energybeT,and letthekinematical conditions due tothenon- holonomic constraints beexpressed bytherelations A.kdq,+A^dq 2+...+Ankdqn+Tkdt=(k=1,2,...,m). *Ifthese relations were integrable,itwould bepossibletoexpress some ofthecoordinates (&lt;?i&gt;92&gt;11n)iQterms oftheothers, andthencoordinates would therefore notbeindependent: which iscontrarytoourassumption. 87] Non-holonomicSystems. Dissipative Systems 215 Now itisopentouseithersimplytoregard thesystemassubjectto these kinematical conditions, orinplaceofthese toregardthesystemas acted onbycertain additional external forces, namelytheforces which have tobeexertedbytheconstraints inorder tocompelthesystemtofulfil the kinematical conditions; weshall forthepresenttake thelatterpointof view. Let Qi8qi+Q*Bq t+...+Qn*q* bethework done onthesystem bythese additional forces inanarbitrary displacement (Sq^ 8q2,...,Sqn)(whichisnownotrestricted tosatisfythe kinematicalconditions), and let Ql8q 1+Q2Sq2+...+QnSqn bethework done onthesystem bytheoriginalexternal forces inthis dis placement. Since thesubstitution ofadditional forces forthekinematical relations hasmade thesystem holonomic, wecanapplytheLagrangian equations ;wehave therefore dd astheequationsofmotion ofthesystem. The forces Q/&gt;Qz,&gt;Qnareunknown :buttheyaresuch that, inany displacementconsistent with theinstantaneous constraints, theydonowork. Itfollows thatthequantity Qidq l+Q2dq2+...+Qndqn iszero for allvalues ofthe ratiosdq l:dq 2:... :dqnwhichsatisfythe equations A*dq l+Atitdqa+...+Ankdqn=0; hence wemust have Qr=\An+\a4+...+\mArm (r=1,2,...,n), where thequantities \l}\2,...,\mareindependentofr.Wethushave altogetherthe(n+m)equations ddT\ dT ^ifcffi+A2kq2+...+Ankqn+2^=(k=l,2, ...,m), and these aresufficienttodetermine the(n+m)unknownquantities &lt;1\,fc,&gt;qn,^i, )u&gt; ,X.Theproblemisthusreduced tothesolution ofthis setofequations*. Theextension ofLagrangesequations tonon-holonomic systemsisduetoFerrers, Quart. Journ. Math. xn.(1871), p.1 :C.Neumann, Leipzig Berichte, XL.(1888), p.22:andVierkandt, Monatshefte furMath. u.Phys. in.(1892), p.31. 216 Non-holonomic Systems. Dissipative Systems [CH.vin 88.Equations ofmotionreferredtoaxesmovinginanymanner. Themethodgivenintheprecedingarticledepends essentiallyonthe reduction ofthenon-holonomicsystemtoaholonomicsystem byintroducing theforces due tothenon-holonomic constraints. Inpractice,this isoften mostconvenientlydonebyforming separatelytheequationsofmotion of each ofthebodies ofthesystem.Itismoreoverfrequently advantageous touseaxes ofreference which arenotfixed either inspaceorinthebody, andweshallnow findtheequationsofmotion ofarigid bodyreferred to axeswhich have theiroriginatthecentre ofgravityofthebody,andare turningabout itinanymanner*. LetGbethecentre ofgravityofthebody,and letGxyzbethemoving axes. Let(u,v,w)bethecomponentsofvelocityofthecentre ofgravity resolvedparalleltothese axes,and let(0l,0&lt;2,#3)bethecomponentsof angular velocityofthesystemofaxesGxyzresolvedalongtheaxesthem selves; further let (&&gt;j,w2,&&gt;3)bethecomponentsofangular velocityofthe body,resolvedalongthesame axes. Then(64)themotion ofGisthesame asthatofaparticleofmassM,equaltothat ofthebody,acted onbyforces equaltotheexternal forces which actonthebody (includingallforces of constraint, exceptthemolecular reactions between theconstituentparticles ofthebody);let(X,Y,Z)bethecomponents paralleltotheaxesGxyzof these external forces. ThecomponentofvelocityofGparalleltoGx isu,andconsequently (17)thecomponentofitsacceleration inthisdirection isuv03+wQ2;we have therefore theequation which canbewritten whereTdenotes thekineticenergyofthebody, expressedinterms of (u, v,w, a&gt;!,&lt;w2&gt;&&gt;3);andsimilarequationscanbeobtained forthemotion ofGparalleltotheaxesGyand Gz. Consider next themotion ofthebodyrelative toG,which(64)is independentofthemotion ofG;from 62,63,weseethat theangular momentum ofthebodyabout theaxisGx isdT/dwi,sothat therate of increase ofangular momentum about anaxis fixed inspace and in stantaneously coincidingwithGx is Af^.\-B 9 *Intheapplicationsofthismethod, theaxes areusually chosen subject tothecondition thatthemoments andproductsofinertia ofthebody with respecttothem donotvary;butthis condition isnotessential. 88,89]Non-liolonomic Systems. Dissipative Systems217 IfL,M,Ndenote themoments oftheexternal forces about theaxes Gxyz,wehave therefore(40) d dT dT_ 25=**j andtwosimilarequations. Hencefinallythemotionofthebodyisdetermined bythesixequations l*.(W\-aW ffd_T_Y d_fdT^_.dT,dT r dt\du) d.(W\_dt dt\dw/2duldv dt\dwj2 9o&gt;i3h Itwillbeobserved thatthese arereally Lagrangian equationsofmotion interms ofquasi-coordinates,andcould havebeen derived byuseofthe theorem of 30. Example.Iftheoriginofthemovingaxes isnotfixed inthebody,let(w1}u2,a) bethecomponentsofvelocityoftheoriginofcoordinates, resolvedparalleltothe instantaneous positionoftheaxes; let(0l5 2,3)bethecomponentsofangular velocity ofthesystemofaxes, resolved along themselves;let(vi,v2,v3)bethecomponentsof velocityofthatpointofthebody which isinstantaneouslysituated attheoriginof coordinates; and let (&&gt;!,o&gt;2,o&gt;3)bethecomponentsofangular velocityofthebody,also referred tothemovingaxes. Shew that theequationsofmotion canbewritten in theform tfiJ^-^W+e^X, (VL\-fi^J+eM=dt\c dT ^OV 00)2dT7- 5 00)3 dT .dT .dT+u3-0!^+3JT-=. s OVi Oo&gt;3 Oo&gt;i dT dT dT dT where(JT,F,Z,Z,J/,^)arethecomponents andmoments oftheexternal forces with reference tothemovingaxes. 89.Applicationtospecialnon-holonomic problems. Weshallnowconsider someexamplesillustrative ofthetheoryofnon- holonomicsystems. Example1.Sphere rollingonafixed sphere. Let itberequiredtodetermine themotion ofaperfectly rough sphereofradius aand mass TOwhich rollsonafixedsphereofradius6,theonlyexternal force being gravity. 218 Non-JwlonomicSystems. Dissipative Systems [CH.vm Let(b,0, &lt;)bethepolar coordinates ofthepointofcontact, referred tothecentre of thefixedsphere, thepolar axisbeingvertical. Wetakemoving axesGABC, whereGis thecentre ofthemoving sphere,GC istheprolongationofthelinejoining thecentres of thespheres, GA ishorizontal andperpendiculartoGC,andGB isperpendiculartoGA andGC,inthedirection of6increasing. With these axeswehave, inthenotation ofthelastarticle, 0i=-P, 0-2=-&lt;j&gt;si"0, 03=cos0, {5 and ifF,Fdenote thecomponentsoftheforce atthepointofcontactparalleltoGA andGBrespectively, wehave X=F, Y=mgsm0 +F, L=Fa,M=-Fa, N=0. Theequations ofmotion ofthelastarticle become therefore m(uv03)=F=fam(o&gt;2 lw3+3i), in(v-{-u0s) mgsin=F1=gam(o&gt;j 3 a&gt;2+0201s), 0)3"*"C/2O)l~|~UIO)2w. Moreover, thecomponents paralleltotheaxesGA,GBofthevelocityofthepointof contact areu ao&gt;2andv-fao)t,andconsequently thekinematical equationswhichexpress thecondition ofnoslidingatthepointofcontact are Eliminating F,F,a^,o&gt;2,wehave |-t703-f0i3=0, &lt;v+u0 3fa02(&gt;) 3J*-gsin$=0, I0)3=0. The lastequation givesa&gt;3=n,where nisaconstant; whilesubstitutingforu,v, 1} 2, inthe firsttwoequations their values interms of0,6,0,wehave ( d (a+b}-r(0sin0)+(a+6)0&lt;f&gt;cos-fanf&gt;=0,dt (a+b]0-(a+b)&lt;p?cos0sin +%an&lt;f&gt;sin0-%gsiu=0. Theformer ofthese equations canbeintegratedatonce after multiplying throughout bysin0,andgives (a+6)0sin2$+fcmcos0=k, where kisaconstant. Moreover, multiplyingthesecond equation throughout byand the firstequation by (/&gt;sin0,andadding, weobtain anequation which canbeatonce integrated, giving n2^+-2- cos=h, where hisaconstant;this isreallytheequationofenergyofthesystem. Eliminating between these twointegral equations, wehave (a+fe)2sin2 .&=-(k-fancos(9)2-^-g(a+b)sin2cos+h(a+6)2sin2 ; andonwriting cos#=#,thisequation becomes 89] Non-holonomic Systems. Dissipative Systems219 Thecubic polynomialinxontheright-handsideofthisequationispositive when x=+oc,negative when#=1, positiveforsome realvalues of6,i.e.forsome values ofx between -1and 1,andnegative when#=-1;ithastherefore oneroot greater than unity, andtworoots between 1and-1;weshall denote these rootsby coshy,cos/3,cosa, where cosj3&gt;cosa;andwethenhave (t+f)=j{(x-coshy)(x-cos|8)(x-cosa)} J \7a+b) where eisaconstant ofintegration. Writing .14a+b 7A(a+6)2+^a ,+i(coshy +cos/3+cosa)=-*+3,v(a+6) theequation becomes or 2= where thefunction isformed withtheroots e2~14(a+6){CS/3" 30&lt;7(a+i) J thesequantities e^e2,e3areallreal,andsatisfytherelations ei+62+63=0,el&gt;e 2&gt;e3. Now a*isreal forrealvalues oftand(since xisreal)liesbetween cosaandcos3 ; sozisrealand liesbetween ezand e3;hence theimaginary partoftheconstant einthe argumentofthe^-functionisthehalf-period correspondingtotheroot e3,which weshall denote by&lt;B;therealpartofemaybetaken tobezerobysuitably choosingtheorigin oftime :andtherefore wehavefinally 14&lt; Thisequation gives thevariable interms ofthetime :theother coordinate (/&gt;ofthe centre ofthemoving sphereisthen obtained byintegratingtheequation thisintegration canbeeffected byaproceduresimilar tothatused(72)toobtain the Eulerian angles which define thepositionofatopspinningonaperfectly rough plane. Example2.Arough sphererolls incontact with theoutside ofafixed rough sphere, undergravity;ifz2,z3bethegreatest andleast heightsofitscentre, duringthemotion, and zbetheheight atatime tfromaninstant when zwasequalto22,provethat (*2~z} [|f&gt;(0-e2]=(*2~23) (&lt;?i-e2), where el,e2,e3(=-el-e2)arerealquantitiesindescendingorder ofmagnitude. (Coll. Exam.) 220 Non-holonomic Systems. Dissipative Systems [CH.vm Example3.Sphere rollingonamoving sphere. Consider nowthemotion ofarough sphereofradius aandmassmwhich rollsunder gravity onanother sphere,ofradius bandmassM,thelatter sphere beingfreetoturn about itscentre 0,which isfixed. Let(6,0)bethepolarcoordinates ofthepointofcontact referred toaxes fixed inspace with thefixed centre asorigin,theaxisfromwhich 6ismeasured beingvertical. Toobtain theequationsofmotion ofthespheiem,wetake(asinthelastexample) moving axesOABC ofwhichGC istheprolongationofthelineOGfjoiningthecentres of thespheres, andGA ishorizontal. Let(01,$2,3)denote thecomponents ofangular velocityofthecoordinate-systemresolved alongitsownaxes,and let(wj,o&gt;2,^3)denote the components ofangular velocityofthespheremalongthesame axes. Then, asinthelast example, wehave T=\111UZ+V-+W2+ ((Bj2+(B22+0&gt;32 ) , and ifF,Fbethecomponentsoftheforceacting onthespherematthepointofcontact paralleltoOAandGBrespectively, wehave L^Fa,M=-Fa, N=Q, sotheequationsofmotion become (1)3+^3Wi).................. (1), (02+62013).................. (2), a)3- &lt;92an+0j0)2=...................................................... (3). Todetermine themotion ofthesphere M,wetakemoving axesparalleltotheaxes (JABC, butwith theiroriginat0;let(Q1?Q2,Q3)denote thecomponentsofangular velocityofthesphere resolvedalongthese axes. Then forthesphereMwehave and itsequationsofmotion are 3)=F .............................. (5), Q3-(92Qi+&lt;9iQ 2=0 ................................. (6). Theconditions ofnoslidingatthepointofcontact are u-aa&gt;2=bQ2,v+aa&gt;l=-bQl..............................(7). Inorder tosolve this setofequations wemultiply equations (3)and(6)byaand b respectively, andadd;thus, using (7),wehave aw3+bQs+u&i+v6%=0, or ai&gt;3-f&Q3=0. Integrating, wehave a&lt;a3+bQ3=an, where nisaconstant. 89,90]Non-holonomic Systems. Diasipative Systems221 Moreover, from equations (4)and(7)wehave -fM(u- ao&gt;2-b6iQ 3-3v-6zaa&gt;^=F. Eliminating ^and o&gt;2+ #30&gt;ibetween thisandequations (1),wehave ... (u-63v)= d ..,Nrnr1 Similarly from equations (5)and(7),wehave M(-va&iud3+0.0%&lt;w2+b62Q3)=F . Eliminating Fand^-#3co2between thisandequations (2),wehave ,..b(M+ni)., f-(v+M03)=andz+L 2j/9sin6, 5(M+m}qsin d an2M . . (9cos&lt;9- .- _*-rr=----r.r-rr.4&gt;sme.........(B). or (9-&lt;i2sin Now theequations (A)and(B),fromwhich 6and$aretobedetermined interms oft, areofessentiallythesame character astheequationsfound forthedetermination of6 and intheprevious example:theformer equations beinginfactderivable from the present onesbymakingMvery large comparedwith m.Theintegration therefore proceeds exactlyasintheformer case. Example4.Auniform sphererolls onaperfectly rough horizontalplane, under forces whose resultantpasses throughitscentre. Shew that themotion ofitscentre isthesame asthatofaparticleacted onbythesame forces reduced intheratio 5 :7. Example5.Form theequationsofmotion ofaperfectly rough sphere rolling under gravityinside afixedrightcircularcylinder,theaxisofwhich isinclined tothevertical at anangle a;andshew that,ifthespherebesuch that&2=^a2 ,abeingitsradius andk theradius ofgyration about anydiameter, and ifitbeplacedatrestwith theaxial plane throughitscentre making anangle /3with the vertical axialplane,thevelocityof thecentreparalleltotheaxis,when thisangleis6,is {sin\6arccosh(cos6seci/3)+cosi(9arccos(sin\dcosecA/3)}, where b+aistheradius ofthecylinder. (Camb. Math. Tripos, PartI,1895.) Forother examplescf.Woronetz, Math. Ann. LXX.(1911), p.410. 90. Vibrationsofnon-holonomicsystems. We shall next consider thesmallvibratorymotions ofanon-holonomic system:itwillappearthat sofarasvibrations aboutequilibriumarecon cerned, thedifference between holonomic andnon-holonomicsystemsis unimportant. Forconsider thevibrations aboutequilibriumofanon-holonomicsystem withnindependentcoordinates and(nm)degreesoffreedom, inwhich theconstraints areindependentofthetime. LetTbethekinetic andVthe potential energy,sothat forthevibrational problem Twillbesupposedtobe ahomogeneous quadraticfunction of(qltq,...,qn),andFtobeahomogeneous 222 Non-holonomic Systems. Dissipatlve Systems [CH.vui quadraticfunction of(ql,q.2,...,q n],the coefficients inboth cases being constants. There aremequationsofthetype Alkq!+Aaq*++Ankqn=(=1,2,..., m), whichexpressthenon-holonomic constraints :andtheequationsofmotion are(87) d/?T\ dV -r.(^- }=-^-+\jA n+\zAr2+...+\mArm(r=l, 2, ...,&gt;at\cqrj oqr From theseequationsitisevident that\l}X2,...,\mareingeneralsmall quantitiesoftheorder ofthecoordinates;andtherefore forthevibrational problem onlytheconstantpartsof^.n,A12,...,Anmneedbeconsidered. The vibrational motion istherefore thesame asifthecoefficients An,A12,...,Anm were constants independentofthecoordinates;butinthiscasetheequations Alkql+Askq2+...+Ankqn=Q (k=I,2,...,i) canbeintegrated;infact,they give Alkql+A^q, +...+Ankqn=(A=1,2,...,m), theconstants ofintegration beingzerosince thevalues qi=0,q2=0,...,qn= representapossible positionofthesystem. Itfollows thatthevibratorymotion ofthegivennon-holonomic systemis thesame asthat oftheholonomic systemforwhich theequationsofcon straint areexpressibleintheintegratedform Alkql+A2kq.2+...+A nkqn=(k=1,2,....m) ; wecantherefore determine thevibrations byusingthese equationstoelimi natemofthecoordinates(ql}qz,...,q n)fromTandV;weshall thenhave aholonomicsystemwith(nm)degreesoffreedom, thekinetic andpotential energies being expressedinterms of(nm)coordinates andthecorre spondingvelocities :thevibrations ofthissystemcanbedetermined bythe usualmethod described inthepreceding chapter. Asanexample, weshall consider thefollowing problem*. Aheavy homogeneous hemisphereisrestinginequilibriumonaperfectly roughhorizontal planeitith itsspherical surface downwards. Asecond heavy homogeneous hemisphereis restinginthesamewayonaperfectly rough plane face ofthefirst,thepoint ofcontact beinginthecentreoftheface.Theequilibrium being slightly disturbed,itisrequired tofindthevibrations ofthesystem. Take asaxes ofreference (1)Arectangularsetofaxes Z&gt;&gt;xyzfixed intheupper hemisphere,theorigin being itscentre ofgravityZ2. *Due toMadame Kerkhoven-Wythoff, Nieuw Archiefvoor Wiskunde, Deel iv.(1899). 90] Non-holonomic Systems. Dissipative Systems 223 (2)ArectangularsetofaxesZl^^fixed inthelower hemisphere, theorigin being itscentre ofgravity Z\. (3)Arectangularsetof-axesRlmn fixed inspace, theoriginRbeing theequi libriumpositionofthepoint ofcontact ofthelowerhemisphere andtheplane. Wefurther define these axesbysupposing that intheequilibrium position theaxes Z&lt;iZ,Z\,andRnarevertical and thereforecoincident, while theaxesZzx,Z^,Rl areparallel, theaxesZzy,Z^,andRmbeingtherefore alsoparallel. Suppose that attime tthecoordinates ofapoint referred tothese different sets ofaxes areconnected bytheequations =a+n The24coefficients inthese transformation-formulaecompletely specifythepositionof thesystematanyinstant. Ashowever thesystem hasonlysixdegreesoffreedom, there must be18equations connecting these coefficients ortheir differentials. Ofthese, 12are theordinary conditions ofthetypes whichexpress theorthogonal character oftheaxes;theremaining 6aretheconditions of contact androlling, which weshallnow find. LetRl,R2betheradii ofthelower andupper hemispheres respectively, and/j,12 thedistances ofthecentres ofgravity from their plane faces, soli=%Ri,^2=f#2-The coordinates ofthepointofcontact oftheupper hemisphere with thelower are x2=-lizyi,yi=-Rtf*, zzh-R-iyz t theconditions that thispointshallbeatrestrelative tothelowerhemisphere are =0. The lastofthese equations gives 7+^273=0,which isthedifferentiated form ofthe equation ^-7-73^2= -RZ, anequation whichexpressesthecondition ofcontact ofthetwohemispheres:while the firsttwooftheequations give a-atR2y^-azR^y-i+a3(^2-R-zy^=0, -0!Rayi-&RW+/33(12--R2y3,=0, andtheseexpress thecondition ofrollingoftheupperonthelowerhemisphere. These equations giveasafirstapproximation 224 Non-holonomic Systems. Dissipative Systems [CH.vm andtherefore onintegration Similarlythecondition ofcontact ofthelower hemisphere andthehorizontalplaneis andtheconditions ofrolling are Wehave thusnowobtained the18equations connecting the24coefficients:taking a2)/3s?y\ia 2&gt; ^3&gt;ciasthe6independentcoordinates ofthesystem, andsolvingfor theother 18coefficients interms ofthese,wefindwith thenecessary approximation -l-$(a22+71- ), 3=-yi- =1- n .-/sip, U=i-i(yi2+332 ). Thepotential energyofthesystemis F=J/j^c+M2g(c+cia+c2/3+c3y), or,retaining onlysmallquantitiesofthesecond order, IfnowweexpressthecoordinatesI,m,nofany particleoftheupperorlower hemisphereinterms ofitscoordinates relative totheaxesZ2xyzandZl^respectively, andform thesumi2m(P+m2+h2 )foreachhemisphere, neglectingterms above thesecond order ofsmallquantities, andrememberingthat theprincipal moments ofinertia of ahemisphereofmassMandradiusRatitscentre ofgravityare%MR2 , R*,wefindforthekinetic energyofthesystemthevalue T,where ^^^^ +C!2{^Wi+l/2(1J7? 22+f^2+^ Theequationsofmotionevidently separateintothree distinctsets, consistingof (i)Equationsforthecoordinates a2anda2:these coordinates giverisetonoterms inF,anddonotcorrespondtovibrations inthestricter sense;infact,theequilibrium isnotdisturbed ifeither ofthehemispheresisturned through anyangleabout itsaxisof revolution. Wecanthereforeneglecttheseequations. (ii)Equations involvingthecoordinates b3and3. (iii)Equationsforthecoordinates c1andyl;these areexactlythesame asthe equationsfor63and/33,soweneed consideronlythe latter. 90] Non-holonomic Systems. Dissipative Systems225 Theequationsfor63andj33are,inextenso, +g(|RM-^2J/2)63-1^2^2/3 3-|^3 +^/33=0. Thecorresponding determinantal equationforX,whereSTT/V/Xisaperiod,is =0. This isaquadratic equationinX :itiseasily found that itsroots arepositiveif and this istheconditionforstability oftheequilibrium. Thevibrations ofnon-holonomicsystemsabout astate ofsteadymotion aremostconvenientlydiscussed byuseoftheequationsofmotiongiven in 88.Themethod willbeillustrated bythefollowing example. Example. Asolidofrevolution hasanequatorial plane ofsymmetry, and isrolling withangular velocity nround itsaxis insteady motion onaperfectly roughhorizontal plane,theequatorial plane ofthesolidbeingvertical. Thismotionbeing slightly disturbed, tofindtheperiod ofavibration. LetGbethecentre ofgravityofthesolid, and let(C,A)beitsmoments ofinertia about theaxisandabout alinethrough Gperpendiculartothe axis. Take asmoving axes ofreferenceGxyz, where Gzistheaxisofthesolid,Gyisperpendiculartotheplane through Gzandthepointofcontact(soGyishorizontal), andGx isnormal totheplane Gt/z. LetF,F,Rbethecomponentsoftheforceacting onthesolid atthepoint ofcontact,Fbeingintheplane Gxz,Fbeing paralleltoGy,andRbeing normal tothe plane. Let(6l,2,#3)and(wj,w2,co3)denote asusual thecomponentsofangular velocityoftheaxesand ofthebody respectively, and let(u,v,w)bethecomponents of thevelocityofG,paralleltothemovingaxes. Further,letpbetheradius ofcurvature ofthemeridian ofthesolid attheequator, atheradius ofitsequatorial circle, 6the anglemade byGzwith thevertical, and theangle between Gyand itsundisturbed direction. Thenwehave di= o&gt;!=- 4&gt;sin6,$2=co2=#,#3=$cos#, andthekineticenergyis T=$M(u2+v2+708)+1A (a&gt;!2+co22 )+\&lt;7o&gt;32 . Theequationsof18thereforegive,ifPisthepoint ofcontact, PATtheperpendicular from thispoint ontheaxis,andGNtheperpendicular fromGonthehorizontalplane, M(u-w93+w62}=Fuo6-(R- Mg)sin6, ai-Ato&lt;i0 3+Cas6Z=-F .GK, 2- &lt;7co36^+A^e-^-F.GN-R. XP, .3 =F.PK. Intheseequations, (rAandj!\Tare measuredpositively paralleltothepositive direction oftheaxisofzandthehorizontalprojection ofthisdirectionrespectively. w.D. 15 226 Non-holonomic Systems. Dissipative Systems [CH.vm Theconditions ofnoslidingatPare sin6GN. o&gt;2=0, .O! =o, andthecondition ofcontact ofthebodyandplaneis wcos6-11sin6= -^(-OKcos+PKsin6). These equations determine themotion inthegeneral case,when thedisturbance from steady motion isnotsupposedtobesmall. When this latterassumptionismade, wehave wherex,"&, *)aresmall; andF,F,u,w,wj,o&gt;2,QI,#2,$3aresmall, whileRisnearly equaltoMg. Moreover wehaveNP=(p-a)x-Theequations therefore become M(u+an63)=-R+Mg, Mr, =F, =0, Cw =Fa, waa&gt;2 =0, ,77+atzr =0, whereo&gt;i=6l=^ W2=#2=x&gt; QZ=- Eliminating F,F,R,andreplacing61,62,3,a&gt;1}o&gt;2bytheirvalues, theequations become =0, JCw w =ax, =-aw. From thethirdand fifth ofthese equations weseethatwand)arezero,andtherefore orand77areconstants. Theother three equations give,oneliminating w, andtherefore theequationforthedetermination ofxi-s A(A+Ma2 )x+{MgA (P-a)+Cn*(C+Ma2 )}x= ; thisequationshews thattheperiodofavibration is 2?r (MgA (p-a)+Cn2(C+Ma2 )) 91. Dissipative systems ;frictional forces. Wenowproceedtotheconsideration ofsystemsforwhich theprincipleof conservation ofdynamical energyisnotvalid, theenergyofthesystem being 90,91]Non-holonomicSystems. Dissipative Systems 227 continually changedintosome other form(e.g.heat) which isnotrecognised indynamics. Weshall firstconsiderfrictional systems. Iftworigidbodies which arenotperfectly smooth areincontact, the reaction between them atthepointofcontact mayberesolved intoacom ponent alongthecommon normal totheir surfaces atthepoint,which is called thenormalpressure, andacomponentinthecommontangent-plane, which iscalled thefrictional force. The frictional force isdetermined by thefollowing law*, which hasbeen establishedexperimentally:Thebodies will notslide oneach other, providedthefrictional force required forthe prevention ofslidingdoes notexceed/j,times thenormalpressure, where/A isaconstant called the" limiting coefficient offriction"ivhichdepends only onthematerialofwhich thesurfacesincontact arecomposed. Ifonthe otherhand thefrictional force requiredtoprevent slidingisgreater thanfj, times thenormalpressure,there will beslidingatthepoint ofcontact, and thefrictional forcecalled intoplaywill be/*times thenormalpressure. Painleve haspointed outthatthefourhypotheses (1)thattheabove laws offriction hold, (2)thatthere existrigid bodies, (3)thatthenormalpressure between bodies cannot benegative, (4)that allaccelerations andtensions arefinite takentogether lead insome cases tocontradictions ofthefundamental laws ofdynamics. Foradiscussion onthis subject,cf.Comptes Rendus, CXL.(1905), pp.635, 702,847: ibid. CXLI.(1905), pp.310,401, 546; Zeitschrift furM.u.P.LVIII.(1909), p.186. Thefollowing examplesillustrate themotion ofsystems involving frictional forces. Example1.Motionofaparticleonarough fixedplanecurve. Consider themotion ofaparticle which isconstrained tomove onarough fixed tube ofsmallbore,intheform ofaplane curve, under forces whichdepend solely onits positioninthetube. Letf(s)andg(s)denote thecomponents offorceperunitmass acting ontheparticleindirection ofthetangent andnormal tothetube,where sisthe distance oftheparticle fromsome fixedpoint ofthetube,measuredalong thearcinthe direction inwhich theparticleismoving ;and letRbethenormal reaction perunitmass, andpthecoefficient offriction. Since thecomponentsofacceleration oftheparticle along thetangent andnormal are vdv/ds andv2 /p,where visthevelocityoftheparticle andptheradius ofcurvature ofthe tube,wehave S-/0-tt Eliminating R,wehave dv* 2u__+JV=2 Integrating, wehave where (f&gt;=$ds/p, and cisaconstantdepending ontheinitial circumstances ofthemotion. Thediscovery that the friction isproportional tothenormalpressure wasmade by G.Amontons, Paris Mem., annee 1699, p.206. 152 228 Non-holonomic Systems. Dissipative Systems [CH.vm Theright-handside ofthisequationisaknown function ofs,say=F(s). Then wehave v2 sotherelation between sand tis tt-t=/{F(s)}~2ds. Thisequation represents thesolxition oftheproblem. Example2.Acircularhoop ofmassMstands onrough ground, andaparticle of massmisattached totheendofthehorizontal diameter. Tofindwhether thehoopwill roll orslide. Letusinvestigate therolling motion, assumedpossible, andsodetermine whether the frictionrequiredtoproducethismotionis,orisnot,greater thanthemaximum friction actually available,i.e.p.times thecorresponding normalpressure. Let6betheangle turnedthrough bythehoopfrom thecommencement ofthemotion, and letxandybe thecoordinates ofthecentre ofgravityofthesystem,referred tohorizontal andvertical (downward) axesthroughitsown initialposition,sothat amati a\ maax=a6 ,f (1cos#), y= ,, sin6.M+m^9M+m where aistheradius ofthehoop. Thekinetic andpotential energiesare T=Ma2ft+ma?ft(1-sin0), V=mgasin6. TheLagrangian equationofmotion istherefore -T-[22 {M+m(l-sin0)}]+ma2ftcos=mgacos0. Cvt Fortheinitialmotion,thisequation gives soinitially wehave ma_/J-7nf__ _ftA ~ ~ But ifFbethefrictioual forceandRthenormalpressure, wehave F=(X+m), R=(M+m)( soinitially wehaveF x m(M+m) R-y+g2 Thehoopwilltherefore rollorslideaccordingasthecoefficient offriction isgreater or lessthan m(M+m) Example3.Aparticle moves undergravity onarough cycloid whoseplaneis vertical andwhose base ishorizontal :ifbetheinclination ofthetangentatanypoint tothehorizontal, sothattheequationofthecycloid canbewritten s=4asin &lt;, and iftan ebethecoefficient offriction, shew thatthemotion isgiven bytheequation tar ce where cisaconstant. 91,92]Non-holonomic Systems. Dissipative Systems 229 92.Resisting forces dependingonthevelocity. Adifferenttypeofdissipative systemisillustratedbythemotion of aprojectileinthe air,astheresistance oftheairdependsonthevelocityof theprojectile. Nogeneralrulecanbeformulated forthesolution ofprob lemsinvolvingforces ofthiskind :acase ofpractical interest, however, namelythemotion ofaprojectileunder theinfluence ofgravityaridofa resistance varyingassomepoweroftheprojectilesvelocity,canbeintegrated inthefollowingmanner. Forlow velocities (below lOOft./sec.) theresistance oftheairtoaprojectileisnearly proportionaltothesquareofthevelocity.Forhighvelocities(say2000 ft./sec.) the resistance isapproximatelyalinear function ofthevelocity. Attime tletvbethevelocityoftheprojectile,kvntheresistanceper unitmass, 6theinclination ofthepathtothehorizontal, andptheradius of curvature ofthepath.Thecomponentsofacceleration oftheprojectile alongthetangentandnormal toitspatharevdv/ds andv-/p;andhence the equationsofmotion are (vdv/ds=gsin6kvn , \v2 /p=gcos 6. Dividingthe firstequation bythesecond, weobtain 1dvtan6 k v"+~ld0 ti"~= gcos0 d(1\1d nk or-T7jI-^+-=-(nlogsec0)= sec0.dv\vnjvndog Integrating,wehave (I/Osecn+Constant =-(nk/g)fsecn+10d0. Thisequation givesvinterms of0.Toobtaint,theequationv2=pgcos gives gt=Ivsec0d0, andasvisaknown function of0,thisequation givestasafunction of0. Therectangularcoordinates (x,y)oftheparticlecannowbefound from the equations x=Ivcos0dt, y Thesolution oftheproblemisthusreduced toquadratures. Resistingforcesproportional respectivelytov,vz ,andav+bvzwere consideredby Newton, Principia, Book n.1,2,3.Thecase ofaresistance proportionaltoanypower ofthevelocity wasthenexamined byJohn Bernoulli* in1711. *Opera,i.p.502. 230 Non-holonomicSystems. Dissipative Systems [CH.vm DAlembert*shewed that ifgudenotes theratio oftheresistance tothemass ofthe projectile,theintegration canbeeffected inthefourcases u=a+bvn , u=a+blogv, u=a(log v}n+Rlogv+b, wherea,b,narearbitrary constants andRisanother constant depending onthem. Siaccit obtained manymore integrable cases, ofwhich thefollowing maybe mentioned : [ {du du logIvdu ^c I -icr j.- -r-+C, 8 J^ Jl+a(u-\}cl+b(u+l)c where a,b,c,Carearbitrary constants :thisequationdefines vinterms ofu,the number ofterms involved beingfinitewhen cisrational. Poisson pointed outin1806J that thetheoryofsingularsolutions ofdifferential equations hasapplicationsinDynamics, notablyinthecaseofaparticle under aresisting force. Ifaparticleismovinginastraightlineunder aresistingforce varyingasthe squarerootofthevelocity,theequationofmotion is dv/dt=av2. The initialvelocity beingc2 ,themotion isrepresented bythegeneral integral v=(c^at)z solongast&lt;2c/a,afterwhich itisrepresented bythesingularsolution v=0. Example1.Aheavy particlefallsvertically from rest attheorigininamedium whose resistance variesdirectlyasthevelocity. Shew that thedistance traversed intime tis 9*ff,9e-*t, -- 1-- 5KPHI? where pvistheresistance perunitmass. Example2.Aheavy particlefallsvertically from rest attheorigininamedium whose resistance varies asthesquareofthevelocity:shew thatthedistance traversed in time tis -logcosh(Jgii. t), wherep.v2denotes theresistance perunitmass. 93.Rayleigtis dissipation-function. When asystemissubjecttoexternalresistingforces which aredirectly proportionaltothevelocities oftheirpointsofapplication,itispossibleto expresstheequationsofmotion ofthesystemingeneralcoordinates interms ofthekinetic andpotential energiesandofasingle new function. For lettheenergylosttothesystem bytheaction oftheresistingforce which isappliedtoaparticleraofthesystem, whose coordinates are(x,y,z), inanarbitrary displacement (Sac, By,Bz)be kzz8z, *Traite deVequilibreetdumouvement desfluides, Paris, 1744. tComptes Rendus, cxxxn.(1901), p.1175. IJournal deVEcolePolyt.vi.(Cahier 13), p.60. 92,93]Non-holonomic Systems. Dissipative Systems 231 where kx,kv,kzarefunctions ofx,y,zonly. Theequationsofmotion ofthe typical particlerawilltherefore be (mx=kxx+X, &lt;my=-k,Jy+Y, \mz=kzz+Z, where X,Y,Zarethecomponentsofthetotal force(external andmolecular) ontheparticle, excepttheforce ofresistance. Now letafunction Fbedefinedbytheequations where thesummation isextended over alltheparticles ofthesystem;so that F,which iscalled thedissipation-function, represents halftherate at whichenergyisbeinglosttothesystem bytheaction oftheresistingforces; and let(qltq.2,...,qn)becoordinatesspecifyingtheconfigurationofthe system. Multiplyingtheequationsofmotion oftheparticlerabydx/dq r,dy/dq r, dz/dq r,respectively, andsummingforalltheparticlesofthesystem, wehave dx ,..dy ,..dz\ ^/,.dx dy dz\ &gt;+y^-+z==-S(kxx=--hkvy^- -fk,z~ } dqry dqrdqj \x dqryy dqrT dqj -.^-^ s . oqr dqr dqrj Asin26,wehave .dx..dy .,dz\ d/dT\ dT ;^+y^-+^r-= --^ ,dqr*oqrdqjdt\dqrjdqr whereTisthekineticenergy;and -a--*5-- "^, r&gt; oqr dqr dqrj whereQ^+Q28q2+...+QnSqndenotes thework donebytheexternal forces(excludingtheresistances)inanarbitraryinfinitesimaldisplacement: whilewehave &+M oqr d _dF dqr Itfollows that theequations ofmotionofthesystemintermsoftheco ordinates(qltq2,...,qn)canbewritten intheform ddT\dTdF 232 Non-holonomic Systems. Dissipative Systems [CH.vm Example.Iftheresistingforces dependontherelative (asopposedtotheabsolute) velocityoftheir pointsofapplication,sothattheforces actingontwoparticles (x^yt,zt) and(#2 &gt;3/2?gz)have thecomponents -kx(xi-x& -kv(yi-fa\ -*,(*i -z) and -**(#2-*l)&gt; -*(&-&) -*(*2-*l) respectively,shew that, theequationsingeneralcoordinates canbeformed with the expression \2{kx(x,-xtf+ky(ft-ytf+*,fr-z,)2 } asadissipation-function. 94. Vibrations ofdissipative systems. Ifadynamical systemisspecified byitskinetic energy function, potential energy function, anddissipation function, methods similar tothose of ChapterVIIcanbeappliedinorder todetermine thenature ofthesmall vibrations ofthesystemabout anequilibrium-configuration. Forsimplicityweshall consider asystemwithtwodegreesoffreedom. Asin 76,wefindthat forthevibrational problemthekineticenergyand dissipationfunction canbetaken ashomogeneous quadraticfunctions ofthe velocities, andthepotential energyasahomogeneous quadraticfunction of thecoordinates, thecoefficients inthese functions beingconstants. Taking ascoordinates those variables which would benormal coordinates ifthere werenodissipation function, wecanwrite these three functions intheform where\andX2willbesupposed positive,sothattheequilibriumwould be stable ifthere werenodissipativeforces. Theequationsofmotion are drtT\dT+W+d_V=(r=l,2),dt\dqj dqrdqr9&lt;?- or q\+aql+hq^+\lq}=0, &lt;/2+ A&lt;jl+&?2+^2?2=0. Ifweattempttofindaparticularsolution oftheseequationsintheform onsubstitutingthese values inthedifferential equationswehave A(p*+ap+XO+Bhp=0, Ahp+B(p*+bp + X&gt;)=0, fromwhich itfollows thatpmust bearootoftheequation (p2+ap+\0(p2+bp+X2)-A2?2=0. 93,94]Non-holonomic Systems. Dissipative Systems233 Weshallsupposethedissipativeforces tobecomparatively small, sothat squaresofthequantities a,h,bcanbeneglected;onthissupposition,the roots ofthelastequationarereadilyfound tobe Correspondingtotherootp^wehave, from thesecond oftheequations connecting AandB, B_ihVXj Aparticularsolution ofthedifferential equationsisthereforegiven by ^=(Xj-X2)e~^at (cos N/X]t+isinx/Xjtf), fa=h*J\le~^at (icos VXxi-sinVx^), andasecondparticularsolution isobtained bychangingito iinthese expressions.Itfollows thattwoindependentrealparticularsolutions ofthe differential equationsare r x=(Xj-X2)e~^atcosVXjtf f(?!=(Xj-X2)e~^atsinVx^, and -{=hVX.e~talnns v"X .f andtherefore themostgeneralrealsolution involvingePitis (qt=(Xj-X2)Ae~$atsin(\/M+e), sn where -4and earerealarbitraryconstants. Thisrepresentsoneofthenormal modes ofvibration ofthesystem. Addingtothisthecorrespondingsolution inep **,wehavefinallythegeneralsolution ofthevibrationalproblem, namely 1=(\-X2)Aeatsin(\/M+e)+Axa5e~ sin2++7 sinVx^++e+(X2-X:)Be~btsin(*/\ 2t+7), where -4, -B, &lt;;,7arefourconstants which must bedetermined from the initial circumstances ofthemotion. Nowwesupposethedissipativeforces such thatenergyisbeingcon tinuallylosttothesystem,sothatFisapositivedefinite form, andtherefore aand 6arepositive. The lastequationstherefore shew thatthevibration graduallydiesaway,onaccount ofthepresenceofthefactors e~^atand e^1 : theperiodsofthenormal vibrations are(neglecting squaresofu.h,6)the same asifthedissipativeforces were absent;and inanormal vibration, the amplitudeofoscillation ofoneofthecoordinates issmallcomparedwith the amplitudeofoscillation oftheother coordinate, while thephasesofthe vibration inthetwocoordinates atanyinstant differ byaquarter-period. 234 Non-1 lolonomic Systems. Dissipative Systems [CH.vm Asimilaranalysisleads tocorrespondingresults forsystemswithmore thantwodegreesoffreedom;supposingthatthedissipativeforces aresmall andthatthedissipationfunction andpotential energyarepositivedefinite forms, wefindthat theperiodsofthenormal vibrations are(neglecting squaresofthe coefficients inthedissipation function)unaltered bythe presenceofthedissipative forces, butthatthevibrationgraduallydiesaway: and if(qltq2,...,qn}arethenormal coordinates ofthesystemwhen the dissipativeforces areabsent, there isanormal vibration ofthesystemwhen thedissipativeforces arepresent,inwhich theamplitudeofthevibrations in &lt;?2, q-s,,qnissmall comparedwith theamplitudeofthevibration inq1} andthephaseofthevibrations inq2,q3,...,qndiffers byaquarter-period from thephaseofthevibration inqlf Example.Discuss thevibrations ofasystem which isacted onbyperiodicexternal forces which have thesame periodasoneofthenormal modes offreevibration ofthe system ;shewing theimportanceofdissipativeforces (evenwhere small)inthis case. 95.Impact. Another mode inwhichenergy maybelost* toadynamical systemisby the collision ofbodies which belongtothesystem;acollisiongenerally results inadecrease ofdynamical energy. Theanalyticaldiscussion ofcollisions isbased onthefollowing experi mentallaw-f-. When twobodies collide, thevalues oftherelativevelocity ofthe surfacesincontact (estimated normallytothesurfaces)atinstants immediately beforeandimmediately aftertheimpactbearadefiniteratio toeach other : thisratiodepends onlyonthematerialofwhich thebodies arecomposed. This ratio will ingeneralbedenoted bye.When eiszero, thebodies aresaid tobeinelastic. Thegeneral problemofimpactreduces therefore toaprobleminimpulsive motion inwhich theunknownimpulsiveforce atthepointofcontact ofthe bodies istobedetermined bythecondition that thechangeinrelative normalvelocityofthebodies satisfies theabove law. 96.Lossofkinetic energyinimpact. Weshallnow findthe lossofkinetic energywhen twoperfectlysmooth bodies impingeoneach other. Letmtypifythemass ofaparticleofeither body,and let(u,v,w)and (u, v,w)denote itscomponentsofvelocitybefore and after theimpact,and *I.e.losttothesystemconsidered asadynamical system:theenergyisnotannihilated, but appearsinsome other manifestation, e.g.heat. tThelaws ofimpact were discovered in1668 byJohn Wallis(Phil.Trans. No.43,p.864) andChristopher Wren (ibid. p.867). 94-97] Non-holonomic Systems. Dissipative Systems 235 let(U,V,W)bethecomponentsofthetotalimpulsiveforce(external and molecular) onthisparticle. Theequationsofimpulsivemotion(35)give m(uuti)=U, m(v-v)=V, m(w-w)=W. Multiplyingtheseequations by(u+eu9),(v+evQ),(w+ewQ)respectively, adding, andsummingforalltheparticlesofboth bodies, wehave 2w{(u- -MO)(u+eu)+(vv)(v+evn)+(w-w}(w+ew)} =${U (u+eu)+V(v+ev)-fW(w+ew)}. Now sofarasmolecularimpulsesareconcerned, wehave 2(Uu+Vv+Ww)=0,and2(Uu+Vv+Ww)=0, since theimpulsiveforces whichcorrespondtoeach other invirtue ofthelaw ofAction andReaction willgivecontributions tothesesums whichmutually destroyeach other. Also, since thepartof(u+eu)due tothenormalcomponentofvelocity hasthesame value foreach oftheparticlesincontact atthepoint where theimpact takesplace (invirtue ofthelawofimpact)itfollows that theimpulsiveforce between thebodies does notcontribute tothesum %U(u+eu,),andsimilarlydoes notcontribute tothesums^V(v+ev ] and2TF(w+ew). Wehave therefore 2{U(u +eu)+V(v+ev)+W(w+ew)}=0, andconsequently 2m{(u-M)(u+eu)+(v-v)(v+ev)+(w-w)(w+ew)}=0, or Thisequation canbeexpressed bythestatement that thekineticenergy lostintheimpactis(1 e)/(l+e)times thekineticenergy ofthatmotion which would have tobecompoundedwith themotion attheinstantbeforethe impactinorder toproduce themotion attheinstantaftertheimpact. 97.Examples ofimpact. Theimpulsive changeofmotionconsequentonthecollision oftwo free rigidbodies inspace canbemostsimply determinedbythefollowing considerations. Themotion ofeachbodybefore orafterimpactisspecified bysix quantities (e.g.thethreecomponentsofvelocityofitscentre ofgravity and thethreecomponentsofangular velocityofthebodyabout axesthroughits centre ofgravity). The totalnumber ofequations requiredtodetermine the 236 Non-holonomic Systems. Dissipative Systems [OH.vm impulsive changeofmotion istherefore twelve. Ofthese, sixareimmediately furnishedbythecondition that theangular momentum ofeachbodyabout anyaxisthroughthepointofcontact isunchanged (since theimpulsiveforces actatthispoint);anotherequationisobtained from thecondition that the momentum ofthesysteminthedirection normal tothesurfaces incontact isunchanged (since thenormalimpulsiveforces onthetwobodies atthe pointofcontact areequal andopposite), andanotherbytheexperimental lawofimpact.Ifthebodies areperfectly smooth, theremainingfour equationscanbederived from thecondition that thelinearmomentum of eachbodyinanydirectiontangentialtothesurfaces incontact isunchanged (since there isnotangential impulseifthebodies aresmooth):ifonthe other hand thebodies areperfectlyorimperfectly rough,thecondition that thelinear momentum ofthesysteminanydirectiontangentialtothe surfaces incontact isunchanged givestwoequations;ifthebodies are perfectly rough,thecondition that therelativevelocityofthebodies in anytangentialdirection after theimpactiszerogivestheother two :while ifthebodies areimperfectly rough,thecoefficient offriction between the surfaces incontactbeing y-t,theremainingtwoequationsaregiven bythe conditions that (a)therelativevelocityinanytangentialdirection iszero after the impact, providedthetangential componentoftheimpulse requiredforthis does notexceed//,times thenormalcomponentoftheimpulse; (/3)ifthelastcondition isnot satisfied, there isatangential impulse equalto//,times thenormalimpulse between thebodies. Inallcases, therefore, therequiredtwelveequationscanbefound. Ifthemotion takesplaceinaplane,orifoneofthebodies isfixed, this procedureisstillvalid aftermakingsome obvious modifications. Thefollowing examplesillustrate theseprinciples: Example1.Aninelasticsphere ofmassmfalls withvelocity Vonaperfectly rough inelastic inclinedplane ofmassMandangle a,which restsonasmooth horizontalplane. Sheiv that theverticalvelocity ofthecentreofthesphere immediately after theimpactis Fsin2a n,. r-=- . (Coll. Exam.)omsin*a LetUbethevelocityoftheplaneafterimpact, uthevelocityofthesphere parallelto andrelative totheplane,&lt;Btheangular velocityofthesphere,aridaitsradius. Theequationofhorizontal momentum gives TO(ucosa-U~)=MU. Thekinematical condition atthepoint ofcontact isacau. Thecondition that theangular momentum ofthesphere about thepointofcontact shall bethesame before andafterimpactis mVasina=gma-o&gt;+ma(u7cosa). 97] Non-liolonomicSystems. Dissipative Systems 237 These threeequations give,oneliminatingeoandU, 5(M+m) Fsin2a~ ~1M+-2m+5msin*a which istheresult stated. Example2.Asphere ofradius arotatingwithangular velocityQ,about anaxis inclined atanangleatothevertical andmoving,intheverticalplane containingthat axis, withvelocity Vinadirection making anangleawith thehorizon, strikes aperfectly rough horizontalplane. Iftheplanebetangentialltj inelastic, findtheangle which thevertical plane containingthenew directionofmotion makes with theold. Takerectangular axesOxyz, where isthepointofcontact, Ozisvertical, andyOzis the initialplaneofmotion; and let o&gt;!and o&gt;2bethecomponentsofangular velocity about OxandOyrespectivelyafter theimpact, andMthemass ofthesphere. Equating theinitial and finalangular momenta about Ox,wehave MaFcos a= |-Ma2 a&gt;i. Equating theinitial and finalangular momenta aboutOy,wehave |Ma2Qsina=IJ/a2co2. Thetangent oftheinclination ofthenewplaneofmotion totheplane yOzis(on account oftheperfect roughnessoftheplane) 0)2/0)!, andthis isthereforeequalto \Ma?Q.sina MaVcosa orfa(Q/F)tan a. Example3.Aperfectly roughcircular discofmassMandradius cimpinges upon arodofmassmandlength 2acapable ofturning freelyabout apivotatitscentre.If thepoint ofimpactisdistant bfromthecentre oftherod,and thedirectionofmotion ofthe centreofthediscmakesangles a,ftwith therodbefore andafter collision, shew that 2(3Mb2+ma2 )tan/3=3(ema2-3Mb2 )tan a.(Coll.%Exam.) LetFdenote the initialvelocityofthedisc,and letvdenote itsfinalvelocity andQ itsfinalangular velocity. Since there isnoslidingatthepointofcontact, wehave vcos j Denoting bycothe final angular velocityoftherod,andby/thenormalimpulse between therodanddisc, theequationofthemotion oftherod is Ib=\ma2 u&gt;. Theequationofimpulsive motion ofthediscinthedirection normal totherod is M(vsin(i+^sina)=7, andthelawofimpact gives therelation vsinft+ 6o&gt;=eVsina. Equating theinitial and finalangular momenta ofthediscabout thepointofcontact, wehave Fcos a=vcosft-\cO. Eliminating v,Q,/,o&gt;from theseequations, wehave 2tanft(3Mb2+ma2 )=3tana(mea2- which istheresult stated. 238 Non-holonomic Systems. Dissipative Systems [CH. Example4.Acircularhoop,inmotion without rotation initsownplane, impingesona rough fixed straight-edgedobstacle intheplane.Thevelocity ofthecentreofthehoop before impactisV,inadirection making anangleawith theedge,and thecoefficient offrictionisp..Tofindtheimpulsive change ofmotion. Letuand vdenote thecomponentsofvelocityofthecentre ofthehoopafter the impact, parallel andperpendiculartotheedge,and let cobetheangular velocity,Mthe massandatheradius ofthehoop. Equatingtheangular momenta about thepointofcontact before andafter theimpact, wehave -Ma-&lt;a+Mau=MVacosa. Thelawofimpact givestheequation Since theplaneisrough, u+a&lt;o iszero after theimpact, providedthe frictional impulse requiredforthisdoes notexceed/xtimes thenormal impulse:but ifthis condition isnot satisfied, thefrictional impulseisp.times thenormalimpulse. LetFbethefrictional andRthenormal impulse:thenwehave M(u- Fcosa)= -F,M(v+7sina)=-ft,Ma?v=-aF. Wehave therefore R=M(\+e)7sina, and ifu+aa&gt;iszero,weshallhave Thequantityu+awwilltherefore bezero after theimpact, provided and ifp.doesnotsatisfythisinequality, weshallhave F=pM(l+e)7sino. Thusfinally,if/x^cota/2(l+e),themotion isdetermined bytheequations while ifp.&lt;cota/2(1+e),themotion isdetermined bytheequations 7sina. MISCELLANEOUS EXAMPLES. 1.Aperfectly rough sphereofradius aismade torotate about avertical diameter, which isfixed, with aconstant angular velocityn.Auniform sphereofradius bis placedonitatapointdistant aafrom thehighest point: investigate themotion anddetermine inanypositiontheangular velocityofthesphere. Shew that thesphere willleave therotating sphere when thepointofcontact isatanangulardistance 6from thevertex, where 10 4a2w2sin2a COS=^ ^-- JT- . 17 119(a+o)g (Camb.Math. Tripos, Part I,1889.) 2.Arough sphereofradius arollsunder gravityonthesurface ofacone ofrevolution which iscompelledtoturnabout itsvertical axis with uniform angular velocity n, itsvertex being uppermost;ifabethesemi-vertical angleofthecone,rsinabethe distance ofthecentre ofthespherefrom theaxis ofthecone,^betheangle turned vm] Non-holonomic Systems. Dissipative Systems 239 through, relativelytothecone,bythevertical plane containingthecentre ofthesphere, and o&gt;3betherateofrotation ofthesphere about thecommon normal, prove that ,14cosa (1^ Qn}r-=A, where A,B,(7aredeterminate constants. (Camb. Math.Tripos, PartI,1897.) 3.Ahomogeneous solid ofrevolution of.massMwith aplane circular base of radius crollswithoutslipping with itsedgeincontact with arough horizontalplane.Shew that6, o&gt;,Oaredeterminedbytheequations Mac~(acos26)-Mc*& cos26=(C+J/c2 )cos6^, {A(C+Mc2 )-M*a?#}~(acos26)+C(C+Mc2 )o,cos6-MacCacos2=0, (-4+Me2 )&+AQ?cos26-ZMacvQ. cos6+(C+Me2 )or+2% (asin&lt;9+ccos &lt;9)=Constant, where 6istheinclination oftheaxisofthebodytothehorizon, Qtheangular velocity of theverticalplane containingitsaxis,o&gt;theangular velocityofthebody about itsaxis,Athemoment ofinertia ofthebody about adiameter ofitsbase,Cthemoment of inertia ofthebody about itsaxisandathedistance ofthecentre ofgravity from the base -(Camb. Math.Tripos, PartI,1898.) 4.Awheel with 4spokes arranged symmetricallyrollswith itsaxishorizontal ona perfectly rough horizontalplane.Ifthewheel andspokes bemade ofafineheavy wire, prove thatthecondition forstabilityis 3 where aistheradius ofthewheel andVitsvelocity. (Coll. Exam.) 5.Abodyrollsundergravity onafixed horizontalplane.Ifthisplane betaken as planeofyz,shew that 2m{(y-.yA)z- (z-zA)y}=Constant, where(x,y,z)arethecoordinates ofaparticlemand(.r^,yA,ZA)ofthepoint ofcontact, andthesummation isextended over i\lltheparticles ofthebody. (Neumann.) 6.Oneportion ofahorizontalplaneisperfectly smooth aridtheotherportionis perfectly rough. Auniformheavy ellipsoid ofsemi-axes(a,b,c)has its6-axis vertical and moves withvelocityvinthedirection ofitsa-axisalong thesmoothportionoftheplane towards therough. Shewthat,if theellipsoidwillreturn tothesmoothportion, kbeing theradius ofgyration about thec-axis, andthatthemotion willthen consist ofanoscillation about asteadystate of motion.* Inthespecial casea=26,shew that after thereturn oftheellipsoid tothesmooth portion, the6-axis cannever make anangle with thevertical which isgreater than arctanVf(Coll. Exam.) 240 Non-holonomic Systems. Dissipative Systems [CH. 7.Ashell intheform ofaprolate spheroidwhose centre ofgravityisatitscentre contains asymmetrical gyrostat,which rotates withangular velocitytoabout itsaxisand whose centre and axis coincide with those ofthespheroid. Shew that inthesteady motion ofthespheroidonaperfectly roughhorizontal plane, when itscentre describes acircle ofradius cwithangular velocity Q,theinclination aoftheaxistothevertical isgiven by {Mbc (acota+6)-Abcosa+C(asina+c)}O2+CbaQ-Mgb (a-bcota)=0, whereMisthemass oftheshellandgyrostat, Athemoment ofinertia oftheshelland gyrostat together about alinethroughtheir centre perpendiculartotheir axis, (?,C those ofthe shellandgyrostat respectivelyabout the axis, athedistance measured paralleltotheaxis ofthepointofcontact oftheshellandplanefrom thecentre and bitsdistance from the axis. (Camb.Math. Tripos,PartI,1899.) 8.Auniform perfectly rough sphereofradius astartingfrom rest rollsdown under gravitybetween twonon-intersecting straightrods atright anglestoeach other whose shortest distance apartis2candwhich areequallyinclined atanangleatothevertical. Ifpo,poaretheoriginaldistances ofthepointsofcontact from thepointswhere the shortest distance intersects therodsandp,ptheir distances atasubsequenttimewhen thevelocityisV,shew that *--/ __ .,.i6c4(p*- andthat (Camb. Math. Tripos,PartI,1889.) 9.Aparticle moves undergravityonaroughhelixwhose axis isvertical. Ifabethe radius andytheangleofthehelix, shew that thevelocityvandarcdescribed scanbe expressedinterms ofaparameter6bytheequations _cc = cosy.._ y+2siny)} 10.Aparticleisprojected horizontally withvelocity usoastoslideonaroughinclined plane. Investigatethemotion. Prove that if 2^2p,cota&gt;l, theparticle approaches asymptoticallyalineofgreatest slopeatdistance u2 2p,cosa g 4/x2cos2a-sin2a wherep,isthecoefficient offriction, andaistheinclination oftheplane. (Coll. Exam.) 11.Arough cycloidaltubehas itsaxis vertical andvertex uppermost.Ifabethe radius ofthegeneratingcircle andaparticlebeprojectedfrom thevertex withvelocity \/4agsina,shew that itwillreach thecuspwithvelocity equalto [4a ffcos2a(1-2sina~(Jir~a)tana }]*, where aistheangleoffriction. (Coll. Exam.) vm] Non-1 lolonomic Systems. Dissipative Si/stenis241 12.Aheavyrodoflength2ismovinginaverticalplanesothatoneend isincontact witharoughvertical wallandtheother endmoves alongtheground supposedtobeequally rough;andthecoefficient offriction foreach oftheroughsurfaces istan e.Shew thatthe inclination oftherodtothevertical atanytime isgiven by 0(&2+a2cos2f)-a2(92sin2e==a#sin(0-2f). (Coll. Exam.) 13.Athinsphericalshell restsupon ahorizontal plane andcontains aparticleoffinite masswhich isinitiallyatitslowestpoint. The coefficient offriction between theparticle andtheshell isgiven, thatbetween theshellandtheplane being practicallyinfinite. Motion intwodimensions issetupbyapplyingtotheshellanimpulse whichgivesitan angular velocityQ.Obtain anequationfortheangle through which theshell hasrolled when theparticle beginstoslip. (Coll. Exam.) 14.Acircular disc ofradius aisplacedinaverticalplane touching auniform rough (ft.)board which canturnfreely about ahorizontal axis intheuppersurface oftheboard throughitscentre ofgravity,thepointofcontact ofthediscbeingatadistance bfrom this axis.Astring, paralleltothesurface oftheboard, isattached tothepointofthe discfurthest from theboard andtoanarmperpendiculartotheboard attheaxis,and rigidly connected totheboard. Thecentre ofgravityoftheboard andarm liesinthe axis. Thesystemstarts from rest inthat positioninwhich thecentre ofthedisc liesin thehorizontal plane through theaxis. Shew thatslippingwilltakeplace between thedisc andtheboard, when theboard makes anangle Qwith theverticalgiven by tan= whereAisthemoment ofinertia oftheboard about theaxisdivided bythemass ofthe disc.(Coll. Exam.) 15.Ahoopisprojected withvelocity Vdown aplaneofinclinationa,thecoefficient offrictionbeing p(&gt;tana).Ithasinitially suchabackwardspinOthat after atime^it starts moving uphill andcontinues todosoforatime t2,after which itoncemore descends. Shewthat,ifthemotion takeplaceinaverticalplane atright angles tothe given inclinedplane, then (ti+t2)ffsina=aQ.-V.(Coll. Exam.) 16.Aringofradius aisfixedonasmooth horizontal table;asecondringisplaced onthetable inside the firstand incontact withit,and isprojected withvelocityT7 ,but withoutrotation,inadirectionparalleltothetangentatthepointofcontact. Find the time thatelapses beforeslippingceases between theringsifthecoefficient offriction between them isp,andprovethatthepointofcontact willinthistime describe anarc oflength (alog2)//*. Discuss themotion that willensue ifatthemomentslipping ceases thefixedringbe released and leftfree tomove, andprove thatduring thetime thattheinner rin- rolls halfround theouter onethecentre ofthelatter willbedisplaced adistance where m,Marethemasses oftheinner andouterringsandbistheradius oftheinner ring- (Camb. Math.Tripos, PartI,1900.) 17.Inthevertical motion ofaheavy particle descendinginamedium whose resistance varies asthesquare ofthevelocity, shew thatthequantity -ka. e+e where kv2istheresistance, andaand/3arethedistances described intwosuccessiveequal intervals roftime, depends onlyonTand isindependentoftheinitialvelocity. (Coll. Exam.) w.D.16 242 Non-holonomic Systems. Dissipative Systems [OH. 18.Prove thataheavy particle,letfallfrom rest inamedium inwhich theresistance varies asthesquareofthevelocity,willacquireavelocity Uta.nh(gt/U),anddescribe aspaceU2logcosh(gtjU)lginatimet,whereUdenotes theterminal velocityinthe medium. Shew alsothat,forthecomplete trajectoryofaprojectileinsuch amedium, theangle6 between theasymptotesisgiven by Uz jT72=arcsinh cot6+cot6cosec0, where Visthevelocity when theprojectilemoves horizontally. (Coll. Exam.) 19.Shew thatthehorizontal andvertical coordinates(x,y)ofaparticle moving under gravityinamedium ofwhich theresistance isRsatisfytheequation dx3v*cos3 (j) vbeingthevelocity and theinclination ofthetangenttothehorizontal. (Coll. Exam.) 20.Aparticleismoving, under gravity,inamedium inwhich theresistance varies asthevelocity. Shew thattheequationofthetrajectoryreferred tothevertical asymptoteandalineparalleltothedirection ofmotion when thevelocity was infinite, can bewritten intheform y=blog(xjo). (Coll. Exam.) 21.Prove that inthemotion ofaprojectile througharesisting medium which causes aretardation kv3 ,where kisverysmall andtheparticleisprojected horizontallywith velocity F,theapproximate equationofthepathis(neglecting 2~.2- theaxisofxbeinginthedirection ofprojectionandtheaxisofyverticallydownwards. (Coll. Exam.) 22.Aparticlemoves inastraightlineunder noforces inamedium whose resistance is(^_vslogs)/*,where visthevelocity and sthedistance from agiven pointintheline. Shew thattheconnexion between sand tisgiven byanequationoftheform where aand careconstants. 23.Aparticleismovinginaresisting medium under acentral attraction;shew that, ifRbetheretardation duetotheresistance ofthemedium, andvthevelocity,therateof descriptionofareasbytheradius vector tothefixed centre offorce varies as ,-f^ctt (Coll. Exam.)vJv 24.Prove that inaresisting medium, aparticlecandescribe aparabolaunder the action ofaforce tothefocus which varies asthedistance, providedtheresistance at apoint,where thevelocityisv,bek{v(v-v )fi;where visthevelocityatthevertex. Determine k. (Coll. Exam.) 25.Aparticlemoves inaresisting medium under aforcePtendingtoafixed centre. IfRbetheresistance, shew that df7 &gt;odr\ _ -\Pp*-T-\=-2%2 ,ds\*dp) rbeingtheradius vector andptheperpendicularonthetangent. vrn]Non-holonomic Systems. Dissipative Systems243 IfM=l/r,P=p.u2 ,andR=kv2 ,andweneglectPandhigher powers, shew that the differential equationtothepathis P^2~31,7ecu \ctu \~ -i / Abeing acertain constant. (Coll. Exam.) 26.Aparticleismoving under acentral force&lt;(?)repellingitfrom theorigin,in aresisting medium which imposesaretardingforce equaltoktimes thevelocity.Prove thattheorbit isgiven bytheequations r-6=/ie~Jct ,r-^-kr A-;~3e~2fc*=((f), where Aisaconstantquantity. (Coll. Exam.) 27.Aparticleismovinginacircle under aforce ofattraction toaninteriorpoint varyingasthedistance;theresistance ofthemedium isequaltoitsdensity multiplied by thesquareofthevelocity. Shew that thedensityatanypointisproportionaltothe tangentoftheangle between thelinesjoiningittothecentre offorceandthecentre of thecircle. (Coll. Exam.) 28.Arodoflength aisrotating about oneextremity,which isfixed, under the action ofnoforces excepttheresistance oftheatmosphere. Supposingtheretarding effect oftheresistance onasmall element oflength dxtobeAdx .(velocity)2 ,shew that theangular velocityatthetime tisgivenby11 Aa*. whereMkzisthemoment ofinertia about thefixed extremity, andQisaconstant. (Coll. Exam.) 29.Asmooth oval disc ofmass M,turning onasmooth horizontal table with angular velocity butwithout anytranslationalvelocity,strikes asmooth horizontal rod ofmassmatitsmiddlepoint.Prove that theangular velocityisdiminished in theratio where eisthecoefficient ofelasticity, xthedistance ofthecentre ofgravity from the normal atthepointofimpact andktheradius ofgyrationabout avertical axisthrough thecentre ofgravity. (Coll. Exam.) 30.Two rods, each oflengthaandmassTO,arejointed togetherattheirupper ends andthesystemfallssymmetrically,with itsplane vertical, ontoasmooth inelastic plane. Just before impactthejointhasavelocity Vandeach rodhasanangular velocity 12,tendingtoincrease itsinclination atothehorizon. Shew thattheimpulse between each rodandtheplaneis m(P+c2sin2a)(F+aQcosa)/{2+c2+a(a-2e)cos2 a}, where cisthedistance ofthecentre ofgravityofeachrodfrom thejointandmk2isthe moment ofinertia ofeachrodabout itscentre ofgravity. (Coll. Exam.) 31.Three equal uniform rodsAB,BC,CD,each oflength 2a,andhingedatBandC, areinonestraightlineandmoving with agiven velocityinahorizontal planeat right anglestotheirlengths. TheendsAandDmeet simultaneously two fixed inelastic obstacles, reducing AandDtorest. Determine when theywillform anequilateral triangle, andshew that 1oftheoriginal momentum isdestroyed bytheimpacts. (Coll. Exam.) 162 244 Non-holonomic Systems. Dissipative Systems [CH.vm 32.Asmooth uniform cube isfreetoturnabout ahorizontal axispassing through thecentres oftwoopposite facesand isatrestwithtwofaces horizontal;anequal and similar cube isdropped withvelocity uandwithout rotation soastostrike theformer along alineparalleltothefixed axisandatadistance cfrom thevertical plane containing it,prove thattheangular velocity impartedtothelower cube is (\+e)cu where aistheinclination tothehorizon ofthelower face ofthefalling cube, 2aisthe lengthofanedge, ktheradius ofgyration and ethecoefficient ofrestitution. Find alsothemotion oftheupper cubeimmediatelyafter theimpact. (Coll. Exam.) 33.Aperfectlyelastic circular discofmassMandradius cimpinges without rotation upon arodofmassmandlength 2awhich isfreetoturnabout apivotatitscentre, the pointofimpact beingatadistance bfrom thepivot. Prove that ifthecomponentofthe velocityofthecentre ofthediscnormal totherodbehalved bytheimpact, Mb2=ma?,the frictionbeingsufficient toprevent sliding. (Coll. Exam.) 34.Aperfectly rough sphereofradius aisprojected horizontally with avelocity V from apointataheight habove ahorizontalplane. Thesphere has alsoinitially anangular velocity Oabout itshorizontal diameterperpendicular totheplaneofits motion. Shew that before itceases tobound ontheplaneitpasses over ahorizontal distance where eisthecoefficient ofelasticity, andthedistance isreckoned from the firstpoint of contact. Compare the final with the initial kineticenergy. (Coll. Exam.) 35.Ahomogeneouselastic sphere (coefficient ofelasticity e)isprojected against aperfectly roughvertical wall sothat itscentre moves inaverticalplaneatright angles tothewall. Ifthe initial componentsofthevelocityofitscentre areuand v,and itsinitial angular velocity (Q)isabout anaxisperpendicular totheverticalplane, find thesubsequent motion after impinging onthewall,andshew that ifitscentre returns toitsoriginal positionthecoordinates ofthepoint ofimpact referred tothispoint are , g~ where aistheradius ofthesphere. (Coll. Exam.) CHAPTER IX THEPRINCIPLES OFLEAST ACTION ANDLEAST CURVATURE 98.Thetrajectories ofadynamical system. The chiefobjectofinvestigationinDynamicsisthegradual changein time ofthecoordinates(qltqa,...,qn)whichspecifytheconfigurationofa dynamical system. When thesystemhasthree (orlessthanthree) degrees offreedom, there isoften againinclearness whenweavail ourselves ofa geometrical representationoftheproblem:ifapoint betaken whose rect angular coordinates referred tofixed axes arethecoordinates(q1}q2,q3)of thegiven dynamical system,thepathofthispointinspacecanberegarded asillustratingthesuccessive states ofthesystem.Inthesamewaywhen n &gt;3wecan stillregardthemotion ofthesystemasrepresented bythepath ofapoint whose coordinates are(qltq2,...,qn)inspaceofndimensions; thispathiscalled thetrajectoryofthesystem, and itsintroduction makes itnatural tousegeometrical terms such as"intersection,""adjacent," etc., whenspeakingoftherelations ofdifferent states ortypesofmotion inthe system. 99.Hamilton sprinciple, forconservative holonomicsystems. Consideranyconservative holonomicdynamical system whoseconfiguration atanyinstant isspecified bynindependent coordinates(q1}q.2)...,qn\and letLbethekineticpotential which characterises itsmotion. Letagiven arcABinspaceofndimensionsrepresent partofatrajectoryofthesystem, and letCDbepartofanadjacentarcwhich isnotnecessarilyatrajectory: itwould however ofcourse bepossibletomakeCDatrajectory bysub jectingthesystemtoadditional constraints. Let tbethetime atwhich therepresentative point (q1}q.2&gt;...,qn)occupies anyposition PonAB:we shallsuppose eachpoint onCDcorrelated tosome value ofthetime, so that there willbeapointQonCD(oronthearcofwhichCD isaportion) whichcorrespondstothesame value tasPdoes. AsthearcCD is described, thecorrelated value oftwillbesupposedtovary continuously inthesame sense.Amoving point which describes thearcCDwillthere forepassthrough positions correspondingtoacontinuoussequenceofvalues ofqltqi,...,q n,t,andconsequentlytoeachpoint onCDthere willcorrespondasetofvalues ofq1} q&lt;&gt;,...,qn. 246 ThePrinciples ofLeast Action[OH.ix Weshalldenote by8thevariation bywhich wepassfrom apointofAB tothatpointofCDwhich iscorrelated tothesame value ofthetime,and shall denote byt,tltt+A,^+A^thevalues oftwhichcorrespondtothe terminal points A,B,C,Drespectively,andbyLRthevalue ofthefunction LatanypointRofeither arc. Ifnowweform thedifference ofthevalues oftheintegral F ((& q*,,qn,qi,q, iqn,t)dt, takenalongthearcsABandCDrespectively, wehave Ldt- CDLdt=L&t1-L&t +SLdt AB ,-LAA(o+ 64,+,,dt .r= r byLagrangesequations, ,dt at\rmi r=l r/s \r=l But if(Ag r)Bdenote theincrement ofgrinpassingfromBtoD,wehave andsimilarlyif(Agv)^denote theincrement ofqrinpassingfrom J.toC, wehave 1 andconsequently fLdt- fLdt=S-^r-Agr jcx&gt; J^# L*=I dq? SupposenowthatCcoincides withA,andDcoincides with B,andthat thetimes correlated toCandDare tandt,respectively,sothatA^, Ag2&gt; ;A(?n,A,arezeroat^an(^-^ :then thelastequationbecomes =0, which shews that theintegralILdthasastationaryvalue foranypartofan J actual trajectory AB, ascomparedwith neighbouring pathsCDwhich have thesame terminal pointsastheactualtrajectoryandforwhich thetimehas thesame terminal values. This result iscalled Hamilton sprinciple*. *Hamilton, Phil. Trans. 1834, p.307;ibid. 1835, p.95. 99,100] andLeast Curvature 247 Ifthekineticpotential Ldoes notcontain thetimeexplicitly, wecan evidently replacethecondition that thetime istohave thesame terminal valuesbythecondition thatthetotal time ofdescriptionistobethesame ngZ forABasforCD,since Sqr^-L,whichrepresents thetotalenergyof r=lOqr thesystem,isinthiscaseconstant. Helmholtz, J.filrMath. c.p.151,remarked that theconditions forastationary value of 2,..., n,ql,...,qn)+2~(f,-*r)ldt rVr } (where theqsand ffsareregardedasindependent variables) are sothatweagain obtain Lagrangesequations. 100. Theprinciple ofLeast Actionforconservative holonomicsystems. Suppose now that thedynamical system considered issuch that the kineticpotentialdoes notinvolve thetimeexplicitly,sothat theintegral ofenergy 4dLT I2qrr-L=hrl dqr exists. TakingasbeforeABtobepartofatrajectory andCDtobepartof anyadjacent arc,tothesuccessivepointsofwhich values ofthetime areso correlated astosatisfy anequationoftheform *dL-4r^~- L=h+AA, r=l OQV where AA isasmallconstant, wehave .dL\fi.dL\ .2qr\dt- 2qr~}dt=ioqrJ JAB\r=\ dqJ = |(h+A/0dt- !hdt+!Ldt- tLdt JCD JAB JCD JAB =2 Iftherefore wesuppose thatCcoincides withAandDcoincides with B, andthat A/& iszero,weshallhave 248 ThePrinciples ofLeast Action[on.ix //n "bL\ [2qr^-r\dthasastationaryvalueforany \r=i oqr/ part ofanactualtrajectory,ascomparedwith neighbouring pathsbetween the same terminiforwhich thetime iscorrelated tothecoordinates insuchaway astosatisfythesameequation ofenergy.This iscalled theprinciple of Least Action, theintegral beingcalled theAction. Innaturalproblems,forwhichListhedifference ofakineticenergy T, homogeneousoftheseconddegreeinthe velocities, andapotential energy F,independentofthevelocities, wehave(41) andthestationary integralcantherefore inthiscasebewritten iTdt. ThePrincipleofLeast Action originatedinMaupertuis attempt (Mem. deIAcad., 1744, p.417)toobtain forthecorpuscular theoryoflight atheorem analogoustoFerrnat s " PrincipleofLeast Time."Maupertuis principle wasestablished byEuler (Addit. n. p.309ofhisMethodus inveniendi lineascurvas, 1744)forthecaseofasingle particle under acentralforce, andbyLagrange (Miscell. Taurin. II.(1760-1), Oeuvres,I.p.365) formuch moregeneral problems. Example1.Shew that theprincipleofLeast Action canbeextended tosystemsfor which theintegral ofenergy does notexist,inthefollowing form. Lettheexpression 2qr~^r-Lbedenoted byh;then theintegralr=i vqr n .dL dh\ ,2qr+1-j-}dtc?rdt) hasastationaryvalue foranypartofanactualtrajectory,ascompared with otherpaths between thesame terminalpointsforwhich hhasthesame terminal values. Example2.Ifadynamical system which possesses anintegralofenergyisreduced to asystemoflower order asin42,shew that theprincipleofLeast Action forthe original systemisidentical withHamilton sprincipleforthereducedsystem. 101. ExtensionofHamilton sprincipletonon-conservative dynamical systems. We shallnow extend Hamilton sprincipletoholonomicdynamical systemsinwhich theforces arenolonger supposedtobeconservative. n LetTdenote thekineticenergyofsuch asystem, and let2Qr$qr r=l- denote thework done onthesystem bytheexternal forces inanarbitrary displacement (Bqi,&q, ...,&qn),theequationsofmotion ofthesystemare therefore d/dT\ dT -j.U-.--5=Qr (r=l, 2,..., ?i).dt\dqr/dqr 100-102] andLeast Curvature 249 Letadenote apartofatrajectoryofthesystem, and letftbeanadjacent archavingthesame terminals, thetimes correlated tothepath /3atthe terminalsbeingthesame asthevalues tand^ofthetime attheterminals inthetrajectory a;then if8denotes thevariationbywhich wepassfrom a positiononatothecontemporaneous position on/3,wehave */n\ (8r+ !,**)dt- t,r=i(oqr &lt; =0. This result QrSqrdt r=l is(like thetheorem of 99,which isreallyaparticularcaseofit)known as Hamilton sprinciple. 102. ExtensionofHamilton sprinciple andtheprinciple ofLeast Action tonon-holonomicsystems*. Weshallnowshew thatHamilton sprinciple, whensuitably formulated, istrueeven fordynamical systems which arenotholonomic. Consider anon-holonomic conservativesystem,inwhich thevariations ofthencoordinates(ql}q2,...,qn)areconnectedbymnon-integrable kinematicalequations Altdq1+A&dq a+...+Ankdqn+Tkdt=(k=l, 2,...,m) whereAu,A12,...,Anm ,Tl,...,Tm,aregiven functions ofql}q.2,...,q n:so that ifLdenotes thekineticpotential, themotion isdetermined(87)by thenequations dfdL\ dL dt(df r)~ dq~r=**An+**A&lt;*+~+^nA rm (r=1,2,...,n), togetherwith theabove kinematicalequations; theunknownquantities being &lt;?i&gt; $2&gt;,&lt;Jn, ^-i, ^-2) ,^m- LetABbepartofatrajectoryofthesystem, and letCDbeapath derived fromABbydisplacements consistent with theinstantaneous kine- *Cf.Holder, Gott.Nach. 1896, p.122,andVoss, Gntt.Nach. 1900, p.322. ThePrinciples ofLeast Action[OH.ix maticalequations,i.e.theabove kinematicalequationswith thetermsTkdt omitted;thispathCDwillnotingeneralbeitself apathwhose continuous description wouldsatisfythekinematical conditions, soCD isreallyakine- matically impossible path. Itmay naturally beaskedwhywedonottakeCDtobeakiriematically possible path: theanswer towhichis,that inthatcasethedisplacementsfromABtoCDwould notbe displacements consistent with thekinematical equations:forinnon-holonomicsystems, iftwoadjacent possible configurationsaregiven, thedisplacement fromonetotheother isnotingeneral apossible displacement ;there areinfinitely morepossible adjacent positions than there arepossible displacements from thegiven position. ProceedingasintheproofofHamilton sprinciple givenin|99,Bdenoting asusual adisplacement from apointofABtothecontemporaneous pointon CD,wehave Ldt-l Ldt=LB^-LA&t -fPI(~Bqr+~Bqr]dtCD JAB Jttr=i\oq r oqr/ ...+\mArm)Bqrldt. ) Since thedisplacements obeytherelations Alk8ql+A2kBq2+...+AnkBqn=0, itfollows that theterms ofthetype\gArsBqrintheintegralannul each other, sowehave T TJ. T7 TAi TA/ -0\V44 ^.WfV*&lt; \cv 7,Left- Z/&lt;ft=LjAj-L^A^o 4-2L.Bqr+ ,- (.--Bqr\&lt;dt. &gt;J^ Jfcr-ll0$r dt\dq rj1 ] From thispointtheproof proceedsasin99.Wethus obtain theresult thatHamilton sprinciple appliestoeverydynamical system,whether holonomic ornot.Ineverycase thevariedpathconsidered istobederived fromthe actual orbitbydisplacements which donotviolate thekinematicalequations representingtheconstraints; but itisonlyforholonomicsystemsthat the varied motion isapossiblemotion;sothatifwecomparetheactual motion withadjacentmotions whichobeythekinematicalequations ofconstraint, Hamilton sprincipleistrueonlyforholonomicsystems. Thesame remarksobviously applytotheprincipleofLeast Action, and toHamilton sprincipleasappliedtonon-conservativesystems. 103. Are thestationary integralsactual minima ?Kineticfoci. Sofarwehaveonlyshewn that theintegralswhich occur inHamilton s principleandtheprincipleofLeast Action arestationaryforthetrajectories ascomparedwithadjacent paths. Thequestion now arises, whetherthey areactually maxima orminima. 102,103] andLeast Curvature 251 Weshall select forconsideration theprincipleofLeast Action, and for convenience ofexpositionshallsupposethenumber ofdegreesoffreedom inthedynamical systemtobetwo,themotionbeingdefinedbyakinetic energy T=^a n(ql,qz)q?+a12(ql,qs)q,qa andapotential energy Thediscussion canbeextended withoutdifficultytoHamilton sprinciple,and tosystems withanynumber ofdegreesoffreedom. Theprincipleof Least Action, asappliedtotheabovesystem,is(100) thattheintegral F I(a n&lt;?i2+2a, 2gr 152+ 22&lt;7"2 )dt J hasastationaryvalue foranactualtrajectoryascompared with otherpaths between thesame termini forwhich dtisconnected with thedifferentials of thecoordinates bythesameequationofenergy T+V=h. This latterequation gives or dt={2(h i/r)}~2(alldq12+2a1.2dq 1dq+a^dq^, sothestationary integralcanbetaken tobe whereq2stands fordq2/dq1;thisintegralistobetaken between terminals, ateach ofwhich thevalues ofq1andq.2aregiven. Writingthisequation weshall discuss thediscrimination ofitsmaxima andminima(which was first effectedbyJacobi) byamethodsuggested byCulverwell*. Consideranynumber ofpaths adjacenttotheactualtrajectory. These pathswillbesupposedtohave thesame terminals, and tobecontinuous, but their directions mayhaveabrupt changesatanyfinite number of points. Forsuch apathlet(qltq2+&q2)beapoint correspondingtoa point (ft,ga)ontheactualtrajectory; weshallfrequentlywrite a&lt;for8q2, where aisasmall constant theorder ofwhich determines theorder of magnitudeofthequantities wearedealing with,and &lt;/&gt;iszeroattheterminal points. *Proc. Lond. Math. Soc.xxm.(1892), p.241. 252 ThePrinciples ofLeast Action[CH.ix Lettheexpansionofthefunction f(q 1}qa+a&lt;f&gt;, g2+&lt;) inascending powersofabe let57denote thetermsinvolvingainthe firstdegreein and let82/denote theterms ina2 . When therangeofintegrationissmall, and itsterminals arefixed, the value of &lt;atanypointislargecomparedwith thevalue of &lt;.Forsince &lt;/&gt; iszero attheterminals, wehave ..p wherePandRdenote theterminals. Ifthereforeftbethenumerically greatestvalue of (f&gt;between PandR,itfollows that &lt;f&gt;cannever exceed (gKJZ) &lt;?i(p))/3, andconsequently bytakingtherange sufficientlysmall the ratio of &lt;to &lt;/&gt;canbediminishedindefinitely. Thus iftherangeisvery small, themostimportantterm in82Iis Un(j)2dq1;andasthesignofthis isalwaysthesame asthatofUn(thesign- 1 ofdqtistaken tobepositive), weseethat forsmallranges,/isamaximum orminimumaccordingasUuisnegativeorpositive. Now ^\2J? Un=~=(h-^(an+2alsqt+a^_q. 22)~*(ana^-a122 ), " and this ispositive,since thekineticenergyisapositivedefinite formand therefore ana&a122ispositive. Wethushave theresult thatforsmall rangestheAction isaminimumfortheactualtrajectory. Now consider anypointAonanactualtrajectory,and letanother actual trajectorybedrawnthrough Amakingaverysmallanglewith the first. If thisintersects the firsttrajectory again, sayatapoint B,then thelimiting positionofthepointBwhen theanglebetween thetrajectoriesdiminishes indefinitelyiscalled thekineticfocusofAonthe firsttrajectory,orthe point conjugatetoA. We shallnowshew that forfiniterangestheAction isaminimum, providedthefinalpointisnotbeyondthekinetic focus ofthe initialpoint. For letPandQbetheterminals;wehave seen that ifQisverynear toP,thequantityS2/isalways positiveandoforder a2comparedwith the value of/forthelimitsPandQ.Itistherefore evident that asweremove 103,104] andLeast Curvature 253 Qfurther from P,thequantity8-1cannot becomecapableofanegative value until afterQhaspassed throughthepointforwhich B2Icanvanish forasuitably chosen value of (/&gt;. Suppose then thatPBQisanarcofanactualtrajectory, Qbeingthe first pointforwhich itispossibletodraw avaried curvePHQforwhich2/iszero; weshallshew thatthevaried curvePHQ must itself beatrajectory. For if itisnotatrajectory between twoofitsownpointsAandC(supposed near eachother),letatrajectory ADC bedrawn between thesepoints. Then the integral takenalongADC islessthan thattakenalongAHC, sotheintegral takenalongPADCQislessthan thatalongPHQ, whichbyhypothesisis equaltothatalongPBQ. Hence frlalongPADCQisnegative, andthere foreQcannot bethe firstpointforwhich, asweproceed from P,thevariation ceases tobepositive;which iscontrarytowhat hasbeenproved.Itfollows thatPAHCQisatrajectory,andQisthekinetic focus ofP.Hence the Action isatrueminimum, providedthat inpassing alongthetrajectorythe final pointisreachedbeforethekineticfocus oftheinitialpoint. Lastly weshall consider thecase inwhich thekinetic focus oftheinitial pointisreached before wearrive atthefinalpoint. Suppose,withthenotation just used, thattheinitial and finalpointsarePandR;and lettwopointsE andFbetaken, theformer onthecurvePHQ andthelatter onthearcQR ; thesepoints being taken soclosetogetherthatthetrajectory EGFjoiningthemgivesatrueminimum. Since theintegral takenalongEGF isless than thatalongEQF,itfollows thattheintegral takenalongPEGFR isless than thatalongPEQR ;butthelatter isequaltothatalongPBQR, since bothintegralsareequalfromPtoQ;andtherefore theintegral alongPBQR isnotaminimum;but itisnotamaximum, since theintegraltakenalong anysmallpartofitisaminimum. Hence when thekineticfocus oftheinitial pointisreachedbefore wearrive atthefinal point,theAction isneither a maximum noraminimum, Asimple exampleillustrative oftheresults obtained inthis article isfurnishedbythe motion ofaparticle under noforces onasmoothsphere. Thetrajectories aregreat- circles onthesphere, andtheAction takenalong anypath (whether atrajectoryornot) isproportionaltothelengthofthepath. The kinetic focus ofanypointAisthe diametrically opposite pointAonthesphere, since anytwogreat-circles through A intersectagain atA .Thetheorems ofthis article amount therefore inthiscase tothe statement thatanarcofagreat-circle joining anytwopointsAandBonthesphereis theshortest distance fromAtoBwhen(and onlywhen) thepointAdiametrically oppositetoAdoes not lieonthearc,i.e.when thearciuquestion islessthan half agreat-circle. 104.Representation ofthemotionofdynamical systems bymeansof geodesies. TheprincipleofLeast Action leads toaninterestingtransformation of themotion ofnaturaldynamical systems withtwodegreesoffreedom. 254 ThePrinciples ofLeast Action[CH.ix Letthekineticenergyofsuch asystembe and letitspotential energybe^(g 1}q.2).By 100,theorbitscorresponding tothatfamilyofsolutions forwhich thetotalenergyisharegiven bythe condition that r I(an9i2+Zouqiqi +a^q*}dt isstationaryforanypartofanactual orbit, ascomparedwithanyother arc between thesame terminals forwhich dtisconnected with thedifferentials ofthecoordinatesbytherelation Theintegral r i I(h tyyz(audq*+2a12dqldq2+ J isthereforestationary. But thisintegral expressestheprincipleofLeast Action forthemotion ofaparticleunder noforces onanysurface whose linear element isgiven bytheequation ds2=(h i|r){andq^+%aizdq^dq z+aZ2dq./), and istherefore thedefiningcondition ofthegeodesiesonthis surface. Consequentlytheequations oftheorbits inthegiven dynamical systemarethe same astheequations ofthegeodesiesonthissurface. 1.Shew that theparabolicorbits ofafreeheavy projectile correspond tothegeodesies onacertain surface ofrevolution. Example2.Shew thattheorbits described under acentral attractive force &lt;p(r)ina plane correspondtogeodesiesonasurface ofrevolution, theequationofwhose meridian- curve isz=f(p),where andwhere randpareconnected bytherelationp2=r2 { &lt;p 105. Theleast-curvatureprinciple ofGauss andHertz. Weshallnowdiscuss aprinciple which, likeHamilton sprinciple,canbe used todefine theorbits ofadynamical system,butwhich does notinvolve thesignofintegration. Inanydynamical system (whether holonomic ornon-holonomic)let (xr,yr,zr}bethecoordinates ofatypical particle mrattimet,and (Xr,Yr,Zr)thecomponentsoftheexternal forcewhich actsontheparticle. Consider thefunction ..Xr\2 /..Fryxr-- - )+(yr-- } mrjVmj 104,105] andLeast Curvature 255 where thesummation isextended over alltheparticles ofthesystem, and where(xr,yr,zr)refer toanykinematically possible pathforwhich the coordinates andvelocities attheinstant considered arethesame asinsome actualtrajectory.This functionsubstantially represents whatwascalledby Gauss theconstraint andbyHertz (who however consideredprimarilythe case inwhich theexternal forces arezero) thecurvature* ofthekinematically possible pathconsidered. Inwhat follows Hertz sterminologywillbeused. We shallshew thatofallpaths consistent with theconstraints(luhich aresupposedtodonowork), theactualtrajectoryisthatwhich has the leastcurvature^. Inthesimple case ofasingle particle moving onasmooth surface under noexternal forces, this resultclearly reduces tothestatement that thecurvature inspace (inthe ordinarysense oftheterm) oftheorbit istheleastwhich isconsistent with thecondition thattheparticleistoremain onthesurface. Toestablish this result, lettheequations whichexpresstheconstraints (usingasrtotypify anyoneofthethree coordinates ofany particle) be ?.xkrdxr=(k=l,2, ...,m),r where the coefficients xkraregiven functions ofthecoordinates. Differ entiatingthese relations, wehave OTi*xrxs= X Letxrbeatypical componentofacceleration inthepath considered (whichissupposedtobekinematically possible, but isnotnecessarilythe actualtrajectory), and letxrobethecorresponding componentofacceleration intheactualtrajectory. Subtractingthepreceding equation, considered as relatingtotheactualtrajectory, from thesameequation, considered as relatingtothekinematically possible path,wehave(since thevelocities are thesame inthetwopaths) Safe.(xr-xn)= (A?=1,2,. .,m).r Thisequation shews thatasmalldisplacement ofthesystem,inwhich thedisplacement8xrofthecoordinate xrisproportional to(xr-xm),iscon sistent with theequationsofconstraint, i.e.isapossible displacement. Thecomponentsoftheforces exercisedbytheconstraints aretypified by * Strictly speaking, thesquare root ofthisfunction, andnotthefunctionitself, wascalled thecurvature byHertz. tGauss, Crelle sJournal, iv.(1829), p.232; Werke, v.p.23.Gauss measured theconstraint by"thesum ofthemasses oftheparticles, eachmultiplied bythesquare ofitsdeviation from unconstrained motion." Theaboveanalytical expression for itwas firstgiven byH.Scheffler, Zeitschrift filrMath. in.(1858), p.197. Hertz stheoryisgiven inhisMecltanik. 256 ThePrinciples ofLeast Action[OH.ix (mrxnXr):andinanypossible displacementtheforces ofconstraint dono work.Wehave therefore 2(mrxro-Xr)(xr-xro)=0, r anequationwhich maybewritten intheform ..Xr\2~ /..Xr\*~ ...xr--- )=2,mrxro---- )+2,??i,.(xrXro)2 , or(revertingtotheuseofysandzs) F, ^ \/xr\*(..Fry =2,ra r&lt;scn---+n---+m +Sl r{(r-^ro)2+(j/r~ j/ro)2+(^r~^Vo)2 }- r Since theterms inthe lastsummation ontheright-handside are all positive,itfollows that .,Xr\s ,xr--+yr--+zr-mjVmj \mt, ^ (/..Zr\2/..Fr\2/. &gt; 2&lt;m r]x,n--- +yn---+(zn r (\f &gt;nj \mrj \ which establishes theresult stated.Zr 106. Expression ofthecurvatureofapathintermsofgeneralised coordinates. Lipschitzhasshewn* thatthecurvature ofakinematically possible path inaholonomicdynamical systemwith ndegreesoffreedom canbeexpressed interms ofthederivates ofthenindependentcoordinates which define the positionofthesystem. Let (&lt;?!,q2,...,qn)bethecoordinates; let(q1}q2,...,qn)betheaccelera tions ofthese coordinates inany kinematically possible path, and let (ry10, &lt;/20,...,qno)betheaccelerations intheactualtrajectorywhich corre spondstothesame values of(ql}q2,...,qn&gt;q1}q.2&gt;...,qn).Usingxrtotypify- anyoneofthethreerectangularcoordinates ofanyparticle mr,andXrtotypify thecorresponding componentofforce, theGauss-Hertz curvature ofthepath is^nir(xrXr/mr)2 ;and ithasbeenshewn inthelast article that thiscan r bewritten intheform (xroXr/mr}2+"2.mr(xrxr0)~. *Journal farMath. LXXXII.p.323. Cf.alsoWassmuth, Wien. Sitz. civ.(1895);and for further work connected with theprinciple ofLeast Curvature seeLeitinger, Wien. Sitz. cxvi. (1908), p.1321andSchenkl, Wien. Sitz.cxxn.(1913), p.721. 105,106] andLeast Curvature 257 The firstofthese summations isthesame forallthepaths considered, since itdepends onlyontheactualtrajectory:wecantherefore omit itwithout causingthewholeexpressiontolose itsminimum-property,andwecancall theremaining summation 2w,.(xrxro)2thecurvature ofthepath.r Letthekineticenergybe T=^Za klqkql} k I where thequantities ak{aregivenfunctions of(ql}q2,...,qn);letDdenote thedeterminant formed ofthequantities a kt&gt;and letAMdenote theminor of akiinthisdeterminant. From theequation 2,mrx;r2=2tStakiqkql r k I I r-*\jdjrp \}JUywehave au=2mr~ - . roqkdqi XT Vfar-Vv^XrNow av=2~ &lt;fc+22~-qkqt,toq* k idqrfqi* andconsequently,since thecoordinates andvelocities arethesame forallthe paths considered, wehave .._..j_v3a?r... .. .xrx r&lt;) 2i_(qkqko). koqk But ifwewrite _dsdT\ dT dxr "*3* 155" )""5^5~Xr (k=1,2,...,n),at\dqkjdqk ,.dqk since thisexpressioniszero fortheactualtrajectory, wehave Sk=thedifference ofthevalues of-=- (- )forthepath considered andat\dqk) theactualtrajectory, or8k=2aw(qt-qlo) (k=l,2, ...,n), whence wehaveqk-qko=^^AklSt (k=1,2,..., n) ; andconsequently Thecurvature, 2mr(xr-a^)2 ,istherefore orraSS^k Iij W.D.17 258 ThePrinciples ofLeast Action[CH.ix Butbyawell-knownpropertyofdeterminants, wehave 22 ik andthereforefinallythecurvature canbeexpressedintermsofthecoordinates (q\&gt; q2,,qn)and their derivates intheform 107. AppelVs equations. TheGauss-Hertz lawofLeast Curvature isthebasis ofaform inwhich Appellhasproposed*towrite thegeneraldifferentialequationsofdynamics. This form, aswillbeseen, isequally applicabletoholonomic andnon- holonomicsystems. Consider anydynamical system;let Alkdq,+A2kdq2+...+Ankdqn+Tkdt=(k=l, 2,...,ni) bethenon-integrable equations connectingthevariations ofthegeneralised coordinatesql,q2,...,qn;inholonomicsystemstheseequationswillofcourse benon-existent. LetSdenote thefunction^m k(xk2+i/k2+zk2 ),wheremktypifiesthemass k ofaparticleofthesystem, whoserectangular coordinates attime tare (%k, Vk, Zk)-Bymeans oftheequations which define thepositionofthe particlesatanytime interms ofthecoordinates(q1}q2,...,qn),itispossible toexpress 8interms of(qltq2,...,qn)andthe firstandsecond derivates of these variables withrespecttothetime. Moreover, byuseoftheequations ofconstraint wecanexpressmofthevelocities(ql}q2,...,qn)interms ofthe others :letthecoordinatescorrespondingtothese latter bedenotedby(pi,p 2, ,Pn-m)- Bydifferentiatingthese relations wecanexpress q1}q2,...,qn interms ofthequantities p1}p.2,...,p n_m ,plyp2,...,pn_m,qltq2,...,qn&gt;and henceScanbeexpressedinterms ofthis lastsetofvariables. Nowanysmalldisplacement which isconsistent with theconstraints canbedefinedbythechanges (Sp lt8p2,...,Bpn_m)inthequantitiesm (Pi,Pz,,Pi-m);let2Pr&prdenote thework donebytheexternal forces r=l insuch adisplacement. Asin 26,wehave Journal fUrMath. cxxi.(1900), p.310. andLeast Curvature 259 Lettheequation whichexpresses thechangeinockinterms ofthechanges in(PI, pa, ...,pn-m)be where(TTJ,7T2,..., 7rn_m)areknown functions ofthecoordinates: the equationsofthistypeareofcoursenon-integrable. From thiswehave 9#/9p r*Birr,and sotheequationwhichexpresses xkinterms of \pi&gt; PZ&gt;&gt;pnrn) willbeoftheform nm 2irrpr+a, r=l where adenotes some function ofthecoordinates.Differentiatingthis equation, wehave n-mfj rjn .-^U7rr . ftGC ^ ctxicwhence*=7rr= dpr Itfollows that n^.. k ....Pr=2mtxk^+ykf*+zk k .. ..xk~+yk+zk- Vdpr 9r 3 andtherefore /*eequations ofadynamical system, whether holonomic ornot, canbeexpressed intheform dS dJr=Pr (r=1,2,..., n-m), where^8denotes thefunction &mk(xk*+yk2+zk2 ),and(p,,p2,...,p n-m}are coordinatesequalinnumber tothedegrees offreedom ofthesystem*. Itisevident that theresult isvalid even ifthequantities plt...,pn_m arenottruecoordinates, butarequasi-coordinates. Example. Obtain fromAppelPs equations theequations A a&gt;i (BC) a&gt;2(03=L, a&gt;2(CA)0)30)!=JJ/, Co&gt;3(A-B) &lt;i&lt;o2=-Ar , forthemotion ofarigidbody oneofwhosepointsisfixed; where(o^,co2,o&gt;3)are thecomponents ofangular velocity ofthebody resolvedalongitsownprincipal axes ofinertia atthefixedpoint, (A,B,C]aretheprincipal moments ofinertia, and(L,M,N)arethemoments oftheexternal forces about theprincipal axes. *Ontheconnexion ofthese equations with thePrinciple ofLeast Action, cf.H.Brell, Wien. Sitz.cxxn.(1913), p.933. 172 260 ThePrinciples ofLeast Action[OH.ix 108. Bertrand s.theorem. Atheorem inimpulsive motion, whichbelongstothesamegroupof results astheleast-curvatureprincipleofGauss and Hertz,isdue to Bertrand* andmaybestated thus :Ifagivensetofimpulsesisappliedto different points ofasystem (whether holonomic ornon-holonomic)inmotion, thekineticenergy oftheresulting motion isgreaterthan thekinetic energy ofthemotion which thesystemwouldacquireunder theaction ofthesame impulses andconstraints andofanyadditional constraints due tothereactions ofperfectlysmooth orperfectly rough fixed surfaces,orrigidconnexions betweenparticles ofthesystem. For letmbethemass ofatypical particleofthesystem,and let(u,v,w), (u,v,w),(ulyvlfWi)denote thecomponentsofvelocityofthisparticlebefore theapplicationoftheimpulses,after theapplicationoftheimpulses,andin thecomparison motion, respectively. Let(X,Y,Z)denote thecomponentsoftheexternal impulse actingon theparticle:(Xf ,Y,Z)thecomponentsoftheimpulsedue tothecon straints ofthesystem:and(X+XltY+YltZ+ZJthecomponentsof theimpulse due totheconstraints inthecomparisonmotion. Theequationsofimpulsivemotion are m(v-v)=Y+Y ,m(w-w)=Z+Z, ,m(v 1-v)=Y+Y +Y 1,m(w,-w)=Z+Z+Zl. Subtracting,wehave m(u 1u)=Xl,m(v 1v)=Y 1,m(w lw}=Zl. Multiplythese lastequations byultv1}w1}respectively, add,andsum for alltheparticlesofthesystem;wethushave 2m|(MI-u)MJ+(v1-v)V!+(w 1-w)wi]=2(X^+Y&+Z^). Now from thenature oftheconstraints,itfollows that finite forces actingonalltheparticlesofthesystemandproportionaltotheimpulsive forces(XltFa,Zj),would onthewhole donowork inadisplacementwhose componentsareproportionaltothequantities (u l,v1}w^):andtherefore we have I,(X 1u1+Ylv1+Z1w1)=Qi or 2m{(ttju)u^+(Vjv)v1+(w lw}w^\= ; thisequationcanbewritten intheform 2m(u2+v2+w2 )-2m +v?+w,2 )=2m{(u-utf+(v-v,)2+(w-w,)2 }, *Bertrand snotes toLagrangesMec. Anal.;andLiouville sJournal(1),vn.(1842), p.166. 108] andLeast Curvature 261 which shews that 2m(V2+v2+w2 )&gt;\2m(V+vf+wf), andsoestablishes Bertrand stheorem. Thetheorem may readilybeextended tothecasewhen theforces arenot impulsive butcontinuous :inthis case theincrease ofkineticenergy per unit oftime isdiminishedbytheintroduction offresh constraints thatdo notaffect thepotential energy. Thefollowing result, duetoLord Kelvin andgenerally known asThomson stheorem*, caneasily beestablished byaproof ofthesame character astheabove :Ifanynumberof points ofadynamical systemaresuddenlysetinmotion withprescribed velocities,the kineticenergy oftheresultingmotion islessthan thatofanyotherkinematicalty possible motion which thesystem cantakewith theprescribed velocities, theexcessbeingtheenergy of themotion which must becompoundedwith either toproducetheother. LordRayleigh hasremarked +thatthetheorems ofThomson andBertrand mayboth becomprehended inthestatement thattheintroduction offresh constraints increases the inertia, ormoment ofinertia, ofasystem. Example. Aframework of(n-1) equal rhombuses, each with onediagonalinthe same continuousstraight line,andtwoopen ends, each ofwhich ishalf ofarhombus,is formedby2nequal rodswhich arefreely jointedinpairsatthecorners ofallthe rhombuses.Impulses Pperpendiculartoandtowards the line ofthediagonals are applied tothetwo free extremities ofoneopen end;shew that the initialvelocity, paralleltothediagonal,oftheextremities oftheotheropenend is 3P sinacosa TOcos2a+/i2sin2a wheremisthemass ofeach rod,and2aistheangle between each pair ofrods at thepointsofcrossing. (Camb. Math.Tripos,Part I,1896.) MISCELLANEOUS EXAMPLES. 1.Iftheproblemofdetermining themotion ofaparticle onasurface whose linear element isgiven bytheequation dsz=Edu2+ZFdu dv+Gdv\ under theaction offorces such that thepotential energyisV(u, v),canbesolved, shew that theproblemofdetermining themotion ofaparticle onasurface whose linear element isgiven by ds*=V(u,v)(Edu2+2Fdudv +Gdv2 ), under forces derivable from apotential energy I/V(u, v),canalsobesolved. (Darboux.) 2.Ifintwodynamical systemsinwhich thekineticenergies arerespectively SatjtjtjiandS&ijtjijfr,andthepotential energies arerespectively7andF,thetrajectories *Thomson andTait sNaturalPhilosophy, 317. tTheory ofSound, Vol. i.p.100. 262 ThePrinciples ofLeast Action[CH.ix arethesame curves, thoughdescribed with differentvelocities, sothat therelations between thecoordinates(q\,y2,...,qn)arethesame inthetwoproblems, shew that - wherea,/3,y,8areconstants, andthat "S,bikdqidqk=(yU+ 8)2aikdqidq k.(Painleve.) 3.Ifallthetrajectoriesofaparticleinaplane,described under forces such thatthe potential energyoftheparticleisV(x,y\with avalue Aoftheconstant ofenergy,are subjectedtoatransformation x=$(X, Y\ y=+(X, Y\ where$and^areconjugate functions of(#,y\shew thatthenewcurves soobtained are thetrajectoriesofaparticleacted onbyforces derivable from thepotential energy with azerovalue oftheconstant ofenergy. (Goursat.) 4.IfTandVdenoterespectivelythekinetic andpotential energiesofadynamical system, shew that differs from i(/...9rv ./...an2 ./ ...arv _m byaquantity which doesnotinvolve theaccelerations;andhence that - isamaximum when theaccelerations have thevaluescorrespondingtotheactualmotion, ascompared with allmotions which areconsistent with theconstraints andsatisfy thesameintegralofenergy, andwhich have thesame values ofthecoordinates and velocities attheinstant considered.(Forster.) CHAPTER X HAMILTONIAN SYSTEMS ANDTHEIR INTEGRAL-INVARIANTS 109. Hamilton sform oftheequations ofmotion. Weshallnowobtain forthedifferentialequationsofmotion ofacon servative holonomicdynamical systemaform which constitutes thebasis ofmost oftheadvancedtheoryofDynamics. Let(q1}q2,...,qn)bethecoordinates andL(q ltq2,...,qn,q1}q2,...,qn&gt;t) thekineticpotentialofthesystem,sothat theequationsofmotion inthe Lagrangianform are d fdL\ dLjiiaH-a -0 (r=l, 2,...,n\at\oqr/dqr OT Write ^=Pr (r=l,2, ..., TO), 09V- O 7" sothat Pr=(r=l,2,..,n). d&lt;/r From theformer ofthese setsofequations wecanregardeither ofthe setsofquantities (fa,q2,...,qn)or(p1}p2,...,pn)asfunctions oftheother set. If8denote theincrement inanyfunction ofthevariables (qltq2,...,qn,PI,p2,...,p n)or(q1}q2,...,qn,ql}qz,...,qn) duetosmallchangesinthearguments, wehave T -STft** 5- ,^-^ (&gt;oL=2,-oqr+^- o&lt;7r r=i\dqr oqr r=l *% ,4?/* =62&lt;prqr+2,(proqr q,r=l r=l or 82p^r-L\=2 (r=l )r=l 264 Hamiltonian Systems and[CH.x n Thus ifthequantity ^prqrL,whenexpressedinterms of r=\ \Ct Cl Cl *Y) "/) f) ft bedenotedbyH,wehave *\ \_is*n ^40. &H=2(qrpr-pr^qr) (1), &gt;f&gt;}&lt;l** r=1%8^ dprdH . or *-= =iT*=12 -??ii2i7- &lt;-\j- /.*-\ \&*# T*I i**t*l/a^dpr aidgr IViemotionofthedynamical system mayberegardedasdefined bythese equations,which aresaid tobeintheHamiltonian orcanonical form;the dependentvariables are(qltq2,...,qn,p1}p.2,...,pn),andthesystem consists of2nequations,each ofthe first order; whereas theLagrangian system consists ofnequations,each ofthesecond order. TheHamiltonian formwasintroduced byHamilton in1834*. Inparthehadbeen anticipated bythegreat French mathematicians: forPoisson in1809t hadtaken thestep ofintroducing afunction 2pr^r-Tr=l andexpressingitinterms of(ql,q2,...,qn,p1}...,pn),andhadactually derived half of Hamilton sequations:while Lagrangein1810Jhadobtained aparticular setofequations (forthevariation ofelements)intheHamiltouianform,thedisturbing functiontaking the placeofthefunction H.Moreover thetheoryofnon-linearpartialdifferentialequations ofthe firstorder hadledtosystemsofordinarydifferential equations possessingthisform : for,aswasshewn byPfaff in1814-15 andbyCauchy||in1819(completing theearlier work ofLagrange andMonge), theequationsofthecharacteristics ofapartial differential equation f(xi,x z,-..,x n,pi,pv ...,j0 n)=0, *-&gt; o$where p.=^, dxldxz_ _dxn dpl dpn ~ ~~ ~ ~ Hamilton sinvestigation wasextended tothecaseswhen thekineticpotential contains thetime,etc.byOstrogradskylFin1848-50 andbyDonkin** in1854. Theequation (1)above isoften called theHamiltonian form ofthe equation ofvirtual work. Itmaybewritten inthemoresymmetrical form 8(Iprdqr-Hdt)=d(5prSqr-HSt), r=l /=! *Brit. Ass.Rep. 1834, p.513; Phil. Trans. 1835, p.95. tJournal deVEcole polyt.vin.(Cahier xv),(1809), p.266. JMem. deVlnst. 1809, p.343. Berlin Abhand. 1814-15, p.76. ||Bull. soc.philomath. 1819, p.10. HMelanges deVAcad. deSt.-Pet. Oct.1848;Mem. deVAcad. deSt.-Pet. vi.(1850), p.385. **Phil. Trans. 1854, p.71. 109,110]their Integral- Invariants 265 whichdirectly suggeststheimportanceofthedifferential form n prdqrHdt r=l inconnexion with thedifferentialequationsofdynamics:cf.137below. When thekineticpotential Ldoesnotinvolve texplicitly,theHamiltonian functionHwillevidentlylikewise notinvolve texplicitly, andthesystem willpossess (41)anintegralofenergy, namely n _)2Oro^--L=h, r=l Oqr where Aisaconstant. Thisequationcanbewritten H(q1}q2,...,q n,pi,p 2,...,pn)=h, and this istheintegral ofenergy, which ispossessed bythedynamical system when thefunctionHdoesnotinvolve thetimeexplicitly. Fornaturalproblems, itfollows atoncefrom 41thatHisthesum ofthekinetic andpotential energiesofthesystem. Example. Shew thattheequationsofmotion ofthesimple pendulum are &gt; __ \ dt~dp&gt;dt where ff=2-l andwhereqdenotes theanglemadebythependulum with thevertical attimet,Iis thelengthofthependulum, andthemass ofthebob istaken asunity. 110.Equations arising fromtheCalculusofVariations. From thepreceding chapteritappearsthatthewhole science ofDynamics canbebased onthestationarycharacter ofcertainintegrals, namelythose which occur inHamilton sprincipleandtheprincipleofLeast Action : similarlythedifferentialequationsofmostphysical problemscanberegarded asarisinginproblemsoftheCalculus ofVariations. Thus, theproblemoffindingthestate ofthermalequilibriuminanisotropic conducting body,when thepointsofitssurface arekept atgiven temperatures, canbe formulated asfollows :tofind,amongallfunctions Vhaving given values atthesurface, thatonewhich makes thevalue oftheintegral M8PV,/3F\*/8F\2) T )+(T 1+(-5- )\dxdydz./vy/\// integrated throughout thesurface, aminimum. We shallnowshew that allthedifferential equations which arisefrom problemsintheCalculusofVariations, withoneindependent variable, can be expressedintheHamiltonian form*. *Cf.Ostrogradsky, Mem. deVAcad. deSt.-Pet. vi.(1850), p.385. 266 Hamiltonian Systems and[CH.X Suppose,forclearness, thatthere aretwodependentvariables;theproof isequally applicabletoanynumber ofvariables. (m) (n) LetL(t,y,y,y,...,y,z,z,z,...,z)beafunction oftheindependent variablet,thedependentvariablesy,z,andtheir derivates uptoorders m,n, respectively. Theconditions thattheintegral f(m) (n) \L(t,y,y,...,y,z,z, ...,z)dt maybestationary, can,bytheordinary procedureoftheCalculus ofVaria tions, bewritten intheform dt Now writeodz dt dt dL P.m= by Pm+2dL+(-i)w dt- Pm+n= andwrite &= Then if(m-l) (n-D y&gt;Qm+i==ziQm+2=zi &gt;(fm+n=z- +pm-lqm+pmy +Pm+ni (fm+nTPm+nzt 110,111]their Integral-Invariants 267 (whereHissupposed expressedasafunction of(t,qlt...,qm+n ,Pi,,Pm+n), (m) (n)") thequantities yandzbeingeliminated byuseoftheequations pm=vLfdy, pm+n=dL/dz), and if8denote anincrement due tosmallchangesinthe arguments qltq2,...,qm+n ,pltp2,...,pm+n,wehave (&gt;,-rj- ^i\JJ-J J,\J-LJQ -^UJJ r\ (_/-/-/ r\ r=0dv dv?=0dz dz 2pr8qr+l+pm8y r=l r=l m-\-n \ (n)m+n 1(n) 2pr&?r+i+j?H*&* +^qr+i&Pr +Z$pm+n. r=m+l rm+1 Usingtherelations dL dL dL m+n m+n thisbecomes 8H=2p,.8q r+2q,-8pr. r=l r=l Thus, if^fisexpressedinterms ofthevariables (t,PI,PZ,...,pm+n ,qi,q2,...,qm+n)&gt; dqrdH dprdHwehave ^7=^ j7=-^r~ (r=l, 2,...,m+n),atdpr atdqr andthedifferential equations oftheproblemarethusexpressedintheHamil- tonianform. Thesystemsofdifferentialequations which arise intheproblemsofthe Calculus ofVariations areoften calledisoperimetrical systems. 111.Integral-invariants. Thenature ofHamiltoniansystemsofdifferentialequationsisfunda mentallyconnected with thepropertiesofcertainexpressionstowhich Poincare*hasgiventhenameintegral-invariants. Consider anysystemofordinarydifferentialequations dx\-Y ^*-Ydxn_ dt~ dt= dt= An&gt; where JTj,Xz,...,Xnaregivenfunctions ofxl}x2,...,xn,t.Wemayregard theseequationsasdefiningthemotion ofapoint whose coordinates are (xlfxz,...,xn}inspaceofndimensions. *ActaMath. xm.(1890). 268 Hamiltonian Systems and[OH.x Ifnowweconsider agroupofsuchpoints,whichoccupya^-dimensional region fatthebeginningofthemotion, theywillatanysubsequenttime t occupy anotherp-dimensional region A_p-tuple integraltaken over fis called anintegral-invariant,ifithasthesame value atalltimes t;the number piscalled theorder oftheintegral-invariant. Thus, inthemotion ofanincompressible fluid, theintegralwhichrepre sents thevolume ofthe fluid,when theintegrationisextended over allthe elements offluidwhich were containedinitiallyinanygiven region,isan integral-invariant;since thetotalvolumeoccupied bythese elements does notvarywith thetime. Example1.Consider thedynamical problemofdetermining themotion ofaparticle inaplane under noforces :let(x,y}bethecoordinates oftheparticle, and(u,v)its components ofvelocity. Theequations ofmotion may bewritten x=u,y=v,w=0,v=0. Thequantity 7=I(8x-t8u), where theintegrationistaken, inthefour-dimensional spaceinwhich(x,y,u,v) arecoordinates, along thecurvilinear arcwhich isthelocus attime tofpoints which were initially onsome given curvilinear arcinthespace,isanintegral-invariant. Forthe solution ofthedynamical problemisgiven bytheequations u=a,v=b,x=at+c,y= where a,6,c,dareconstants: andtherefore wehave 7= and this isindependentof t. Example2.Intheplane motion ofaparticle whose coordinates are(x,y]andwhose velocity-components are(u,v),under theinfluence ofacentre offorce attheorigin whose attraction isdirectly proportionaltothedistance, shew that l(u8x x8u) isanintegral-invariant. 112. Thevariationalequations. Theintegral-invariantsofagiven systemofdifferentialequationsfurnish integralsofanothersystemofdifferentialequations which canbederived from these. For letthegiven systemofequations be doc -j=Xr(ar.ltx2,...,xn,t) =1,2,...,w). Let(X,x2,...,xn}and(^+Bx1}#2+&2,...,xn+&cn)bethevalues of thedependentvariables attime tintwoneighbouringsolutions ofthissetof equations;where (Sx^8x2,...,xn)areinfinitesimalquantities. Thenwehave -j-(xr+Sxr)=Xr(xl+&%!,x2+8x2,...,a;n+Sa;n,t) (r=1,2,...,n), 111-113]theirIntegral-Invariants andconsequently, djdXrj. 3X,, _ 9Xr. (r-1,2, ..., These lastnequations, togetherwith theoriginalnequations, maybe regardedasasetof2nequationsinwhich(xl}x2,...,xn,8xlt8x2,...,8xn) arethedependent variables. Now if denotes anintegral-invariantoftheoriginal system,thequantity -^ -j2Fr(z\,a.-2,...,xn}8acr- must, since thepathofintegrationisquite arbitrary,bezero invirtue of preciselythisextendedsystemofdifferentialequations ;andtherefore HFr(x-i ,x.2,...,xn}8xr=constant r must beanintegraloftheseequations:sothat toanintegral-invariant of order oneoftheoriginal system ofequations therecorresponds anintegral of theextendedsystem ofequations, and viceversa. Ifaparticularsolution(a?1}x2,...,Xn)oftheoriginal equationsisknown, wecansubstitute thecorrespondingvalues(ar1}#2,....xn)intheextended differentialequations, andsoobtain nlinear differentialequations todeter mine(&BJ,&fc2,...,&&n),i.e.todetermine thesolutions oftheoriginal equations which areadjacenttotheknownparticularsolution. These nequationsare called thevariationalequations. 113. Integral-invariants oforder one. Letusnow findtheconditions tobesatisfied inorder that .Sx,+M2Sx2+...+MnSxn), where(MltM2,...,Mn)arefunctions of(x1}#2,...,xn,t),maybeanintegral- invariant oforder oneofthesystemofdifferentialequations dxrjdt=Xr(X,#2,.,.,xn,t) (r=1,2,...,n). Wemust have 2+...+MnSxn)=0, where thederivates of(8x 1}8x2,...,8xn)are tobedeterminedbythe 270 Hamiltonian Systems and[OH.x extendedsystemofdifferentialequationsintroduced inthelastarticle;and therefore 11 /rl /l/i /"/A i"N2(^&v+Jfr^F)-0( j.=i \u/c at i or 5?Xtfcv+Mr2-r ft*=0. Since (&E I;&r2,,&)areindependent,thecoefficient ofeachquantity $xrinthisequationmust bezero :andconsequentlytheconditions for integral-invariancyare Corollary1.Ifanintegralofthedifferentialequations, say jF^u;r2, &gt;^i=constant, isknown, wecanatoncedetermine anintegral-invariant. Forwehave adF\ a/a^\ a^azfc a+a^\^a^azfc a/aF a^ \ ^ Afc+Zr^5- ="^I-^T-+2trAA;I OBr/ k=\ox kdxrdxr\dt k=iox k J- dt\dxj k-\0& .JL(*dxr\dt =0, andtherefore theexpression /VvdF2^Ar-l9r isanintegral-invariant. Corollary2.Theconverse ofCorollary1isalso true, namelythatif n8U \S^Sorjisawintegral-invariant ofthedifferential equations, whereUis r=luxr agiven function ofthevariables, thenanintegral ofthesystem canbefound. Forwehave dxr)k=idxk\dxrjk=\3x k()xr andconsequentlytheexpression dt k=idxk 113,114]their Integral-Invariants 271 which isagivenfunction of(#1}x2,...,x n,t),isindependentof(x^,x2,...,xn); letitsvalue be &lt;f&gt;(t):this isaknownquantity. Thenwehave dU/dt=$(t), r or U I (f)(t)dt=constant; andthis isanintegralofthesystem. 114. Relativeintegral-invariants. Hitherto wehaveonlyconsidered thoseintegral-invariants which have theinvariantiveproperty when thedomain oftheinitial values, overwhich theintegrationistaken, isquite arbitrary ;these aresometimes called absoluteintegral-invariants. Weshallnowconsiderintegrals which have the invariantiveproperty onlywhen thedomain overwhich theintegrationis taken isaclosed manifold(usingthelanguageofw-dimensionalgeometry); these arecalled relativeintegral-invariants. Thetheoryofrelativeintegral-invariantscanbereduced tothatofabsolute integral-invariantsinthefollowing way. Let\(M lSx,+M,8x2+ ...+Mnxn} bearelativeintegral-invariantoftheequations dxr/dt=Xr (r=l, 2,..., ?i), where(M ltM2,...,Mn,X1}X 2&gt;...,Xn)arefunctions of(xltx.2,...,xn,t); sothat thisexpressionisinvariable withrespecttotwhen theintegrationis taken, inthespaceinwhich(xltxz,...,xn)arecoordinates, round theclosed curve which isthelocus attime tofpointswhich wereinitiallysituated on some definite closed curve inthespace. ByStokes theorem, thisintegralisequivalenttotheintegral where theintegrationisnowtaken overadiaphragm boundedbythecurve; thisdiaphragmcanbetaken tobethelocus attime tofpoints which were originallysituated onadefinitediaphragm boundedbytheinitialpositionof theclosed curve :andsince thediaphragmisnotaclosed surface, thisintegral isanabsoluteintegral-invariantoforder twooftheequations. Similarly, byageneralisation ofStokes theorem, anyrelativeintegral invariant oforderpisequivalenttoanabsoluteintegral-invariantof order (p+1). 272 Hamiltonian Systems and [en.x 115.Arelativeintegral-invariant which ispossessed byallHamiltonian systems. Consider nowthecase inwhich thesystemofdifferentialequationsisa Hamiltoniansystem,sothat itcanbewritten dqr=dH dpr=_3H(r=i2..n)dtdpr dtdqr whereHisagivenfunction of(q1}q2,...,qn,p1}p2,...,pn,t). Forthissystemlet denote Hamilton sintegral,sothatListhekineticpotential ;let (!,O2 &gt;&gt;an,fii,/32, ,ftn) betheinitial values ofthevariables (ql,q2)...,qn,pi,p2,...,p n) respectively, and letSdenote thevariation from apointofoneorbit tothe contemporaneous pointofanadjacentorbit. By 99,wehave n n oil=2proqr2f3rootr. r=l r=l LetCdenote anyclosed curve inthespaceof2pdimensions inwhich (q1}q2,...,qn,PI,pz&gt;...,pn)arecoordinates, and letCdenote theclosed curve which isthelocus attime tofthepointswhich areinitiallyonG . Integratingthe lastequationround thesetoftrajectorieswhichpassfrom GOtoC,wehave rn rn 2pfBo r=I2/3r&ctr, rn andthisequationshews that thequantityI2pr&qrisarelativeintegral- J&gt;=! invariant ofanyHamiltoniansystem ofdifferential equations. 116. Onsystems whichpossesstherelativeintegral-invariant Weshall nextstudytheconverseproblem suggested bytheresult of the last article, namelythat ofdeterminingallthesystemsofdifferential /n equations whichpossessthe relativeintegral-invariantI2pr8qr,where (#!&gt; &lt;li&gt;,qn)arehalfthedependent variables, and(pi,p 2, ,Pn)arethe other half. 115,116]their Integral-Invariants273 Consider then asystemofordinarydifferentialequationsoforder 2nin which thevariables canbeseparatedintotwo sets,(&lt;/,,q2,...,qn)and (pi,p2,...,p n},such that isarelativeintegral-invariantoftheequations, andconsequently byStokes theorem J isanabsoluteintegral-invariant. Letthesystemofdifferentialequations be t-fc.-P, (r-l.2,..,), where (QlfQ2,...,Qn,PltP2,...,Pn)aregiven functions of (&lt;?!,&,..-,qn,pi,p 2,...,p n,t). Asthedomain ofintegrationoftheabsoluteintegral-invariantisoftwo dimensions, wecansupposethateachpointinitisspecified bytwoquantitiesXandp,which donotvarywith thetime butarecharacteristic ofthe trajectoryonwhich thepointinquestionlies. The absoluteintegral- invariant cantherefore bewritten intheform [[(v9(qt,PJ)\- , //(^3,1^ d\dfj,,JJ\i=i (X, (A)J andasXand/xdonotvarywith thetime,wemust have _ dtM9(A,, (Qi.Pi)a, or.._ 8(X,/i) 3pt8(x,/*)"*"8^8(X,/*)^8^8(x, / Owingtothecomplete arbitrariness ofthedomain ofintegration and thechoice ofXand^thecoefficients of^d fi,^^,and8^^*inthisdXdyttdXd/u, dX8/u, equation must vanishseparately. Wethus obtain dpif dpi W.D.lg 274 Hamiltonian Systems and[CH.x Theseequationsshew that afunction H(q ltqz,...,qn,p,p-2,...:pn, t) exists such that Qr=dH/dp r,Pr=-dH/dq r (r-I,2,...,); andthuswehave theresult thatifasystem ofequations dqr_ dpr_p (r=l 2 n)~dt~Qrdt~L possessestherelative integral-invariant f I(PiSg,+p-2S?2++Pn&g), then theequationshave theHamiltonian form dqr=dH_ dPr=_dH_ dt dpr dtdqr this istheconverse ofthetheorem ofthelast article. Corollary.If I(pi8?i+p 2^q2+...+pnfyn) isarelativeintegral-invariantofasystemofequations dqr/dt=Qr,dpr/dt=Pr (r=1,2,...,k), where kisgreaterthan n,itfollows inthesamewaythattheequationsfor (qi, q*&gt;-&gt;qn:pi, P*&gt;~.,pn)form aHamiltonian system ^^ _ dtdpr dt dqr whereHisafunction of(qltqz,...,qn,p,,pz,...,p n,only,notinvolving (qn+i,qn+2,--,qk,p n+i,...,pk)- 117. Theexpression ofintegral-invariantsinterms ofintegrals. Ifthesolution ofasystemofdifferential equations isknown, theabsolute andrelativeintegral-invariantsofthesystem may easily beconstructed. Thus, let where cl}c2,...,cnareconstants, benintegralsofthesystem;theabsolute integral-invariantsoforder oneareevidently given bytheformula 116-118]theirIntegral- Invariants 275 where (N1}K2,...,Nn)areanyfunctions of(y1,y2,...,yn)which donot involve t:andtherelativeintegral-invariantsoforder onearegiven bythe formula ,By,+N2Sy2+...+NnSyn+SF), r whereFisanyfunction of(xlta?9,...,xn,t},since thetermSFvanishes when thedomain ofintegrationisclosed. Itfollows from thisthatanysystem ofdifferential equations possesses an infinite numberofabsolute and relativeintegral-invariants ofthefirstorder. 118. ThetheoremofLieandKoenigs. Theprecedingresults enable ustoestablish atheorem duetoLie*and Koenigs fonthereduction ofanysystemofordinarydifferentialequations to theHamiltonian form. Let llf=Zr(r-1,2,...,*) bethegiven systemofequations, and let beanyrelative orabsoluteintegral-invariantoforder oneofthissystem, where,, ,,..., j~karegivenfunctions ofthevariables :wehave seen inthe lastarticle thataninfinite number ofsuchintegral-invariants exist. Now letthedifferential form bereduced tothecanonical form (PI,pa,,pn,qi, ft, -,qn,ty areindependent functions of(a^,a?a,...,xk\innumber notgreater than k, andwhereHmaybezeroJ.Let(u^ ,u2,...,uk.m)beasetofother functions of(a?!,a?2,...,a?t),such that(u,,u2,...,uk_,n,qltqz,...,qn,pltp*,...,p n)are asetofkindependent functions of(x1}x.,, ...,xk);andsuppose that the *ArchivforMath, ogNatur. n.(1877), p.10. tComptes Eendus, cxxi.(1895), p.875. JTheproof ofthepossibility ofthisreduction (which however requires ingeneral the solution ofanumber ofordinary differentialequations)willbefound inanytreatise onPfaff s problem. 182 276 Hamiltonian Systems and[OH.x systemofdifferential equations,whenexpressedinterms ofthese kfunctions asindependentvariables, becomes dqr/dt=Qr, dpr/dt=Pr (r=1,2,...,n), du,/dt=U8 (s=1,2,...,k-2w), where (&,Q2,....Qn,PI,P*,&gt;Pn,U1}U,, ...,Uk^n)arefunctions ofthe new variables. Theexpression isanintegral-invariant (relativeorabsolute)ofthissystem,sinceintegral- invariancyisapropertyunaffected bysuch transformations ashave been performed:andconsequentlyitfollows(116)that the first2nequations have theform dq_r=dH_ dpr=_3H^(r=l 2 n) dtdpr dt~ dqr whereHisafunction of(qltq2,....qn,PI,p*,&gt;pn&gt;t)only.Thegiven system ofdifferential equationsisthusreduced toaHamiltonian system of order 2n,togetherwith the(k 2ri)additional equations 119. TheLast Multiplier. Beforeproceedingtodiscussintegral-invariantsofhigherorder than those hitherto considered, weshall introduce theconception,introduced byJacobi* in1844, oftheLast Multiplierofasystemofequations. dxldx2 dxn_dx where(XltX 2&gt;...,Xn,X)aregivenfunctions ofthevariables (xl,xz,...,x n,x), beagiven systemofequations:andsupposethat (n l)integralsofthis systemareknown, say f(Tv T &lt;r\n &lt;Y=12 n1)Jr\Xi, X%,...,tCn}ICJU/r \i J.,^,.,n From these equationslet(xltx2,...,xn^}beexpressedasfunctions ofxn andx :then there remains onlythesolution oftheequationofthe first order dxn_dx TT /~Y~A-n**- tobeeffected; inwhich accents areused todenote that (xl}x z&gt;...,a;n_1)have beenreplacedinXnandXbythevalues thus obtained. *Crelle sJournal, xxvn. p.199,xxix.pp.213, 333. 118,119]their Integral- Invariants 277 Weshallshew that theintegral ofthisequationis fjf )7(XdxnXndx)=constant, whereMdenotes anysolution ofthepartial differential equation +(MX.) +(*JO=o, anc?Adenotes theJacobian Thefunction .M"iscalled the asMultiplierofthesystemofdifferential equations. Fortheproofofthistheorem, weshallrequirethefollowing lemma : Ifasystemofdifferentialequations dxrjdt=Xr (r=l, 2,..., ri) istransformedbychangeofvariables intoanothersystem dyr/dt=Y r (r=l, 2,...,n), a8then 2-5 r=1c whereDdenotes theJacobian o(aSi ,xz,...,xn) Toprove this,wehave 2 *^r=1xrr=ldxr\k=l dyk 3v -^ifc5 /fc=l 3 _vvv - -"^-&lt;vvvy(vd*x &gt;-^YkdxA -"^-&lt;^J.fcr^--h =. =i3=1k=ioxr\dy.,dy kdysdykl Inthisexpression thecoefficient ofdYk/dy sisI^^,which iszero ,.=i doordyk orunity accordingassisdifferent from, orequal to,k.Also3y,/9a? r=Ars/D, where ^lrsdenotes theminor ofdxr/dy sinthedeterminant D :sothecoefficient ofYkintheaboveexpression, which is =ls=l 278 Hamiltonian Systems and[CH.x maybewritten or-?5(^1, #2,~Dr=isriT8 dysdy 18Z) ory:^.Ddyk Wehave therefore Idxr_|ajj %Y^ r=idxrk=ldykk=ikDdyk which establishes thelemma. Now intheoriginal problemwrite /y/&gt; Cirx* dIT ciA andconsider thechangeofvariables from bythelemma, wehave sothequantity M,which isasolution oftheequation Mdt satisfies theequation All a8/XA which shews thattheexpression istheperfectdifferential ofsome function ofxnand a; ;thisestablishes the theorem oftheLastMultiplier. Boltzmann andLarmor shydrodynamical representation oftheLastMultiplier. Thetheorem oftheLast Multiplier mayalsobemade apparent byphysicalcon siderations. Forsimplicity weshall takethenumber ofvariables tobethree, sothatthe differential equations maybewritten dxdy_dz U~ VW* 119,120]their Integral-Invariants 279 where(u, v,w)aregivenfunctions of(#,y,z) ;andthe lastmultiplier Msatisfies theequation j-(Mu)+^(Mv} +^(Mw)=0. 3#xdyoz This equationshews that inthehydrodynamical problemofthesteady motion of afluid inwhich(u,v,w}arethevelocity-componentsatthepoint (#, ?/,z),theequationof continuityissatisfied whenMistaken asthedensityofthefluid atthepoint (#,y,z}. Now let(f)(.v,y,z)=C beanintegralofthedifferential equations; then theflow willtake place between the surfacesrepresented bythisequation;thuswecanconsiderseparatelytheflow inthe two-dimensional sheet between consecutive surfaces Cand(7+8(7. Theflowthrough the gapbetween anytwogiven pointsPandQonCmust bethesame whatever bethe arcjoiningPandQacross which itisestimated :andsince theflowacross arcsPRandRQ togetheristhesame asthatacross PQ,weseethattheflowacross anarcjoiningPandQ must beexpressibleintheformf(Q)-f(P).Soifdsdenotes anelement ofthisarc,and rthe(variable)thickness ofthesheet, sothat r={(d^/dx)2+(B&lt;/8?/)2+(90/9z)2 }~* .8C,and ifdenotes thevelocity-component perpendiculartods,wehave sothatMrds istheperfectdifferential ofafunction ofposition.But itiseasilyseen that thisexpressioncanbewritten intheformMSC(vdx-u dy)/(d&lt;t&gt;/dz);andconsequently M(vdxudy} isaperfectdifferential;this isthetheorem ofthe lastmultiplierforthecasecon sidered. Wereadilyfind fordsthevalue sothetheoremreallystates that equationdz; uv ID &lt;f&gt;x$y 4&gt;z isanintegrating factor ofthe dxdydz=0. u vw (f)x(t&gt;y&lt;&gt;z This, aswasremarked byAppell (Comptes Rendus, CLV.(1912), p.878),isasymmetrical form ofthetheorem oftheLastMultiplier. 120. Derivationofanintegral fromtwomultipliers. Supposenowthattwodistinct solutionsMandNofthepartialdifferential equationofthelastmultiplierhavebeen obtained, sothat /**\ o\ o~v~ ^~v~ o~y o~v I-rrU -r-r -TVU -rrG\ -i rC/-A 1QA.Q C-^- n J\ ,.(X l^+X.,^-+...+Xn~-+X^losrJ/+^1+^r-J-+...+^--+^-=0, V f/T*"uSC ulT fiF/ til* ()/ft(jQC G3C and 3.tr9.vd .vd\i HT .3^1,9^2 , ,dXndX ^-OX-- OXnOX=0. 280 Hamiltonian Systems and[CH.x Subtractingtheseequations, wehave 3-3 v butthis isthecondition thattheequation log(M/N)=constant shallbeanintegralofthesystem Ct/tX/i C&W2Ctt/jj,\AjJU X^-3T2Xn.A. andwehave therefore thetheorem that thequotient oftwolastmultipliers of asystem ofdifferential equations isanintegral ofthesystem. Thereader who isacquainted with thetheoryofinfinitesimal transformations willbe able toprove withoutdifficultythat iftheequation 8/ ,Yf, admits theinfinitesimal transformations then thereciprocalofthedeterminant VV VV A-l At, A.nA. ll^12 ln l isalastmultiplier.fen 121.Application ofthelastmultipliertoHamiltoniansystems:useofa single knownintegral. Ifthesystemofdifferentialequationsconsidered isaHamiltoniansystem, wehaveevidently ^dX r/dxr=0,andconsequently M=1isasolution ofthe r partialdifferentialequationwhich determines thelastmultiplier;sothelast multiplier ofaHamiltoniansystem ofequationsisunity. From thisresult wecandeduce atheorem which enables ustointegrate completely anyconservative holonomicdynamical systemwithtwodegreesof freedom when oneintegralisknown inaddition totheintegralofenergy. Letthesystembe ^2i=^h=dpl dp~2fit djf d_H_dH_dH dpi dp2 dqj. dq2 andinaddition totheintegralofenergyH(qI}q2,PI,PZ) h,letanintegral V(q l}q2,pi,pz)=cbeknown. From thetheorem ofthelastmultiplierit follows that 1(dH ,dH=constant 9(Pi. 120,121]their Integral-Invariants 281 isanotherintegral;where, intheintegrand, plandp2aresupposedtobe replaced bytheir values interms ofq1and &lt;?aobtained from theknown integralsHand V. But ifwesupposethat theresult ofsolvingtheequations H=hand V=cforp1andp2isrepresented bytheequations thenwehaveidentically dpidc""" dp2dc !dcdp2dc andtherefore 8/1 dH/dp, df2_-dH/dPl dc~d(V,H),dcd(V,H} p2) d(p l,p2) sothetheoremofthelastmultiplier canbeexpressed bythestatement that isanintegral. This result leadsdirectlytothetheoremalready mentioned, which may bethus stated* :Ifinthedynamical system defined bytheequations dqr_dH dpr_dH/_io\ dtdpr dt dqr theintegral ofenergyisH(q1}q2,pl,p^=h,andifV(q^., q%,PI,PZ)=c denotes anyotherintegral notinvolvingthetime, then theexpression Pidq1+p2dq2,wherep-^andp2have thevalues found fromtheseintegrals, istheexactdifferential ofafunction6(gnq2,h,c);and theremaining integrals ofthesystemare 7\fi o/3 ^-=constant, and-j=t+constant. dc dh Thisamounts tosayingthat ifany singly-infinite familyoforbits is selected(e.g.theorbits which issue from apoint qi=ct1}q2=2)which have *This theorem isreally anapplication ofthewell-known method forthesolution ofa partial differential equation ofthe first order, theequations ofthedynamical system being theequations ofthecharacteristics ofthepartialdifferential equation. Asadynamical theorem,itwaspublished forasimple case (motionofasingle particle) byJacobi in1836 (Comptes Eendus, in.p.59),and forthegeneral casegiven herebyPoisson in1837 (J.deM. n.p.317)andLiouville in1840(J.deM.v.p.351). 282 Hamiltonian Systems and[CH.x thesameenergy,sothat toanypoint (ft,q2)therecorresponddefinite values ofplandp.2(namelythevalues ofplandp2correspondingtotheorbitwhich passes throughthepoint ql}q2andbelongstothefamily),then thevalue of theintegralIpldql+p 2dq2takenalong anyarcjoiningtwo definitepoints (fto, #20)and(qu,q21)isindependentofthearcchosen. Tocompletetheproof, wehave ondifferentiatingtheequations H=h andF=c, 9ft dpi9ft 9/&gt;29ft andconsequently 9(F,#) 9(F,#) dsimilarlv9/;_3(p 2,ga)iri - ButsinceV=cisanintegral,wehave 3F. 3F. 3F .3F. ^-ft+_PI+^-q2+v-P2=0,9gi 9pi9g2*dp,^ 8(F, .ST) ,3(F, ^T)or(+^7--r=0,9(ft,Pi)9(ft,PS) andtherefore M-^ =0. 9ft oft Thisequationshews thatfidql+f2dq2istheperfectdifferential ofsome function 6(qlyq2,h,c):andtheresult derived above from thetheoryofthe lastmultipliershews thatd&/dc=constant isanintegral. Moreover, wehave andtherefore 3F, 3F 5dq1- -7. 3(F, //) 9(pa,PI) Butobtaining 9/1/3Aanddf2/dhinthesamewayas9/l/9c and3/a/3cwere found, wehave ,df2 dh3(F,J3 9A3(F, 121,122]theirIntegral- Invariants 283 ?)f PI/* Consequentlydt= ~jdq l-fdq.2, Wor t=JTT-+constant,on whichcompletestheproofofthetheorem. Example. Intheproblem oftwocentres ofgravitation (53),if(r,/)denote the radii vectores tothecentres offorce, and(d,6}theangles formed by r,/with the linejoiningthecentres offorce, obtain theintegral r2r20ff-2c(pcos6+/*cosd}=constant, andhence complete thesolution bytheabove theorem. 122.Integral-invariants whose order isequaltotheorderofthe system. Thetheoryofthe lastmultiplierofasystemofdifferentialequationsis connected with that oftheintegral-invariants whose order isequaltothe order ofthesystem. docLet~s=x- (r=1-i*) where (ZnXZ)...,Xk)aregiven functions of(xl}xz,...,xk,t),beasystem ofordinarydifferentialequations ;and letusfindthecondition which must besatisfied inorder that fffi III...MQX, ox,,...6xkJJJ J maybeanintegral-invariant, whereMisafunction ofthevariables. Let (cl5c2,...,ck}beanysetofconstants ofintegrationoftheseequations, sothat,bysolvingtheequations, (xl}xz,...,xk)canbeexpressed interms of(d,c2,..., a*, t).Thenwehave [.I &gt; v\Oj ,G2,..., andtherefore thecondition ofintegral-invariancyis d(M8(^,^2, ...,x ky\ ~JI i-"^~- dt {d M* dt8(c 1(c2)...,ck) r=l d(c 1} or M.,,..., dt8Cc ... a .^ l,c2,...,a =iCXr which shews thatMmust bealastmultiplier ofthesystem ofequations. 284 Hamiltonian Systems and[CH.x This resultgives immediatelythetheorem thatforadynamical system whose motion isdetermined bytheequations dq,._?)H dpr__dH _ dt~dp r dt~dqr whereHisanyfunction of(q1}q2,...,qn,pltp2,...,pn,t),theexpression isanintegral-invariant ;since inthiscaseunityisalastmultiplier.This theorem isofimportanceintheapplicationsofdynamicstothermodynamics. Example.Forasystem withtwodegreesoffreedom,lettheenergy-integral when solved forptake theform H(fh,to,Pi,Ps,A)+_pt=0. Shewthat,fortrajectories which correspondtothesame value oftheconstant of energy,thequantity isindependent oftand also ofthechoice ofcoordinates :andhence shew that the trajectoriesoftheproblem canberepresented asthestream-lines inthesteadymotion ofafluidwhosedensityis 123. Redactionofdifferential equationstotheLagrangian form. Anotherquestiontowhich thetheoryofthe lastmultipliercanbe appliedisthefollowing:Tofindunder what conditions agiven systemof ordinarydifferentialequationsofthesecond order isequivalenttoaLagrangian system d/dL\ dL jlla-H-5 -0 (r=l, 2,...,w),dt\dq rjdqr whereLisafunction of(qltq2,,..,qn ,ql}q2,...,qn,t). Ifthese twosystemsareequivalent,theequations ^(3-o &lt;ik+3./Qk}+^-5. ^=(r 1,2,...,n) k=i\oqrdqk oqrdqkz/dqrdtdqr mustevidentlyreduce toidentities when thequantities qkarereplaced by theexpressions fk;andtherefore therequiredcondition isthatafunction L shall existsatisfyingthesimultaneouspartial differential equations d/ io(r=l,2,...,n), 00dqrotdqr where(qltq2,...,qn,qi,qz, ...,q n,t)areregarded astheindependentvariables. 122-124]their Integral-Invariants 285 When n=1,thequestioncanbesolved interms ofthelastmultiplier. Fortheequationsatisfied byListhen _ dq* dqdq* dqdt dq fromwhich wehave __f-.~ ~ dq\dq2/dq\dqdq dqdt dqj andtherefore ifwewrite d2L/dq2=M,thefunctionMsatisfies theequation dM/%-- butthis istheequation definingthe lastmultiplier Mofthesystemof equations andtherefore wAen w=1,/&lt;edeterminationofthefunction Lreduces tothe determinationofthelastmultiplier ofthesystem. 124. Case inwhich thekineticenergyisquadraticinthevelocities. Whenn&gt;l, themost importantcase isthat inwhich each ofthefunctions fr consists ofapartFrwhich ishomogeneous andoftheseconddegreein(q^,j2,...,qn)and apartGrwhich doesnotinvolve(j1}j2&gt;j0)anditisrequiredtodetermine whether theequations qr=Fr+Or (r=l, 2,...,) areequivalenttoasystemdT where T7 ishomogeneous andoftheseconddegreein(jhj2,-,?)andalsoinvolves the variables(^,^2,-,?),and(&, 2,...,$ft)arefunctions of(g-1?j2,....qn)only. Thevalue ofTisclearly notdependent on(6? 1,^j ...,6-n),andtherefore wecan consider theprobleminwhich (G1}6r2,...,6rM.)arezero, i.e.theproblem offinding afunction Tsuch that theequations ir=Fr (r=l,2, ...,) areequivalenttothesystem d/dT dT Thecondition forthis istheexistence ofafunction Tsatisfying thepartial differential equations " &lt;(r=12- *&gt; SinceFkishomogeneous, wehave 2qsdFk/dqs=2Fk,andtherefore 8=1 * 286 Hamiltonian Systems and[OH. But since dF/dq rishomogeneous,wehave andtherefore k=iOqrdqkk~ s=\cqrVfc=t 9ygdqkj~* "=\9jr3^ Theequationstobesatisfied byTmayconsequentlybewritten nfiF^tiTn(PT dT 1-v*kUJ ,T?uL UJ_f\ cT\/.w9^9^ 9T\,._ or -- 25r+rr (r=1"%)8 andevidentlythesemaybereplaced bytheequations *dFtdTdT ,1,i2~-^+~=0 (r=l, 2,..., ). *k=io^-^k9?r Thus, writing/,,for(Fr+Gr),wehave thetheorem thatifthesystem ofequations qr=fr (r=l, 2,...,), w/tere/rconsists ofapartwhich ishomogeneous ofdegreetwointhevelocities anda partwhich does notinvolve the velocities,isreducible totheform d thenTmust beanintegral ofthesystem il^pl+^O(,.1, 2,...,). 2k=idq rdqkdqr MISCELLANEOUS EXAMPLES. 1.Intheproblemoftwocentres ofgravitation,thedistance between thecentres of force is2c,andthesemi-majoraxes ofthetwoconies whichpassthroughthemoving particleandhave their fociatthecentre offorce are(ql,qz}.Writing shew that theequationsofmotion are dqroHdpr(&gt;H, ,_. F-=^ i-T=^ (r= l&gt;2), a&lt;ojo,.a^ d^ where JT"- 2_ and/niand/u2areconstants. 2.Shew that jfjj where thesummation isextended overthe\n(n-l) combinations oftheindices iandj, isanintegral-invariantofanyHamiltonian systeminwhich(qi,q2,...,qntPi,p^, ...,pn) arethevariables. (Poincare.) x]their Integral-Invariants287 3.Intheproblemdefined bytheequations dqr_dff^ dpr__dl[ dt~3p rdt~ dqr where H=qlpl-q2p2-a?i2+bq&lt;?, f)a batshew that =constant ?i isanintegral ;andhence bythetheorem of121obtain thetworemaining integrals (q^z constant, (logqi=t+constant. 4.IfMisalastmultiplierofasystemofdifferential equations dx-i_d.r2__dxn_dx XlA2 AnX ofwhich theequation zi &gt;xnx=Constant isaknownintegral, and ifanaccent annexed toafunction of#l5#2,...,acn,xisused to indicate thatxnhasbeenreplacedinthefunction byitsvalues found from thisintegral, shew thatM/(cf/cx n)risalastmultiplierofthereduced system dx, dxn dxn_idx-=-,=...= ,=-7- (Jacobi.) 5.IfQI=Constant, 62=Constant, ...,6n=Constant areasetofintegrals ofthe equations dxdxdxz dxn shew that18(^,^ 2,...,^)X0(X 1,#2,---i*) isalastmultiplier. 6.Let(iii,u2,...,un)bendependent variables, and letTj,/2,...,/beasetof linear differential expressions denned bytheequations k=l If(vi,v z,...,vn)arefunctions oftsuch that isanexactdifferential, shew thatthefunctions(vj,v2,...,vn)satisfyasetofnlinear differential equations, which willbecalled thesystem adjointtothesystemoflinear differential equations 7r=(r=l, 2,...,). IfJ1 ^.denotes theexpression whereZisanygiven function of(ql,j2&gt; &gt;?&gt; ?i&gt; ?2&gt; &gt;qn,shew that thesystemof linear differential equations isadjointtoitself. Shew that theconverse ofthis latter theorem isalso true.(Hirsch.) CHAPTER XI THETRANSFORMATION-THEORY OFDYNAMICS 125. Hamilton sCharacteristic Function andContact-Transformations. Wehave seen* that theintegrationofadynamical systemwhich is solublebyquadraturescangenerallybeeffectedbytransformingitinto anotherdynamical systemwith fewerdegreesoffreedom. We shall in thepresent chapter investigatethegeneral theorywhich underlies this procedure, and, indeed, underlies thesolution ofalldynamical systems. Theoriginofthemethod istobefound inacelebrated memoir on optics,which waspresentedtotheRoyalIrishAcademy byHamilton in 1824f:theprinciplesthere introduced were afterwards transferredbytheir discoverer tothefield ofdynamics. Inorder tofollow Hamilton sthought, wemust refer totheconnexion betweendynamics andopticsaconnexion which isperhapslessobvious in ourdaythan inhis,when thecorpuscular theoryoflightwas stillwidely held. Ifarayoflighttraverses anoptically heterogeneousbutisotropic medium, therefractive index atanypoint (a?,y,z}being JJL,thepathof araymaybedeterminedbyFermat sPrinciple J,namelythat theintegral (x,y,z)ds hasastationaryvaluewhen theintegrationistakenalongtheactualray joiningtwogiventerminalpoints,ascomparedwithneighbouring paths joiningthem. Ifontheother handweconsider themotion ofafreeparticle ofunitmass inaconservative field offorce where itspotential energyis &lt;b(IK,y,z},and itsconstant ofenergyish,thepathoftheparticle maybe determinedbythePrincipleofLeast Action(100),which inthiscase asserts thattheintegral ff i* I hasastationaryvalue fortheactualtrajectoryascomparedwithneighbouring paths joiningthesame terminalpoints. Comparingthesetwostatements, we *Cf.Chapter III, 38-42. tTrans. E.Irish Acad. xv.(1828), p.69; xvi.(1830), pp. 4,93;xvn.(1837), p.1. JCf.rayHistory oftheTheories ofAether andElectricity, pp.9-10, 102-3. 125] TheTransformation-Theory ofDynamics 289 seethat thetrajectories oftheparticleinthedynamical problemarethe same asthepaths oftheraysintheoptical problem, providedasuitable correspondence *-&lt;*-* issetupbetween thepotential-energyfunction intheonecaseandthe refractive index intheother. Inthecorpuscular theoryoflight,thiswasregardedasfurnishingthe explanation oftheoptical phenomena,therayoflight beingconceived as aprocessionofrapidly-moving corpuscles. Butthestatement initself is truewhateverhypothesis regarding lightbeadopted:and therefore it suppliesameans ofconnecting dynamics with theundulatory hypothesis. This idea isthestarting-pointofHamilton stheory. When theundulatory hypothesisisadopted, wehave thechoice oftwo different methods ofdiscussingthepropagationoflight mathematically: the first istoconsiderrays,thesecond istoconsiderwave-fronts. The latter method, which wasintroducedbyHuygensin1690, maybethus explained. Consider awave-front, orlocus ofdisturbance inanoptical medium, as itexists atadefinite instantt,havingtheform ofasurface &lt;r.Each element ofthiswave-front mayberegardedasthesource ofasecondary wave, propagated outwards from it;sothat atasubsequentinstant t,the disturbanceoriginatinginanypoint (xty,z)oftheoriginal wave-front will extend over asurface. Toobtain theequationofthis surface, weobserve thatthetimetakenbylighttotravelthroughthemedium fromanarbitrary point (x,y,z}toanotherarbitrary point (x ,y,z)depends onlyonthesix quantities (x,y,z,x ,y,z):letitbedenotedbyV(x, y,z,x,y,z}.This function V(x, y,z,x,y,z}wascalledbyHamilton thecharacteristicfunction forthemedium inquestion. Adisturbance whichoriginatesatapoint (x,y,z}oftheoriginal wave-front attheinstant twill therefore atthe instant textend over the surface whoseequationinthecoordinates (x,y,z}is V(x,y,z, x,y,z)=t-t(1). Nowaccordingtotheprincipleofwave-propagationlaiddownby Huygens,thewave-front whichrepresents thewhole disturbance atthe instant tistheenvelopeofthesecondary waves which arise from the various elements oftheoriginalwave-front. Call thisnewwave-front 2; anddenote thedirection-cosines ofthenormal tothewave-front &lt;rat (x,y,z)by (I,m,n),andthedirection-cosines ofthenormal tothewave- front2atthecorresponding point* (x,y,z}by(V,m,n):these arethe *Thepoint (x ,y,z)issaid tocorrespond to(x,y,z)ifthesecondary wavepropagated from(x,y,z}touches theenvelope Sat(x ,y,z). W.D.19 290 TheTransformation-Theory ofDynamics [OH.xi direction-cosines oftheraysat(x,y,z)and(x,y,z)respectively,since in anisotropic medium therayisnormal tothewave-front*. Then since 2istheenvelopeofthesurfaces Vcorrespondingtopointsona,the equationdV , idV, dV, -=-doc+-^-dy+-^-dz=Q dx dydz must besatisfied byallthose values ofthe ratios dx :dy:dzwhich correspondtodirections inthetangent-planetoa-,i.e.whichsatisfythe relation ldx+mdy+ndz=0. Hence wehave 1^?=1LF=18F (2) Idxmdyndz Moreover, since(I ,m,n)arethedirection-cosines ofthenormal tothe surfaceFatthepoint (x,y,z),wehave = Idxmdyndz Now arayoflightwhichpasses throughthepoint (x,y,z)inthedirection (I,m,n)attime tpasses throughthepoint (x,y,/)inthedirection(I1 ,m,n) attime t :andequations (1), (2), (3),togetherwith theequation p+m/a+n/2=l................................. (4), aresixequations,fromwhich wecandetermine thesixquantities (x,y,z, I,m,n)interms of(x,y,z, I,m,n).Thusbytheseequationsthebehaviour ofraysoflightinthemedium iscompletely specifiedintermsofthesingle function V(x, y,z,x,y,z}.Itwillbeobserved thattheyarenotdifferential equations,butthatthey give directly,intheintegrated form, thechangesin anysystemofraysafter afinite interval ofpropagation throughthemedium. Itisevident therefore that allproblemsinoptics dependonthedeter mination ofHamilton scharacteristic function V(x,y,z,x,y,z}forthe opticalmedium orsystemofmedia throughwhich therays pass. From thepointofview ofPure Mathematics, weregardthechangefrom thesetofvariables (x,y,z, I,m,n)tothesetofvariables (x,y,z,I,m,n), or(toexpressitgeometrically)from thesurfaces&lt;rtothesurfaces 2,as atransformation.Thefunction Visthus toberegardedasdetermininga transformation ofspacewhich changes anysurface&lt;rintoanew surface 2. Itisevident that iftwosurfaces aand &lt;/touch atapoint,thecorresponding transformed surfaces 2and2alsotouch atthecorresponding point:onthis account thetransformation hasbeen called byS.Lieacontact-transformation. Thus anyfunction V(x, y,z,x,y,z)definesacontact-transformation,which *For simplicitywearesupposingthat themedium, though optically heterogeneous,is isotropic.Hamilton considered also themore generalcase ofacrystalline medium. 125] TheTransformation-Theory ofDynamics 291 transforms anywave-front&lt;rinto thewave-front 2which isderived from a-bypropagation throughthemedium intheintervaloftime t t. Asimple exampleofacontact-transformation isthewell-knowngeometrical trans formation known asreciprocation.Inorder tofindthereciprocalofanygivensurface o&gt; withrespecttoagiven surface, wecorrelate toevery point (#,y,z)on &lt;raplane, namely thepolar planeof(x,y\z)withrespecttothequadric. When thepoint (x,y,z)takes all possible positions onthesurfacea,theplane envelopes asurface2,which isthereciprocal of a-.Thetransformation from o-to2isevidentlyacontact-transformation. Inthis caseHamilton sfunction Vislinear withrespectto(x,y,z)and alsowithrespectto (r z}v*1 jyiz) ProceedingnowwithHamilton sproblem, equations (2)and(3)maybe written =Kl =*m = dV 8F 8F r7=M,^, \1H, -^r-f=Ml .ox oyoz where Kand\arequantitiesnotasyetdetermined. Theycanhowever be readilyfound. Fortheequations maybewritten dV=K(ldx+mdy+ndz)+\(ldx+mdy+ndz) (5). Nowbyproceedingasmall distance dsalongtherayat(x,y,z},we increase Vbythetimewhichlighttakes totravelalongds .But ifthe units aresochosen that thevelocityoflightinfreeaether isunity,then thevelocityoflightinthemedium at(x,y,z)is1/X,wherefjfdenotes therefractive index atthispoint. Thus thetimetakenbylighttodescribe dsis[ids ,orp!(I2+m2+n-}ds,or//(Idx+mdy+ndz).Comparing thiswithequation (5),weseethat A.= //,.SimilarlyKJJL,where p denotes therefractive index at(x,y,z}.Thus Hamilton sgeneral formula becomes dV=p(Idx+mdy+ndz) /j,(Idx+mdy+ndz). Ifwewrite fjd= ,/Jim= 77, fin= , yt/,7= , fjim=v,/j!n=" , thistakes theform Thequantities (f, 77, ),(,?? , )were calledbyHamilton thecomponents ofnormal slownessofpropagationofthewave at(x,y,z)and(x,y,z) respectively. Consider nowtheparticularcase inwhich theinterval oftime(t- 1) between thetwopositionsa-and2ofthesame wave-front isverysmall : 192 292 TheTransformation-Theory ofDynamics [OH.xi denote itbyA. Inthis case thecontact-transformation issaid tobe infinitesimal. Write =-MtAf, 7/=77+v&t, =+wA, ...............(7). V=WA*j Thenequation (6)becomes dW.At =(J;+u&t)(dx+da .A)+0?+v&t)(dy+dp.A) +(f+wA) (ek+dy.A)- &lt;&c-vjdy- %dz =u&tdx+v&tdy+w&tdz +%dda.+rjAtd/3+%Atd&lt;y, or dW=udx+vdy+wdz+%da.+ qd/3+%dy, or d(go.+?7/3+7W)=otdj;+/3dtj+yd udx vdywdz. Thus ifwedenote thefunction fa+rj/3+7TFbyiT,andsupposeH expressedasafunction ofx,y,z,g,77,f,wehave dH=adt; +ftdrj+yd^udx vdywdz ...............(8). d dxNowevidently,from(7),inthelimitubecomes ~ ,abecomes -y, ,etc. dt dt Thuswehave dx ,.,dy 7dz-..,d -.dn ,dt -.- sotherates ofincrease ofthesixvariables(x,y,z,%,77, )aregiven bythe equations dx_dHdy_dHdz_ d_H dt d dtdrj dt d /m (Ph dt dx dtdydt dz and this isaHamiltoniansystem ofequations,such asoccurs indynamics. Ourinvestigationshews that itmayberegardedasrepresenting anin finitesimal contact-transformation,that istosay,themotionofawave-front fromonepositiontoaposition indefinitelynear it.Theintegralsofthis Hamiltonian systemaretheequations (1), (2),(3),(4)above :they represent afinite contact- transformation, that istosay,themotion ofawave-front fromonepositiontothepositionwhich itacquiresafter afinite interval of time. Thusweseehowbyusingtheideasoftheundulatory theory oflight, Hamilton wasable toobtain anintegrated form forthedifferential equations ofdynamics, dependingonasingle unknownfunction. 126. Contact-transformationsinspace ofanynumberofdimensions. The restofthepresent chapterwillbeconcerned with theapplicationof Hamilton sideas, described inthepreceding article, tothegeneralcaseofa dynamical systemwithanynumber ofdegreesoffreedom, andtheconnexion 125,126]TheTransformation-Theory ofDynamics 293 oftheresults with certain theorems due toLagrange, Poisson, Pfaff, and Jacobi. We shall first define acontact-transformation inw-dimensionalspace, usingforthispurposeageneralisationofequation (6)ofthe last article. Let(q1}q2,...,qn,p^p.2,...,p n)beasetof2/ivariables, and let be2/iother variables which aredefined interms ofthemby2nequations. Iftheequations connectingthetwo sets ofvariables aresuch that the differential form PidQj+P.2dQ 2+...+PndQn-p^-p2dq2-...-pndqn is,whenexpressedinterms of(qltq2,...,qn,pi, PZ,,pn)andtheir differ entials, theperfectdifferential ofafunction of(q1}q2,...,qn,pl,p2,...,pn), then thechangefrom thesetofvariables(ql,q.2,...,qn,p1}p2,...,pn)tothe other set(Q1}Q2,...,Qn,P1?P2,...,Pn)iscalled acontact-transformation. Itmaybeobserved that this isdifferent informfrom thedefinition which ismost convenient when contact-transformations arestudied withaview totheirapplicationsin geometry andinthetheoryofpartialdifferentialequations:thelatter definition maybe stated thus :acontact-transformation isatransformation from asetof(2w+l)variables (?i&gt; ?2, -,qn,Pi,Pn, -,Pn, z)toanother set(#l5@2,...,Qn,Pl5P2,...,Pn,Z},for which theequation dZ-PdQ l-P2dQ2-...-PndQn=p(dz-p 1dql-p2dq2-...-pndqn) issatisfied, wherepdenotes some function of(q1,q2,...,qn,pltpz,...,pn,z). Ifthenvariables (QlyQ2,...,Qn)arefunctions of(qltq2,...,qn)only, thecontact-transformation from thevariables(q1}q2,...,qn,pl,...,pn)tothe variables (Q 1,Q2,...,Qn,P1}...,Pn)iscalled anextendedpoint-transformation, theequations which connect(q1}q2,....qn)with(Ql}Q2,...,Qn}beinginthis case said todefine apoint-transformation. From thedefinition itisclear thattheresult ofperforming twocontact- transformations insuccession istoobtain achangeofvariables which isitself acontact-transformation. Itisalsoevident that ifthetransformation from (?!, &lt;?2&gt;&gt;qn,pi, &gt;pn)to(Qi,Q 2,...,Q n,PI, ..,Pn)isacontact-trans formation, then thetransformation from(QnQ2,...,Qn,P1}P2,...,Pn)to (qi,q z,,qn,pi,p2, --^Pn)isalsoacontact-transformation; this isgenerally expressed bysayingthat theinverseofacontact-transformationisacontact- transformation. This, together with theforegoing,shews that contact-trans formations possessthegroup-property. Example1.Shew thatthetransformation defined bytheequations isacontact-transformation. 294 TheTransformation-Theory ofDynamics [CH.xi Inthiscasewehave PdQ-pdq=(2qfisinp{(2q)~^cospdq- (2qfisiupdp} -pdq =d(qsinpcospqp), which isaperfectdifferential. Example2.Shew thatthetransformation 1 isacontact-transformation. Example3.Shew thatthetransformation (P=2(1+2^cosp)q^sinp, isacontact-transformation. We shallnow obtain theexplicit analytical expressionofacontact- transformation. Letthetransformation from variables(q1}q2,...,qn,p1,...,pn)tovariables (Qi)Qz, ",Qn,P\, ,Pn}beacontact-transformation, sothat (PrdQr-p rdqr)= r=l wheredW isacompletedifferential. From theequations which define (QltQ2,...,Qn,P1}...,Pn)interms of (qi, q*,,qn,pi,.,pn)itmaybe possibletoeliminate(P 1,P2,...,Pn,p1,...,pn) completely,soastoobtain oneormore relations between thevariables (QnQa, &gt;Qn, qi,,qn), letthenumber ofsuch relations bek,and letthem bedenotedby flrtei, 2,,-.,qn,Qn ....Q)=(r=l, 2,...,A)...(A). Themeaningofthese relations maybeillustrated byrevertingtothegeometrical theoryofcontact-transformations inordinarythree-dimensionalspace, when there are three cases toconsider : (a)Theremaybeonlyasinglerelation between thenewandoldcoordinates, say Q(x,y, s,x,y,/)=0. When(#,y,2)aregiven,thisequation, regardedasthelocus ofapoint (a/,y,z ~), representsasurface;sothateachpoint (#,y,z)istransformed intoasurface, which we maycallanQ-surface :andanyarbitrarysurface &lt;ristransformed intoasurface 2which istheenvelopeoftheQ-surfaces correspondingtotheindividual pointsof er.This isthe general case,and istheonlyoneweconsidered in 125. (/3)Theremaybetworelations ofthiskind, say Q:(as,y,2,x,y , z")=0,Q2 ,y,*,d,/,*)=0. If(x,y,z]aregiven,these twoequationsin(a/,y,z)representacurve :soeachpoint (x,y,z)istransformed intoacurve, which wemaycalla.fiT-curve :andanyarbitrary 12(&gt;] TheTransformation-Theory ofDynamics295 surface cristransformed intoasurface 2which istheenvelopeofthe.ff-curves corre spondingtotheindividual pointsof cr. (y)Theremaybethree relations ofthiskind, say flj(a?,y,z,x,y,z}=0,O2(x,y,z,x,y,z}=0,Q3(.r,y,z,x,y,2)=0, inwhich caseeachpoint (x,y,z)istransformed intoapoint (x ,y,z},andanyarbitrary surface cristransformed intoasurface 2which isthelocus ofthepoints correspondingto theindividualpointsofcr. Since thevariations(dq lfdqz,...,dqn,dQ lt...,dQn)intheequation 2(PrdQr-prdqr)=dW r=l areconditionedonlybytherelations (r=l, 2,...,&), wemust have dW . kan, where (X^Xg, ...,X t)areundetermined multipliersandwhereWisafunction .of(ql}q2,...,qn,Q,,Q2,...,Qn).Theequations (A)and(B)are(2n+k) equationstodetermine the(2/i+k)quantities (Qi,}Qn, PI&gt; Pn&gt; Xi, ..., Xfc) interms of(qlt...,qn,pl,...,pn).Theseequations may thereforeberegarded asexplicitly formulatingthecontact-transformation,intermsofthefunctions (W,ni(n.2,...,n&)which characterise thetransformation. Conversely,if(W,nifH2,...,n4)areany(k+1)functions ofthevariables (&lt;?i,?2,&gt;qn,Qi, -..,Qn),where k^n,and if (Qi,Q2, -,Qn,Pi, ..-,Pn,X1;...,Xfc) aredenned interms of(ql,q2,...,qn,p1}...,pn)bytheequations* 9TF an, 90 -+...+X fc an, an,, ?=-^-Xi^--...X-r- Vr-L &gt;Z n )&gt; dqr dqr dqr *These equations were firstgiveninJacobi sVorlesungeniiberDijnamik (1866), p.470,where their utilityinthetransformation ofpartialdifferential equationsofthe first order(towhich dynamical problems canbereduced) wasindicated. Their placeinthetheory ofcontact-trans- formations waspointed outbyLie. 296 TheTransformation- Theory ofDynamics [CH.xi then thetransformation from (qltqz,...,qn,Pi,...,p w)to(Q1}Q2,...,Qn,P1}...,Pn)isacontact-transformation ;fortheexpression M 2(PrdQr-p rdqr) r=l becomes, invirtue oftheseequations, dW,andsoisaperfectdifferential. Example.If Q=(Zq)k~ cosp,P dW ...P=WP where TF=Q (&lt;2qk-F$2)*-qarccos sothatthetransformation from(^p}to($,P)isacontact-transformation. 127. Thebilinear covariantofageneral differential form. Now let(x1}x2,...,#)beanysetofnvariables, andconsider adifferential form Xldx1+X2dx2+...+Xndasn, where(X J}X2,...,Xn)denote anyfunctions of(x\,oc z,...,#);aform ofthis kind iscalled aPfaffsexpression*inthevariables(x1}xz,...,xn).Letthis expressionbedenoted by0&,andwrite where 8isthesymbolofanindependentsetofincrements. Thenwehave 8dd d6$=&(X 1dxl+X2doc2+...4-Xndxn)d(X 18^+X28x2+...+XnSxn} =8X1dxl+...+8Xndxn+Xl8dxl+...+Xn8dxn dXl8x1...dXn8xnXld8x1...Xnd8xn. Using therelations 8dx r=d8x r&gt;which exist since thevariations dand8 areindependent,andreplacing dXr,8Xrby dXr, dXr,dXr~ dXr~ . -=dxl+...+ dxn, -^-6x1+...+-^-oxnrespectively, OX-^ (]Xfi OX- OXji nn wehave 8ddd0s= 2 i= whereydenotes thequantity dX{/dxj dXj/dxj. Let(y-i, 2/2,...,yn)beanew setofvariables derived from(xltx.2,...,xn} bysome transformation;letthedifferential formwhenexpressedinterms of these variables be Fjefyi+Y2dy2+...+Yndyn, *Pfaff scelebrated memoir onthese expressions waspresentedtotheBerlin Academyin1815 : Abhandl. Akad. derWiss. 1814-15, p.76. 126-128] TheTransformation- Theory ofDynamics 297 and letthequantity 9P&lt;/&lt;?$- dJ}/3y&lt;bedenotedbyby.Then since the expression 86dddshasobviouslythesame value whatever bethevariables interms ofwhich itisexpressed, wehave nn 2SctijdtCiSxj=2Sbijdyibyj. i=lj=l 1=1j=i Theexpression taydxi&K)is,onaccount ofthisequation,called the bilinear covariant oftheform2Xrdxr. 128. Theconditionsforacontact-transformation expressed bymeans of thebilinear covariant. (Qi, Qz, ,Qn,PI, .-.,Pn)bevariables connected with(qltq2,..., n &lt;ln,pi, -.ipn) byacontacttransformation, sothatXPrdQrdiffers from r=i byanexact differential. r=l Itisclearfrom thelastarticle thatthebilinear covariant ofadifferential form isnotaffectedbytheaddition ofanexact differential totheform, since itdepends onlyonthequantities dXifoxj-bX}fixi,which are allzerowhen theform isanexact differential: andwehaveshewn that the bilinear covariant ofaform istransformedbyanytransformation into thebilinear covariant ofthetransformed form. Itfollows thatthebilinear covariants of n n theforms 2PrdQrand2prdqrareequal,i.e.that r=l r=l ^(8PrdQ,-dP r8Qr)=2(8p rdqr-dq r8pr); sothatifthetransformation from (qi,q^, ...,q n,pi,...,^)to(Ql}Qz,...,Qn,P 1,....Pn) isacontact-transformation, theexpression n 2(8p rdqrdqr8pr)r=l isinvariant under thetransformation. Example. Forthetransformation denned bytheequations Q=(2q)*k~4cos wehave rfP-(2g)~ijt*sin ^cospdjzT 298 TheTransformation-Theory ofDynamics [OH.xi Bymultiplication wehave dP8Q-8PdQ=-sin*p(d&lt;28p-8qdp)+cos2p(dp8g-8pdq) =dp8q 8pdq, andconsequentlythetransformation isacontact-transformation. 129. Theconditionsforacontact-transformationintermsofLagranges bracket-expressions. Weshallnowgiveanother form totheconditions thatatransformation from variables(ql}q2,...,qn,plt...,pn)tovariables (QltQ2,...,Qn,P1,...,Pn) maybeacontact-transformation. If(ql,q^,...,q n,pi,...,p n)areanyfunctions oftwovariables(u,v)(and possiblyofanynumber ofothervariables),theexpression 3(dqrtyr_dprdqA r=i\du dv dudv) iscalled aLagranges bracket-expression*,and isusuallydenoted bythe symbol [u,v]. Ifnow(q1}q%, ...,qn,pl}...,p n)areanyfunctions of2nvariables (Qi, Qz, ,Qn,PI, -,Pn),then intheexpression 2(dp rqr-Bprdqr) wecanreplace dprbyr=l andsimilarlyfortheotherquantities;wethus obtain, oncollecting terms, n 2(dpr&qr-Bprdqr)=S[uk,u{](diii8u k-Suidu k), r=l k,I where thesummation ontheright-handside istaken over allpairsof variables (uk,Ui)intheset(Q1}Qz,...,Qn,PI, ,Pn)-- But ifthetransformation from thevariables (ql}q2, ,qn,PI,&gt;Pn) tothevariables (&,Q 2&gt;...,Q n&gt;P1}...,Pn)isacontact-transformation, we have I(dp^r-Bprdq,)=I(dP rSQr-8PrdQr), r=l r=l andthisholds foralltypesofvariation 8anddofthequantities;comparing with theaboveequation, wehave therefore i,P t]=0,[Qi,&]=(i,k=I,2,...,), [Q*,P*]=0 (i,k=l,2, ..., ;i$k), [Qi,PJ=1 (t-1,2, ...,&gt; Lagrange, Mem. deVInstitut deFrance, ann^e 1808: reprinted Oeuvres, vi.p.713. 128-130] TheTransformation-Theory ofDynamics %299 Thesemayberegardedaspartial differential equationswhich must be satisfied by(qltq2,...,qn,p1}...,p n),considered asfunctions of (Qi,Q,...,Q, PI,...,P n) inorder that thetransformation fromonesetofvariables totheothermaybe acontact-transformation. Theseequations representinanexplicitform the conditionsimpliedintheinvariance oftheexpression r=\(dprSqr8prdqr). 130. Poisson sbracket-expressions. We shall next introduce another class ofbracket-expressionswhich are intimately connected with those ofLagrange. Ifuand vareanytwo functions ofasetofvariables(q1}q2,...,qn, Pi,,Pn),theexpression I/8wdv__dudv_\ r=i\dq rdprdprdqj iscalled thePoisson sbracket-expression*ofthefunctions uandv,and is denotedbythesymbol (u,v). Suppose now that(u1}u2,...,um)are2nindependentfunctions ofthe variables(ql}q2,...,gn,plt...,pn),sothatconversely (q1}q2,...,qn,plf...,pn) arefunctions of(u1}u.2,...,u2n).There willevidentlybesome connexion between thePoisson-brackets(ur,us)andtheLagrange-brackets [ur,us]: thisconnexion weshallnowinvestigate. Wehave 2(,,,)[,..]- 22I(2?t*-lSt=\ t=\i=ij=i \oqi dpi dpid Nowmultiplyouttheright-hand side,rememberingthat o t=ioqiout t=lpi areeach zero ifi$jandunityifi=j;andthat ^dutdPj.^dutd2r^andS^t=ioqidut t=ldpid areeach zero;theequation becomes %?/ \r -i2(%,w,)[nt&gt;wjt=i andconsequently 2n 2(wt,ur)[ut,Ug]=when r^s,t=i 2n while 2(wt,wr)[w&lt;tM,.]=1. &lt;=i, *Poisson, Journald$VEcolepoly tech. vin.(Cahier 15),(1809), p.266. 300 TheTransformation Theory ofDynamics [CH.XT Butthese aretheconditions which must besatisfied inorder thatthe twodeterminants !,M,]...[UltU2n] [u2,u^[u2, ,uzn\and (U2,U2)...(U2n ,U2) [U2n ,U-^J \_U2n ,U2n\ (^Ui,Uvn) \U2n , maybereciprocal,i.e.thatanyelement intheoneshould beequaltothe minor ofthecorrespondingelement intheother, dividedbythis latter determinant;theproductofthetwodeterminantsbeing unity;andthus the connexion between theLagrange-brackets and thePoisson-brackets isexpressed bythefact that thedeterminants formed fromthem arereciprocal. Example1.If/, &lt;,^areanythree functions of(jt,qz,...,qn,p1,...,pn\shew that Example2.IfF,$arefunctions of(/l5f2,...,fk),which inturn arefunctions of(?D &lt;?2,&gt;$n,Pi, -..,p),shew that where thesummation istaken over allcombinations fr,fg. 131. Theconditionsforacontact-transformation expressed bymeansof Poisson sbracket-expressions. Now let(QltQ2,...,Qn ,Plt...,Pn)denote 2nfunctions of2nvariables (^i) ^2&gt;&gt;c[n,p\, -,pn)&gt; weshall shew that theconditions which must be satisfiedinorder that thetransformation fromonesetofvariables totheother maybeacontact-transformation maybewritten intheform ( (Pi,Pj)=0,(Qi, Qj]=(,j=1,2,...,n), Forwehave seen in129thattheconditions foracontact-transformation areexpressed bytheequations f [Pi,Pj]=0,[Q{,Qj]=(i,j=1,2,...,n), Hence therelations =0 130-132] TheTransformation-Theory ofDynamics 301 ofthelastarticle become while therelations 2n 2(ut,ur}[u t,nr]= thetheorem isthus established. Example1.If(QltQ2,...,Qn,Pl}...,Pn)areconnected with(gi,q2,...,q n,Plt ...,Pn) byacontact-transformation, shew that sothatthePoisson-brackets ofanytwofunctions and^withrespecttothetwosetsof variables areequal.* Example2.If(&,..., Qn]aregiven functions of(?1 ,?2&gt; ..., ?(l ,ft ,...,p n\and satisfy thepartial differentialequations (Qr,^8)=(r,=l, 2,...,), shew that%other functions (PltP2,...,Pn)canbefound such thatthetransformation from(?] ,?2 ,...,?n ,Pl ,;;.fpn)to(^ 2,...,Qn,plt...,pw)isacontact-trans formation./Lje\ 132. TAesub-groups ofMathieutransformations andextendedpoint- Iransformations. Ifwithin agroupoftransformations there exists asetoftransformations such that theresult ofperforminginsuccession twotransformations ofthe setisalways equivalent toatransformation which alsobelongstotheset,this setoftransformations issaid toform asub-groupofthegroup.Asub-groupofthegeneral groupofcontact-transformations isevidently constitutedbythose transformations forwhich theequation n n SPrdQr=2prdqr r=l r=l issatisfied. These transformations havebeen studiedbyMathieu*. They areessentially thesame asthetransformations called"homogeneous contact- transformations in(ft, g-2,...,?n ,Pl,...,p n)"byLie. Inthis case,weseefrom 126that(&,Q9,...,Qn,plf...,pn)aretobe obtainedbyeliminating (\l}\2,...,\k)from the(2n+k)equations Journal deMath. xix.(1874), p.265. 302 TheTransformation-Theory ofDynamics [CH.xi From theform oftheseequationsitisevident that if(pj}p2,...,p n)are each multiplied byanyquantity //.,the effect istomultiply (P1}P2,...,Pn) eachby yu,;andtherefore (Pj,P2,...,P?l)must behomogeneousofthe first degree (thoughnotnecessarily integral)in(p1}pz,...,p n). Asub-groupwithin thegroupofMathieu transformations isconstituted bythose transformations forAvhich (P :,P2,...,Pn)arenotonlyhomogeneous ofthe firstdegreein(pl}p2,...,p n)butalsointegral,i.e.linear, inthem; so thatwehaveequationsoftheform nPr=2pkfrk(q1}q2,...,qn) (r=l,2,...,n}. k=\ Substitutingintheequation n n SPrdQr2rdgv=0, r=l r=l* andequatingtozerothecoefficientofpk,wehave n dqk (fcl, 2,...,n), r=l so (&lt;?!,qz,...,qn)arefunctions of(Q1}Q2,...,Qn)only,and frk=dqk/dQr (r,k=l,2, ...,n). Itfollows thattransformations ofthiskind areobtainedbyassigning narbitraryrelations connectingthevariables(ql}q2,...,qn)with thevariables (Qi,Q2, ,Qn),andthendetermining (PlfP2,...,P n)fromtheequations Pr=IPtj2*(r=l, 2,...,n). k=lV^r These transformations areextendedpoint-transformations (126). n n Example.If 2PrdQr=2prdqr, r=l r=l -O0rndPr Shewtha,tP*S^= lf*^-P " 133. Infinitesimal contact-transformations. We shallnow consider transformations inwhich thenew variables (Qi, $2,&gt;Qn,PI, -&gt;Pn)differ from theoriginalvariables(q1}q2,...qn, pl}...,p n)byquantitieswhich areinfinitesimal. Letthese differences be denoted by(&q ltAg 2,...,A#n&gt;&plt...,Apn),where andA^isanarbitraryinfinitesimal constant;sothat Qr=qr+&qr=qr+ &lt;f&gt;r 132,133]TheTransformation-Theory ofDynamics 303 andthetransformation isspecified bythefunctions Nowsupposethatthetransformation isacontact-transformation. Then wehave r=l(PrdQr-prdqr)= whereWissome function of(ql}q.2,...,qn,plt....pn);or 2 &gt;=! or Itisevident thatthefunctionWmust contain Aasafactor :writingW=U&t,where ?7issome function of(^:,qz,...,qn,plt...,pn),theequation becomes Hence wehave M / r=l andthereforer=l =-dK(q 1}q2,...,q n,ply...,p n)say, Thus /te mos&lt;general infinitesimal contact-transformationisdefined bythe equations whereKisanarbitrary function of(q1}qz,...,qn,plt...,p n),andAisan arbitrary infinitesimal quantity independent of(q1}q.2,...,qn,p1}...,pw). Theincrement inanyfunctionf(q ltqz,...,qn,p1}....,pn)when itsargu ments(q1}qz,...,qn,p1}...,p n}aresubjectedtothistransformation is _ rdprdq or onthisaccount thePoisson-bracket(/,K)issaid tobethesymbol ofthe mostgeneralinfinitesimal transformation oftheinfinitegroupwhich consists ofallcontact-transformations ofthe 2??,variables(q1}q.2,...,qn,plt...,p n). 304 TheTransformation-Theory ofDynamics [CH.xi 134. Theresulting newmewofdynamics. Thetheorem established inthe last article enables ustoextend toall conservative holonomicdynamical systems,whatever bethenumber ofdegrees offreedom, theconceptionwhich wasformulated attheendof125 for certainsimple systems.Forthemotion isexpressed (109)byequationsof thetype dqrdff dprdH j-r__**_ [fB12 n) dtdpr dtdqr andfrom thelast article itfollows thatwecaninterprettheseequationsas implyingthat thetransformation from thevalues ofthevariables attime t totheir values attime t+dtisaninfinitesimal contact-transformation. The whole courseofadynamical system canthus beregardedasthegradual self- unfolding ofacontact-transformation. This result isreally ageneralisation ofthestatement that thepaths oftheraysinapencil oflightcanbespecified bythegradual propagation ofawave-front. Taken inconjunctionwith thegroup-propertyofcontact-transformations, itisthefoundation ofthe transformation-theoryofdynamical systems. From this itisevident that if(q1}q2,...,qn,plf...,p n)arethevariables inadynamical system,and(al&gt;ot2,...,a.n,{31}..., /3n)aretheirrespective values atsome selectedepocht=tQ,theequationswhichexpress (ql,qz,...,qn, PI,&gt;Pn)interms of(alt 2,...,an,@1}..., j3n,t)(andwhich constitute the solution ofthedifferentialequationsofmotion) expressacontact-transforma tionfrom (!,or2,...,ctn,&, ..., /3n)to(q1}q3,...,qn,plf...,p n);inthis tis regarded merelyasaparameter occurringintheequationswhich define the transformation. 135. Helmholtz sreciprocaltheorem. Since thevalues ofthevariables(ql}q.2,...,qn,plt...,pn)ofadynamical systemattime tarederivable byacontact-transformation from their values (!,cr2)...,an,fil;...,fin)attime t,wehave(128) where thesymbols Aand 8refer toincrements arrived atbypassagesfrom agivenorbit totwodifferentadjacentorbitsrespectively. Nowsupposethat 8refers totheincrements obtained inpassingtothat orbitwhich isdefinedbythevalues (!, 2,..., n,&,/32,...,&._!, r+S@ r,ftr+i,&gt;fin) attime t;and letArefer totheincrement obtained inpassingtothat orbit which isdefinedbythevalues (ql}qt &gt;...,q n,pi, -,PS-I,PS+&Ps&gt; PS+I,,Pn) 134-136] TheTransformation- Theory ofDynamics 305 attime,;then theaboveequation becomes &ps&qs=-8/3,-ky&gt;; sotheincrement inqsdue toanincrement in/3r(whenal,cr2,...,an, /31;..., /3/--1, /3r+1,..., finarenotvaried)isequaltotheincrement (with sign reversed)inarcorrespondingtoanincrement in.ps(when qi,q2,...,q n,pi, ., JOg-i, PS+-L,,pnarenotvaried) equaltotheprevious increment in/3r. This result can formany systemsbephysically interpreted,aswas observedbyHelmholfcz*; forasmallimpulse appliedtoasystemcanbe conveniently measured bytheresulting changeinoneofthemomenta (p-i,...,pn),andthechangeinarduetoachangeinpgcanberealised inthe reversed motion,i.e.themotion which starts fromsomegiven position with each ofthevelocitiescorrespondingtothatposition changedinsign,sothat thesubsequent historyofthesystemisthesame asitsprevious history, but performedinreverse order. Wecantherefore state thetheorembroadly thus :thechange producedinanyintervalbyasmall initialimpulse ofany typeinthecoordinateofanyother (orofthesame) type,inthedirect motion, isequaltothechange producedinthesame intervalofthereversed motion in thecoordinateofthefirst type byanequalsmall initialimpulse ofthe secondtype\. Example.Inelliptic motion under acentre offorce inthecentre, ifasmallvelocity 8vinthedirection ofthenormal becommunicated totheparticle asitispassing through either extremityofthemajor axis,shew that thetangential deviationproduced after aquarter-periodisp. "dtf,where/*istheconstant offorce. Shew alsothatatangential velocity 8v,communicated attheextremityoftheminoraxis, produces after aquarter- periodanequalnormal deviationp.~2dv.(Lamb.) 136. Jacobi stheorem onthetransformation ofagiven dynamical system intoanotherdynamical system. Itappearsfrom 116 that ifaHamiltoniansystemofdifferential equations dqrdH dprdH -nr=^, =-3- (r=l, 2,...,w)dtdpr dtdqr istransformed bychangeofvariables, thesystemofdifferentialequations so obtained will stillhave theHamiltonian form dQr_dK_ dPr__dK dtdPr dt~ dQr:"** providedthenew variables (Ql}Q2,...,Qn,Ply...,Pn)aresuch that isanintegral-invariant (relative orabsolute) oftheoriginal system. *Journal fiirMath. c.(1886). tCf.Lamb, Proc. Loud. Math. Soc. xix.(1898), p.144. w.D. 20 306 TheTransformation- Theory ofDynamics [on.xi Atransformation ofthiskindis,ingeneral, specialtotheproblem considered, i.e. ittransforms thegivenHamiltonian systemintoanother Hamiltoniansystem,but itwillnotnecessarilytransform anyotherarbitrarily chosen Hamiltoniansysteminto aHamiltoniansystem. Amongthese transformations however areincluded transformations which have thepro pertyofconservingtheHamiltonian form ofanydynamical systemtowhich theymaybeapplied:thesemaybeobtained inthefollowing way. Wehave seen(115) that isarelativeintegral-invariantofanyHamiltoniansystem.Let(Ql}Q2,...,Qn, PL ...,Pn)beasetof2nvariables obtained from(qltq2,...,qn,p1}...,p n) byacontact-transformation, sothat PrdQr- prdqr= r=\ r=l wheredWdenotes anexact differential. Theequationswhich define the transformation mayinvolve thetime, sothat(Q1}Q2,...,Qn,P1}...,Pn)are functions of(ql,qz, .,qn,pi,&gt;Pn,5butinthevariation denoted byd inthisequationthetime isnotsupposedtobevaried :iftissupposedto vary,theequationbecomes n n 2PrdQr-2prdqr=dW+ Udt, r=l r=l whereUdenotes some function ofthevariables. Now thevariation denoted bySintheintegral-invariantisavariation from apointofoneorbit tothecontemporaneous pointofanadjacentorbit; iftherefore weregardthevariables asfunctions of(al,a2,...,a2n ,t),where (a1}a2,...,a2n)aretheconstants ofintegrationwhich occur inthesolution of theequationsofmotion, thevariation &isoneinwhich (a1}a2,...,a2n)are varied but tisnotvaried :wehaveconsequently,asaspecialcase ofthelast equation, IPrSQr- 2prBqr=8W, r=l r=l rni andtherefore 2Pr&Qr J}=! isarelativeintegral-invariant;sothetransformed systemofdifferential equations,inwhich (Q1(Qz,...,Qn,P1}...,Pn)aretaken asdependent variables, willhave theHamiltonian formandcanbewritten dQr_dK_dPI__dK dt~dP r dt~ dQr whereKissome function of(QltQ2,...,Qn,Pi, ...,Pn,t). 136,137]TheTransformation-Theory ofDynamics 307 Hence acontact-transformation ofthevariables(qltq2,...,qn, T&gt;\-&gt;&gt;Pn} ofanydynamical systemconserves theHamiltotiian form oftheequations of thesystem*.Inthecaseofanordinary "changeofvariables" inthedynamical system,inwhich (Q1}Q2,...,Qn)arefunctions of(qltq2,...,qn)only,the contact-transformation ismerelyanextendedpoint-transformation. Example. Shew thatthecontact-transformation defined bytheequations q=(2#)4 k~*cosP, p=2Q)%ktsinP, changesthesystem dq__3H_ dp=dff di~dpdtdg where H=\(p*+kzq*\ intothesystem dQ_dK dP__dK_~dt~dP "dt~ dQ where K=kQ. 137.Representation ofadynamical problem byadifferential form. Thereason fortheimportanceofcontact-transformations inconnexion withdynamical problemsismoreclearlyseenbytheintroduction ofacertain differential formwhich isinvariantivelyrelated totheproblem. Letanydifferential formwith (2n+1)independentvariables (X,x2,..., be wehave seen(127) that itsbilinear covariant 2+l2w+l 2SddSaS where atjdenotes thequantity (dXi/dxj dXj/dxi),isinvariantivelyrelated to theform. Ifweequatetozerothecoefficients ofSx1}Sx2,...,Bx^^, we obtain thesystemof(2?i+l) equations Since thedeterminant ofthequantitiesafjisskew-symmetric andofodd order, itiszero, and theseequationsarethereforemutually compatible. Theyareknown asthefirstPfaffssystem ofequations correspondingtothe differential form2Xrdxr,andfrom themode oftheir formation arein- r=l variantivelyconnected with it;that istosay,ifanychangeofvariables is made, thenewvariables (yl}y2,...,ym+1 )being givenfunctions of(ar,,x2,..., #2n+i)&gt; and ifthedifferential formbechanged bythistransformation to 2Yrdyr, r=l *Thisimportant theorem was firstgiven byJacobi, Comptes Rendus, v.(1837), p.61. 202 308 TheTransformation- Theory ofDynamics [OH.xi 2n+l and if2&flcfa/;=0, 2b^dyi=0, ..., 2bii2n+ldyi= 1=1 ^=1 t=i bethe first Pfaff ssystemderived from thedifferential form 2Fretyr, then thissystemisequivalenttothesystem 2n+l iidxi=0, 2UiZdxi=0,...,2;,2?i+ic^i=0. Consider nowthespecialdifferential form inthe(2n+ 1)variables(g1}q2,...,qn,plt...,pn,t),whereHisanyfunction of(ft,q2,&gt;qn&gt;P\&gt;&gt;Pn, t).Formingthecorresponding quantities a^,we findthatthe firstPfaff ssystemofdifferentialequationsofthis differential form is dpr ;rdt=(r=1,2,...,n\ dHUAdqr-^dt=() (r=I, 2,...,n), dpr ~^dit~ Ofthese thelastequationisaconsequenceoftheothers :andtherefore the systemofequationscanbewritten dqrdH dprdH . -* .*-^^ /iv*IV7? i* dtdpr dtdqr butthese aretheequationsofmotion ofadynamical systeminwhich the Hamiltonian function isH. Itfollows that thedynamical system whose HamiltoniahfunctionisHisinvariantivelyconnected with thedifferential form __ Pidqi+p2dq2+...+pndqnUdt, inasmuch astheequations ofmotion ofthedynamical system,interms ofany variables(x l,x2,...,x2n ,T)whatever, are thefirst Pfaffssystem ofthe differential form which isderived fromtheform Pidyi+ptdq t+...+pndqnHdt bythetransformation fromthevariables(qltq2,...,qn,p\,-,pn,t}tothe variables (xltx2,...,x^,, T). 137,138]TheTransformation- Theory ofDynamics 309 138. TheHamiltonianfunction ofthetransformed equations. Theresult ofthelastarticle furnishes anotherproofofthetheorem that theequationsofdynamics dqr_dHdp,. dH119 dtdpr dtdqr conserve theHamiltonian formunder allcontact-transformations of(ql,q.2,..., qn&gt;Pi&gt; &gt;Pn),andmoreover itenables ustofindtheHamiltonian function Kofthesystemthus obtained, ~df= dPr ~dT~~~dQ r(r=l,2,...,n). For letthecontact-transformation bedefinedbytheequations (r=12 k} anxan2 an* -i -,y~\ ~T~ A&gt; ,,.-. -T- ...TA,_.-.^7*J_,A,...,n), (j/j. U*%r dv/ &gt; an, an2 an, X;^ X2^.*.Xj-= (ral, 2.....n). u(/,. dgv a^,. d^ where (n^H2,...,n,,TF)areanyfunctions ofthevariables(q1}q From theseequationswehaveidentically 2p.rdq.r2PrdQr2I-dqr+^~rdQr}2Xs2(-5-^&lt;ferH r=l andhence(thesymbolddenotingavariation inwhich allthevariables, includingt,arechanged) o j^Dj/a ,u "Jj JTIT ,4Zprdqr=iJrrd(^r-f--_aca(r+z w w / aTF ^-27TT7j V" 7^7/^ /TTV ^^prdqrHdt=2PrdQrIH -_-2Xs __& Theperfectdifferential dWontheright-handsidecanbeneglected, since itdoes notaffect the first Pfaff ssystemofthedifferential form :and hence thecontact-transformation transformsthesi/stem ofequations da^=ciHdpr^_d_H12 tothesystem 7~/~ r\T\ j 7,^~"^^ ^s/-^ \A X. ^. .&gt;. /fr/.dtdPr dt dQr where K=H^2Xs-^,ot x=i dt Kbeing supposed expressedintermsof(QltQ2,...,Qn,P1}...,Pntt). 310 TheTransformation- Theory ofDynamics [OH.xi 139. Transformationsinwhich theindependentvariable ischanged. The result of137alsoenables ustodetermine those transformations of thewhole setof(2n+1)variables(ql}q2,...,qn,pl},..,pn,t)tonewvariables (Qi, Qz,&gt;Qn&gt;PI, &gt;Pn,T}bywhichanyHamiltoniansystem dqr_dH dpr_dH _~ . ~dt= dpr ~dt= ~dq^ istransformed intoasystemoftheHamiltonian form dQr_dKdPr_dK dT~dP r~dT~~dQ r For this isthesamethingasfindingthetransformations whichchangethe differential form Pidq-t+p 2dq.2+...+pndqn+hdt, where the variables(qlyq2,...,qn&gt;plt...,pn,t,A)areconnected bythe equation H(qi, qt,&gt;qn,Pi, "-,pn, t)+h=0, intothedifferential form P1dQ l-fP2dQ 2+ ...+PndQn+kdT+aperfect differential, where thevariables (Q1}Q2, ,Qn,PI,PZ, ,Pn,T,k).areconnectedby therelation Butanycontact-transformations ofthe(In-\-2)variables(q1}q2,...,qn,t, plt...,p n,h)tonewvariables (Ql}Q2,...,Qn ,T,PlfP3,...,Pn,k)willsatisfy thiscondition;when thetransformation hasbeenassigned,thefunctionKis obtained bysubstitutingintheequation q*,...,qn,pi,...,p n&gt;t+- thevalues of(qlyq2,...,qn&gt;t,p1}...,p n,h}asfunctions of(Qlt...,Qn,T, Pj, ...,Pn,k),andthensolvingthisequationfork,sothat ittakes theform K(Q l}Q2,...,Qn,P15...,P7l,T)+k=0; therequiredtransformations arethereby completelydetermined. 140.Newformulation oftheintegration-problem. Wehave seen(137) that ifanychangeofvariables ismade inthe dynamical system dqr=3H dpr==_dH dtdpr dt~ dqr thenew differentialequationswillbethe first Pfaff ssystemoftheform which isderived from Pidq!+p 2dq2+...+pndqnHdt bythetransformation. 139,140]TheTransformation- Theory ofDynamics 311 Supposingthatatransformation isfound, definedbyasetofequations which issuch thattheabove differential form,whenexpressedinterms of thenew variables, becomes P,dQ,+P2dQ.+...+PndQn-dT, where dT istheperfectdifferential ofsome function ofthe variables (Qi, Qz, ,Qn,PI, ,Pn,t);thecorrespondingfirst Pfaff ssystemof equationsis dQr=0,dPr=(r- 1,2,....n), andtheintegralsoftheseequationsare Qr=Constant, Pr=Constant(r=1,2,...,n) ; sotheequations qr=&lt;l&gt;r(Qi&gt;Q...,Q n,P 1,...,Pn,t}}\ l constitute thesolutionofthedynamical system, when thequantities (Q1;Q2,..., Qn,PI, ,Pn)areregarded as2narbitraryconstantsofintegration. Theintegration-problemisthusreduced tothedeterminationofatrans formation forwhich thelasttermofthedifferential form becomes aperfect differential. MISCELLANEOUS EXAMPLES. 1.Shew thatthetransformation denned bytheequations !=arctan--arctan- ,P2=\arctan %L isacontact-transformation, andthat itreduces thedynamical system whose Hamiltonian function is^(pi2+p z!i+\~z qiz+\-2 q.iz )tothedynamical system whose Hamiltonian function isQz. 2.If(#1?x2,...,x.2n)denoteanyfunctions of(qlt^ -,?n &gt;Pi, ./&gt;),andif ifmoreover amndenotes dXmldxn-dXnfixm,Ddenotes thedeterminant formed ofthe quantities amn ,Aikdenotes theminor ofaikinD,divided byD,anduandvdenote arbitrary functions ofthevariables, shew that 8v CU 312 TheTransformation- Theory ofDynamics [OH.xi 3.Shew that foranyHamiltonian systemtheintegral-invariants fjj...J8ql8q2...8q n8pl...8p n and lff...J8Ql8Q2...8Q n8Pl...8P n, extended over corresponding domains, areequalif(qi,q2,...,qniPi&gt;)Pn)and (Qit $2&gt; Qn,A;P")areconnected byacontact-transformation. 4.Prove thatthecontact-transformation denned bytheequations =Xj "(24k)cos^i+X2~ (2^2)cosAi =-X!~4 (2i)4cosPl+\2~^(2$a)"oosP|, = changesthesystem dq_r=dH_ dpr= _&lt;W(r=1 2) o? 9prdtdqr where intothesystem dQr_dKdPr_tt dt~dP~ r&gt;dt~~dQ r where JT-X^+X^,. Integratethissystem,andhence integratetheoriginal system. CHAPTER XII PROPERTIES OFTHEINTEGRALS OFDYNAMICAL SYSTEMS 141. ReductionoftheorderofaHamiltoniansystem byuseoftheintegral ofenergy. Wehaveshewn in42howtheLagrangian equationsofmotion ofa conservative holoiiomicsystemcanbereduced inorderbyuseoftheintegral ofenergyofthesystem. We shallrequirethecorrespondingtheorem for theequationsofmotion intheir Hamiltonian form;thismaybeobtained as follows. Consider adynamical systemwithndegreesoffreedom forwhich the Hamiltonian functionHdoes notinvolve thetimeexplicitly,sothat H+h=0, where Aisaconstant, istheintegralofenergyofthesystem. Letthisequationbesolved forthevariable pltsothat itcanbewritten The differential form associated with thesystemis Pidq^+p^dq. 2+...+pndqn+hdt, where thevariables(qltq2,...,qn&gt;PI,p.2&gt;...,pn,h,t)areconnectedbythe lastequation:thedifferential form cantherefore bewritten p.2dq2+p3dq3+...+pndqn+hdt-K(p2,p3,...,pn,q,,...,qn,h)dqlt where wecanregard (q1}q2,...,qn,p.2,...,pn,h,t)asthe(2n+ 1)variables. Butthedifferentialequations correspondingtothisform are(137) dqrdK dprdK -r~=^,d~=~i (r=2, 3,..., ri), dqi dprdq dqr d=d^Kdk_ dqldhdql~ The lastpairofequations canbeseparated from therestofthesystem, since the first (2&gt;i 2)equations donotinvolvet,andhisaconstant. 314Properties oftheIntegrals of [CH.xu Theoriginal differential equations canthereforebereplaced bythereduced system dqr_dK dpr_dK 7 ~\ ) 7 o V* "&gt; )"/) dq1dprdq 1 dqr which hasonly (n 1)degrees offreedom. This result isequivalenttothat obtained in 42,ascanbeshewn by direct transformation. Example. Consider thesystem dt~dp rdtdqr where abeing aconstant;these areeasilyseen tobetheequationsofmotion ofaparticle which isattracted toafixed point with aforcevaryingastheinverse cube ofthedistance : &lt;?2andq1arerespectivelytheradius vector andvectorialangleoftheparticle referred to thecentre offorce. WritingH=h,andapplyingthetheorem given above, theequations reduce tothe system dq&lt;2_dK dp.2_dK dq l~~ dp2dqi dq2 where SinceKdoes notinvolveqlttheequation K=Constant isanintegralofthis last system, andwecantherefore perform thesameprocess again:writingK-k,wehave * andthesystem reduces tothesingle equationddL k- ~T ^~r o s--* dqzokq./ theintegralofwhich(supposing /u&lt;2 )is where eisanarbitrary constant. This istheequation,inpolar coordinates, oftheorbit described bytheparticle. 142. Hamilton spartial differential equation. Iffollows from 138that ifacontact-transformation definedbythe equations pdW dWP= -dQ rPr= Tqr(r-1,2,...,), whereWdenotes agivenfunction of(q1}q.2,...,qn,QltQ2,...,Qn,t),is performedonthevariables ofadynamical systemdefinedbytheequations dqr_m dpr__dHdt~dpr ~dtdqr 141,142] Dynamical Systems315 theresulting systemis dQrdK dPrdK dt=dPr&gt;~di= -dQ r where K=H+dW/dt. IfthefunctionKiszero, thesystemwillbesaid tobetransformed into theequilibrium-problem. Now thefunctionKwillbezero,providedWisa function such that *\ ft i.e.provided TF,considered asafunction ofthevariables (5^ ^2,, &lt;?n&gt;0&gt; satisfies thepartialdifferential equation dW dWdW dW \ ^t+H(qi,q2,..., qn,^,^-,...,?-,*J= This iscalled Hamilton spartial differential equationassociated with the given dynamical system.Itwaspublished byHamilton in1834*, beingthe extension todynamicsofthepartialdifferential equationwhich hehad discovered tenyears previouslyinconnexion withoptics. Supposethata"complete integral"ofthisequation,i.e.asolution con tainingnarbitraryconstants inaddition totheadditive constant,isknown. Let(a,,&lt;*2,..., w)bethesearbitrary constants, sothat thesolution canbe writtenW(qlyqz,...,qn,i, 8,&gt; .0;andperformonthe original dynamical systemthecontact-transformation from thevariables (qlyqz,...,q n, Pi,..., pn)tovariables(ax,a2,...,an,fr,&,..., ),defined bytheequations SinceWsatisfies Hamilton sequation,theHamiltonian function ofthe newsystemiszero,andconsequentlytheequationsofthesystemare sothat(a,, 2,...,an,/8i, ..., y3M)areconstantthroughoutthemotion. It follows thatifWdenotes acomplete integral ofHamilton spartial differential equation, containingnarbitraryconstants(a1}a2,...,yn),then theequations dW dW^=-^ r*&gt;=Wr^=12-W &gt; constitute thesolutionofthedynamical problem,sincetheyexpressthevariables (&lt;7i&gt;&lt;/2&gt;&gt;cLn,p\, "-,pn)intermsoftand2narbitraryconstants (alto2,...,ctn, *Phil. Trans. 1834, p.247; ibid. 1835, p.95. 316Properties oftheIntegrals of [CH.xn &,.. ,/3M)*.Inthiswaythesolution ofanydynamical systemwith ndegrees offreedom ismade todependonthesolution ofasingle partialdifferential equationofthe firstorder in(n4-1)independentvariables. Itshould however beobserved thattheconverse ofthistheorem namelythetheorem that thesolution ofapartialdifferential equation such asHamilton sdependsonthe solution ofasetofordinarydifferential equations (thedifferential equationsofthe characteristics), which inthiscaseareoftheHamiltouian form, hadbeen discovered by PfaffandCauchy (completingtheearlier work ofLagrange andMonge)before Hamilton andJacobiapproachedthesubjectfrom thedynamicalside. Ontheusethatcanbemade ofanincomplete integralofHamilton spartialdifferential equation (i.e.onecontaininglessthannarbitraryconstants besides theadditive constant), cf.Lehmann-Filhes, Astr. Nach. CLXV. (1904),col.209. Itmayberioted thatHamilton spartialdifferential equationisnotapplicableasit stands tonon-holonomicsystems:foranextension tosuchsystems,cf.Quanjel, Palermo Rendiconti, xxn.(1906), p.263. The integration ofHamilton sequation byseparationofvariables isdiscussed by F.A.BallAcqua, Math. Aftn. LXVI.(1908), p.398. Example.Consider thesystem dq_3H dp=_3H dt" dpdt~ ~dq where jf=fc&gt;-,ITg andfj.isaconstant. TheHamilton sequation correspondingtothissystemis 9W acomplete integralofthisequation maybefound inthefollowing way. Assume W-fW+tte), where/and$arefunctions oftheir respective arguments:thenwehave 0=/ (0+H4&gt; (2)}2-F/?- Thisequation canbesatisfied bywriting where aisaconstant;which gives /(0=1*1a, (?)-(2,ia)*arcsin(gr/a)*+{^q (a-?)/a}* TF=fMt/a+(2/xa)*arcsin(g/a)*+{fyq (a-?)/a}*. The solution oftheoriginal problemistherefore given bytheequations /3=- p=dWfdq,where aand/3arethetwoconstants ofintegration. 143. Hamilton sintegralasasolution ofHamilton spartial differential equation. There areaninfinite number ofcomplete integralsofHamilton spartial differentialequation;andeachoneofthem furnishes acontact-transformation *Thistheorem isduetoJacobi, Crelle sJ.xxvu.(1837), p.97andLiouviltfs J.in.(1837), pp.60,161. 142,143] Dynamical Systems 317 from thevariables(q^,q2,...,q n,p1,...,p n)ofthedynamical systemto variables(j,or2,.., n ,@lt..., /3n),(thetransformationinvolving t},such that theequationsofmotion ofthesystem whenexpressedinterms of (oij, 2&gt; -,a n&gt;fii, -,/3W)become theequationsoftheequilibrium-problem, i.e.thequantities (al,0%,...,an,fti, ..., /3n)areconstants. Amongthis infinite number oftransformationsthere^isoneofspecial interest; namelythat inwhich thequantities (al,a2,...,an,/3X,...,/3n)are theinitial values of(q1}qz,...,qn,plt...,pn)respectively,i.e.their values at atime t,which istaken asanepochfromwhich themotion isestimated. In thiscasewecanfindinanexplicitform thecorresponding complete integral ofHamilton spartialdifferentialequation. Forconsider Hamilton sintegral (99) Ldt, whereLdenotes thekineticpotentialofthesystem. Supposethat &denotes avariation duetosmallchanges (S^,S2,...,8an,Sfti, ...,8/3n)intheinitial conditions. Then(99)wehave B[Ldt= I(pr8q,.-l3 rSr\ Jtv r=l ft Itfollows that ifthequantityILdt,when theintegrationisperformed, beexpressedinterms of(ql}q2,...,q n,aly...,a n,t),(wesupposethispossible, i.e.weassume that itisnotpossibletoeliminate(/3 1,/32,...,/3n,ply...(pn) from therelationsconnecting (1}...,an,^, ..., fin,qly...,qn,pl}...,p n),so astoobtain relations between(q1}...,qn,alt...,an))and ifthefunction thus obtained (which Hamilton called thePrincipal Function) bedenotedby W(q l}q2,...,qn,a1;...,an,t),thenweshall have 9JF_^__ andtherefore* thetransformation from (qltq2,...,qn,PI,,pn)to(ajta2,...,an,{3lt...,/9n) ifsacontact-transformation,and theintegral ofthekineticpotentialisthe determining function ofthetransformation. *Hamilton. Phil. Trans. 1834, p.307;ibid. 1835, p.95.Inhisearliest dynamical investiga tions, Hamilton used a"characteristic function" strictly analogous tothecharacteristic function which hehademployed withsuch success inoptics:thisfunction being theActionintegral, expressedinterms ofthefinaland initial coordinates. Hefound however that thisfunction, whenemployedindynamics, involved theconstant ofenergy, andsosubstituted for itthe "principalfunction" described above. 318 Properties oftheIntegrals of [CH.xn Alsowehave dW=dWIfrWdqr dt dt r=idqrdt T ,Vor L=-+2,prqr, ot =i or OL andtherefore theintegral ofthekineticpotential satisfiestheequation dWW dW dW M which isHamilton spartial differential equation. Example.Let(al5a2,...,an,ft, ..., ft()bethe initial values (attimefo)of (qi, ?2,,?MPi, -,Pn)respectively,inthedynamical system represented bythe equations dq,.=cHdpJ1==_dH(r=l,2,...,w). dtdprdtdqr Supposethat from the relations connecting (aj,a2,..., On, ft, ...,ft)with (?i, 2, -,?n,Pi,)P)ifcispossibletoeliminate(ft,ft, ...,ft,jon...,pn) entirely,so thatanumber (saym)ofdistinct relations exist between(qltq2,...,qn,al5...,an);let these besolved for(alsa2,...,a,),soastotaketheform and letVdenote Hamilton sintegral Ft I^ forthesystem, expressedinterms of(q^,qz,...,qn,o?n+i, , &lt;*n)-Establish the equations 9F B/^ Pr=~ 5-- ^A*odarfc=1 9ar where(Xi,X2,...,Xm)arearbitrary;andshew thatthefunction W=V+ 2Xt/t fc=i isanintegralofthepartialdifferential equation 144. TAe connexion ofintegralsurithinfinitesimal transformations admitted bythesystem. LetdqrdH dprdH . .. -= -- ,;.=-x (r 1,2,...,n) 143,144] Dynamical Systems 319 betheequationsofanydynamical system,and let 0(?i, 921.qn,Pi,-~,pn,&lt;)=Constant denote anyintegralofthesystem;weshallshew thattheknowledgeofthis integralenables ustofindaparticularsolution ofthevariationalequations (H2). Forthevariationalequationfor8qris d,d*HR &H &H , &H -T7oqr=,.. o&lt;fr+...+ _ o&lt;?n+djjj+...+~-6n; J *dtj.* j j." j. j.*j. j.*"~ j.~* butwehave 82zz8092/i8082/z80 82-/z80 dqidp rdpi dqndp,-dpndpidp rdq^ 3pndprdqn 3,dpk80",dqjc80\^?)H 82^dJJ 82 A;=I rfi9pfc ftlirf&lt;S^A;/ fc=i9?*9pt9p rk=idpk dqkdpr ct0 90\ct /80\9/90\ 9/&gt;r\^ 9^/ d^V8pr/dt\dpr) =1f?^" dV8pr/ andhence thevariationalequationsfor(Sq 1}Bq2,...,8qn)aresatisfiedbythe values where eisasmall constant.Similarlythevariationalequationsfor (Spi,Sp2)...,8pB) canbeshewn tobesatisfied bythese values;andhence theequations eisasmall constant and isanintegral oftheoriginal equations, constitute asolutionofthevariationalequations. This result canevidentlybestated intheform :Theinfinitesimal contact- transformation ofthevariables(q^q3,...,qn,plt...,p n),which isdefinedby theequations transforms anyorbit intoanadjacent orbit, and therefore transforms the wholefamilyoforbits into itself.Adoptingthelanguageofthegroup- theory, wesaythat thedynamical system admits thisinfinitesimal contact- transformation. Wehave therefore thetheorem thatintegrals ofadynamical 320 Properties oftheIntegrals of [CH.xn system, andcontact-transformationswhichchangethesystemintoitself,are substantiallythesamething; anyintegral 4&gt;(qi&gt; &,"-,qn,Pi, ",Pn,=Constant correspondstoaninfinitesimal transformationwhose symbol (133) isthe Poisson-bracket((f&gt;,f). Itwillbeobserved that theignorationofcoordinates arises from theparticular case ofthistheorem inwhich theintegralisp,.=Constant, whereqristheignorable coordinate;thecorresponding transformation isthatwhich changes qrwithoutchanging anyoftheother variables. 145. Poisson stheorem. The lastresult leads toatheorem discovered byPoisson* in1809, by means ofwhich itispossibletoconstruct from twoknownintegralsofa dynamical systemathirdexpressionwhich isconstantalong anytrajectoryof thesystem,andwhich therefore (whenitprovestobeindependentofthe integrals already known) furnishes anewintegralofthesystem. Let 0(?i ?2,&gt;qn,P\,,Pn, t)=Constant and-\Jr(q-i, &lt;/2,...,qn,PI,--^Pn,=Constant denote thetwointegralswhich aresupposedknown. Consider the in finitesimal contact-transformation whosesymbolisthePoisson-bracket (/, -v/r);sincetyisanintegral,this(144) transformseveryorbit intoan adjacentorbit. Theincrement ofthefunction&lt;under thistransformation ise (&lt;/&gt;, -v/r), where eisasmall constant; butsince&lt;isanintegral, &lt;f&gt;hasconstant values alongtheoriginalorbit andalongtheadjacentorbit: thevalue of (&lt;/&gt;, -v/r) must therefore beconstant throughoutthemotion. Wethushave Poisson s theorem, thatif(f&gt;andtyaretwointegrals ofthesystem,thePoisson-bracket (&lt;f), -v^)isconstant throughoutthemotion. If (&lt;, A/T),which isafunction ofthevariables(qltq2,...,qn,p},...,pn,t), does notreduce tomerelyzero oraconstant, and ifmoreover itisnot expressibleinterms of (/&gt;,^randsuch otherintegralsasarealready known, then theequation (&lt;/&gt;, -^r)=Constant constitutes anewintegral ofthesystem^. Thefollowing examplewillshewhowPoisson stheorem canbeappliedtoobtain new integralsofadynamical system when twointegralsarealready known. *Journal deVEcole polyt.vin. (1809), p.260. tAdiscussion ofthistheorem isgiven byBertrand inNote VIItothethird edition of LagrangesMec. Anal.(1853):cf.Oeuvres deLagrange,t.xi.p.484. Ontheextension ofPoisson stheorem tonon-holonomic systems,cf.Dautheville, Bull, dela Soc.math, deFrance, xxxvn. (1909), p.120. 144-146] Dynamical Systems 321 Consider themotion ofaparticle ofunit mass, whoserectangular coordinates are (?!&gt; ?2,&)andwhose components ofvelocityare(pl,p.2,p3),which isfree tomove inspace under theinfluence ofacentre offorce attheorigin. Theintegrals ofangularmomentum about twooftheaxes are ps?2-5r3j2=Constant, and i~\%=Constant. Letthese betaken asthetwoknownintegrals&lt;/&gt;and\^;thePoisson-bracket (&lt;, \//-), which is r=l becomes inthis case PWi-fcft! and infact, theequation i=Constant isanotherintegral ofthemotion, beingtheintegralofangular momentum about the third axis. 146. Theconstancy ofLagrangesbracket-expressions. Thetheorem ofPoisson has,asmightbeexpected, ananalogueinthe theoryofLagrangesbracket-expressions. Let ur=0r (r=l, 2,...,2n) denote 2nintegralsofadynamical system with ndegreesoffreedom, con stitutingthecompletesolution oftheproblem: thequantities urbeing given functions ofthevariables(qltq2,...,qn,plt...,p n,t],andthequantities ar being arbitraryconstants. Bymeans oftheseequations wecanexpress (&lt;fc,q2,...,qn,pi,--^Pn)asfunctions of(a1;az,...,a2n ,t),andform the Lagrangesbracket-expressions [ar,a,],where arandasareanytwoofthe quantities (a1}a2,...,am). Since thetransformation from thevariables(q1}q2,...,qn,p1}...,p n)at time ttotheir values attimet+dtisacontact-transformation, wehave(128) d atr=i where thesymbols Aand Brefer toindependent displacements from one trajectorytoanadjacent trajectory.Ifnowwetake thesymbol Atorefer toavariation inwhich atonlyisvaried, therestofthequantities (alta2,...,a2n) .remaining unchanged,andtake 8torefer toavariation inwhich a}onlyis varied, thelastequation becomes d3/dqrdprdqrdpr\ dtr=i\daidaj dajdaj w.D.21 322Properties oftheIntegrals of [on.xn or -^[a{,cij]=0, which shews that theLagrange-bracket [a;, a,-]hasaconstant value during the motionalong anytrajectory;thistheorem wasgiven byLagrangein1808. Lagrangesresult, unlike Poissons,does notenable ustofindanynew integrals;forwehave toknow alltheintegralsbefore wecanform the Lagrangesbracket-expressions. 147.Involution-systems. Let(ul}u2,...,ur}denote rfunctions of2nindependentvariables (?l, ?2,,qn,Pi,-,Pn)\ ifitispossibletoexpressallthePoisson-brackets(iii,uk)asfunctions of (wj.Mg, ...,u r),thefunctions(ultu2,...,u r)aresaid toform a,function-group*. Anyfunction of(ultu.2,...,ur)belongstothisgroup. Ifthequantities (uitn^}areallzero, thefunctions(u1}u2,...,ur}aresaid tobeininvolution, ortoformaninvolution-system. Nowsupposethat(i/u 2&gt;...,w,-)arefunctions ininvolution: and let v=Uandw=beanytwoequations which areconsequencesofthe equations u^=0,M2=0,...,ur= ; weshallshew that vandwsatisfytherelation(v,w)=0. Forsince(ultu.2,...,ur)areininvolution, each oftheequations M,=0,u2=0,...,ur= admits each oftherinfinitesimal transformations whosesymbolsare ("I,/), (W2,/),,(Mr,/)5 andconsequentlytheequationv=0,beingaconsequenceoftheseequations, must alsoadmit these transformations;that istosay,wehave ("*,)=0 (k=l,2,...,*), andtherefore each oftheequations ii!=0,u2=0,...,ur= admits theinfinitesimal transformation whosesymbolis(v,f).Since the equation w= isaconsequenceoftheseequations,itfollows that the equation w=must alsoadmit thistransformation, andtherefore wehave (v,iv}=0, which establishes theresult. *Lie,Math. Ann.vm. (1875). p.215. 146-148] Dynamical Systems323 Hence weseethatif(u1}uy,...,ur)areininvolution, andtheequations v}=0,v.20,...,vr= areconsequences oftheequations M,=0,u2=0,...,ur=0, then thefunctions (vltv2,...,vr)areininvolution. 148. Solutionofadynamical problemwhenhalftheintegralsareknown. Theresult which wasestablished forsystemswithtwodegreesoffreedom in121cannowbeextended tosystemswithanynumber ofdegreesof freedom. Thetheorem maybethus stated* :Ifndistinctintegrals &lt;f&gt;r(qi,q*.&gt;qn,Pi, &gt;~,p n,t)=a r(r=l, 2,...,n), where(a1}a2,...,an)arearbitrary constants, areknown forthedynamical system dqrdHdp,._ dH , . ~TT=^&gt; IT~~~ "-5 v&gt;"/dtOprdt dqr whereHisany given function of(q1;q.2,..,qn,pi&gt; &gt;Pn,t), andifthe functions (^,02,&gt;&lt;f&gt;n)areininvolution, thenonsolvingtheseintegrals for (p\,lh, ipn)soastoobtain them intheform Pr=fr(qi, q- -.5r n&gt;ai^2, ,On, (r=l,2,...,n) andsubstituting (/i,/2,...,/n) respectively for(pl,p.2,...,pn)intheexpression pldql+p 2dqt+...+pndqnHdt, thelatterexpressionbecomes aperfect differential:denotingitby dV(q ltq^...,qn,i,a2,...,an,t), theremaining integrals ofthesystemare ^=6, (r=l,2,...,n),oar where(bl}62,&gt;bn)arearbitraryconstants. Forsince thefunctions&lt;f)i-a lt &lt;f&gt;.2a.2,...,&lt; areininvolution,it follows bythelast article that thefunctions Pifi, p^-f*,&gt;pnfnare ininvolution, andtherefore (Pr~fr, PS~fs)=0(r,S=1,2,...,n\ or |i-M=0 (r,-l,2,...,n&gt; dqrdqs *Thistheorem isessentially theapplicationtoHamilton spartialdifferential equation ofthe well-knosvn method forfinding aComplete Integralofanon-linear partial differential equation ofthe first order. Asadynamical theorem itisdue toLiouville, Journal deMath. xx.(1855), p.137. 212 324 Properties oftheIntegrals of [CH.xn dH_dpr_dfr dqrdt dt =dA+|M^i dt s-idqsdt =dfr+|df.dH^ dt s=ityrdps andconsequently dfr=_d_H_IdHdf^ dt dqr s=idp sdqr _dH, dqr where H^stands forthefunctionHwhenexpressedinterms ofthearguments (?1,?2, .?n, i,&gt;ni0- Theequations _ dqrdqs dt dqr shew that/irfg-j+/ 2dq2+...+fndqnH^dt istheperfectdifferential ofsome function V(qltq^,...,qn,0,1, ,an,t) , which establishes the firstpartofthetheorem. Ifnow thesymbol ddenote thetotal differential ofthefunction Vwith respecttoallitsarguments, wehave thereforeWdV=fldq1+/ 2dq2+...+fndqn-H^dt+*2rdar. Inthisequation replacethequantities arbytheir valuescf)r:wethus obtain anidentityin(ql}q2,...,qn,PI,p%, ,pn,i),namely dVdV-2^d(f)r=pldq1+p2dq2+...+pndqn- where ontheleft-hand side oftheequation wesupposethat indVand dV xthequantities (alyaz,...,an)arereplaced bytheir values (&lt;&gt;!, &lt;/&gt;2,...,0n). \)\AJ-f Thisequationshews thatthedifferential form Pidqi+p 2dq^+...+pndqnHdt, whenexpressedinterms ofthevariables(ql}q.2&gt;...,qn, &lt;}&gt;i, &lt;f&gt;z,,0n, t), takes theform andhence thedifferentialequationsoftheoriginal dynamical problemare equivalenttothe first Pfaff ssystemofthis differential form, namely d(dV/da r)=0,d(f&gt;r=(r=1,2,...,n). 148,149] Dynamical Systems 325 Theexpressions dV/da raretherefore constantthroughoutthemotion, i.e. theequations dV/da r=br (r=l, 2,...,n), where(bltb2,...,bn)arenewarbitrary constants, areintegralsofthesystem; thiscompletestheproofofthetheorem. Example.Inthemotion ofabodyunder noforces withonepoint fixed,let($,$, -v|/-) denote thethree Eulerianangles which specify thepositionofthebodyrelative toany fixed axesOXYZ atthefixedpoint, (.4, J5,(7)theprincipal moments ofinertia ofthe bodyatthe fixedpoint,atheconstant ofenergy,attheangular momentum about thefixed axisOZ,anda2theangular momentum about thenormal totheinvariable plane: and let(#n l5^)denote 37730, dT/d&lt;j&gt;, dT/d^ respectively. Obtain the equations Q=arctan{(22 i2-#i2 )/ai} Hence shew that 9d istheperfectdifferential ofafunction F,andthat theremaining integralsofthe systemare oV 37 97 9^~"* ^"* a^T where6,&i,62arearbitraryconstants.(Siacci.) 149. Levi-Civita stheorem. Levi-Civita* hasestablished aconnexion between theintegralsofa dynamical systemandcertain families ofparticularsolutions oftheequations ofmotion. Consider firstasysteminwhich some ofthecoordinates areignorable. Let (&lt;?!,q2,...,qm)betheignorableand(qm+i, -,qn)thenon-ignorable coordinates;and letLdenote thekineticpotential. Theintegrals correspondingtotheignorablecoordinates are dL/dq r=Constant (r=l, 2,...,ra), andcorrespondingtotheseintegralsthere exists aclassofparticularsolutions ofthesystem, namelythosesteadymotions(83)inwhich(ql}q2,...,qm) have constant values which canbechosenarbitrarily,while(qm+i&gt;qm+z,,qn) have constant values which aredetermined bytheequations dLjdq r=0(?=m+1,m+2,...,n); there are oo2inoftheseparticular solutions, since themconstant values of *Bend, dellAce. delLined, x.(1901), p.3.Cf.Burgatti, ibid. xi.(1902), p.309. 326 Properties oftheIntegrals of [CH.xn (qlt 2,...,qm)andtheminitial values of(qi, q-2, ,qm)canbearbitrarily assigned. Thetheorem ofLevi-Civita, totheconsideration ofwhich we shallnowproceed, mayberegardedasanextension ofthis result. dqrdH dprdH Let l= ay/*"-3(-%*. .) betheequationsofmotion ofadynamical system,thefunctionHbeing supposednottoinvolve thetimeexplicitly. Let Fr(ql,qt,...,q n,p1,...,p n)=Q(r=1,2,...,m)...(A) beasystemofmrelations, which when solved for(p1}p2,...,pm)take theform Pr=fr(qi, q*,,qn,pm+i,&gt;Pn) (r=1,2,...,m)...(A 1), andwhich areinvariant relations withrespecttotheHamiltoniansystem, i.e.which aresuch that ifwedifferentiate therelations(Aj)withrespect tot,weobtain relations which are satisfiedidenticallyinvirtue ofthe Hamiltonianequationsand oftheequations (Aj) themselves. These invariant relations include, asaparticular case, integralsofthesystem: inthis case, theywillinvolvearbitraryconstants. Since therelations (A x)areinvariant relations, wehave dH_dfr_$dfrdH|dfrdH o-~T7--- * ~5i*-5oV *&gt;* &gt;"^^ dqrdtj=m+\opjoqj j-\oqjOp) andwriting (F,Tf}= S /-*- thisbecomes thisequation becomes anidentity when foreach ofthequantities (Pi,p2&gt; ,pm)wesubstitute thecorrespondingfunction fr. Moreover, weshallsupposethattherelations (A)or(Aj)areininvolution amongthemselves. This condition isexpressed bytheequations !;-!+i*"- c--- 1.2,...,.). ..(2). LetKdenote thefunction obtained fromHonreplacing (p1}jo2,...,pm) bytheir values(/i,/ 2,...,/m),sothat 3F=8^_I8/7 dpr~dp r.~idp,d ,and5T "a---^=^~ (r=1,2,...,m)...(4). 149J Dynamical Systems 327 From(3)wehave f)ff [H,fr\=[K,fr}+^~ !/,/.} (r=1,2,...,m),*=iwf* andcombiningthiswith(4)wehave Substitutingin(1)thisvalueofdH/dq r+{H,f r},andusing (2),weobtain theequations Weshallnowshew thatthesystemofequations Pr=fr (Pm+i,pm+2 ,...,pn,qlt .,?) (r=1,2,..., isinvariant withrespecttotheHamiltonianequations,i.e.that dfdK\,dfdE\ dt(ty,Jand dt(dq r)(r=m+l,m arezero invirtue ofequations (A), (B), (1), (2), (3), (4), (5). Wehavefrom theHamiltonianequations d(BK\_ f /7/ I57T I~ 1"5 ff-^^^Tdt\oqr/(dqr)g=loqrdq and(5)gives ondifferentiation, using (B), 5T" &gt;/*f= nowtaking account of(B),wehavefrom(3) ___ 9?r *=i9p, dq andhenceequations (6)become-__ dtdp dpsldprdqsdpr&gt; i^r^ .r,dp.ldqrdqs Properties oftheIntegrals of [CH.xn orby(7), ddK\ . d/dK\=0, jrU = / whichprovesthat thesystemofequations (A)and(B)isinvariant with respecttotheHamiltonianequations. Nowfrom theequations (A)and(B),letthevariables (PI, Pa, ,pn,qm+i, ,qn) bedetermined interms of(qltqz,...,qm):from theinvariant character of (A)and(B)itfollows thatonsubstitutingthese values intheHamiltonian equations, weshall obtainmindependent equations, namelythose which express (dqjdt, dq2/dt, ...,dqm/dt)interms of(qltqz,...,qm),theothersbeing identicallysatisfied :andthegeneralsolution ofthissystem,which will contain marbitrary constants, willgiveoomparticularsolutions ofthe Hamiltonianequations. The solution ofthissystem can,bymakinguse oftheintegralofenergy,bereduced tothat ofasystemoforder(m 1): andthusweobtain Levi-Civita stheorem, which canbethus stated :Toany setofTOinvariant relations ofaHamiltoniansystem,which areininvolution, therecorresponds afamily ofoomparticularsolutionsoftheHamiltonian system, whose determination dependsontheintegration ofasystem oforder (m- 1). Iftheinvariant relations (A)areintegralsofthesystem, theywillcontain another setofmarbitraryconstants :andhence toasetofmintegrals ofa Hamiltoniansystem,which are ininvolution, therecorrespondsingeneral afamily ofoo2mparticularsolutionsofthesystem,which areobtainedby integratingasystem oforder(m 1). Example. Forthedynamical systemdefined bytheHamiltonian function H=qipi-q2pz-aqi2+bq^, shew thattheLevi-Civita particularsolutions correspondingtotheintegral (Pi~bqz)lq\=Constant aregiven bytheequations 2i=0,qz=e~t+f ,Pl=ae~t+e ,p,=be~t+e , where eisanarbitraryconstant. 150. Systemswhichpossess integralslinear inthemomenta. We shallnowproceedtotheconsideration ofsystemswhichpossess integralsofcertainspecialkinds. Supposethatadynamical system, expressed bytheequations dqr_dH dpr_dH ~df-dp rdt~dqr.!,*.....&gt; 149,150] Dynamical Systems 329 hasanintegralwhich islinear andhomogeneousin(p1}p2,...,pn),say fipi+fap&lt;t++fnp n=Constant, where(fi,f2,...,/)aregivenfunctions of(qi,q 2,...,qn). Consider thesystemofequations dq\_dq*_ _dqn 7T~72= "/. which isoforder(?? !);supposethatthe(n 1)integralswhich constitute itssolution are Qr(qi, q?,,qn)=Constant (r=l, 2,...,w1); and letQnbeafunction definedbytheequation where intheintegrandthevariables(q2,q3,...,qn)aresupposed replaced bytheir values interms of(qltQ1}Q2,...,Qn-i)before theintegration iscarried out. Then ifthevariableschangeinsuch awaythat(Q x,Q2,...,QM_j)remain constant andQnvaries, itfollows from theaboveequationthat dqi_d&lt;h_ _dqn_jnf f f** *&lt;ln&gt; /i /2 Jn sothat if(Q1}Q2,...,Qn)areregardedasasetofnew variables interms of which((?!,q2,...,qn)canbeexpressed, weshallhave Supposethen thatweconsider thecontact-transformation which isthe extension ofthepoint-transformationfrom thevariables(q1}q2,..,,qn)tothe variables (QltQ2, -,Qn),sothat thenew variables (Pl}P2,...,Pn)are defined(132)bytheequations Pr=i^|f(r=l,2,...,n).*=1 O^r Bythistransformation thedifferentialequationsofthedynamical system arechangedintoanew setofHamiltonianequations dQr_dPr_J&gt;K dtdPr dt bQr andtheknownintegralbecomes Pn=Constant. SincedPn/dt=0,wehavedK/BQ n0,sothefunctionKdoesnotinvolve Qnexplicitly:andthusweobtain theresult thatwhen adynamical system 330Properties oftheIntegrals of [CH.xn possesses anintegral which islinear andhomogeneousin(pltp2,...,pn),there exists apoint-transformation, fromthevariables(ql}q.2,...,qn)tonewvariables (Qi&gt; Qs&gt;&gt;Qn),which issuch that thetransformedHamiltonianfunction does notinvolve Qn.Thesystemastransformedpossessestherefore an ignorable coordinate, andwehave thetheorem that theonlydynamical systems whichpossess integralslinear inthemomenta arethosewhichpossess ignorable coordinates, orwhich can betransformed byanextended point- transformationintosystems whichpossess ignorablecoordinates. Theconverse ofthistheorem isevidentlytrue. This result might have been foreseen from thetheorem(144)that if ?2, -,qn,Pi,--,Pn,=Constant isanintegral ofthesystem, then the differential equationsofmotion admit the infinitesimal transformation wrhosesymbolis (&lt;, /).Forwhen (f&gt;islinear andhomogeneous in(pl,pi, ...,pn},thistransformation is($132)anextended point-transformation:if thispoint-transformationistransformed bychangeofvariables soastohave thesymbol df/3Q n,itisclear that theHamiltonian function oftheequationsafter transformation cannot involve Qnexplicitly. Considering now inparticular systemswhose kineticpotentialconsists ofakineticenergy T(ql}q2,...,qn,ql}...,qn)which isquadraticinthe velocities(ql}q2,...,qn)and apotential energy V(q-i, q2, ,qn)which is independentofthevelocities, weseethat inorder thatanintegrallinear in thevelocities mayexist thesystemmustpossessanignorable coordinate, ormust betransformable byapoint-transformationinto asystemwhich possesses anignorablecoordinate. Butineither casethefunctions TandV evidentlyadmit thesame infinitesimal transformation, namelythetrans formation which, when thecoordinates aresochosen thatoneofthem isthe ignorable coordinate, consists inincreasingtheignorablecoordinate bya smallquantityandleavingtheother coordinates andthevelocities unaltered; andconversely,ifTandVadmit thesame infinitesimal transformation, then there exists anintegrallinear inthe velocities. This result isknown as Levystheorem, havingbeenpublished byLevy*in1878. Example1.Shew that ifthedifferential equationsofmotion ofaparticleadmit an integrallinear inthecomponentsofmomentum, theline ofaction oftheforcemust belongtoalinear complex. (Cerruti,Collect, math, inmem. D.Chelini :cf.P.Grossi, Palermo Rend. xxiv. (1907), p.25.) Example2.Iftheequations d &lt;^ " (,-!,*..., Comptes Rendus, LXXXVI. 150,151] Dynamical Systems331 whereT=% 22aikqiqk,andwhere(ft, Q.2,...,Qn,au,a12,...,ami)aregivenfunctions i=lk=l f (?i&gt; ?2&gt; ?))possessanintegraloftheform Clqi+C2q2+...+Cnqn+C=Constant, where (C l,C.2,...,Cn,C)arefunctions of(q1}q%, ...,qn),shew that itispossibleto displaceaninvariable systeminonedirection fromanyoneofitspositionsinthespaceSn defined bytheform nn=22 i=lk=l Shew that forthisanecessary and sufficient condition isthat theds2canbetrans formed insuchawaythatoneofthevariables becomes absent from thecoefficients. (Cerruti andLevy.) 151. Determination oftheforces actingonasystem forwhich an integralisknown. Beforeproceedingtodiscuss systemswhichpossess integrals quadratic inthe velocities, weshall obtain aresult due toBertrand*, namelythat inthemotion ofadynamical systemofgiven constitution, forwhich however theactingforces areunknown(itbeing supposedthat the forcesdepend solelyonthecoordinates oftheirpointsofapplication, andnotonthe velocities ofthesepoints),wecandiscover theunknown forcesprovidedone integralisknown. Moreover, thisintegralcannot bechosen atrandom, but mustsatisfycertain conditions. Let(q1}q*,,...,qn)bethenindependentcoordinates ofthesystem, Tthe kineticenergy,and(Q1}Q2,...,Qn)theunknown forces, which aresupposed todepend onlyon(qi,q 2, ,qn),sotheequationsofmotion are didT\ dT\8T )-^-=Qr (r=l, 2,...,n).9o, dt\dqr Let &lt;(?i&gt; ?2,,qn,#1,&gt;qn,t)=Constant beanintegralofthesystem;ondifferentiating it,wehave Substitutinginthisequationfor(qlyq2,...,qn)their values asgiven by theequationsofmotion, wehave arelationinvolving (QlyQ.2,...,Qn)linearly. This relation, asitcontainsonlythequantities (qltq2,...,qn,qi9...,qn &gt;t},to allofwhich wecanassign arbitrary independent values, must beanidentity: wecantherefore differentiate itwithrespectto(ql}q2,...,qn),and soform nnewequations which, likewisecontaining (Q1}Q2,...,Qn)linearly,will sufficeingeneraltodetermine theseunknownquantities. Theintegralwill *Journal deMath,(i)xvn.(1852), p.121. 332 Properties oftheIntegrals of [CH.xn relate toanactualsystem onlywhen these values of(QlfQ2,...,Qn)satisfy theequation thecases inwhich theequationsforthedetermination of(Q},Q2&gt;...,Qn)are notindependent,sothat(Q1}Q2,&gt;Qn)areindeterminate, arethose in which theintegraliscommon toseveral distinct dynamical problems. Example.Ifanintegraloftheequationsofmotion ofapointinaplaneiscommon totwodifferent problems, shew that itisoftheform F(&lt;fi, x,y,t}=Constant, where(x,y)arerectangularcoordinates and$isthederivate with respectto tof afunction $(x,y)which, equatedtoaconstant, representstheequationsofaset ofstraightlines. (Bertrand.) 152. Applicationtothecaseofaparticlewhoseequations ofmotion possess anintegral quadraticinthevelocities. AsanapplicationofBertrand smethod, let itberequiredtofindthe nature ofthepotential energyfunction Finorder that theequationsof motion ofaparticlewhich isfree tomove inaplaneunder theaction of conservative forces,dv dvM . /it ._,____ da;y~ dy may possessanintegral (otherthan theintegralofenergy)oftheform Ptf+Qxy+Rif+Sy+Tx+ K=Constant, where P,Q,R,8,T,Karefunctions ofxandy. Differentiatingthe lastequation,andsubstitutingforxandyfrom the equationsofmotion, wehave ^.^~ yxT~ ^ dy dxj \dy dxJ dx .8F .dV\ .8F^- -K- dyJ dKdK.8F^8F /A,^-x+-^-y-Sj--T^ r= .................................... (A). oes dy dyoxn/.-QI&lt;fc*\ Equatingtozerotheterms ofthethirddegreeinxandy,wehave dP dR dPdQ dQdR ^=0,-=0,^+-_--=0,^+^=U,dx dy dyoxdyox fromwhich itisreadilydeduced thattheterms oftheseconddegreeinthe integralmust have theform (ay"+by+c)x2+(ax2+bx+c}y-+(-2axy-by-bx+Cj)xy, where(a,6,c,6,c,c^areconstants. 151,152] Dynamical Systems333 Equatingtozerotheterms ofthesecond degreeinxandyinequation (A),wehave C/7/ ^jOC (jCC (JU from theseequationswededuce 8=mx+p,T=-my+q, where (m,p,q}areconstants. Equatingtozerotheterms independentofxandyin(A),wehave dV,.97 or (mx+p)-d- Thisequationshews that if(m,p,q)aredifferent from zero, theforce is directed toafixed centre offorce, whose coordinates arep/mandqfm ;we shall exclude thissimple particular case,andhence itfollows thatthecon stants (m,p,q)must eachbezero, sothattheintegralcontains noterms of the firstdegreeinx,y. Equatingtozerotheterms linear inxandyin(A),wehave 2P8FQdV+dK-0 ~----v~r-z. w,ox oyox Differentiatingtheformer ofthese equationswithrespecttoy,andthe d*K latter withrespecttox,andequatingthetwovalues of^thus obtained, wehave - dxdy dxdy dy* dydy dxdydxdydx andreplacing P,Q,Rbytheir values asfound above, wehave 87.97 dx dy Darboux* hasshewn that thispartialdifferentialequationforthefunction 7canbeintegratedinthefollowing way. *Archives Neerlandaises, (ii)vi.p.371(1901). 334 Properties oftheIntegrals of [CH.xn Excludingtheparticularcase inwhich theconstant aiszero,wecan always bychangeofaxesreduce thegiven integraltothesimplerform ^(xy yx)2+ex2+cy2+K=Constant, which amounts tosupposingthat tt=i,6=0,6=0,d=0; ifmoreover wereplaceccby^c2 ,thepartialdifferentialequationforV becomes ,,2F .dV _dV (y~x+c)5-31+3y=--3a?3-0.cxdyydx dy Tointegratethis equation, weform the differentialequationofthe characteristics xy(dy2-dx-)+(x2-y2-c2 )dxdy=0. Ifinthisequationwetakex2andyzasnew variables,itbecomes a Clairaut sequation:wethus find that itsintegralis (7?^+1)(mx2y2 }me2=0, wheremdenotes thearbitraryconstant. Byasimple changeofnotation, we canwrite thisintegralintheform x2 y- I&lt;- i a2a2-c2 where thearbitraryconstant isnow a.This formputsinevidence the interestingfactthat thecharacteristic curves ofthepartialdifferential equationaretwofamilies ofconfocal conies. Takingthen asnew variables aand/3theparametersoftheconfocal ellipsesandhyperbolas,sothat *=?,y^^-c^^-F)}*,C C itisknown from thegeneral theorythatthepartialdifferentialequationwill take theform d2V+A^+B^=0, oaofi efunctions of variables wefindda.d/3" da whereAandBarefunctions ofaandy3 ;infact,onperformingthechangeof which canbeimmediately integrated, giving where/and&lt;arearbitraryfunctions oftheir arguments.Itfollows that the onlycasesofthemotion ofaparticleinaplane,under theaction ofconservative 152,153] Dynamical Systems 335 forces,whichpossessanintegral quadraticinthe velocities other than the integral ofenergy,arethoseforwhich thepotential energyhas theform /(a)-$(0) -?T05-. where a.and/3aretheparameters ofconfocal ellipses andhyperbolas. Since bydifferentiation wehave thekineticenergyis T=A(V-#2)P)* andaninspectionoftheforms ofTandFshews that theseproblems areof Liouvilles class(43),andaretherefore integrable byquadratures. 153. Generaldynamical systems possessing integrals quadraticinthevelocities. Thecomplete determination oftheexplicit form ofthemost general dynamical system whose equationsofmotionpossess anintegral quadraticinthevelocities(inaddition totheintegral ofenergy) hasnotyetbeen effected. Itisobvious from 43that all dynamical systems which areofLiouville stype, orwhich arereducible tothistypebya point- transformation, possess suchintegrals:andseveral more extendedtypes havebeen determined *. Example1.Let H(qk) (&,I=1,2,...,n) ben2functionsdepending solely onthearguments indicated, and let 4&gt;=2W*H (1=1, 2,...,%) fc=i denote thedeterminant formed bythese functions. Shew that ifthekineticenergyofa dynamical systemisreducible totheform andthepotential energyiszero, there exists notonlytheintegral ofenergy, butalso(n 1)otherintegrals, homogeneous and oftheseconddegreeinthevelocities, namely A-^f*1 ""1 (*=2,3,...,), where(a1(a2,...,an)arearbitrary constants: and that theproblemissoluble by quadratures.(Stackel.) Example2.Lettheequations ofmotion ofadynamical system withtwodegrees of freedom be d where *Cf.G.diPirro, Annali diMat. xxiv.(1896), p.315. 336 Properties oftheIntegrals of [on. and(a,A,6)areanyfunctions ofthecoordinates (q,q2):and letthissystem possess an integralaq^+2AJ!j2+&?22=Constant, quadraticinthevelocities anddistinct from theequationofenergy, where( ,A,6)are functions ofthecoordinates. IfAandAdenote (afc-A2 )and(a&-A2 )respectively, and if /A\2Jl(_ )(ojji+Zhftqj+bq^\ where &lt;?/stands fordqrjdf,shew that theequations,-, at\cqr/oqr define thesame relations between thecoordinates(g^, q?)astheoriginal equations ofmotion, andthatonesetofequationscanbetransformed totheother bythetrans formation MISCELLANEOUS EXAMPLES. 1.Adynamical systemisdenned byitskinetic energy ni (where $denotes thedeterminant $21 $22 $2rc nl 2......n inwhich theelements ofthethlinearefunctions ofqkonly,and$wdenotes theminor of $w),andbyitspotential energy * * where ^=^11 andthequantity \^fcdenotes afunction ofqkonly. Shew thatacomplete integralofthe Hamilton-Jacobi equation -= nC 2{a1$il+a2$j2+...+a n i=U where(a1?a2,..., a,t)arearbitraryconstants. (Goursat.) 2.If isanintegralofadynamical systemwhichpossessesanintegralofenergy, shew that -^=Constant, ^*f=Constant, etc.,arealso integrals. ct OT xii] Dynamical Systems337 3.Asystemofequations -jf=Ar(qi, ?2, -,qn,PI, -,?*, , (r=l, 2,...,n) d{=Br(3li ?2, Sn.l l, jn, issuch that if&lt;and -v|/-areanytwointegrals whatever, thePoisson-bracket (&lt;, *//)isalso anintegral. Shew thattheequations must have theHamiltonian form .dqrdff dpr_dH ~di-ty r~dt~~dg r (Korkine.) 4.If at=Constant, a2=Constant, ..., a*-=Constant, &=Constant, /32=Constant, ..., /3fc=Constant, eany2integralsofaHamiltonian systemofdifferential equations,thevariables being !)?2,&gt;gn,^!, j/&gt;),shew thatare A,,..., A* O?AI ?A2oqxkp^ p\k isalsoanintegral. (Laurent.) 5.Lettheexpression f-..:B(^,^...,^ n) [JZi. /z.), ....uJ 2r-5-7- -r, i-l&lt;K*Ui ^2t, ,*&lt;) whereHI,H2,...,^narefunctions ofthenvvariables xji(j=l, 2,...,n;il, 2,..., i&gt;) becalled aPoisson-bracketofthenth order. IfG\,G2,...,Ghvare Ai&gt;functions of yn, Vi-2, ,i/hV!^n, ^12, ,AV ;n 2, -, AV&gt;where(h+k=ri),and if denotes allthePoisson-brackets formed fromevery nfunctions G,shew that represents thenecessary and sufficient conditions that thefunctions yt=F*t(xn, #12, ,Xkv", i, 2, -, (i&gt;hv) (*= !&gt;2 &gt; iA;^=152 arisingfrom theequations &lt;?4=(i=l, 2,..., shallsatisfythesimultaneouspartialdifferentialequationsofthefirstorder wherePi(yh ,F}denotes theexpression which isobtained whenwereplace hofthe functions FinPt(Fn )byasmany ys.(Albeggiani.) 6.Aparticleofunitmass whose coordinates referred tofixed rectangularaxes are (.r,y)isfreetomove inaplane under forces derivable from apotential-energyfunction /(#,y\thetotal energy being h.Shew that iftheorthogonal trajectoriesofthecurves 82\+5~~9 Ilg{*/(*! y)}Constantiayy W.D. 22 338 Properties oftheIntegrals ofDynamical Systems [en.xn areorbits, thedifferential equationsofmotion oftheparticle possess anintegrallinear and homogeneousinthevelocities(#,y). 7.Theequationsofmotion ofafreesystemofmparticlesare ^=X8 (s=l, 2,...,3m). Ifanintegralexists oftheform 3m 2fgxs-Ct=Constant, 8=1 where/i,/2,...,/3marefunctions ofA^,#2 &gt;&gt;^sm? an(*C*isaconstant, shew that this integral canbewritten 3m 3n 28.r8+2 of,.g(.,.-xrXg)Ct=Constant, s=l r,s=l where thequantitiesksandargareconstants. (Pennacchietti.) 8.Twoparticles move onasurface under theaction ofdifferent forces depending onlyontheir respective positions:iftheir differential equationsofmotion have in common anintegral independentofthetime, shew that thesurface isapplicable on asurface ofrevolution. (Bertrand.) CHAPTER XIII THEREDUCTION OFTHEPROBLEM OFTHREE BODIES 154. Introduction. Themost celebrated ofalldynamical problemsisknown astheProblem ofThree Bodies, andmaybeenunciated asfollows : Threeparticles attract each otheraccordingtotheNewtonian law, sothat between eachpair ofparticlesthere isanattractiveforce which isproportional totheproduct ofthemassesoftheparticles and theinversesquare oftheir distanceapart: theyarefreetomove inspace, andareinitially supposedtobe movinginanygivenmanner;todetermine theirsubsequent motion. Thepractical importanceofthisproblemarises from itsapplicationsto Celestial Mechanics :thebodies which constitute thesolarsystemattract each otheraccordingtotheNewtonian law,and(astheyhaveapproximately theform ofspheres, whose dimensions areverysmallcomparedwith the distances whichseparate them)itisusual toconsider theproblemofdeter miningtheir motion inanideal form, inwhich thebodies arereplaced by particlesofmassesequaltothemasses oftherespective bodies andoccupying thepositionsoftheir centres ofgravity*. Theproblemofthree bodies cannot besolved infinite termsbymeans ofanyofthefunctions atpresent known toanalysis.Thisdifficultyhas stimulated research tosuch anextent, that since theyear1750 over800 memoirs, manyofthembearingthenames ofthegreatest mathematicians, havebeenpublished onthesubject f.Inthepresent chapter, weshall discuss theknownintegralsofthesystem and theirapplicationtothereduction of theproblemtoadynamical problem with alesser number ofdegreesof freedom. Themotions ofthebodies relative totheir centres ofgravity (intheconsideration ofwhich their sizesandshapesofcourse cannot beneglected) arediscussedseparately, e.g.intheTheory ofPrecession andNutation. Insome caseshowever(e.g. intheTheoryoftheSatellites ofthe Major Planets) theoblateness ofoneofthebodies exercises sogreat aneffect, thattheproblem cannot bedivided inthisway. tForthehistory oftheProblem ofThreeBodies,cf.A.Gautier, Essaihistorique sur le probleme destroiscorps (Paris, 1817):B.Grant, History ofPhysical Astronomy from theearliest agestothemiddle ofthenineteenthcentury (London, 1852):E.T.Whittaker, Report onthe progress ofthesolution oftheProblem ofThree Bodies(Brit.Ass.Bep. 1899, p.121):and E.0.Lovett, Quart. Journ. Math. XLII.(1911), p.252,whodiscusses thememoirs oftheperiod 1898-1908. 222 340 TheReduction ofthe[CH.xm 155. Thedifferential equations oftheproblem. LetP,Q,Rdenote thethreeparticles, (ra1;m2,m3)their masses, and (TIB,? si&gt;^12)their mutual distances. TakeanyfixedrectangularaxesOxyz, and let(qltq2,q3),(q4,q5,q6),(q7,qs,q9),bethecoordinates ofP,Q,R,respec tively. Thekineticenergyofthesystemis T=m,++ 32 theforce ofattraction between m^andm2isTc1mlmzrl^z ,where k2isthe constant ofattraction :weshallsupposetheunits sochosen that&isunity, sothat this attraction becomes m^m^r^ ,andthecorrespondingterm in thepotential energyism1m.2r12~1 .Thepotential energyofthesystem istherefore _m.2m3m3??z1 {(q,-q7}2+(qs- q*)*+(q6-q9) -m.m, {(q7-qj*+(q8-qj2+(q,-q3)-}"* -w,w a{(q,-qJ2+(q2-q 5)2+(q 3-q 6Y}~^ Theequationsofmotion ofthesystemare mkqr=-dV/dq r (r=l, 2,...,9), where kdenotes theinteger partof^(r+2). Thissystemconsists of 9differentialequations,each ofthe2nd order, andthesystemistherefore oforder 18. Writing mkqr=pr (r=1,2,...,9), 9n2 and H=S-g^+V,r=i2mjc theequationstaketheHamiltonian form dqr_oH dpr_oE _ ~r, ^ jj7~ ~~r~ \~-1 "&gt; 9hatdpr at oqr andthese areasetof18differentialequations,each ofthe1storder, forthe determination ofthevariables(ql&gt;q.2,...,qa,pi,p2, ,^9)- Itwasshewn byLagrange*that thissystemcanbereduced toasystem which isonlyofthe6thorder. That areduction ofthiskindmustbepossible maybeseenfrom thefollowingconsiderations. Inthe firstplace,sincenoforces actexceptthemutual attractions ofthe *Eecueil despieces quiontremportelesprixde IAcad. deParis, ix.(1772). Lagrangeof course didnotreduce thesystemtotheHamiltonian form. Cf.Boblin, Kongl. Sv.Vet.-Handl. XLII. (1907),No. 9,foranimproved Lagrangiau reduction. 155] Problem ofThree Bodies 341 particles, thecentre ofgravityofthesystem moves inastraightlinewith uniformvelocity. This fact isexpressed bythe6integrals m3q7-(p, j+m2q-3+m3qs-(p2+p s+p 8)t=a4, +m.2q6+m3q9-(p3+p 6+p 9)t=at, where alta.2,...,a6areconstants. Itmaybeexpected that theuseofthese integralswillenable ustodepress theequationsofmotion from the18th to the12th order. Inthesecondplace, theangular momentum ofthethree bodies round each ofthecoordinate axes isconstantthroughout themotion. This fact isanalytically expressed bytheequations qiP-2- q-2pi+q^p*-q5p4+q7ps-qap7=a,, pe-qsp5+q8p,-q9pa=03, p4-qtp6+q9p7-q7p9=a9, where a-,a8)a9areconstants. Byuseofthese threeintegrals wemay expecttobeable todepress further theequations ofmotion from the 12th tothe9thorder. Butwhen oneofthecoordinates which define the positionofthesystemistaken tobetheazimuth&lt;ofoneofthebodies withrespecttosome fixed axis(saytheaxisofz\andtheother coordinates define theposition ofthesystem relative totheplane havingthisazimuth, thecoordinate &lt;isanignorable coordinate, andconsequentlythecorre sponding integral (which isone oftheintegralsofangular momentum above-mentioned) canbeused todepress theorder ofthesystem bytwo units;theequations ofmotion cantherefore, asamatter offact,bereduced inthiswaytothe8th order. This fact(though containedimplicityin Lagrangesmemoiralready cited) was firstexplicitly noticedbyJacobi* in 1843, and isgenerally referred toastheeliminationofthe nodes." Lastly,itispossible againtodepress theorder oftheequations bytwounits asin 42,byusingtheintegralofenergy andeliminatingthe time. Sofinallytheequations ofmotion maybereduced toasystem ofthe 6thorder. *Journ.fiirMath. xxvi.p.115.From thepoint ofview ofthetheoryofPartial Differential Equations, wemayexpress thematter bysaying thattheintegrals ofangular momentum giverisetoaninvolution-system, consisting oftwofunctions which areininvolution witheach otherandwithH :andhence theHamilton-Jacobi partial differential equation with 6independentvariables canbereduced toapartial differential equation with 6-2or4independent variables : this willbetheHamilton-Jacobipartial differential equation forthereduced system. 342 TheReduction ofthe[CH.xm 156. Jacobi sequation. Jacobi*,inconsidering themotion ofanynumber offreeparticlesinspace,which attract each other accordingtotheNewtonian law,hasintroduced thefunction 1 2. i,J wheremtandm,arethemasses oftwotypical particlesofthesystem, r^isthedistance between them attimet,Misthetotalmass oftheparticles, andthesummation isextended over allpairsofparticlesinthesystem.This function, which hasbeenused inresearches concerningthestabilityofthesystem,willbecalled Jacobi sfunction anddenoted bythe symbol*. Weshallsupposethecentre ofgravityofthesystemtobeatrest;let(#,-,yf, z&gt;)bethe coordinates oftheparticle mtreferred tofixed rectangularaxeswiththecentre ofgravity asorigin. Thekinetic energyofthesystemis r=Umi (#,*+&* +**), i andconsequently wehave=(2?H;)X2m;(X But (2ii)x2niiX?-(^m txt}2=2mt%(xt-%)2 , i i i i,3 where thesummation ontheright-handside isextended overevery pairofparticlesinthe system:andwehave S.mixi=Q,invirtue ofthepropertiesofthecentre ofgravity. Thuswehave T=^2TO^- {(*-%)2+(&-&)a+&- &gt;)2 } *- "i,j ^,3 where v^denotes thevelocityoftheparticle TOJrelative to?ny. Inthesamewaywecanshew that IfnowFdenotes thepotential energyofthesystem,thearbitraryconstant inVbeing determined bythecondition thatVistobezerowhen theparticlesareatinfinitely great distances from each other, wehave F=- sm*my . . .V.. 1,1 IJ Theequationsofmotion oftheparticlemfare 8F 8F 8F w=-^, **-~5,**--5. Multiplythese equations by#f,y{,zf,respectively,addthem, andsum forallthe particlesofthesystem:sinceFishomogeneousofdegree-1inthevariables, wethus obtain 27/1;faSt+ytifi +ziZi)= &gt; i or 2mi( or^2 This iscalled Jacobi sequation. *VorlesungeniiberDyn., p.22. 156,157] Problem ofThree Bodies 343 157. Reduction tothe12th order, byuseoftheintegrals ofmotionofthe centreofgravity. We shallnowproceedtocarryoutthereductions which have been described*. Itwillappearthat itispossibletoretain theHamiltonian form oftheequations throughoutallthetransformations. Takingtheequationsofmotion oftheProblem ofThree Bodies inthe form obtained in155, dqr_dH dpr__W dtdpr dtdqr wehave first toreduce thissystemfrom the18th tothe12th order, byuse oftheintegralsofmotion ofthecentre ofgravity. For thispurpose we performonthevariables thecontact-transformation definedbytheequations dW whereW=p.q,+p2q2+p.q 3+p 4q4+p5q&lt;?+p6q6+(p,+p4+p7)q7 +(P*+ps+Ps)qs+(ps+p6+p s)q9. Interpretingtheseequations,itiseasilyseen that (&lt;?/,q2,qs)arethe coordinates ofm1relative toms,(?/,qs,qs}arethecoordinates ofm2relative tora3,(q7,q8,q)arethecoordinates ofms,(pi,p2,p3)arethecomponents ofmomentum ofwilf(/&gt;/,pf,p6)arethecomponentsofmomentum ofw2,and (P?, PS,Pa)arethecomponentsofmomentum ofthesystem. The differentialequations nowbecome(138) &lt;%=a#rfp/__w ,_12dtdpr"dt~^qr1,4. ..,), where, onsubstitution ofthenewvariables fortheold,wehave [PiP*+P*P*+P*P*+\Pi*+\Ps2+ 3&gt;ps"2-p7(pS+p 4) K?/-?/ *Thecontact- transformation used in157 isduetoPoincare, C.R. cxxin.(1896);thatused in158 isduetotheauthor, andwasoriginally publishedinthe firstedition ofthiswork(1904). Itappears worthyofnotefrom thefactthat itisanextendedpoint-transformation, which shews thatthereduction could beperformed ontheequations intheir Lagrangian (asopposed totheir Hamiltonian) form, bypurepoint-transformations. Thesecond transformation inthealternative reduction(160)isnotanextendedpoint-transfurmation. Another reduction oftheProblem of Three Bodies canbeconstructed from thestandpoint ofLie sTheory ofInvolution-systems and Distinguished Functions :cf.Lie,Math. Ann. vin.p.282. Cf.alsoWoronetz, Bull. Univ.Kief, 1907,andLevi-Civita, AttidelR.1st.Veiieto, LXXIV.(1915), p.907. 344 TheReduction ofthe[CH.xm Since(//,qs,q9arealtogetherabsent from H,theyareignorable coordinates :thecorresponding integralsare p7=Constant, pa=Constant, p9=Constant. Wecanwithout lossofgenerality supposethese constants ofintegration tobezero, asthisonlymeans that thecentre ofgravityofthesystemis taken tobeatrest :thereduced kineticpotentialobtainedbyignorationof coordinates willtherefore bederived from theunreduced kineticpotential byreplacing p7,p8,p9byzero,andthenew Harniltonian function willbe derived fromHinthesameway. Thesystem ofthe12th order, towhich the equations ofmotionoftheproblem ofthree bodies havenowbeenreduced, may thereforebewritten(suppressingtheaccents totheletters) wherer__ r__ _ dtdpr dtdqr [q* {(q,-qtf+(9,-g5)2+(q3- Thissystem possessesanintegralofenergy,H=Constant, andthreeintegralsofangular momentum, namely whereAl}A2,A3areconstants. 158. Reduction tothe8thorder, byuseoftheintegrals ofangular momentum andeliminationofthenodes. Thesystemofthe12th order obtained inthe last article mustnowbe reduced tothe8thorder, byusingthethreeintegralsofangular momentum andbyeliminatingthenodes. Thismaybedone inthefollowing way. Applytothe variables the contact-transformation denned bythe equationsdW dW/ /1C* /\ q= Wr^= Wr(r=l&gt;2,...,6), where W=Pi (&lt;?/cosql~q*cosqtfsinqs)+p2(g/sinq,+q2cosq6cosqs)+p 3q2sinqt- +P* (q-Acosqs-q4cosg/sin qs)+ps(q3sinqs+q4cosg6cosq5)+p 6qtsinqgf . Itisreadilyseen thatthenew variables canbeinterpreted physicallyas follows : 157,158] Problem ofThree Bodies 345 Inaddition tothefixed axesOxyz,take anew setofmovingaxesOxyz; Ox istobetheintersection ornode oftheplane Oxywith theplaneofthe three bodies, Oyistobealineperpendiculartothis intheplaneofthe three bodies, andOzistobenormal totheplaneofthethree bodies. Then (&lt;?/,q^)arethecoordinates ofmlrelative toaxesdrawnthrough???3parallel toOx,Oy;(q3,q)arethecoordinates ofm.2relative tothesame axes;qs istheanglebetween OxandOx;q6istheangle between OzandOz;p{ andp2arethecomponentsofmomentum ofm1relative totheaxesOx,Oy ; p3andptarethecomponentsofmomentum ofm2relative tothesame axes; psandpsaretheangular momenta ofthesystemrelative totheaxesOz andOxrespectively. Theequationsofmotion interms ofthenewvariables are(138) dq^JbH_ dp^__dj[ dtdpr"dtdqr where, onsubstitution inHofthenewvariables fortheold,wehave Pi 1r1~PiP+Pp4~ -jf&gt;--T-, sL (q-2q3- qiq*/V* IB*^***" j.-./.A *a* j.* j.*&gt;/ . isq\qi) +p,qtcosecqt+ ,_oo)8KPifc-^V+PS?/- /&gt;/& )tfacot?/ +j35^2cosecg/+^sV}2 ^ )2 -Pa qi+Paq*-P*q)q*cotqa+psq4cosecq9+p 9q3} -Piqi+p3qt-ptq,)qscotqK+p,q2cosecq+p6q}} NOWT q5doesnotoccur inH,and istherefore anignorablecoordinate;the corresponding integralis PS=k, where A;isaconstant. Theequation dqa/dt=dH/dk canbeintegrated byasimple quadrature when therestoftheequationsofmotion havebeenintegrated;theequations forq5andp5willtherefore falloutofthesystem, which thusreduces tothe systemofthe10th order =- ^ (r=1,2, 3,4,6), where /&gt;/istobereplaced bytheconstant kwherever itoccurs inH. Wehavenowmade useofoneofthethreeintegralsofangular momentum (namely ps=k)andtheelimination ofthenodes :when theother two 34(3 TheRedaction ofthe [CH.xm integralsofangular momentum areexpressedinterms ofthenew variables, theybecome (paqi-piq3+Ptqs-pq*)sinqcosecq-ksin?cotq+P*cosqs=Ai, (pzqi-pi q*+Kofs/-jpsV)cos&cosec ?"+^cos?cotq+Psin?=^2- Thevalues oftheconstants J.!and J.2dependonthepositionofthefixed axesOxyz ;weshall choose theaxisOztobethelineofresultantangular momentum ofthesystem,sothat(cf.69)theconstants AlandAzarezero : thespecial a;?/-planethus introduced iscalled theinvariableplaneofthe system.Thetwo lastequationsthengive kcos(/=p*qi-Piq.2+^V-#V p.=0. Theseequationsdetermine q&lt;?andpainterms oftheother variables, and socanberegardedasreplacingtheequations dq_dH dpi__9-ff dt dp6"dt d&lt;/6" inthesystem.Thesystemthusbecomes dqr_dH&lt;W=_d# rr-12^4^ dt~dp r"dt~ dqr where H= +_^_ 2V2m12m3/j/1 (q3q*-qlq,)2 iq*-p*qi+psq*-^V)cot(?+&cosecW}2 ,t,^4 . {(Piq*-Pa qi+ps q*-ptqs)cotq6+&cosec^6}2 fl4/ iV-piqi+psq*-Piq-t)cot^+^cosec^}2 andwhere, afterthederivates ofHhave beenformed, q6istobereplaced by itsvalue found from theequation kcosqt=piqi-pfq*+pfq*-p-iq. Now letHbethefunction obtained when thisvalue ofq6issubstituted inH;then ifsdenotes anyoneofthevariablesg/,q2,q3,qt,p^,p2,p3,p4, wehave = ds~ ds dq6ds 158,159] Problem ofThree Bodies 347 ButsincepK=0,wehavedH/dq6= /&gt;=0,andtherefore ds fa inother words, wecanmake thesubstitution forq6inHbefore formingthe derivates ofH;andthus(suppressingthe.accents)theequations ofmotionof theProblemofThree Bodies arereduced tothesystem ofthe8thorder dqr_dff dpr_dH dtdpr dtdqr where H I*" ",- T^^ m2ms(q^+qf)2msm1(q^+q%2 )*mlm2{(&lt;?i qs)2+(q2q4J2 }- . Manyofthequantities occurringinHhavesimple physical interpretations: thus(qzqqiq 4)istwice thearea ofthetriangleformedbythebodies: and 2m1m2mA(/1 1\/I 1\ isthemoment ofinertia ofthethree bodies about theline inwhich the planeofthebodies meets theinvariableplane throughtheir centre of gravity. Itisalsotobenoted that thisvalue ofHdiffers from thevalue ofHwhen kiszeroby terms which donotinvolve thevariables pl,p.2,p3,pi.these terms inkcantherefore be regarded aspart ofthepotential energy, andwecansaythatthesystemdiffers from the corresponding systemforwhich kiszeroonlybycertain modifications inthepotential energy.Itmay easily beshewn thatwhen kiszerothemotion takesplaceinaplane. 159. Reduction totheQthorder. Theequationsofmotion cannowbereduced further from the8thtothe 6thorder, bymakinguseoftheintegralofenergy H=Constant, andeliminatingthetime. Thetheorem of141shews that inperforming this reduction theHamiltonian form ofthedifferentialequations canbe conserved. Astheactual reduction isnotrequired subsequently,itwillnot begiven here indetail. TheHamiltoniansystem ofthe6thorder thus obtainedis,inthepresent stateofourknowledge,theultimate reduced form oftheequations ofmotionof thegeneral ProblemofThree Bodies. 348 TheReduction ofthe[CH.xin 160. Alternative reduction oftheproblem fromthe18th tothe6thorder. We shallnowgiveanother reduction* ofthegeneral problemofthree bodies toaIlarailtonian systemofthe6thorder. LettheoriginalHamiltonian systemofequationsofmotion(155) betransformed bythecontact-transformation where W=Pi(q*~ ft)+p*(ft-ft)+P*(ft-ft) nil i^2 / Theintegralsofmotion ofthecentre ofgravity,whenexpressedinterms ofthenew variables, canbewritten qr=q*=q*=p?=p*=p*f= &gt; andconsequentlythetransformed systemisonlyofthe12th order: sup pressingtheaccents inthenew variables,itis dqr=dHdp^=_&lt;^_(r=l,2,...,6), dt dpr dt dqr where H=~(p?+pf+P/)+ 2~~,(PS+P?+Pe2 )-mlm.2(qf+q,2+q/} -f 2 *M \2)~2 2 .2 ,+m,/ f ?42 tli" *t&lt;2 Thenew variables maybeinterpreted physicallyinthefollowing way: LetGbethecentre ofgravityofm,andw2.Then(qltq2,qs)arethe *Due toKadau, Annales de IEc.Norm. Sup.v.(1868), p.311. 160] Problem ofThree Bodies 349 projectionsofm^n^ onthefixed axes, and(g4,q5,qs)aretheprojections ofGm3ontheaxes. Further ^=Pr(r-1,2,3),andl*^=pr(r4,5,6). ThenewHamiltoniansystem clearly representstheequationsofmotion oftwoparticles,oneofmasspatapoint whose coordinates are(qltq2,q3\ andtheother ofmass/*atapointwhose coordinates are(g^,qs&gt;q6);these particles being supposedtomovefreelyinspaceunder theaction offorces derivable from apotential energy represented bytheterms inHwhich areindependentoftheps.Wehave thereforereplacedtheProblem of Three Bodies bytheproblemoftwobodiesmovingunder thissystemof forces. This reduction, though substantiallycontained inJacobi s*paperof 1843, was firstexplicitlystatedbyBertrandf in1852. Weshallsupposetheaxes sochosen thattheplaneofxyistheinvariable planeforthemotion oftheparticles fj,andfjf,i.e.sothat theangular momentum oftheseparticlesaboutanylineintheplane Oxyiszero. LettheHamiltoniansystemofthe12th order betransformedbythe contact-transformation which isdefinedbytheequations dW dW^=W^= Wr 0-^-V-X where W=(p2sinqs+p1cosg5)q^cosq3+q,sinq3{(p2cosqs-plsinq,)2+p 32 }% +(pssinqs+p 4cosqs)q2cosqt+qzsinqt{(pscosq*-ptsinqt)*+p^. Thenew variables areeasilyseen tohave thefollowing physical inter pretations:qiisthelengthoftheradius vector from theorigintothe particle //,, q.,istheradius from theoriginto^,q3istheangle between &lt;// andtheintersection(ornode)oftheinvariableplane with theplane through twoconsecutivepositionsofq^(which weshall calltheplane ofinstantaneous motion of/A),&lt;/4istheanglebetweenq.2andthenode oftheinvariableplane ontheplaneofinstantaneous motion ofpf,q^istheangle between Ox andtheformer ofthese nodes, q^istheangle between Oxandthelatter of these nodes, pSispq\,pis^q^psistheangular momentum ofpround theorigin, plistheangular momentum of///round theorigin, psisthe angular momentum of/Around thenormal attheorigintotheinvariable plane,andpsistheangular momentum of/Around thesame line. Theequationsofmotion intheirnewform are(138) __dH_ dtdpr"dtdqr Journal furMath. xxvi.p.115. fJournal demath. xvn. p.393. 350 TheReduction ofthe[CH.xm whereHissupposed expressedinterms ofthenew variables. Let this systembetransformed bythecontact-transformation dW,dWpr= W"**57(r~12"-6)&gt; where W= q*"(p*-p*)+ &lt;i(p*+#)+qipi+q*"p*+q*"p*+qtp*. Theequationsofmotion nowbecome dqr-_dHdPr "_dH dtWdt" ~3q r" ButHdoes notinvolveq6",asmaybeseen either byexpressing H interms ofthenew variables, orbyobservingthatqfi" dependsonthe arbitrarilychosenpositionoftheaxisOx,while none oftheother coordinates dependonthisquantity. Wehave therefore p6"=- dH/dq,"=0, sop6"=k, where A;isaconstant;this isreallyoneofthethreeintegralsofangular momentum. Substitutingkforp6"inH,theequation q6"=dH/dk canbeintegrated byaquadraturewhen therestoftheequationshavebeen solved :sotheequationsforp6"andq6"canbeseparatedfrom thesystem, which reduces tothe10th ordersystem dqr"_~ ( 12_, dt~dp7"dt" dqr" Wehave still tousethetworemaining integralsofangular momentum; these, whenexpressedinterms ofthenew variables, arereadilyfound tobe represented by 25"=90, Jcp5"=p s"*-pr-; noarbitraryconstants ofintegration enter, owingtothefactthattheplaneof xyistheinvariable plane. Thesystem maytherefore bereplaced bythese twoequationsandthe equations dqr"_dHdfr" __3# , , "dt~dpr"dt dqr" where, inthis last set,qs"canbereplaced by90before thederivates ofH have been formed, andp5" istobereplaced by(p3"2p4"2 )/kafter the derivates ofHhavebeen formed. LetHdenote thefunction derived from Hbymakingthis substitution forp5",and letsdenote anyoneofthe variablesq", #&gt;",qa",q", p", p",pa",p"\thenwehave 8^_877dHdp,"=dH ..dp,"=dH ds dsdp&"ds ds ds ds 160,161] Problem ofThree Bodies 351 and itistherefore allowable tosubstitute for p"inHbeforethederivates of Hhavebeen formed. Theequationsofmotion arethusreduced toasystem ofthe8thorder, which(suppressingtheaccents) maybewritten intheform _ dtdpr dt~ dqr where, effectinginHthetransformations which havebeen indicated, wehave H- (p+ )+(p*+1)- ""*-1 .2ra2&lt;j/ kz-p,?-p?. . 5*--- *smg 3smg* ,2m!ftga/2-p,--p 4* . .\ ~m&lt;,ms-\q^+- Mcosftcosft --^--^-sinftsinft+7 (*ml+m^\ 2p,p t **J ( Theequationsofmotion mayfurther bereduced toasystemofthe 6thorderbythemethod of141,usingtheintegralofenergyHConstant andeliminatingthetime. Asthereduction isnotrequired subsequently,it willnotbegivenindetail here. 161. Theproblem ofthree bodies inaplane. Themotion ofthethreeparticles maybesupposedtotakeplaceina plane,instead ofinthree-dimensionalspace;this willobviously happenifthe directions oftheinitial velocities ofthebodies areintheplaneofthebodies. This case isknown astheproblem ofthree bodies inaplane:weshall nowproceedtoreduce theequationsofmotion toaHamiltoniansystemof thelowestpossibleorder. Let (&lt;/!,qz)bethecoordinates ofm1}(q3,q4)thecoordinates of ???2,and (^5. &lt;7e)thecoordinates ofms,referred toanyfixed axes Ox,Oyintheplane ofthemotion; and letpr=mkqr)where kdenotes thegreatest integerin ^(r+1).Theequationsofmotion are(asin155) dqr_*ff dpr__dH dtdpr dt~dqrl,-,..., Oj, where ^~to +jtf)+0tf+i*fH(^ g3- tfj)2+(ft- ft)2 ]~-M!wia{(q,-q3Y+(ft-g4)2 }" Theseequationswillnowbereduced from the12th tothe8thorder, by usingthefourintegralsofmotion ofthecentre ofgravity. Perform onthe variables thecontact-transformation definedbytheequations dW dW 352 TheReduction ofthe[OH.xm where W=plq1+p2q*+pqa+P*q*+(Pi+P* +P*) ft+(P*+P*+P) ti ltiseasilyseen that (&lt;//, ft)arethecoordinates ofmlrelative toaxes through Wgparalleltothefixed axes, (qatqf)arethecoordinates ofm2 relative tothesame axes, (qsfq)arethecoordinates ofw3relative tothe original axes, (pf, p.2)arethecomponentsofmomentum ofm1}(p3,pt)are thecomponentsofmomentum ofw2,and(p6,p6)arethecomponentsof momentum ofthesystem. Asin|157,theequationsfor &lt;?5,q6,p5,pidisappearfrom thesystem; and(suppressingtheaccents inthenew variables)theequationsofmotion reduce tothesystemofthe8thorder, dqr9# dpr_dH . dt= tyr -dl- Wr where -m2m3(q/+ft2 )~?-m,m, (q?+q22)~+h* {(?i- ?s)a+(q*~qtf]" Next,weshallshew that thissystem possessesanignorable coordinate, which willmakepossibleafurther reduction throughtwounits. Perform onthesystemthecontact-transformation defined bytheequa tionsdW ,dW .^=WrP^Wr where W=ptficosqt+paqisinqS+ps (qjcosq4f-qasm^)+P*(ftsinqt+ftcosg/). Thephysical interpretationofthistransformation isasfollows :g/isthe distance m1ms;ftandqiaretheprojectionsofm2m3on,andperpendicular to, m1w3;q4istheanglebetween mzmlandtheaxisofx;piisthecomponent ofmomentum ofm1alongm3m, ;p*andp3arethecomponentsofmomentum ofm.2parallelandperpendiculartom.^ ;andptistheangular momentum ofthesystem. The differential equations,whenexpressedinterms ofthenew variables, become dt dpr"dt where P-\f*v~w ik-*7+rt~*. 161,162]Problem ofThree Bodies 353 Sinceqtisnotcontained inH,itisanignorablecoordinate;thecorre sponding integralisp4=k,where kisaconstant;thiscanbeinterpretedas theintegralofangular momentum ofthesystem.Theequation q4=dH/dpi canbeintegrated byaquadraturewhen therestoftheequationshavebeen integrated;andthustheequationsforptand &lt;?/disappearfrom thesystem. Suppressingtheaccents onthenew variables, theequationscantherefore bewritten qr= _ Pr=__(r=l,2, 3),dtdpr dt dqr where H- &lt;?,-#2)2 -I-9s2 i This isasystemofthe6thorder;itcanbereduced tothe4thorderby theprocessof 141,makinguseoftheintegralofenergyandeliminating thetime. 162. Therestricted problem ofthree bodies. Anotherspecialcase oftheproblemofthree bodies, which hasoccupieda prominent placeinrecent researches, istherestricted problem ofthree bodies; thismaybeenunciated asfollows : Two bodies $andJrevolve round their centre ofgravity, 0,incircular orbits, under theinfluence oftheir mutual attraction. Athirdbody P, without mass(i.e.such that itisattractedbySandJ,butdoesnotinfluence their motion), moves inthesameplaneas8and &lt;/;therestrictedproblem ofthree bodies istodetermine themotion ofthebody P,which isgenerally called theplanetoid. Letmlandm2bethemasses of8andJ,andwrite m, m.2 SPJP TakeanyfixedrectangularaxesOX,OF,through 0,intheplaneofthe motion :let(X,Y)bethecoordinates, and(U,V)thecomponentsofvelocity, ofP.Theequationsofmotion are dt*~dX dt*~dY orintheHamiltonian form, dX=dH d7dH&lt;W__dH dV__3H dtatr dtdv dt"dx}~dt~ ar1 where H=%(U2+7s )-F. \v.D. 23 354 TheReduction ofthe [OH.xm SinceFisafunction notonlyofXandFbutalsooft,theequation H=Constant isnotanintegralofthesystem. Perform onthevariables thecontact-transformation which isdenned by theequations v_dW V_3W _^W_ _3EX~dU ~3V= dx= dy where W=V(xcosnt-ysinnt)+V(aesinnt+ycosnt), andnistheangular velocityof8J.Theequationsbecome dx_dK &lt;fy_c_Kdu=_dKdv=_dK eft~ 8wT ~dt~~ ~dv ~di~ dx dt~ fy dW where(138) K=H--^ =I(u2+v2 }+n(uy-vx)-F; itisatonce seen thatxandyarethecoordinates oftheplanetoidreferred tothemovinglineOJasaxis ofas,andalineperpendiculartothisthrough asaxisofy.Fisnowafunction ofxandyonly,soKdoesnotinvolve t explicitly,andK=Constant isanintegralofthesystem;itiscalled theJacobian integral*oftherestricted problemofthree bodies. Another form oftheequationsofmotion isobtained byapplyingtothe lastsystemthecontact-transformation dW dW 3W dW x= -^&gt;y= W&gt;p^^P2=^ where W=ql(ucosqz+vsinq2). Thenewvariables maybedefineddirectly bytheequations andtheequationsofmotion become dq-r_dHdp,.__dH ,=12) ~dt~ dp,.dt= dqr where H=(pc+~-np2-F. Another formf isobtained byapplyingtothese equationsthecontact- transformationdW ,dW , -,. Pr=Wr ^^Wr^ /.(222 1)where W=p2q2+ Au^n p7*\dv " .jpt\P,-(P^-P^}( Pl ) *Jacobi, Comptes Rendus, in.(1836), p.59. tAdopted byPoincare inhisKouvelles methodes delaMec. Celeste. 162] Problem ofThree Bodies 355 where udenotes acurrent variable ofintegration. Theseequations maybe written J)/22(7 Q~\* g/=arccos-iJV--^+~4*--^Vj,#2=a2-arccos2p. and itiseasilyseen that &lt;?/isthemeananomalyoftheplanetoidinthe ellipsewhich itwould describe about afixedbodyofunitmass at0,if projected from itsinstantaneousposition with itsinstantaneousvelocity ;q2 isthelongitudeoftheapseofthisellipse, measured fromOJ;pisa*,and p2is{.(!-e2)p,where aisthesemi-majoraxisand eistheeccentricityof thisellipse.Hdoes notinvolve texplicitly,soH=Constant isanintegral oftheequationsofmotion, which arenow dq^=aff dprf_3ff dt~dp r"~dT~~dq?1}2^ Ifwetakethesum ofthemasses of8andJtobetheunitofmass, and denote these masses by1- /u,andprespectively, wehave ~SP This isananalytic function ofp^,p2,q,f ,qat^which isperiodicinqfandq2, with theperiod2?r. Moreover, tofindthetermindependentoffj,inH,we suppose ptobezero];sinceSPnowbecomesqltwehave 1\11 Thusfinally, discardingtheaccents, theequations ofmotionoftherestricted problem ofthree bodiesmaybetaken intheform dqr_dHdpr_dH dt~~dp r ~dt~~~3^ whereHcanbeexpanded asapower-series inpintheform whileHltH2,...areperiodic inq1andq2,with theperiod2?r. Theequationsofthis4thordersystem maybereduced toaHamiltonian systemofthesecond orderbyuseoftheintegral#=Constant andelimina tionofthetime, asin141. 232 356 *TheReduction ofthe[CH.xm 163. Extension totheproblem ofnbodies. Manyofthetransformations which havebeen used inthepresent chapter inthereduction oftheproblemofthree bodies canbeextended soasto applytothegeneral problemofnbodies which attract each otheraccording totheNewtonian law. Intheiroriginal form, theequationsofmotion of thenbodies constitute asystemofthe6?ith order;thiscanbereduced to the(Qn 12)th order, byusingthesixintegralsofmotion ofthecentre of gravity,thethreeintegralsofangular momentum, theintegralofenergy,the elimination ofthetime, andtheelimination ofthenodes. Thereduction hasbeen performed byT.L.Bennett, Mess, ofMath.(2)xxxiv.(1904), p.113. MISCELLANEOUS EXAMPLES. 1.Ifintheproblemofthree bodies theunits aresochosen thattheenergy integralis1111 i(vi2+V+*V) =++--- , ?*23r3lr12r where r12isthedistance between thebodies whose velocities are Viandv2,and ifrisa positive constant, shew thatthegreatest possible value oftheangular momentum ofthe systemabout itscentre ofgravityisf\2r. (Camb. Math.Tripos,PartI,1893.) 2.Intheproblemofthree bodies,let*beJacobi sfunction,letQbetheangle between anyfixed lineintheinvariable planeandthenode oftheplaneofthethree bodies ontheinvariableplane,letibetheinclination oftheplaneofthethree bodies tothein variableplane, and let77bethearea ofthetriangle formed bythethree bodies. Shew that dQ,_k &lt;fc "* j_di=fM_n* dt where kistheangular momentum ofthesystemround thenormal totheinvariable plane. (DeGasparis.) 3.Lettheproblemofthree bodies bereplaced bytheproblemoftwobodiesp.and/* asin160 :letqiandq2bethedistances ofp.andp.from theorigin:letq3andq4be theangles madebyqland &lt;?2respectivelywith theintersection oftheplane through the bodies andtheinvariable plane:letplandp2denotepqiandpfq-2respectively;and letp3 andpibethecomponentsofangular momentum ofp.andp.respectively,intheplane throughthebodies andtheorigin. Shew thattheequationsofmotion maybewritten dqr_dH dpr_dff } dityS ~di~~dq r whereH=Constant istheintegralofenergy. (Bour.) 163JProblem ofThree Bodies 357 4.Applythecontact-transformation defined bytheequations qi={(V*-?7)2+(?5-?8)2+(26-?9)2 }*, 92={(?7- &lt;?i)2+ (&lt;? 2- 26)2A 6=mlq+m =" &i(?i+iq$+b2(g-4+t Pr=sX- (r-0, 1,8,...,8), A;=0 O^r (wheretstands forV^landi, 2, s&gt;& i&gt;^2,bs,cl5c2,c3areanynine constants which satisfytheequations a1+2+3= ) ^1+^2+^3=0,^+02+03=0,a2&3-a 3&2=l), totheHamiltonian systemofthe18thorder which(155)determines themotion ofthe three bodies. Shew thattheintegralsofmotion ofthecentre ofgravityare q&= q-i=q=p&=PI=pa=- Shew further thatwhen theinvariable planeistaken asplaneofxy,thevariablep5is zero,andthattheintegralofangular momentum round thenormal totheinvariable plane is piqi=k, where kisaconstant. Hence shew thattheequationsreduce tothe8thordersystem dq,!_dff dpr_dff ~dt~-ty? ~dt~ fy? where Reduce thistoasystemofthe6thorder, bythetheorem of 141.(Bruns.) CHAPTER XIV THETHEOREMS OFBRUNS ANDPOINCARE 164. Bruns theorem. (i)Statementofthetheorem. Wehave seen(155)thattheproblemofthree bodiespossesses10known integrals:namelythe sixintegralsofmotion ofthecentre ofgravity,the threeintegralsofangular momentum, andtheintegralofenergy;these are generallycalled theclassicalintegralsoftheproblem.Each ofthem isan algebraic integral,i.e.isoftheform /(&lt;7i, &lt;?2,.,q9,Pi,P*&gt;-&gt;Pa,t)=Constant, where/isanalgebraicfunction ofthecoordinates(q1}q2, ,$9,PI, -,^9) andof t. Efforts havefrequentlybeenmade toobtain otheralgebraic integralsof theproblemofthree bodies independentofthese(i.e.notformed bycombina tions ofthem), butwithout success; and in1887Bruns* shewed thatno suchnewalgebraic integralsexist;inother words, theclassicalintegrals are theonlyindependent algebraic integrals oftheproblem ofthree bodies. Itmayberemarked +thatthenon-existence ofalgebraic integralsdoesnotnecessarily imply great complexityinasystem. One ofthesimplestofdifferential equations, namely thelinear differential equationwithconstant coefficients hasnoalgebraic integral except when^i//n 2isarational number,inwhich case the first integral-^=Constant canbetransformed intoanalgebraic integral. (ii) Expression ofanintegralintermsoftheessential coordinatesofthe problem. We shallnowproceedtoaproofofBruns result, consideringfirstthose integralswhich donotinvolve texplicitly. *Berichte derKgl. Sachs. Ges. derWiss. 1887, pp. 1,55;Ada Math. xi.p.25. Cf.also Forsyth, Theory ofDifferential Equations,Vol. in.(1900),Ch.xvn. tCf.K.Bohlin, Astron. lakttagelser ochUnders. aSfockholms Observ. ix.(1908), Nr1. 164] TheTheorems ofBruns andPoincare 359 Theequationsofmotion oftheproblem may (160)bewritten inthe form dqr_dH dpr_dH W-fa ~dt~ Wr where H=T-U, T= 2P(P*+p*+P^+5?(^+p^+P^ )-* _+q3q6)+ \ m3(m l+m2)m3 Weshall write ===/, ^= yu,5= //,6= y 6 sothat T=2 Letthecoordinates ofthethree bodies be(g/,g/,^ (g/,qs,q6),(g/,gg,g/), and letmkqr=pr,where kdenotes thegreatest integercontained in(r+2): theintegrals whose existence weproposetodiscuss areoftheform &lt;(qi,q*, -,q9,Pi,,p9)=a, where aisanarbitrary constant and (j&gt;isanalgebraicfunction ofits arguments. The formulae of160enable ustoexpressthevariables (&lt;?/, q*,,qLp\, ,p&)aslinear functions of(qltq2,...,q6,p1}...,p6):we shall therefore, onmakingthese substitutions intheintegral,obtain an equation /(?i, q2,-..,q,pi, ...,p6)=a .....................(2). Iftheintegral&lt;iscompoundedoftheintegralsofmotion ofthecentre of gravity,/willevidently reduce toaconstant;ifnot,/willbeanalgebraic function ofthevariables(q1}...,q6,pl,...,ps).Wehave toenquireintothe existence ofintegrals, such as(2),oftheequations (1). (iii)Anintegral must involve themomenta. WTeshall firstshew thatanintegral such as(2)must involve some ofthe quantities p,i.e.itcannot beafunction of(5-^q2,...,^6)only. Forsuppose,ifpossible, thattheintegral, say doesnotinvolve(pltp.2,...,p6).Differentiating withrespecttot,wehave Q_ j.df._4d/jy 360 TheTheorems ofBruns andPoincare[CH.xiv andtherefore theequations J^=(r~l,2, ...,6) dqr must besatisfiedidentically;that is,/does notinvolve(ql}q2,...,q6),and soisamere constant. (iv) Onlyoneirrationality canoccur intheintegral. Asthemutual distances ofthebodies are irrational functions of (qltg2j q6}}thefunction Uwillbeanirrational function ofthese variables. Denoting bysthesum ofthethree mutual distances, itiseasily seen thatthemutual distances canbeexpressedasrational functions ofthe sevenquantities (ql}q2,...,q6,s);inother words, theirrationalities involved inthemutual distances areallcapableofbeing expressed bymeans ofthe irrationalityofs;wemaythereforesupposethatUisexpressedasarational function of(q1}q.2,...,q6,s). Now thefunction /isalgebraic,butnotnecessarily rational, inthe variables(q1}...,q s,pi, ...,p6),lettheequation (2)berationalised, and let theresulting equationbearrangedinpowersofa,sothat itbecomes am+am-1 (f)l(ql,q,,...,q e,pi, ...,p6)+aw~2 &lt;/&gt;2(gi,&gt;9s,PI,...,p*)+... + &lt;/&gt;m(?i, ...,q 6,pi, ...,2&gt;6)=...(3), wherefa,fa,&gt;fanarerational functions of(qlt...,qK,plt...,p6).Ifthis equationisreducible inthevariables(qlt...,q6,plt...,p6,s},i.e.ifitcanbe decomposedintootherequations,each oftheform a1+al~l^(ql,...,q 6,pl,...,pe&gt;s)+...+^i(q1,...,q6,p1,...,p 6,s)=0...(4), where^,tyz,...,&gt;|rarerational functions of(ql,...,qfi,plt...,p6,s),thenone ofthese lastequationswillgivethevalue ofawhichcorrespondstoequation (2),andweshall consider thisequationinstead of(3).Asthetypeof equation represented by(4)includes thetype represented by(3)asa particularcase,weshallsupposeatobegiven byanequationoftheform(4), irreducible in(q1}...,qs,plt...,p6,s). Differentiatingwithrespecttot,andusing equations (1),wehave al-l(^H} +a^(^,H) +...+(^l,H)=Q(5), where (tyr ,H}denotes asusual thePoisson-bracket oftyrandH. We shall firstsupposethat theexpressions (//&gt;,H},which arerational functions of(q1}...,q6,pl}...,p6,s),arenot allzero. Thenequations (4) and(5)have oneormorecommon roots a,andconsequently equation (4)is reducible in(q1} &lt;?2 &gt; ,q&,PI,,Pe,*);butthisequationisirreducible, and therefore thishypothesisisinadmissible, andthequantities (-v/rr,H}areall zero. Thisimpliesthat allthecoefficients(^r l,ifr2&gt;..., tyi)inequation (4) areintegralsoftheequations (1):andhence theintegral fcan becom pounded algebraically fromotherintegrals,which arerational functions of (?i,...,% Pi, ,&&gt;*)- 164]TheTheorems ofBruns andPoincare (v)Expression oftheintegralasaquotient oftworealpolynomials. Weneed therefore henceforthonlyconsiderintegralsofthetype ?,... q,pi, ...,p,s)=a.....................(6), where /isarational function oftheargumentsindicated. Theform of/can befurther restricted bythefollowingobservation. Ifintheequationsof motion wereplace qr,pr,tbyqrk2 ,prk~l ,andtk3 ,respectively,where kisany constant, theequationsareunaltered. Iftherefore these substitutions are made inequation (6),thisequation must stillbeanintegralofthesystem, whatever kmaybe. Now/isarational function ofitsarguments:itcantherefore beex pressedasthequotientoftwofunctions, each ofwhich isapolynomialin (qi, $2,&gt;(?e&gt;PI, &gt;P&lt;&gt;&gt;s)-When inthesepolynomials wereplace qr,pr,s byqrk2 ,prk~l ,sk-,respectively,thefunction /will(onmultiplyingits numerator anddenominator byanappropriate powerofk)take theform where (A,Alt...,Bq)arepolynomialsin(q1}...,qe,plt...,p 6,s).Since df/dtiszero,wehave "f^Efr*-** fk"+ Now kisarbitrary,sothecoefficients ofsuccessivepowersofkinthis equationmust bezero;andtherefore /74 /-77?U/-iln .LLJJnn_ dA dA, dB dB, =5,^8adB These(q+p+l)equationsareequivalenttothesystem _ __ _Adt~A, dt~Ap~dT~B~dt fromwhich itisevident thateach ofthequantities5grfi A2 B B 362 TheTheorems ofBruns andPoincare[OH.xiv isanintegral:andthuswehave theresult thatanyintegralsuch asfcanbe compounded fromotherintegrals,which areoftheform where eachofthefunctions6r1?G.zisapolynomialinitsarguments, and is merely multiplied byapower ofkwhen thevariablesqr,pr,sarereplaced respectively byqrk2 ,prk~l ,sk2 .Weneed thereforeonlyconsiderintegralsof thisform. Itmayfurther beobserved thatthefunctions GlandG2may,without loss ofgenerality,betaken tobefreefromimaginaries.For ifPandiQdenote therealandimaginary partsofanintegral P+iQ=Constant, dP .dQ.., .. ,,wehave-j-+i-~=0,identically.at ctu Since thedifferentialequationsarefreefromimaginaries,itfollows that dP/dt anddQ/dt arefreefromimaginaries:andsodP/dtanddQ/dt must be zeroseparately. HencePandQarethemselvesintegrals,andevery complex integralcanbecompoundedfrom realintegrals. Weshall therefore hence forthassume thatGi/G zisfreefromimaginaries. (vi) Derivationofintegrals fromthenumerator anddenominatorofthe quotient. Itmaybethecase that thefunction GIisresoluble intoaproductof irresolublepolynomialsin(pltp2,...,p 6),thecoefficients inthesepolynomials beingrational functions of(qltq2,...,q,s).Let^besuch apolynomial,and supposethat itisrepeatedA,times inGl:and letxdenote theremaining factors ofG1}sothatG^^ X. WhenGlisirreducible, weshall ofcourse have GI=ty,and%=1. Theequation T^\TT}= at\Cr2/ \dilr 1d-y 1dG2. -r-TTH--- -JTTTJT=U, vrdt dtG2dt dG. 2 Nowd^fr/dtispolynomialin(plt...,p 6),and-v/risalsopolynomialin (p1}...,p6),oforder lessbyunitythan theorder ofd-ty/dt. Also, tyhasno factor incommon withG2or%.Hence weseethat 1dG21dx \G2dt\dt 164] TheTheorems ofBruns andPoincare 363 must beapolynomialin(p1,...,p6),oforderunity:denote thispolynomial byco :thenwehave Itmaybeshewn inthesamewaythateach oftheother irreducible factors ofGlsatisfies anequationofthiskind. Denote thevarious factors ofGby ^,i/r",...,sothat and lettheequations they satisfybe JL^-&lt;tyVdt""Y thenwehave 1dGlndty vd^r" where &&gt;isapolynomialin(plt...,pe),oforderunity, and rational in (ft,...,q 6,s).Thus G!satisfies theequation dGl= andtherefore(since GJGzisanintegral), G2also satisfies theequation ~dt~ AsGlandG2satisfythesame differentialequation, weshall infuture use todenote either ofthem: so9isarealpolynomialin(p1}...,p 6,ft,...,q G,s), which satisfies theequation 9= o&gt;&lt;. Now &lt;f&gt;ismerely multiplied byapowerofkwhenft.,pr,sarereplaced respectively byqrk2 ,prk~\sk2 :since m= ~I~TT=^T I^1~^^ )&gt;9acr=i9V^ft- A6oproqr/ weseethat &&gt;ismultiplied by^~3when thissubstitution ismade. Itfollows that &)cannot contain atermindependentof(plf...,p6),since suchaterm would bemultiplied byanevenpowerof&;&&gt;istherefore oftheform where each ofthequantitieswrishomogeneousofdegree1inthequantities (ft,...,?,). Further, letoneoftheterms in &lt;/&gt;beofordermin(plt...,p6)andoforder nin(ft, ...,q&,s),while another term isofordermin(pl}...,p 6)andof order nin(ft, ..., &lt;?6,s):since these terms aremultiplied bythesame powerofkwhen theabove substitution ismade, wehave m+2n=m+2ri, sommisanevennumber. Hence &lt;/&gt;canbearrangedintheform 364 TheTheorems ofBrans andPoincare[CH.xiv where &lt;/&gt;denotes theterms ofhighestorder in(plt...,p6),&lt;.2denotes terms oforder lessbytwounits in(pl,...,p s)than these, andsoon:andeach of thequantities&lt;f)risapolynomialin(pl}...,ps,ql}...,q K,s),homogeneousin (pi,"-, Pe)andalsoin(#1,...,qt,s}. We shallnowshew thatwhen &lt;/&gt;does notinvolve s,&lt;canbemade intoan integral bymultiplyingitbyanappropriaterationalfunction of(q l9...,&lt;?). Forsupposethat &lt;/&gt;doesnotinvolve s:theequation tty or-+-+ ......=wxp!+w2p24-...+ ft gives,onequatingtheterms ofhighest degreein(pl&gt;...,p6), Ip,.^0= ^ &gt;+ r=l^roqr Now &lt;^&gt;maycontain p6asafactor :inorder totakeaccount ofthis case, write &lt;f)= p&lt;f &lt;j&gt;o,where &lt;f&gt;does notcontain p6asafactor, andwhere asa specialcasewemayhave k=0,$=$.Substituting p6k &lt;j&gt;for&lt;inthe differentialequation,itbecomes Let ^&gt;"denote those terms in &lt;which donotinvolvej;6;equatingthe terms which donotinvolve p6onthetwosides ofthisequation,wehave Itmaybethat &lt;/&gt;"isamere function ofqltq2,...,qK,sayequaltoR;in thiscasewehave 13R D / no -\~^=a)rR(r=1,2,...,o) /ardjV 19^2x_. , orfjira)r=-^^- (r=l,2,..., o),XIC/gr 3o andtherefore(/n.ra&gt;r)=(/*&)) (r,s=1,2,...,5). d^g c;^ Supposingnext that&lt;"does involve some ofthequantities (p1}...,p 5), itmayinvolve psasafactor: totake account ofthis case,wewrite &lt;f&gt;"=p^fa" ,where &lt;"does notinvolve p5asafactor. Theequationnow becomes r=\ 164] TheTheorems ofBruns andPoincare 365 Let&lt;ivdenote theterms in$"which donotinvolve p5:equatingthe terms which donotinvolve psonthetwosides ofthisequation, wehave r=l f^rOqr Proceedinginthisway,weultimatelyarrive atthealternatives, that either orelseafunctiontyexists, which ispolynomialinqlf...,q6,PI,p2&gt;homo geneousinql}...,q6andalsoinpl}p2,and isfreefromanyfactors which are merepowersofplandp2,andwhich satisfies thedifferentialequation^+2a^ /MI3^ frdq-2 Now letty=apj+bp2l+cp^p.* + ...; equatingcoefficients ofp^+1andp2l+lonthetwo sides ofthe lastequation, wehave Ida 1dbwl=-- "^&gt;W1=-7^ fra d^j /A2o0^2 Thequantities a,b,c,...arepolynomialsin(q1}q.2,...,q6):theymayhave acommonpolynomialfactorQ,sothat aaQ, b=bQ,etc. Let-v//=apj-f&X+cp^ps + ..., sothat-^r=Q\Jr. Then where/^ ^ )ft Q/AJ9g( 1/ Ji^j+6)2j52,say, 1da 1 &&gt;!= Pi^p2-\/ /\iso-/-+--g-= (&&gt;!p1+ a&gt;,p. 2}i/r. /*i3^ y^29g2 The left-hand side ofthisequationisapolynomialin(c^, q.2,...,qK, Pi,pz))tut ifa/contains9^then tw/contains a,orsome factor ofit,as adenominator. Hencetymust contain a,orsome factor ofit,asafactor. But this isinconsistent with thesuppositionthata,b,...havenocommon factor. Hence acannot involveq1;andtherefore &&gt;/iszero.Similarly wziszero. 366 TheTheorems ofBruns andPoincare[CH.xiv rp,IdQIdQThus &&gt;,=-7^ ^r- ,to2 andtherefore^d which isthesame asthealternativepreviouslynoted :sothisequationis true inanycase. Similarly wecanshew ingeneralthat ^&lt;^-^.OM*X andhence wemaywrite whereRissome rational function of(q1}q2,...,q6). Thuswehave 4pr1dR a}1p1+o)^p2+...+co6p6=Z- -p~ &gt;=].pr"^(?r r=l.R3gV* (/&gt;dRdt andtherefore ^=Constant. Thus&lt;can 6etransformedintoaconstant, bymultiplyingitbyan appropriaterational function ofqltq.2,...,q6,namely 1/.R;which isthe requiredresult. Iftherefore theterms &lt;/&gt;inGland G.2donotinvolves,wecantransform GlandG2intointegrals, bymultiplyingthembyappropriaterational functions of(q\, ^2.&gt;9e);andhence, ifitcanbeshewn thattheterms&lt;inG1andG2 donotinvolve s,weshall have theresult thatanyalgebraic integralof theproblemofthree bodies canbecompoundedfromintegralswhich are polynomialin(pltp2,...,p6)and rational inq1}q2,...,q6,s. (vii) Proofthat &lt;f&gt;does notinvolve theirrationalitys. Thecase inwhich&lt;involves sisnotincluded intheaboveinvestigation. Weshallhowever nowproceedtoshew thatnorealfunction&lt;,which satisfies anequation caninvolve s;andhence that thefunctions &lt;/&gt;occurringinourproblemdo notinvolves,sothattheabove result isquite general. 164] TheTheorems ofBruns andPoincare 367 Forsupposethatafunction&lt;exists, which involves sand satisfies the above differentialequation. When the8values ofsaresubstituted suc cessivelyin &lt;f&gt;,&lt;nwilltakeanumber ofdistinct values;letthese values be denoted by&lt;/&gt;, &lt;$&gt;&lt;&gt;&gt; Jthey satisfy equationsoftheform ^prd(f)Q , ,4prd(f&gt;" 2,---=to9o, 2,- -5=a),..., r=lPr3qr r=lPrfyr where a/, fa",...arethevalues of o&gt;when thevalues ofscorrespondingto &lt;&lt;&gt; , &lt;f&gt;", respectivelyaresubstituted in it. Let 4&gt;= &gt;&lt;&gt; ".... Thenwehave | | &lt;f&gt;o"fyr i .//i =&)+&&gt; 4-... mQ, where IIisalinear function of(pi,p 2,...,p 6),thecoefficientsbeingrational functions of(q1}q2,...,q6). Now O,from themanner ofitsformation, isarational function of (qlfqz,...,q K),notinvolvings:and itisclearlyapolynomialin(pltp2,...,p s). Sowecanapplyto &lt;l&gt;theresultsalready obtained, which shew that (on multiplying bysome rational function ofqltq2,...,ge)Oiszero,and therefore that &lt;f&gt;satisfies theequation This isapartialdifferentialequationfor &lt;& :there are6independent variables, and 5independentsolutions can atonce befound, namelythe quantities f^1-fc&V.,(Ml_^|.Itfollows that 3&gt;isafunction VPi (**/ \Pi PKJ onlyofthequantities ,,P, ....p.. Now thefactors of &lt;I&gt;differ from each otheronlyinthat different roots s areused intheir formation :sowhen such arelation exists between (&lt;?!,q2,...,q6)thattwo ofthese roots sbecomeequaltoeach other, then twofactors of willbecomeequaltoeach other;hence if &lt;l&gt;=beregarded asanequationinjo1}atleast two roots willbecomeequaltoeach other. When this relation f(&lt;h, q-2,...,9)= exists between(q1,q2,...,q6),weshall therefore haved^/dp^=0;andsimilarly ,...,3&lt;f&gt;/9p 6willeachbezero. 368 TheTheorems ofBruns andPoincare[CH.xiv Since&lt;E&gt;ishomogeneousin(p1}p2, .,p6),theequation Pi -&gt;HPioI-...+p6K-= isequivalentto &lt;I&gt;=:so &lt;&=doesnotconstitute anequation independent oftheequations 94&gt;/8pj=0,...,d&lt;&/dp6=0. Ifsmall variations aregiventothevariables whichsatisfytheequation &lt;t&gt;=0,their increments areconnected bytheequation 2 r=1 but if(qlt?.., ?6,Pi, Pe)satisfytheequations d$&gt;/dp r=0,thisequation becomes 6 andthisrelation between theincrements 8^rmust therefore beequivalentto therelation 6df2f-Bqr=0. r=ioqr Hence theequations 67)r3&lt;J) . areconsequencesoftheequations d&lt;$/dp r= ;and so,since2g~iszero, wehave (forsets ofvalues ofqltq2,...,q,plt...,p6whichsatisfythese equations) Theequations f=and2^r~= arethereforealgebraicallyderivable r=1 from theequations 94&gt;/3p r=0.Now theactual values of(g^....,g 6)are ofnoimportanceinthisalgebraical elimination; sowecanreplace qrby pr)ina11tneequations:andthusweseethat theequations 2 l+!t....,,+,. 8, r=1/^dty arealgebraical consequencesoftheequations 7\ ft) 3&gt;(qlt...,q8,p1,...,p 6)--^T 164] TheTheorems ofBruns andPoincare 369 Hence theresult ofeliminatingtbetween theequations AH must beanalgebraiccombination oftheequations (r=l, 2,...,6). Now onesuchalgebraiccombination oftheseequationsis &lt;(?!, ...,q 6,p!, ...,#,)=(); foritcanbederived bymultiplyingtheequations by(p l,...,ps)inturn,and addingthem.Weshallshew that itistheeliminant which hasjustbeen mentioned. Forlettheeliminant inquestionbedenotedbyW;then theequation 6/axir, /Ui.15-8gr r=iV3gr- must beacombination oftheequations r=lo and I1#a1.+i .-8,,+8^+ S(=0. Since thelatterequationinvolves Si,weseethat itcannot enter intothe combination :andsowehave Theidentityoftheseequationswith those which havealreadybeen found for &lt;J&gt;shews thattheequations&lt;!&gt;= and^= areequivalent.Hence &lt;t&gt;= istheeliminant oftheequations and .3* Now theequations f(q ltq2, ,^e)=0,which aretheconditions thatthe equationforsmayhaveequal roots, caneasilybewritten down :and this result enables usthen tofind allpossible polynomials&lt;1&gt;,andhence, by factorisation of &lt;J&gt;,tofind allpossible polynomials&lt;/&gt;. w.D. 24 370 TheTheorems ofBruns andPoincare[OH.xiv Theeightroots saretheeightvalues oftheexpression +rlr^r 3, where r1}r2,r3denote themutual distances: sowemayhave two roots s equalasaresult ofanyoneoftheequations ?i=0,r2=0,r3=0,rz=r3,r3=r1}i\= r.2,1\r.2r3=0. Theequationrx=gives &lt;tf+q^+q:?=0; andtheeliminant of p3t\2 +=0 =8-fi&fc&Y sothevalue of &lt;I&gt;arisinginthisconnexion is yu-3 7x2 thisexpressionisnotresoluble into real factors, andtherefore norealpoly nomials &lt;/&gt;canarisefrom thissource. Asimilar result canbededuced inconnexion with theequationsr2= andr3=0. Consider next theequation r,=r3; itcanbewritten intheform -I+1, Im7T-9*M*M. in f/tj"T~fits \*^1i or 2(9lgr4+?2^5+?3?6)-^^T(?!2+^+q*)=0. //tj ~| ^2 Replacing 5-,.by(qr+prt/f^r),andformingthediscriminant withrespect totoftheequationthus obtained, wefind =2(ql9t+q*q5+qsq6)+^^(ft"+?+ ^1^4,^2^5_,P3-- 1---- 1--~( o~r o"i~ Ifml+m 2\/j, l-^2 pf/) This expansioncannot befactorised intopolynomials&lt;/&gt;,linear in {p\,P-2, --ipe),sonofunctions ^&gt;canarise from this source. TheTheorems ofBrims andPoincare 371 Similarlyitmaybeshewn thatnopolynomials &lt;/&gt;canarise inconnexion with theequationsrs=rr,rl=+rz. Lastly,therationalised form oftheequations 1\r2r3= is(r32-r?+r*}*-4rfa*=0. Wheni\iszero, thiscasereduces tothatwhich was lastdiscussed :and since thepolynomial&lt;l&gt;isnotresoluble inthisspecial case, itcannot be resoluble inthegeneralcase. Thusfinally,norealpolynomials $ ,involving s,can exist. Summarisingthe results obtained hitherto, wehaveshewn thatany algebraic integralofthedifferentialequations, which does notinvolvet,is analgebraicfunction ofintegrals &lt;/&gt;,each ofwhich canbewritten inthe form &lt;f&gt;0+ &lt;/&gt;2+ &lt;/&gt;4+, where&lt;isahomogeneous polynomialinthevariablesp,sayofdegree k, andahomogeneous algebraicfunction ofthevariablesq,sayofdegreeI: &lt;/&gt;2isahomogeneous polynomialinthevariablesp,ofdegree (k-2),and ahomogeneous algebraicfunction ofthevariablesq,ofdegree(1);&lt;4is ahomogeneous polynomialinthe variablesp,ofdegree (k-4),anda homogeneous algebraicfunction ofthevariablesq,ofdegree (I2) ;and soon. (viii) Proofthat &lt;/&gt;isafunction onlyofthemomenta and theintegrals ofangular momentum. We shallnowproceedtoshew thatanintegral &lt;f&gt;,characterisedbythese properties,isanalgebraic function oftheclassicalintegrals. Theequation g-o,ator givesonreplacing &lt;f&gt;by&lt;4-&lt;2+$4+,andequating terms ofequal degree, _r , ^5 h5 ~, r=\oqrHr dp, dq,. r **~~~ r=l _ dqr 242 372 TheTheorems ofBruns andPoincare[CH.xiv The firstofthese equationsisalinearpartialdifferential equationfor&lt; which canatoncebesolved, andgives &lt;=/(P2,P3,...,P6,pi,Pz,..,p), , p Qrpl PrQl &lt;V9^ fi\where "r=---(iAo,..., 01. /Ltj fj,r Lettheexpressionof &lt;2interms ofthevariablesft,P2,...,P6,PI,.P* be 4&gt;2=/2(ft,P 2,PS,...,P s,pi,...,p), 3/28&lt;f&gt;24d&lt;b2dqr frP r ,wehave /-= a+2aF ^"wheregr=+ _- dqlr~-2 or ^r Mi8 Integratingwehave sothere canbenologarithmic terms in^Xdqlywhere X=S^-^,expressedinterms ofqltP2,...,Pe,Pi&gt;&gt; =, ., ._*8Pr IfFdenotes theexpressionofUinterms ofthevariables qP,P6,pi, ...,pe, wehave -= ^-rj (r&gt;1)and *=~S5^5- . Theterms inXwhich maygiverisetologarithmicterms in IXdq^are nowseen tobe sotheterms which maybelogarithmicin jXdq^are Vdqt+. *r=2=29 pl8 164] TheTheorems ofBrims andPoincare 373 NowVisasum ofthree terms, each oftheform(A+Bql+Cq^y-. Takingeach ofthese termsseparately, wehave forthetranscendentalpart ofthe lastexpression 49/0Pr !V-(7arcsm fprps185.2&lt;7o, *..... Qyvcnn * &lt;ML V&gt;Olll 8/0p,1dB 2Cq, &gt;tZ-- fl,rr.sinarcsm Thus foreach ofthefractions (A+Bq l+Cq^^, wemust have gr=2 r ./A, r Now forthe firstofthese fractions, namely (q^+q.*+q/)~^, wehave sothe firstofthethreeequationswillbe r=2 8P2^28PS or(since /ix= yu.2= yu,3) andonsolvingthisequation weseethat/isafunction of Pi,P*, -,Pe,P*,Ps,(p^ p^qt), and(p^qs psqt). Since thethreeexpressions (A+Bq1+Gq^)arelinear functions ofthe threequantities (q,2+q22+qs2 ),(q.q.+q^+q.q,), (qt*+qf+q,*),wecan for ourpresent purpose replace thembythese threequantities:sothesecond expression (A+Bq^+Gq^)maybetaken tobe(q^ 4+q2q5+q3qK},or (/iP4 /J,pt\fftP2P^q\\ ff^Ps /^PsQi\ if^Ps PaQi\ /AtPfiH /?iI+ I I h,1+ I h^ 11-+ PIPP1 I\pi PiI\Pi fjfpl} \Pi pi soforthisexpression wehave Pi2VPS Pi* 374 TheTheorems ofBruns andPoincare[OH.xiv andthecorresponding equationis -P. - l- ,-...(B). 0-*5\/* /0*i \/* The thirdexpression (.4+.B^+Cijj*)maybetaken tobe &lt;?42-fqs~+q*~; thecorresponding equation provestobethesame asequation (A),andmay consequentlybeneglected. Wehave thereforeonlytoconsiderequations (A)and(B):simplifying (B)bymeans of(A),theymaybewritten -P,Pi+P-2+PS=I theseequationsareobviously algebraically independent;andtheJacobian conditions ofexistence aresatisfiedidenticallyforthem, since thecoefficients r)f ofthederivates^donotinvolve thequantitiesP.These twoequations therefore form acomplete system,with 5independentvariables P2,P3,P4, P5,P6:sothere must be52=3independent solutions, andanyother solution willbeafunction ofthese three solutions andofpltp2,...,pti. Itiseasilyverified thatthreeindependentsolutions are 2+P5p6-P6ps, +P 6p4-P4p6, or -q3p2+qsp6-q6ps, where M=q^-q,p3+q6p,-q,p 6, \ andthethreeequations L=Constant, M=Constant, N=Constant arethethreeintegralsofangular momentum ofthesystem. Wehave therefore theresult that &lt;wfunction ofL,M,N,plyp2,...,pKonly. (ix)Proofthat&lt;isafunction ofT,L,M,N. Since&lt;,whenexpressedinterms ofql,q2,...,q6,p1}...,p 6,isapoly nomial inpltp2,...,ps,itisclear that$isapolynomialinitsarguments L,M,N,pi, ...,p6.Weshall write 164] TheTheorems ofBruns andPoincare 375 sowehave &lt;^.=||G&lt;fe=|8GW. ar=i9pratf r=i9pr9gv andtheequation for/2is whereF,.stands fordU/dq r,supposed expressedinterms ofqltP2,....P, pl}...,p6.Wehave therefore Pir=l& r9F =V+- ^a/h ,-tapr Pi where thesymbol 2indicates summation over thethree values ofthe A expression (A+Bql+Cqf). Now theterm%(P 2)...,P6,p1}...,p 6)cannotgiverisetotermsinvolving (^4+Bql+Cqi2 )inthedenominator :sothequantities multiplyingeach of theexpressions (A+Bq^+Cq^ must themselves have thesame character as&lt;2,i-e.theymust bepolynomialin(plt...,p6)whenexpressedinterms of (&lt;?!&gt; &lt;?2&gt; $6,PI, ,pz)-Weseetherefore thattheexpression dB 95iR9^1 dA KdG ]g|/dGWdG\ dPr2gl9Pr"*^gp;4Cgl9P; Pi9/iW r^2\9p rHrpidpJ B2~^AC must beapolynomialin(p^ ...,pfi\whenexpressedinterms of(j9u...,p 6, qlt...,J6).TakingfirstA+%+Cqf=q*+q.22+q32 ,thisexpression becomes 2 or(omittingafactor/a) 376 TheTheorems ofBruns andPoincare[CH.xiv 1dG or - Pioft ^^ The lastfraction must thereforerepresentapolynomialinp1,p2,...,p sothedenominator must beafactor ofthenumerator. &gt;.-n iru-w(dG PiP?dG\if^G ISowGisapolynomialinL,M,N,soU--S*-=_and--- Vdp 2topiop!/ \dps arepolynomialsinZ, -/If, J\randinvolveqltq2,q3onlybymeans ofL,M,N, soeither theycontain noterms inqltqz,qsinwhich casethedenominator cannot beafactor ofthenumerator orelsetheycontain some terms free fromqltqz,q3inwhich case alsothedenominator cannot beafactor ofthe numerator. Thecondition canthereforeonlybesatisfiedbysupposingthat 8^_PiPa?= 8^_Pip 33G= dp2p2pidpj. dp3/x3^1dp l Asmightbeexpectedfrom considerations ofsymmetry,theconditions arisingfrom theother sets ofvalues ofA,B,Cgive dGfi.prdG fc-l&fc~(=4,5,6). Thefunction Gtherefore satisfies these fiveequations, which areevidently acomplete systemoffiveindependent equations with sixindependent variables, andconsequently possess onlyoneindependent solution; this solution iseasilyfound tobe 6n2 2^,orT. =i2/, Thefunction Gtherefore involves (p l,...,pe)onlybymeansoftheexpression T : andsinceGispolynomialin(plt...,p6),itmust alsobepolynomialinT. Since$ishomogeneousin(qltq2,...,q6),andalsoin(pl}p2,...,p 9),and theexpressions (L,M,N)areeach linear in(ql,...,q6),whileTdoesnot involve(qlt...,q6)and isofdegree2in(plt...,p6),itisclear that ifTis involved in &lt;f&gt;atall, itmust beasafactor :sowecanwrite &lt;f&gt;=h(LtM,N)Tm , where hisahomogeneous polynomialinitsarguments. (x)DeductionofBrims theorem, forintegrals which donotinvolve t. Theequation which determines thefunction fzis 164] TheTheorems ofBruns andPoincare ButwehaveKWKWW PJ.dp,p1dTdp, andtherefore /2=%(P2,...,Pt,plt...,p.)-mh(L, M,N)T-&gt; U. Thus Theintegral&lt;cantherefore becompounded fromtwootherintegrals, namely: 1theintegral h(L,M,N}(T- U)m ,which isitselfcompounded from theclassicalintegrals, and2theintegral $,where &lt;/&gt;= &lt;-f&lt;/+ 4+ ... and &lt;=x(P2,...,P6,plf...,p6), &=fc-m( ^"1}h(L,M,N)T~* U*, 0;=^+?^2) h (L&gt;M&gt;N)Tm-3 U3&gt; But ((&gt;isanintegralofthesame character as &lt;f&gt;,exceptthat itshighest term,&lt;f&gt;,isoforder twodegreeslessin(plt...,p 6)than thehighest term, &lt;f&gt;,of &lt;j).Nowwehaveshewn that &lt;canbecompounded from theclassical integrals together with theintegral&lt; .Similarly&lt;canbecompounded from theclassicalintegrals togetherwithanintegral&lt;"which hasthesame character as &lt;,but isoforder lessbyfourunits than &lt;inthevariablesp. Proceedinginthisway,weseethat (f&gt;canbecompoundedofthe classical integrals together withanintegral&lt;(n) ,whose order in(pl}...,_p 6)iseither unityorzero. If &lt;(n &gt;isoforderunityin(plt...,p6);then intheequation &lt;l&gt;M=&lt;l&gt; w=h(L,M, N)Tk wemustevidently have &=0;inthis case, therefore,&lt;&lt;n &gt;iscompoundedof theclassicalintegrals.If &lt;/&gt;&lt;n &gt;isoforder zero in(plt...,p6),itisafunction of(ql,...,qG)only:butwehavealready shewn thatsuchintegralsdonotexist : and soinanycase &lt;/&gt;canbecompounded algebraicallyfrom the classical integrals. Hence wehaveBruns theorem, thatevery algebraic integral of thedifferential equations oftheproblem ofthree bodies, which doesnotinvolve thetime, canbecompounded bypurely algebraic processes fromtheclassical integrals. 378 TheTheorems ofBruns andPoincare[CH.xiv (xi)Extension ofBruns result tointegralswhich involve thetime. Wenowproceedtoconsider thosealgebraic integralsoftheproblemof three bodies which involve thetimeexplicitly. Forthispurposeweshall taketheequationsofmotion asasystem (155) ofthe18th order: wehave therefore toinvestigate integralsoftheform where/isanalgebraicfunction ofitsarguments, andaisaconstant. The function /isnotnecessarilyrational initsarguments.Letthelast equationberationalised, sofarasthevariable tisconcerned, sothat itmaybe arrangedintheform am+am ^(f&gt;1(ql,...,q,plt...,p 9,t)+am~2^(q^ .,.,q^p,, ...,p 9,t)+... + &lt;t&gt;m(qi,&gt;q,pi, &gt;#,&lt;)=o, where thefunctions&lt;arerational functions oftandalgebraicfunctions of their other arguments (qlt...,qs,pl}...,p9).Thisequation maybesupposed irreducible int,i.e.such that itcannot befactorised intoother equationswhich areoflowerdegreeinaandarerational int:forifitisreducible, wecan supposeitreplaced bythatoneofitsirreducible factors whichcorrespondsto theoriginal equation f=a. Differentiatingwithrespecttot,wehave a^dp+a^d^+...+d^=0. dt at at Now thequantities d&lt;f&gt;r/dt,whenexpressedinterms of(glfqa,..., &lt;?9, plt...,p 9,t),arerational functions oft:sothattheprevious equationwould bereducible intifthisequationdidnotvanishidentically.Itfollows that thisequationdoesvanish identically:that is, Theexpressions&lt;f)raretherefore themselves integrals:andhence the integral fcan becompounded fromother integrals &lt;j),which arerational functions oftandalgebraic functions of(qlt...,q^ypi, ,pa)- Letsuchanintegral&lt;beresolved into factors linear int:sothat itmay bewritten where(P,^!,^,...,^,*^!,...,^*)arealgebraicfunctions of(q1}...,q 9,pi, ,&) Since thisexpressionisanintegral,wehave 1d-tm&gt;i [~d(pi\ni^if {-a&lt;p]f\ n^i-. -ri~77+T1~~"+- +T T" I* 37" J~ 4 IT, \~Pdt^^^A dt)^ ^t-&lt;}&gt; k{dtJt-^\ dt) ni dtJ 164]TheTheorems ofBruns andPoincare 379 Wh-f-f--*f ;-1arerePacedb ytheir ralues (P *&gt; (&lt;&gt;!, //), ..., (tyi,H),thisequationmustbecome anidentity:but thiscan happen onlyif , ,.,, ..... , eft cfa eft d at i.e.ifeach oftheexpressions P,t -&lt;/&gt;!,t-$2,...,t-fa,t-yjr lt...,t-^i isanintegral.Hence am/algebraic integral oftheproblem ofthree bodies which involves tcanbecompounded (1)ofalgebraic integralswhich donotinvolve tand(2)ofintegrals oftheform t (f&gt;==Constant, ivhere &lt;f&gt;isanalgebraic function of(qlt&lt;&gt;&gt; ...,q9,PI,...,&gt; 9). Now itisknown that i_^gi+fr!M? =Constant ^1+^4+^7 isanintegral:hence anyalgebraic integraloftheproblem,which involvest, canbecompoundedof (1) algebraic integralswhich donotinvolve t; (2) integralsoftheform _miqi where&lt;isanalgebraicfunction of(q1}...,q9,p^, ...,p9);and (3) theclassicalintegral _?fti9i Buttheintegralsinclasses(1)and(2)arealgebraic integralswhich do notinvolve thetime;andhence, bytheresultalready obtained, theyare combinations ofthe classicalintegrals. Thusfinally every algebraic integral ofthedifferential equations ofthe problem ofthree bodies, whether itinvolves thetime ornot,canbecompounded fromtheclassicalintegrals. Bruns theorem hasbeenextended byPainleve*, whohasshewn thatevery integralof theproblemofnbodies which involves thevelocitiesalgebraically (whether thecoordinates areinvolved algebraicallyornot)isacombination oftheclassicalintegrals. *Bull. Astr. xv.(1898), p.81. 380 TheTheorems ofBruns andPoincare[CH.xiv 165. Poincare stheorem. Weshallnext establish another theorem onthenon-existence ofacertain typeofintegralsintheproblemofthree bodies, which isinmany respects analogoustothat ofBruns, andwasdiscovered in1889byPoincare*. (i)Theequations ofmotion oftherestricted problem ofthree bodies. Intherestrictedproblemofthree bodies, theequationsofmotion ofthe planetoidcan(162)bewritten intheform dqr_dH dpr__dH dtdpr dtdqr where H=H+p.H l+fj?H 2+..., 1 2p^~UP2 andHl,H2,...areperiodicinqlyq2,withperiod27r. TheHessian(r=l, 2), isevidentlyzero :asthiscircumstance wouldproveinconvenient intheproof ofPoincare stheorem, weshallmodifytheform oftheequationssoastoobtain asystemforwhich thecorrespondingHessian isnotzero. WriteH2=K,and letH=hbetheintegralofenergy;thenwehave dqr1dK dpr1dK , -\a\ ~rr=oio&gt;~JT=~~o7^~ v.***.*)*dt2hdp,-dt 2hoqr and therefore, takinganew function HequaltoK/2h, wecanwrite the differentialequationsoftherestrictedproblemofthree bodies intheform dqr=dH^ dpr=_dH(r=l 2} dtdpr dt dqr where forsufficientlysmall values of/*,Hcanbeexpandedasapower-series intheparameter n, and theHessian ofHisnotnow zero,and(H l,Hz,...)areperiodicinqltq2,with period2?r. *ActaMath. xm.(1890), p.259;Xouv. Meth. delaMec. Gel. i.(1892), p.233. 165] TheTheorems ofBruns andPoincare 381 (ii) StatementofPoincare stheorem. Let &lt;3&gt;denote afunction of(q:,q^,p1}p2,JJL)which isone-valued and regularforallrealvalues ofq1andq.,,forvalues ofpwhich donotexceed acertain limit, and forvalues ofplandp2which form adomain D,which maybeassmall asweplease;andsupposethat &lt;J&gt;isperiodicwithrespectto qlandqz,havingtheperiod2?r.Under these conditions thefunction &lt;f&gt;can beexpandedasapower-seriesin/j,,say where &lt;I&gt;,^,&lt;I&gt;2,...areone-valuedanalyticfunctions of(q1}qz,pi,p^), periodic in^andq2.Poincare stheorem isthatnointegral oftherestrictedproblem ofthree bodies exists(excepttheJacobianintegral ofenergy andintegrals equivalenttoit),which isoftheform 4&gt;=Constant, wliere &lt;J&gt;isafunction ofthischaracter. Theproofwhich follows isapplicable toanydynamical system whoseequationsofmotion areofthesametypeas those oftherestrictedproblemofthree bodies. Thenecessary and sufficient condition that =Constantmaybean integralisexpressed bythevanishingofthePoisson-bracket (H, &lt;I&gt;) ;sothat andtherefore (H ,&lt;I&gt;)=0, 0ra,4g+(jr,,t)-a (iii)Proofthat ^&gt; **notafunction ofHQ. Weshall firstshew that 4&gt;cannot beafunction ofH .Forsuppose a relation oftheformO=ty(H )toexist. From theequationH=H(pltp2) wehaveonsolvingforplanequationoftheformp^=6(H,p2),andBwill beaone-valued function ofitsargumentsunlessdHo/dp^iszero inthe domain D.Replacing plbyitsvalue 6inthefunction ^&gt;(q1}q^,p-i,p^), we haveanequationoftheform andas 4&gt;isaone-valued function ofitsarguments tywillbeaone-valued function of(qltqz,H,p2);butbyhypothesis,thefunctiontydepends only onH .Itfollows that^risaone-valued function ofH,solongasthe variablespi,p zremain inthedomain D,andprovided ^H^fdp^isnotzero inD; ormoregenerally provided oneofthederivatesdH^/dp! anddH/dp 2isnot zero inD,acondition which isevidentlysatisfied ingeneral.Sincet/ris aone-valued function, theequation -/r(H)=Constant willbeaone-valued integralofthedifferentialequations, and therefore&lt;- -v/r(H)=Constant will alsobeaone-valuedintegral, and willbeexpansibleasapower-series 382 TheTheorems ofBruns andPoincare[CH.xiv infj,:itwillmoreover bedivisibleby //,,since 4&gt;ty(#)iszero. Ifthen wewrite &lt;X&gt;-^(H)=p , theequation&lt;!&gt;=Constant willbeaone-valuedanalytic integral:writing thefunction&lt;I&gt;willnotingeneralbeafunction ofH :ifhowever itisa function ofH,weperformthesameoperation again,thusarrivingatathird one-valuedanalytic integral,whosepartindependentof/Awillnotingeneral beafunction ofH;and soon. Itisevident that inthiswayweshall ultimatelyobtain anintegralwhich doesnotreduce toafunction ofHwhen IJLiszero, unless &lt;$&gt;isafunction ofH,inwhich casethetwointegralsHand &lt;I&gt; arenotdistinct. If,therefore, there exists anintegral&lt;which isone-valued andanalytic anddistinct fromH,butwhich issuch that &lt;l&gt;isafunction ofH,wecan alwaysderive from itanotherintegral,ofthesame character butsuch that itdoesnotreduce toafunction ofHwhenpvanishes. Wecantherefore always supposethat &lt;E&gt;isnotafunction ofH . (iv)Proofthat &lt;&Qcannot involve thevariablesq1}&lt;?,. Ifthefunction&lt;involves thevariablesq,q2,then since itisperiodicin these variables wecanwrite 4&gt;=2A ,,2 _ . i,i where m,andm2arepositiveornegative integers,idenotes v 1,the quantitiesAmm^arefunctions ofp1}pz,andrepresentstheexponential co-factor ofAmm.SinceHdoesnotinvolveq1}q2,wehave HIj,ill.-) r -Q) o - dp, dq, dp2oq* Butwehave d3&gt;/dqr=2imrAmm^%,sotheequation (H ,4&gt;)= m, ,m-i becomes 1dPl* c andtherefore (asthisequationisanidentity) ij-=-+m2-=0. dp, dpj Hence wemust have either Ami,mo= or&gt;idHo/dpi +m2dH(lfdp 2= ; butthelatter alternative ispossible onlywhen m,andm2areboth zero, or when theHessian ofHiszero,which isnotthecase. Itfollows that all thecoefficients Am)OToarezero, exceptA0t;andconsequently&lt;I&gt;does not involve thevariablesq,andqz. 165] TheTheorems ofBruns andPoincare 383 (v)Proofthat theexistenceofaone-valuedintegralisinconsistent ivith theresultof(iii)inthegeneralcase. Consider nowtheequation - =dprdqr r!Tidprdqr AsthefunctionsHland 4&gt;xareperiodicwithrespecttoq1}q2,theycan beexpandedinseries oftheform 2B ei(m iqi+m,q,)=2B ,say, m, , !,m., wherem1andm2arepositiveornegative integers,andthecoefficients Bm andC^,^ depend onlyonpltp2.Wehave therefore ^=i2Bmnmrt,d^=i 2CmnmrS,dqr mi,m,m "m2dqrWi&gt;w,2"^^ 4a^a^aa^a^sotheequation 2r5-=-^ 2? ^= r=i9jor9^ r=ldprdqr becomes 2Bf(!m,^)-2Cf(2m,3^)=0, OT,,m 2 V=l Opr/ m^m., \r-l Opr/ or(since thisequationisanidentity) Thisequationisvalid forallvalues ofpl}p2:andtherefore forvalues of plandp2whichsatisfytheequation dH. dH &lt;1^--- f-W12^r=0, opi 3jp2 wemust have either B m,,m2- &gt;ormi9^o/3^i +m&gt;t9^0/3^2=0. We shallsaythatacoefficient Bm^m^becomes secular whenpj,p 2have values such thatmldH/dp 1+mzdHfdp 2=0. AsHisagiven function, thecoefficients Bm&gt;maregiven. Inthegeneral caseofdynamical systems expressed bydifferentialequationsofthekindwe areconsidering,nooneofthese coefficients willvanish when itbecomes secular, andweshall take thiscase first: sothat theequation mxd^o/dp, +m^d/dp 2= isaconsequenceoftheequation mldffo/dp, +m2dH/dp=0. Now letk-i,k.2betwointegers:suppose thatwegivetop^andp2values such thattheequation 384 TheTheorems ofBruns andPoincare[CH.xiv issatisfied. Wecanfindaninfinite number ofpairsofintegers mltm.,such thatm1k1+m zkziszero: and foreach ofthesesystemsofintegersthe expression m1dH/dp l+m2dH/dp 2iszero,andconsequently m-!BOo/a^j +m2d$&gt;/dp 2 iszero. Comparingthese twoequations,wehave sotheJacobian d(H0&gt;&lt;)/3(p1}p2)iszero forallvalues ofpltp2forwhich dHfdplanddHjdp zarecommensurable witheach other. Thus inanydomain, however small, there areaninfinite number ofsystemsofvalues ofpltp2for which thisJacobian iszero :astheJacobian isacontinuous function, itmust therefore vanishidentically,andconsequently&lt;J&gt;must beafunction ofH . But this iscontrarytowhat wasprovedin(iii),andtherefore thefunda mental assumptionastotheexistence oftheintegral&lt;&must beerroneous; that is,theHarniltonianequations possessnoone-valuedanalytic integral other thanH=h,providednooneofthecoefficients Bm^mvanishes when it becomes secular. (vi)Removal oftherestrictions onthecoefficients B mi&gt;. Wehavenow toconsider thecase inwhich atleastoneofthecoefficients B vanishes when itbecomes secular. We shallsaythattwopairsof indices (raj,m2)and(m/, ?n/)belongtothesame classwhen they satisfythe relation mj/m/=ra2/m2,andthat inthis case the coefficients Bm^mand R&lt;belong tothesame class. wij,m&lt;2 Weshall firstshew thattheresult obtained in(v)astothenon-existence ofone-valued integralsistrueprovidedthat ineach oftheclasses there isat least onecoefficient Bmmwhich does notvanish onbecomingsecular. For supposethat thecoefficient B^^iszero,butthecoefficient Bm^m^isnot zero. Ifpi,pzhave values such that m-idHftpi+m2dHQ/dp 2iszero,wehave i+W3H/dp 2=0,andconsequently andalthoughtherelation n^3&lt;&/9.Pi+ 2d3&gt;/dp2=cannot beinferred from theformer oftheseequations,itcanbeinferred from thelatter :theproofis inotherrespectsthesame asin(v). Now aclass iscompletelydefined bytheratio oftheindicesmltra2;let Xbeanycommensurable number, and letCbetheclass ofindices forwhich mj/w 2=X-We shallsayforbrevitythat this classGbelongstoagiven domain, orisinthisdomain, ifasetofvalues ofpl}p2canbefound inthis domain such that dH 9//o_nA,^r--1-_ V. 165] TheTheorems ofBruns andPoincare 385 Weshallshew thatthetheorem isstilltrue ifineverydomain B,however small, which iscontained inD,there areaninfinite number ofclasses for which not allthecoefficients oftheclass vanish whentheybecome secular. Fortakeanysetofvalues ofpl,p2,such that forthese values wehave dH 3/f_A.-T U. 9pi op* Supposethat A,iscommensurable, andthat fortheclasswhichcorresponds tothisvalue ofX,allthecoefficients oftheclass donotvanish whenthey become secular :thepreceding reasoningthenappliestothis setofvalues, andsoforthese values ofplandp2theJacobian d(H ,4&gt;)/3(p 1}p.z}iszero. But,byhypothesis,there exists inevery domain8,however small, which is contained inD,aninfinite number ofsuch sets ofvalues ofpltp2.The Jacobianconsequentlyvanishes atallpointsofD,andtherefore Oisafunc tionofH;so,asbefore, there exists noone-valuedintegraldistinct fromH (vii) DeductionofPoincare" stheorem. Inthefourpreceding sections, wehave consideredequationsofthetype dqr_dH dpr_dH dtdpr dt~ dqr inwhichHcanbeexpandedintheform where theHessian ofHQwithrespecttop^andp2isnotzero,Hdoesnot involveqlandq2,andHltH2,...areperiodic functions ofqltq2:andwehave shewn thatnointegraloftheseequations exists which isdistinct from the equationofenergy and isone-valued andregularforallrealvalues ofq1and q2,forvalues of/*which donotexceed acertain limit, and forvalues ofpland pzwhich form adomain D;provided that inevery domain, however small, contained inD,there areaninfinite number ofratiosm^m^forwhich not all thecorrespondingcoefficients Bm^m^vanish whentheybecome secular. This result canbeappliedatonce totherestrictedproblemofthree bodies :forwehave seen in(i)that theequationsofmotion inthisproblem areofthecharacterspecified, andondeterminingthefunction H^byactual expansion wefindthat thelastcondition issatisfied. Poincare stheorem is thus established. Poincare stheorem establishes thenon-existence ofintegrals uniformwithrespectto theKeplerian variables, whichimplies uniformityintheneighbourhood ofallthetra jectories which have thesameosculating ellipse. Thishowever does notexclude the existence ofintegrals which areuniform indomains ofadifferent character. Cf.Levi- Civita, ActaMath. xxx.(1905), p.305. Thetheorem hasbeen extended byPoincare" tothegeneral problemofthree bodies : cf.Nouv. Meth. delaMec. Cel. i.p.253;ithasalsobeenextended byPainleve,C.R.cxxx. (1900), p.1699. w.D. 25 CHAPTER XV THEGENERAL THEORY OFORBITS 166. Introduction. Weshallnowpasstothestudyofthegeneralformanddispositionofthe orbits ofdynamical systems.Forsimplicity weshall inthepresent chapter chieflyconsider themotion ofaparticlewhich isfree tomove inaplane under theaction ofconservative forces, butmanyoftheresults obtained can bereadilyextended tomoregeneral dynamical systems. Ithasalreadybeen observed (104) thatthedetermination ofthemotion ofaparticlewithtwodegreesoffreedom under theaction ofconservative forces isreducible totheproblemoffindingthegeodesiesonasurface with agiven line-element; anaccount ofthepropertiesofgeodesies might therefore beregardedasfallingwithin thescopeofthediscussion. Manyof thesepropertiesarehowever ofnoimportanceforourpresent purpose:and asthetheoryofgeodesiesisfullytreated inmanyworks onDifferential Geometry,weshallonlyconsider those theorems which areofgeneral dynamicalinterest. Theprincipalresults which havebeen obtained hitherto relate toperiodic orbits(167-171),tothestabilityofagivenorbit(especiallyofaperiodic orbit) with respecttosmall displacementsfrom it(172-176), and tothe stabilityofagiven groupoforbits withrespecttothetime, i.e.thequestion astohow fartheorbitspreservetheirgeneralcharacter after thelapseofa very greattime(177-179). 167. Periodic solutions. Great interest hasattached inrecentyearstotheinvestigationofthose particularmodes ofmotion ofdynamical systemsinwhich thesame con figurationofthesystemisrepeatedatregularintervals oftime, sothatthe motion ispurely periodic.Suchmodes ofmotion arecalledperiodicsolutions. Thetermperiodicsolution isalsoused incases where arelative rather than anabsoluteconfigurationisperiodically repeated:thus intheproblemof three bodies, asolution issaid tobeperiodicifthemutual distances ofthe 166-168] TheGeneral Theory ofOrbits 387 bodies areperiodicfunctions ofthetime, althoughthebodies maynot necessarily have thesame orientation attheend ofaperiodasatits beginning. Considering speciallythemotion ofaparticleinaplaneoronafixed smooth surface under theaction ofconservative forces, itisevident that a familyofperiodicorbits will exist intheneighbourhoodofeachpositionof stableequilibriumoftheparticle, namelytheorbitscorrespondingtothe normal vibrations oftheparticle about thisequilibrium position.Ifthe positionofequilibriumisunstable, itmayhappenthattheperiodsofboth themodes ofnormal vibration areimaginary,inwhich casenoperiodicorbits exist inthevicinity, or_thattheperiodofoneofthemodes ofnormal vibration isreal, inwhich case these realnormal vibrationsgiveafamilyof periodicorbits :these orbits willevidently however beunstable, whereas the orbits intheneighbourhoodofthepositionofstableequilibriumarestable. 168. Poincare snormal variablesforaknownperiodicorbit. Theequations which define aperiodicorbit aremostconvenientlyex pressedinaformdue toPoincare *. Letthemotion ofthedynamical systemconsidered bedefinedbythe equations dq,=dHdpr__dH dtdpr dt= dqr where thefunctionHdoesnotinvolve thetime texplicitly;and let fc= 0l(0&gt; ?2=&lt;k(0&gt; Pl= ^l(t)&gt; P*=fa(i) betheequations which define aknownperiodicorbit ofthissystem. There isclearlynolossofgeneralityifwesupposethecoordinates(ql}q2,p^p2)to besuch that after thelapseofaperiodthevariablesqltqz,plresume their initialvalues, whilep.2increasesby2?r. From theseequationstcanbeeliminated :lettheresult oftheelimination bewritten intheform sothatthefunctions 61}0.2,6Shave theperiod2-rr. Perform onthesystem thecontact-transformation definedbytheequations where ( 8*(,)- *Nouvelles Methodes delaMec. Cel.n.p.369. 252 388 TheGeneral Theory ofOrbits[en.xv Theequationsofthistransformation canbewritten Q,-9,-*.(ft)+{*-ft(ft))-fft-*3(ft)} Theequationsofmotion ofthedynamical system,interms ofthenew variables, are dtdPr dt0& andfromtheabove equationsoftransformation itisevident that theperiodic solution isnowdefined bytheequations Q1=0,Q,=0,P1= 0,"P2=f2(0- Thisform oftheequationsoftheorbit willbecalled Poincare snormal form. 169.Acriterion forthediscovery ofperiodicorbits. Weshallnowshew thattheexistence andpositionofperiodicorbits can bedetermined byatheorem* analogoustothose theorems which furnish the positionoftheroots ofanalgebraic equation byconsiderations dependingon thesignofexpressionsconnected with theequation. Weshall forsimplicity supposethedynamical problemconsidered tobethat ofthemotion ofa particleofunitmass inaplaneunder theaction ofconservative forces: the result canbeextended tomoregeneral systemswithoutdifficulty!- Let(as,y}bethecoordinates oftheparticleattimet,referred toanyfixed rectangularaxes intheplane,and letV(x, y)beitspotential energy function, sothat theequationofenergyis l(#+f)+V(a&gt;,y)=K, where histheconstant ofenergy. The differential equationsofmotion oftheparticleform asystemofthe fourth order, and theirgeneralsolution consequentlyinvolves fourarbitrary constants. One ofthese constants is,however, merelytheconstant additive tot,which determines theepochinthe orbit, sothere areonlyoo3really distinct orbits. Thistriple infinityoforbits canbearrangedinsets,each containingadoubleinfinityoforbits, byassociating togetherthose orbits for which theconstant ofenergyhasthesame value h :such asetofoo2orbits *Whittaker, MonthlyNotices R.A.S. LXII.(1902), p.186. Cf.A.Signorini,Rend. d.Lincei, xxi. (1912), p.36;Rend. d.Palermo, xxxin. (1912), p.187; L.Tonelli, Rend. d.Lincei,xxi. (1912), pp.251, 332. fFortheextension totherestricted problemofthree bodies,cf.MonthlyNotices R.A.S. LXII.(1902), p.346. 168,169] TheGeneral Theory ofOrbits 389 maybedefinedanalytically bytheprincipleofleast action(100),namely thattheorbitbetween twogiven points (x ,y)and(xlyy^)issuch astomake thevalue oftheexpression stationaryascomparedwith other curvesjoiningthegiventerminalpoints Oo, ?/)and (&gt;,, 2/a)*. Consider anysimpleclosed curveCintheplaneofxy;and letanother simpleclosed curveGbedrawn, enclosing Canddiffering only slightlyfrom it.WemayregardC"asdefinedbyanequationoftheform where8pisthenormal distance between thecurves CandC(measured out wards from C,andconsequently always positive) and7istheinclination of thisnormal distance totheaxisofx.Then if/bethevalue oftheintegral when theintegrationistaken round thecurve C,and if/+81denote the value ofthesameintegral when theintegrationistaken round thecurve C" (sothatthesymbol8denotes anincrement obtained inpassingfromGtoC), wehave 81={(dxf+(dy)^8[h-V(x, y)}?+ [{h-V(x, y)}?8{(dx?+ Butwehave iarr/M-*/8F 9F V,, -i{A-V(x, y}\I ^j-cos7+y-sin7)8p, and 8{(dxf+(dy}=8p.dy={(dx? wherepistheradius ofcurvature ofthecurveCatthepoint (x,y). Thuswehave S/=f{(&lt;fe)+(dy)}*{A-r^^J( p oxoy } Thisequation shews that ifthequantity h-V(x,y} dV dV --Acos7^--*sin7-7=r- PldxT dy *Following Painleve, Liouville sJournal,x.(1894),itiscustomarytocallafamily oforbits which have thesame constant ofenergy anaturalfamily. 390 TheGeneral Theory ofOrbits[OH.xv isnegativeatallpointsofC,then81isnegative,and sotheintegralIhas itsvalue diminished when anycurve surrounding CandadjacenttoCis taken instead ofCasthepathofintegration. Nowsupposethatanother simpleclosed curveDcanbedrawnenclosing C,andsuch that atallpointsofDthequantity h-V /3F dV\*cos7-,.--+sin7^r P2V 3 dy/. ispositive. Then, inthesame way,itcanbeshewn that theintegral/is diminished when anysimpleclosed curveD,enclosed byDandadjacentto D,istaken instead ofDasthepathofintegration. When, therefore, weconsider theaggregateofallsimpleclosed curves situated inthering-shaped spacebounded byCandDwhich isassumed to contain nosingularityofthefunction V(x, y)itisclear that thecurve which furnishes theleast value ofIcannot beCorD,andcannot coincide withCorDforanypartofitslength.There exist, therefore, amongthe simpleclosed curves ofthisaggregate,oneormore curvesKforwhich the value of/islessthan forallother curves oftheaggregate.SinceKdoes notcoincide withCorDalong anypartofitslength,itfollows, that the curves adjacenttoKare allmembers oftheaggregateinquestion,andhence that thecurveKfurnishes astationaryvalue of/ascomparedwith allthe curves adjacenttoit.ThecurveKistherefore anorbit inthedynamical system. Wehave thus arrived atthetheorem: //onedosed curve be enclosed byanother closed curve, andifthequantity h-V(x,y) dV.8Fv?-icos7-~ *sin7-r- p*dx dy benegativeatallpoints oftheinner curveandpositiveatallpoints oftheouter curve, theninthering-shaped spacebetween thetwocurves there exists aperiodic orbit ofthedynamical system, forwhich theconstant ofenergyish.Asthe quantity h-V(x,y).dV.dV^ "-icos7^\sin7-5- pox dy canbecalculated immediatelyforevery pointonthecurves CandD,de pendingasitdoesonlyonthepotential-energyfunction and thecurves themselves, thisresult furnishes ameans ofdetectingthepresenceofperiodic orbits. 170. Lagrangesthreeparticles. Weshallnowconsiderspeciallycertainperiodicsolutions oftheproblem ofthree bodies. Auseful summaryoftheorems relatingtofamilies ofperiodicorbits intherestricted problemofthree bodies isgiven by.K.Moulton, Proc. Inter. Cong, ofMath. Cambridge, 1912,II.p.182. 169,170] TheGeneral Theory ofOrbits 391 The relation ofperiodicorbits toorbits ofejection and collision, inwhich twoof thebodies occupyinthesame position atthesame time,isstudied byMoulton, Proc. L.M.S.(2),xi.(1912), p.367. Acategoryofnon-plane periodicorbits intheproblemofthree bodies isdiscussed by Pavanini, Annali diMat.(3),xin. (1906), p.179. Lettheequationsofmotion oftheproblembetaken inthereduced form obtained in160,and letusfirstenquirewhether theseequations admit ofa particularsolution inwhich themutual distances ofthebodies areinvariable throughoutthemotion. Themutual distances are f2m2ftft/k2-p s--pf. . \ &lt;7i ,-\Q-- cosqscosq,--- -sinqssino4+11 (*~ml+m2\^2p3p, */ ,(2m1o1go/ k2-p 3--pf. .\andW*+ cosftcosft---g- -smftsinft1+*m1+m,\ 2, ta) itfollows that, intheparticularsolution considered, thequantities , k2- p&lt;?-p 42 . ft, ft,and cosftcosft---^-sinftsin^ must beconstant, andhence thefunctions U,dU/dq^ dU/dq 2must beconstant, whereU=^m1m2r12~l . Theequations .dHp1.dHp.2 =ft=5= ,=03=^=^, 9^i /* 3^2 A* shew thatplandp.2must bepermanentlyzero :while theequations shew thatp3andp4must beconstant. Moreover, theequations A. aff a#= /&gt;3=-^-, 0=^4=-^ 3ft 3ft shew thattheexpressions 3 9ftaCS?4 fjfW*2__yj^2^.^^J2 \ and^cosftcoso4-- ^4sino,sin .) 3ft\2jo3p4 arezero, sowehave Pi2+K-&2 tan^3cotft=cotfttanft=^^ , *ppt andthereforep*+^42-^2=+2p3p4, or ^=(/,3p4)2 392 TheGeneral Theory ofOrbits[en.xv anequation which shews that theinstantaneousplanesofmotion ofthe bodiesfj,and//coincide with theplane throughthese bodies andtheorigin: inother words, themotion of//,and //takesplaceinaplane:andtherefore themotionofm1,m2,mstakesplaceinaplane. Itfollows that, thecentre ofgravityofthesystem being supposedat rest,theparticles m^m2,ra3(which weshall denotebyP,Q,R)mustmove incircular orbits round 0.Wehavenow toseeifsuch amotion ispossible. One condition which mustobviouslybesatisfied isthat theresultant attraction ofanytwooftheparticlesonthethirdmust actinthelinejoining thethirdparticletothecentre ofgravity.This condition issatisfied ifthe threeparticlesareinthesamestraightline. Iftheyarenotinastraight line, itgives *ffl sinPRO=^~-2sinQRO, andtwosimilarequations. Butsince isthecentre ofgravityoftheparticles, wehave mlsinPRO_sinQPR_QR ra2sinQRO~ sinPQR~P and thiscombined with thepreceding equation givesPR=QR:similarly wefindPR=PQ. Hence either thebodies must becollinear, orelsethetriangle formed by themmust beequilateral. Consideringfirstthecollinear case, letthedistances ofthebodies from their centre ofgravity (measured positivelyinthesamedirection) beal}a.,, asrespectively:weshallsupposethatal&lt;a2&lt;as,which does notlessen thegeneralityofthediscussion. Since theforceactingonPmust bethat correspondingtocircular motion round 0,wehave n?a=m2(a2 i)~2ni3(asaj)~2 , where nistheangular velocityofthelinePQR ;andsimilarly n2a2=m3(asa2)~2+n^(a2- a^"2 ,n2a3=ml(as o^)"2+m2(a3a2)~2 . From theseequations wereadilyfind rn^fc {(1+k)3-1]+ma(1+k)2(&-1)+m,[ks-(1+k)3 }=0, where &denotes theratio(asa2)/(a 2a-^. This isaquintic equationink,with realcoefficients. Since theleft-hand side oftheequationisnegative when kiszero,andpositive when k=+oo, there isatleastonepositiverealroot; sucharootdeterminesuniquelyreal values fortheratios a1:a2:a3;and ifnisgiven,thedistances altaz,a3can becompletelydetermined. Itfollows that there areaninfinite numberof solutionsoftheproblem ofthree bodies, inwhich thebodies remainalwaysina straightline atconstant distances fromeach other;thestraightline rotates 170]TheGeneral Theory ofOrbits 393 uniformly,andwhen itsangular velocityhasbeen (arbitrarily) assigned,the mutual distances ofthebodies aredeterminate. Consideringnext theequilateral case, letabethelengthofonesideof thetriangleformed bythebodies, and letnbeitsangular velocity.Since theforceactingonm3isthatwhichcorrespondstoacircular orbitround 0, wehave -1cosPRO+~2cosQRO=n* .OR, a? a2 acondition which reduces to Theconditionsrelatingtothemotion ofQandofRreduce tothesame equation: andhence amotion ofthekind indicated ispossible, providedn andaareconnectedbythis relation. Hence there areaninfinite numberof solutionsoftheproblem ofthree bodies, inwhich thetriangle formed bythe bodies remainsequilateral andofconstant size,and rotates uniformlyinthe plane ofthemotion: theangular velocity ofitsrotation canbearbitrarily assigned, and thesizeofthetriangleisthendeterminate. Thetwoparticular typesofmotion which havenowbeen found willbe calledLagrangescollinearparticlesandLagrangesequidistant particles respectively*. Formore than acenturyafterLagrangesdiscovery,itsinterest wassup posedtobepurelytheoretical. But in1906 anewminorplanet,588Achilles, wasfound tohave amean distanceequaltothat ofJupiter:and itwassoon realised thattheSun, Jupiter,andAchilles constitute, approximatelyatany rate, anexampleoftheLagrangian equilateral- triangular configuration. Shortlyafterwards came thediscoveryofthree other Asteroids, 617Patroclus, 624Hector, and659Nestor, which areinthesame caset- Ofthis"Trojan group,"Patroclus isinlongitude Jupiter 60,andtheother three inlongi tudeJupiter +60. Example. Shew thatparticularsolutions oftheproblemofthree bodies exist, inwhich thebodies arealwayscollinear oralways equidistant, althoughthemutual distances are notconstant butareperiodicfunctions ofthetime. These areevidently periodicsolutions oftheproblem, andinclude Lagrangesparticles asalimitingcase. *They were discovered byLagrangein1772 :Oeuvres deLagrange,vi. p.229. For references toextensions ofthese results totheproblemofnbodies,cf.myarticle iuthe Encyklopadied.math. Wiss. vi.2,12,p.529; tothepapersthere mentioned maybeadded E.0.Lovett, Annali diMat.(3),xi.(1904), p.1;W.R.Longley, Bull. Amer. Math. Soc.xm. (1907), p.324,andF.R.Moulton, Annals ofMath. xn.(1910), p.1. tCf.F.J.Linders, Arkiv forMat. iv.(1908), No.20. 394 TheGeneral Theory ofOrbits[CH.xv 171.Stability ofLagrangesparticles:periodicorbits inthevicinity. Ithasbeen observed(167)that intheneighbourhoodofanyconfigurationofstable equilibriumorsteady motion there exists ingeneral afamilyofperiodic solutions, namely thenormal vibrations about thepositionofequilibrium orsteady motion. Weshallnow applythisidea tothecase oftheLagrange s-particlesolution oftherestricted problemof threebodies, andtherebyobtain certain families ofperiodicorbits oftheplanetoid. LetSandJbethebodies offinite mass,m1andm2their masses,their centre of gravity, ntheangular velocityofSJ,xandythecoordinates oftheplanetoid Pwhen istaken asorigin andOJasaxis ofx.Theequations ofmotion oftheplanetoid are(162) d^_ dJK dy_dK du__dK dv___%& ~di~du ~di~~dv ~di~~dx dt~ 8y where K=\(uz+v*)+n(uy vx)-mi/SP m2/JP. Let(a,b)denote thevalues of(x,y)inthepositionofrelativeequilibrium con sidered,sothat forthecollinea,r casewehave 6=0,and fortheequidistantcasewehave a=^(m lm2}l/(m l+m2),b=\*J%1,where Idenotes thedistance SJ,sothat(46) Thevalues ofu,vinthepositionofrelative equilibrium areeasilyseen tobenband narespectively. Write #=a+, yb+rj,u=nb +Q,v=na+ (f&gt;, where,77,6,&lt;paresupposedtobesmallquantities:neglecting aconstant term,wehave Onexpanding andretaining onlyterms ofthesecond order inthesmallquantities, weobtain anexpressionforKwithwhich theequationsforthevibrations about relative equilibrium canbeformed :weshall fordefiniteness consider vibrations about theequi distantconfiguration:inthiscasetheexpressionforKbecomes Theequationsofmotion are dK dK ,dK _dK~Wn~W ~3|~ *--frj- Solving these equationsinthemanner described inChapter VII,wefindthattheperiod ofanormal vibration is27T/X, where Xisaroot oftheequation X4-n*\*+(*i-#)n*=0,where k=- .. .4mi+m2 Thetwovalues ofX2given bythisequationwillbepositive provided theyarereal, since (fi~^2 )ispositive:andtheywillberealprovided4(fJ&2 )&lt;1,or(wij+?n2)2 &gt;27mjra2; arelation which issatisfied providedoneofthemasses S,Jissufficiently largecompared with theother. When thiscondition issatisfied,there exist twofamilies ofperiodicorbits oftheplanetoidinthevicinity ofitsequidistant configuration ofrelativeequilibrium:the periods are,toafirstapproximation, STT/XJand27r/X 2,where X^andX22aretheroots ofthe equationinX2 , 171,172] TheGeneral Theory ofOrbits 395 Asimilar discussion leads totheresult that thecollinearLagrange s-particle configurations areunstable; buttheequation fortheperiods ofnormal modes ofvibration hasalwaysone real root,andconsequentlyintheneighbourhood ofaposition ofrelativeequilibrium ofthe planetoidonthelineSJthere exists afamily ofunstableperiodicorbits*. Example. Shew that, foroneofthemodes ofnormal vibration oftheplanetoidinthe vicinityoftheequidistant configuration,theconstant ofrelative energyisgreaterthan in theconfigurationofrelative equilibrium,while fortheothermode theconstant islessthan intheconfigurationofrelative equilibrium. (Charlier.) 172. Thedifferential equation ofthenormaldisplacement from anorbit. Weshallnowproceedtoconsider thestabilityoforbits ingeneral. Supposethatsomeparticularsolution ofthemotion ofaparticleofunit mass inaplane,under theaction offorces derived from agiven potential energyfunction V,isknown;andconsider asolution which isimmediately adjacenttothisknown solution, and forwhich theconstant ofenergyhas thesame value. LetPandQbethepositionsoftheparticleintheknown andadjacent orbitsrespectivelyattime t.DrawQNperpendiculartotheknown orbit, and letPN= ,NQ=u;let beafixedoriginontheknown orbit :let arcOP=a,arcON=s,sos a-= ;and letpbetheradius ofcurvature of theorbit atP.We shallregardthepositionofanypointontheadjacent orbit asspecified bythequantities (u,s). Thekineticenergyoftheparticle whendescribingtheadjacentorbit is T=i^+i(l +W/p)2*2 . and itsLagrangian equationsofmotion aretherefore , p/p ou u\-..2/,u\. /.,u\uszdp3F 1+- )s+-1+-)us-1+--f=- -JT-p) p\p/\p/p-as os theseequations possessaknownintegral, namelytheintegralofenergy (M\2 1H )s2+V=h,where hisaconstant. The firstLagrangian equation andtheintegralbecome dp \\dujp*\dudajp \du+u *Forfurther workonthesubject oforbits intheneighbourhoodoftheLagrange s-particle solutions, cf.thememoirs referred toonpage 530ofmyarticle intheEncyjtlopadle, and also Lovett, Astr.Nach. CLIX.(1902), p.281;Stromgren, Astr.Nach. CLXVIII.(1905), p.105;Moulton, Math. Ann. LXXIII.(1912), p.441. 396 TheGeneral Theory ofOrbits[CH.xv Since du and thetwopreceding equations become Eliminating(&lt;T fir),weobtain theequation f/92F\ So-2 ) or(takingsinstead oftastheindependent variable, andwritingvforcr) d2u1dvdu flid2V ds2vdsds(v2\du2 and this isthedifferential equation oftheadjacentorbit From thisequation wecan atonce deduce consequences relatingto thestabilityoftheknown orbit. ForbySturm stheorem*,ifwehaveany differential equationoftheform d2u where foracertainrangeofvalues oftthequantity /(t)liesbetween two positiverealquantitiesa-andb-,thenanysolution uwhich iszero for avalue twithin therangewillbezeroagainforsome value twithin the range,where(tt)liesbetweenir/aandirjb, providedtherangeis sufficiently largetocomprehendthis interval. Itfollows that ifthequantity (32F/9w2)p+3v2 /p2ispositiveatallpointsoftheknown orbit, thisorbit will bestable^,i.e.anyadjacentorbitwhich intersects itonce willnotdiverge greatlyfrom it,but willintersect itagain infinitelyoften. Thisexpression cantherefore becalled thecoefficient ofstabilityfortheorbit. *Cf.Darboux, Th.gen. desSurfaces, Vol. in. fInthediscussion ofstabilityin172-176, allpowersofthedisplacement above thefirstare neglectedinforming thedifferential equations oftheadjacentorbits. The effect oftheneglected terms onthestability hasbeen studied byLevi-Civita, Annali diMat. v.(1901), p.221,whohas found thattheneglected terms giverisetoinstabilityincertain cases which appeartobestable when only first-order terms areconsidered :thishappens whenaT/2iriisacommensurable number, where aisthecharacteristic exponent (175)andTistheperiodofthesolution. Cf. alsoA.E.Cigala, Annali diMat. xi.(1904), p.67. 172,173] TheGeneral Theory ofOrbits 397 173. Kortewegstheorem. Suppose now that theknown orbit, withrespecttowhich thenormal displacementuismeasured, isaperiodicorbitwhoseperimeterisS :then if u= &lt;f&gt;(s)istheequationofanadjacent orbit, itisevident thatu= &lt;f&gt;(s+nS), where nisanyinteger,isalsotheequationofanadjacentorbit :theorbits represented bythesetwoequationsareinfactcongruent,butthecorresponding tracing pointsareseparated byoneormoreperiods. Inanyorbitadjacenttotheknown orbit letun,un+1 ,un+2(where nisan integer)denoterespectivelythenormaldisplacementsinthenth,(n+ l)th, and(n+2)th period,atthesameplaceintheorbit, sothatwecanwrite un= &lt;f&gt;(s+(n- 1)S), un+l= &lt;(s+nS), un+2= (f&gt;(s+(n+1)S), where u= &lt;f&gt;(s)isasolution oftheequation d2 u, 1dvdu(1/9-F\ 3 ds2vdsds\v*\du2 /, Since un,un+l ,un+2arethree solutions ofthislinear differentialequation, theymustsatisfyarelation oftheform where kand k^areindependentofs. Weshall firstshew thatthese constants kand&xareindependentofthe choice oftheadjacentorbitandofthenumber n,sothattheywillbethesame foranyother set um=ty(s+(m 1)S),um+l=ty(s+mS), um+2=ty(s+(m+1)$). Forumisalinear function ofthetwosolutions unandun+1 ,say andtherefore onadding periodstotheargument s,wehave Butfrom theequations Un+2== "Un+ii"iUn , ^n+3=foUn+2" "a^n+i&gt; wehave c^un^+c2un+s=k(c1un+l+c2wn+2)+kt(dun+c2un+l\ andtherefore um+2=kum+1+k^um) which shews that theconstantsoccurringinthelinear relation between um+z,um+l ,umarethesame asthoseoccurringinthelinear relation between un+2 ,un+l ,un. Next, weshall findthevalue oftheconstant k^.From theequations d2un 1dvdun[1/S2F\ 31 ds-vds dsl*\8tt*/ p*)n~ 1dvdun-L f1/92F\ 3n+l+~--/t-f-ljIf " ~J~~ /I1~ o\~^\ nds2vdsds 398 TheGeneral Theory ofOrbits[CH.xv //S?/ /"/".I/"1/-/^i/ /YJ/ /7IItt -til I//tv7)4-1 Lt-tt// W/tlfrIX/I**]wehave un. andhence, onintegrating, aun aun+ic 7un j-=- ,where cisaconstant. as as v Changingstos+S,wehave ds ds ds fii/ /riti riii^ /7 7 \^^71-t-l /7wW/Tj.-^!*IX/M/TJ,, cun+l+klUn)-diun+1^/c-^- .^-jg-J aWn+j dtO ~dT-1ds sothat &ihasthevalue 1.Wethushave thetheorem* thatifun,un+l ,un+ denote thenormaldisplacementsinanorbitadjacenttoaknownperiodicorbit inthree consecutive revolutions, theratio k=(un+2+un)/un+1hasaconstant value, which isthesameforalladjacentorbits. 174. Theindexofstability. Theconstant ratiok=(un+z+un)/un+1,where un,un+1 ,un+2arethenormal displacementsfrom aperiodicorbit inthree consecutive revolutions, iscalled theindexofstabilityoftheperiodic orbit, forreasons which willnowappear. Thenature oftheintegralofthedifference-equation ^n+2 rCUn+i~^~Un=" depends,asiswellknown, ontherealityornon-realityoftheroots ofthe quadratic equation \*-k\ +I=0, i.e.itdependsonwhether k &gt;2or jk j&lt;2. Supposingfirstthatkispositiveandgreaterthan 2,write k=2cosh a ; then theroots ofthequadraticareeaande~a ,andweknow thattwoinde pendentsolutions ofthedifference-equationareoftheform as_as u=e^&lt;f&gt;(s) and u=e" tt1 (s), where&lt;(s)and -v|r(s)arefunctions ofswhich have theperiod S :choosing these functions soastomake thesolutions usatisfytheequation d*uIdvdu (1/82F\ 3 TT+~j--3-+11T^4 as vasas[&\ou* J *Korteweg, Wiener Sltzungsber. xcm.(1886). 173,174] TheGeneral Theory ofOrbits 399 (which giveslinear differentialequationsofthesecond order forthefunctions (f&gt;and-vjr),wehavetwoindependent particularsolutions ofthelatterequation: thegeneralsolution isalinear combination oftheseparticular solutions, and consequentlythegeneral equationoftheorbitsadjacenttotheknown orbit, when k &gt;2,isoftheform a* a*u-Kl*4&gt;(*y+Kte~~s+0), whereKlandK2arearbitrary constants, and &lt;j&gt;(s)and^(s)have theperiodS. Similarlyifk &lt; 2,writingk=2cosh a,thegeneral equationofthe orbitsadjacenttotheknown orbit isofthesame form as_os u=K^ &lt;f&gt;(s)+Kze~^^r (s\ whereKlandK2arearbitrary constants, andwhere &lt;f&gt;and$*arefunctions ofs whichsatisfytheequations Nextsupposethat k &lt;2,sothat 2 &lt;k &lt;2 :letk=2cos a.Inthe samewaywenow findthat thegeneral equationoforbitsadjacenttothe known orbit is whereKandAarearbitraryconstants andwhere &lt;f&gt;and^rarefunctions ofs with theperiodS. From these resultsimportant consequences relative tothestabilityof theknownperiodicorbit canbededuced. For ifk &gt;2,itfollows from the character oftheexpressions obtained foruthat thedivergence from the periodicorbit (orif &lt;and^rhave real zeros, theoscillation aboutit)becomes continually greaterassincreases;while ifk&lt;2,thenormaldisplacement isrepresented bycircular functions ofrealarguments, andconsequentlywill remain within fixed limits. Wethus obtain thetheorem thataperiodicorbit isstable ornot,accordingastheassociated indexofstabilityislessorgreater (inabsolutevalue) than two. The results ofthepresent articleagree with, andmaybededuced from, thetheorem thatthegeneral solution ofadifferential equation ofthetype (Pu ,(2?rs . 47T \ p+(n+ icos--+ 2cos_+ .1 isoftheform u=ae$(s}+be~C8^(s), wherea,barearbitrary constants,cisadefiniteconstant, and$and^areperiodic with periodS.Of.Whittaker andWatson, ModernAnalysis, Chapterxix. 400 TheGeneral Theory ofOrbits[CH.xv Example. Discuss thetransitional case inwhich theindex ofstability hasoneofthe values 2 :shewing thattheequation oftheadjacentorbits isofoneoftheforms u=KI{&lt;p(s)+s\l/-(s)}+K,,\js(s), where$and &gt;//either have theperiod orsatisfytheequations andthattheknown orbitmaybeeither stable orunstable.(Korteweg.) 175. Characteristicexponents. Thestabilityoftypesofmotion ofmoregeneral dynamical systems may bediscussedbytheaidofcertain constants towhich Poincare hasgiventhe name characteristicexponents*. Consideranysetofdifferentialequations where (X^,X2,...,Xn)arefunctions of(a^,a?2,...,xn}andpossiblyalsooft, havingaperiodTint;andsupposethataperiodicsolution oftheseequations isknown, definedbytheequations wherefa(t+T)=fa(t) (i=1,2,...,n). Inorder toinvestigatesolutionsadjacenttothis,wewrite where (1, 2,...,n)aresupposedtobesmall, andaregiven bythevariational equations (112) 7,^^cjfc "^ (^==J-)^ &gt;...,n).fit ,*rtf,^ \Juv K=1^K Asthese arelinear differentialequations, with coefficientsperiodicinthe independentvariablet,itisknown from thegeneral theoryoflinear differential equations thateach ofthevariables &willbeoftheform where thequantities S{kdenoteperiodicfunctions oftwith theperiod T,and thenquantitiesakareconstants, which arecalled thecharacteristicexponents oftheperiodicsolution. Ifallthecharacteristicexponentsarepurely imaginary,thefunctions (fi,&,&gt;fn)canevidentlybeexpressedassums andproductsofpurely *AdaMath. xin.(1890), p.1;Nouv. Meth. delaMec. Gel. i.(1892). Onthegeneral problem ofstability thereader should consult theextensive memoir ofA.Liapounoff ,originally published in1892bytheMath. Soc. ofKharkow, andtranslated intoFrench byE.Davaux, Annales de Toulouse(2),ix.(1907), p.203. 174-176] TheGeneral Theory ofOrbits 401 periodic terms; while this isevidentlynotthecase ifthecharacteristic exponentsarenot allpurely imaginary. Hence theconditionfor stability oftheperiodicorbit isthat allthecharacteristicexponents must bepurely imaginary. We shallnow find theequation which determines thecharacteristic exponentsofagivensolution. Inoneoftheorbitsadjacenttothegiven periodic orbit, let({31}/32,...,@n) denote theinitial values of(, %%,..., n)and let/3j+fabethevalue of,- after thelapseofaperiod. Asthequantities (fa^faz, ...,fa n)areone-valued functions of(&,$,,...,&), which arezerowhen (&, /32,...,/3n)areallzero,we havebyTaylorstheorem(neglecting squares andproductsof&,j82,...,@n) Ifakisoneofthecharacteristicexponents,oneoftheadjacentorbits will bedefinedbyequationsoftheform andconsequentlyasetofvalues of&,/32,...,@nexists forwhich theequations (i=l, 2,...,n) aresatisfied :thequantityakmust therefore bearootoftheequationina vfai -i O.T 9^1 9"^=^- ^-+l-ea r - OPi Op2 f)ft dfa2 dfa 2 . aT 9-^ T/ie characteristicexponents arethereforetherootsofthisdeterminantal equation. 176.Properties ofthecharacteristicexponents. When tisnotcontainedexplicitlyinthefunctions (XltX.,, ...,Xn\itis evident that if Xi=fa(t) isasolution oftheequations, then e) W.D.26 402 TheGeneral Theory ofOrbits [OH.xv isalsoasolution, where eisanarbitraryconstant. Theequations &-*(*+) (t-1,2,...,*) therefore define aparticularsolution ofthevariationalequations;butas d(j)i(t+e)/deisevidentlyaperiodicfunction oft,itfollows that the coeffi cient eaktreduces inthiscase tounity:andhence when tisnotcontained explicitlyintheoriginal differential equations,oneofthecharacteristic exponents ofevery periodicsolution iszero. Supposenext thatthesystem possessesanintegraloftheform F(#! ,x2,...,x^=Constant whereFisaone-valued function of(oc^,x2,...,xn)anddoes notinvolve t. Inthenotation ofthelast article, wehave F {&lt;/&gt;&gt;(0)+t++i}=F (&lt;/&gt;;(0)+&}, where forbrevityF(#;)iswritten inplaceofF(i,x.2,...,xn).Differentiating thisequationwith respectto/3fjwehave dFty ,dFd+ 2W^ n_^m+tezWi*+ teW&lt; where indF/dx 1}dF/dx 2,etc.,thequantities (x1}cc2,...,xn)aretobereplaced byfa(0),&lt;j)2(0),..., (j&gt;n(0).From theseequationsitfollows that either the Jacobian 8(&lt;f lfi/r2,..., A/r,l)/a(/81,&, ..., )iszero, orelsethequantities dF/dx,, dF/dx 2,...,dF/dx nare allzerowhen =0.Ifthelatter alternative iscorrect, weseethat(sincetheoriginoftime isarbitrary)theequations =0,dF/dx,=0,...,dFldae n= must besatisfied atallpointsoftheperiodicsolution: this isevidently avery exceptional case,andtheformer alternative must beingeneralthe trueone :butwhen theJacobian iszero, thedeterminantal equationforthe characteristicexponentsisevidentlysatisfied bythevalue eaT=l,i.e.by =:sothat oneofthecharacteristicexponentsiszero. Thusifthe differential equations possessaone-valuedintegral,oneofthecharacteristic exponentsiszero. Acomparisonof 173,174with thetheoryofcharacteristic exponents shews that inthemotion ofaparticleinaplaneunder theaction of conservative forces, thecharacteristic exponentsofanyperiodicorbit are (0,0,a, a),where thecharacteristic exponentaisconnected with theindex ofstabilitykandtheperiodTbytheequation k=2coshaT; theorbit isstable orunstableaccordingasaispurely imaginaryornot. 176,177] TheGeneral Theory ofOrbits 403 Example1.Ifthe differential equations donotinvolve thetimeexplicitly, and possess pone-valuedintegrals /\, ...,Fpwhich donotinvolvet,shew that either (p+l) characteristic exponentsarezero, orthat allthedeterminants contained inthematrix dF 9^-(i=l,2,...,^;=1,2,...,), arezeroatallpointsoftheperiodic solution considered.(Poincare.) Example2.Ifthedifferential equations form aHamiltoniansystem, shew that the characteristic exponentsofanyperiodicsolution canbearrangedinpairs,theexponents ofeachpairbeing equalinmagnitude butoppositeinsign. (Poincare.) 177. Attractive andrepellent regions ofafield offorce. Thegeneralcharacter ofthemotion ofaconservative holonomicsystem isillustrated byatheorem which wasgiven byHadamard* in1897. For simplicity, weshallsupposethat thesystemconsists ofaparticleofunit mass, which isfreetomove onagiven smooth surface under forces derivable from apotential energyfunction V;asimilar result willreadilybeseen to hold formorecomplex systems. Let(u,v)betwoparameterswhichspecifythepositionoftheparticleon thesurface, and lettheline-element onthesurface begiven bytheequation ds*=Edu*+2Fdudv +Gdv* where(E,F,G)aregivenfunctions ofnand v.The kineticenergyofthe particleis T=\(Eu?+ 2Fui&gt;+Gtf}, andtheLagrangian equationsofmotion are _ .__ dt\du du du dt\dv dv dv which canbewritten -^--^du dv \duzdu2dv - du dv 1ff11V " "\ I"\*~^ ff"^~~~ ~7*o Iw if)uj \dv*dv-duj Wehave,bydifferentiation, T&gt;37. dV.V=-~-u+ v,ou dv dV.. dV...,_. -^w+-^--y+-u2+2^-uv+-v.du dv du2dudv dv* *Journ. deMath.(5),in.p.331. 262 404 TheGeneral Theory ofOrbits[CH.xv Substitutingforuand vtheir values from thepreceding equations, wehave where 8F/ .,9^ z.+-=-^^^ dvV3^ drdG+FdG (*Gdu+*F dv- dv (FZFEZG ZG dv\dv2dv 9tt Thequantities occurringinthisequationcanbeexpressedinterms of deformation-covariants*. Theprincipaldeformation-co variants connected with thesurface whose line-element isgiven bytheequation ds*=Edu-+ZFdudv +Gdv" arethedifferentialparameters ,f)=(EG-F*r\B%&-F&&+%df} dvdv \dudv dvdu/1du A,W=(EG-F*rB- 2F +G j dudv where&lt;and-fyarearbitraryfunctions ofthevariables uand v. With thisnotation, thepreceding equation becomes Utilisingtheequationofenergy Eu2+2Fuv+Gv*=2(h- V), andobservingthattheexpression 3&gt;(u,v)&lt;(dV/dv,-dV/du) Eu&gt;+2Fuv+Gv* E(dV/dvf-2F(d V/dv) (dV/du)+G(dV/du)2 Thedefinition ofadeformation -covariant isgiveninthefootnote onpage 111. 177] TheGeneral Theory ofOrbits 405 contains thequantity (udV/du +vdV/dv)asafactor, wecanwrite 8(-V)/_r=-A1(F&gt;+Al(F)+(XU+^)F; where \andyu,contain intheir denominatorsonlythequantity andwhereIvdenotes theexpression 3&gt;(dV/dv,-dV/du)/(EG-F*) ; wereadilyfindthatIvcanbeexpressedintheform Consider, ontheorbit oftheparticle,apointatwhichVhasaminimum value;atsuch apointViszeroandVispositive:asA,(F)isessentially positive (since theline-element ofthesurface isapositivedefinite form),it follows that/F^O, theinequality becominganequality onlywhen A,(F)is zero, i.e.atanequilibrium-positionoftheparticle. Astheparticle describesanytrajectory,thefunction Fwilleither have aninfinite number ofsuccessive maxima andminima(thisisthegeneral case) or(inexceptional cases) thefunctionwill, afterpassing somepointofthe orbit, vary continuallyinthesame sense.Supposefirst that theformer of these alternatives isthetrueone :then ifwedivide thegivensurface into tworegions,inwhich 7rispositive andnegative respectively,itfollows from what hasbeenproved above thattheformer oftheseregionscontains allthe pointsoftheorbit atwhich Fhas aminimum value, i.e.itcontains ingeneral aninfinite number ofdistinctpartsoftheorbit, each offinitelength;whereas intheotherregion,forwhichIvisnegative,theparticle cannot remainper manently. These twopartsofthesurface areonthisaccount called the attractive andrepellent regions. Each oftheseregionsexists ingeneral,for itiseasily found thatanyisolatedpointofthesurface atwhichFisaminimum(i.e.anypoint where stableequilibriumispossible)isinanattractive region, andanypointatwhich Fisamaximum isinarepellent region. Itisinterestingtocompare thisresult with thatwhichcorrespondstoitinthemotion ofaparticle withonedegree offreedom, e.g.aparticle which isfreetomove onacurve under theaction ofaforcewhichdepends onlyontheposition oftheparticle. Inthiscase theparticle eitherultimately travels anindefinite distance inonedirection oroscillates about aposition ofstableequilibrium. Theattractiveregion, inmotion withtwodegreesoffreedom, corresponds totheposition ofstableequilibrium inmotion withonedegree of freedom. Consider next thealternativesupposition, namelythat aftersome definite instant thevariation ofFisalwaysinthesame sense. Weshallsuppose that thesurface hasnoinfinite sheets and isregularatallpoints, andthatFis aneverywhere regular function ofposition onthesurface;sothat, since the 406 TheGeneral Theory ofOrbits[CH.xv variation ofVisalwaysinthesame sense,Vmust tendtoward some definite finite limit,VandVtendingtothelimit zero.Consideringtheequation weseethat ifAx(F)isnotvery small, Xand/j,arefinite andthe lastterm ontheright-handside oftheequationisinfinitesimal; andconsequently either there exist values oftaslargeaswepleaseforwhich Ivispositive (in which casethepartoftheorbit described intheattractiveregionisoflength greater thananyassignable quantity)orelseAj(F) tends tozero. But Aj (F)canbezeroonlywhen dV/duanddV/dvarezero;iftherefore(asisin generalthecase) thesurfacepossesses onlyafinitenumber ofequilibrium positions, theparticlewilltend tooneofthesepositions, with avelocity which tends tozero.Apositionofequilibrium thusapproached asymptotically must beapositionofunstableequilibrium:fortheasymptotic motion re versed isamotion inwhich theparticle, being initiallynear theequilibrium positionwith asmallvelocity,does notremain intheneighbourhoodofthe equilibrium position ;andthis isinconsistent with thedefinition ofstability. Thusfinally weobtain Hadamard stheorem, which maybestated as follows :Ifaparticleisfreetomove onasurfacewhich iseverywhere regular andhasnoinfinite sheets, thepotential energy function being regularatall points ofthesurfacearidhaving onlyafinite numberofmaxima andminima onit,either thepartoftheorbit described intheattractiveregionisoflength greater thananyassignable quantity,orelsetheorbit tendsasymptoticallyto oneofthepositions ofunstableequilibrium. Example.Ifallvalues oftfrom-xto+ooareconsidered, shew thattheparticle must forpartofitscourse beintheattractiveregion. 178.Application oftheenergy integraltotheproblem ofstability. Asimplecriterion fordeterminingthecharacter ofagivenform ofmotion ofadynamical systemisoften furnished bytheequationofenergyofthe system. Consideringthecaseofasingle particleofunitmasswhich moves inaplaneunder theinfluence offorces derived from apotential energy function V(x, y),theequationofenergycanbewritten ^(x-+y2 )=h-V(x,y). Now thebranches ofthecurve V(x, y)=hseparatetheplaneintoregions forwhich[V(x, y) h]isrespectively positiveandnegative;butas(&-+y*) isessentially positive,anorbit forwhich thetotalenergyishcanonlyexist in theregionsforwhich V(x, y)&lt;h.Ifthen theparticleisatanytime inthe interior ofaclosed branch ofthecurve V(x,y}=h,itmustalways remain within thisregion. Thewordstabilityisoftenappliedtocharacterisetypes ofmotion inwhich themoving particleisconfined tocertain limitedregions, andinthissensewemaysaythatthemotion oftheparticleinquestionis stable. 177-179]TheGeneral Theory ofOrbits 407 Theabove method hasbeen usedbyHill*, Bohlinf, andDarwinJ,chiefly inconnexion with therestrictedproblemofthree bodies. 179. Application ofintegral-invariantstoinvestigations ofstability. Thetermstability wasappliedinadifferent sense byPoisson toasystem which, in thelapseoftime, returnsinfinitelyoften topositions indefinitelynear toitsoriginal position, theinterveningoscillations being ofanymagnitude.Ithasbeenshewn by Poincare thatthetheoryofintegral-invariants maybeappliedtothediscussion ofPoisson stability. Considering asystemof.differential equations forwhich III...Ifta?&B ...8x I...I isanintegral-invariant, weregard these equations asdenningthetrajectoryinndimen sions ofapointPwhose coordinates are(xv,#2,-.,%)Ifthetrajectories have no branchesrecedingtoaninfinite distance from theorigin,itmaybeshewn that ifanysmall regionRistaken inthespace, there existtrajectories which traverse IIinfinitelyoften : and,infact, theprobabilitythatatrajectory issuing from apointofRdoesnottraverse thisregion infinitelyoften iszero,however smallRmaybe.Poincare hasgivenseveral extensions ofthismethod, andhasshewn thatunder certain conditions itisapplicablein therestricted problemofthree bodies. MISCELLANEOUS EXAMPLES. 1.Shew thatthemotion ofaparticleinanellipse under theinfluence oftwo fixed Newtonian centres offeree isstable.(Novikoff.) 2.Aparticleofunitmass isfree tomove inaplane under theaction ofseveral centres offorcewhich attract itaccordingtotheNewtonian lawoftheinverse square of thedistance :denoting theresulting potential energy oftheparticle byV(x,y\shew that theintegral I/I(1-^)log{h-V(a-%; where theintegrationistaken overtheinterior ofanyperiodic orbit forwhich theconstant ofenergy hasthevalue h(thecentres offorcebeing excluded from thefield ofintegration bysmall circles ofarbitraryinfinitesimalradius), isequaltothenumber ofcentres offorce enclosed bytheorbit, diminishedbytwo.(MonthlyNotices R.A.S. LXII.p.186.) 3.Letafamilyoforbits inaplane bedefinedbyadifferential equation where(#,y)arethecurrentrectangular coordinates ofapoint onanorbit ofthefamily ; and let8ndenote thenormal distance from thepoint (,v,y)tosome definiteadjacent orbit ofthefamily. Shew that8nsatisfies theequation *Amer. J.Math. i.(1878), p.75. fActaMath. x.(1887), p.109. +ActaMath. xxi.(1897), p.99. Poincare, ActaMath. xin.(1890), p.67;Nouv. Meth. in.Ch.xxvu. 408 TheGeneral Theory ofOrbits[CH. r I-,fdy\2 }(3$dyS0\where /= 41+-/ )M-^+-f-^-)+{(b(x, y)}2 , and isavariable defined bythegquation *:ldt \dx (Sheepshanks Astron. Exam.) 4.Aparticle moves under theinfluence ofarepulsiveforcefromafixed centre :shew thatthepathisalwaysofahyperbolic character, andnever surrounds thecentre offorce; thattheasymptotes donotpassthrough thecentre inthecaseswhen thework, which has tobedone against theforce inorder tobring theparticle toitspositionfromaninfinite distance, hasafinite value;butthatwhen thiswork isinfinitely great,theasymptotes passthroughthecentre, andtheduration ofthewhole motion maybefinite. (Schouten.) 5.Shew that inthemotion ofaparticle onafixedsmooth surface under theinfluence ofgravity,thecurve ofseparation between theattractive andrepellent regionsofthe surface isformed bytheapparent horizontal contour ofthesurface, together withthelocus ofpointsatwhich anasymptotic direction ishorizontal. 6.Aparticle movesfreelyinspace under theinfluence oftwoNewtonian centres of attraction;shew thatwhen itsconstant ofenergyisnegative,itdescribes aspiral curve round thelinejoining thecentres, remaining within atubular region bounded bytwo ellipsoidsofrotation andtwohyperboloidsofrotation, whose fociarethecentres offorce : andthatwhen theconstant ofenergyiszeroorpositive, theparticledescribes aspiral pathwithin aregion which isbounded byanellipsoid andtwo infinite sheets ofhyper boloids ofthesame confocalsystem. (Bonacini.) 7.Thenecessary and sufficient condition inorder thatatwo-parameter familyof curves defined byadifferential equation y"=&lt;i&gt;(x,y, /) maybeasystemoforbits with thesame constant ofenergyisthat shall bealinear homogeneous perfect differential. Thepotential energyisthen a constant multipleof .-*f(8-&)*(P.Frank.) 8.Inthemotion ofaparticleinaplane under forces which depend onlyonitsposition, aone-parameter familyoftrajectoriesisobtainedbystarting particlesatagiven pointina givendirection with allpossible velocities. Shew thatthelocus ofthefocioftheosculating parabolasisacirclepassing through thepoint.Iftheinitial direction isnowvaried, shew thatthelocus ofthecentres ofthe oo1circles obtained isaconichaving thegiven point asfocus :and iftheforces areconservative,thisdegeneratesintoastraightlinecounted twice. 9.Inorder thatasystemofoc5space-curves,ofwhichx1passthrough every point inevery direction, maybeidentifiable with thesystemoftrajectoriesofaparticleinan arbitrary positionalfield offorce,itisnecessary (but notsufficient) that thesystem should have thefollowing properties: (a)Theosculating planesofthe oo2curvespassing through agiven point form apencil:thatis,alltheplanes passthrough afixed direction. xv] TheGeneral Theory ofOrbits 409 (/3)Theosculating spheresofthe oc*curvespassing through agiven pointinagiven direction form apencil:their centres thus lieonastraightline. 10.Shew thatthe on2curves ofanatural family which meetanysurface orthogonally areorthogonal tooc1surfaces, thatis,form anormal congruence. (The surfaces in question arethesurfaces ofequal Action.) (Hamilton.) 11.Shew that theproperty referred toinEx.10belongs exclusivelytonatural families. 12.Inorder that afamilyofoc4curves inspacemay constitute anaturalfamily of orbits, twopropertiesarenecessary andsufficient, viz. : (a)Iftheosculatingcircles ofthose curves ofthefamily whichpassthrough agiven pointpareconstructed atthatpoint, theyhave asecondpointPincommon, andthus formabundle. Consequently,three ofthecircles insuchabundle havefour-point contact with thecorresponding curves. (/3)These threehyperosculating circles willbemutually orthogonal. 13.Theonlypoint-transformations which convert everynaturalfamilyintoanatural familyarethosebelongingtotheconformalgroup. [Examples 8,9,11,12,13above aretaken frommemoirs byE.Kasner intheTrans actions oftheAmer. Math. Soc., 1906-1909. Forfurther work inthisdirection thereader isreferred toProfessor Kasner sPrinceton Colloquium lectures onDifferentialGeometric Aspects ofDynamics.] 14.Two setsofoolcurves inaplane, which formanorthogonal system,areorbits in acertain conservative field offorce. IfUdenote theAction atanypoint (x,y~]ofa particle considered asmoving ononeofthe first setoforbits, andVdenote theAction at (x,y)when theparticleisconsidered asmoving ononeofthesecond setoforbits, shew thatUandVareconjugate functions ofx-andy:andthat thefamilies ofcurves U=constant, Vconstant, areidentical with the orbits. (P.G.TaitandK.Ogura.) CHAPTER XVI 180. Theneedforseries whichconverge forallvaluesofthetime; Poincares series. Wehavealreadyobserved(32)thatthedifferentialequationsofmotion ofadynamical systemcanbesolved interms ofseries ofascending powers ofthetime measured fromsome fixedepoch;these seriesconvergein generalforvalues of twithin some definite circle ofconvergenceinthe -plane, andconsequentlywill notfurnish thevalues ofthecoordinates exceptforalimited interval oftime. Bymeans oftheprocessofanalytic continuation* itwould bepossibletoderive from these series successive sets ofotherpower-series,which wouldconvergeforvalues ofthetime outside this interval;buttheprocessofcontinuation istoocumbrous tobeofmuch use inpractice,andtheseries thus derivedgivenoinsightinto thegeneral character ofthemotion, orindication oftheremote future ofthesystem. The efforts ofinvestigatorshave therefore been directed totheproblemof expressingthecoordinates ofadynamical system bymeans ofexpansions whichconvergeforallvalues ofthetime. Onemethod ofachievingthis resultj-istoapplyatransformation tothe-plane. Assumingthat the motion ofthesystemisalways regular (i.e.that there arenocollisions or other discontinuities, andthatthecoordinates arealways finite), there willbe nosingularitiesofthesystematpointsonthereal axisinthe-plane, and thedivergenceofthepower-seriesint1after acertain interval oftime must therefore beduetotheexistence ofsingularitiesofthesolution inthe finitepartofthe-planebutnotonthereal axis.Supposethatthesingu laritywhich isnearest totherealaxis isatadistance hfrom therealaxis; and letTbeanewvariable defined bytheequation 2hI+T t~t=log. 7T 1T Aband which extends toadistance honeither sideofthereal axis inthe -plane evidently correspondstotheinterior ofthe circleJT=1 inthe *Of.Whittaker andWatson, ModernAnalysis,55. tDue toPoincare, ActaMath. iv.(1884), p.211. 180,181] Integration byTrigonometricSeries 411 r-plane ;thecoordinates ofthedynamical systemaretherefore regular functions ofTatallpointsintheinterior ofthis circle, andconsequently theycanbeexpressedaspower-seriesinthevariable T,convergentwithin this circle. These series will therefore convergefor allreal values ofT between 1and 1,i.e.forallrealvalues oftbetween ocand+oc .Thus these series arevalidforallvaluesofthetime. 181. Theregularisation oftheProblem ofThree Bodies. Inthelastarticle wemade thereservation thatthere aretobenocollisions orother discontinuities forrealvalues of t.Theimportanceofcollisions in themathematicaltheoryoftheProblem ofThree Bodies was firstindicated byPainleve *,whoshewed thatthemotion ofthebodies isregular (i.e.their coordinates areholomorphicfunctions oft)foralltime, providedtheinitial conditions arenotsuch that after afinite interval oftimetwoofthebodies collide. Therelations -which must subsist between theinitial values ofthe variables inorder that acollision may ultimately happenbetween twoof thethree bodies havebeen discussed byLevi-Civitaffortherestricted problem ofthree bodies (when there isonesuch relation) andbyBisconcini\forthe general problem, when there aretworelations :these relations areanalytic, buttheyareexpressed bysomewhatcomplicatedinfinite series, andarenot directly applicable except when theinterval oftimebetween theinitial instant andthecollision issufficientlyshort. Aconsiderable advance wasmadewhen K.F.Sundman shewed thatthe singularityofthedifferentialequationswhichcorrespondstoacollision oftwo ofthebodies isnotofanessential character, andthat itmayinfactbe removedaltogether bymakingasuitable changeoftheindependentvariable : that istosay,itispossibletochoose thevariables whichspecifythemotion, andtheindependent variable, insuchawaythatthedifferential equationsof motion areregularevenwhen two-ofthethree bodies occupycoincident positions ||.Itisthuspossibletoobtain arealprolongationofthemotion after thecollision IF:thecoordinates canbespecifiedforallvalues ofthetime t from ooto+oo,whether collisions takeplaceornot :andapositivelower bound /canbeassignedtothetwogreaterofthemutual distances. There *Lemonssurlatheorie anal, deseq.diff .,Paris, 1897, p.583. fAnnali diMat. (3)ix.(1903), p.1;Comptes Rendus, cxxxvi. (1903), pp.82,221. *ActaMath. xxx. (1905), p.49. Cf.alsoH.Block, Medd. franLunds Obs., Series n.,No.6 (1909); Arkivf. Mat. Astr. ochFys.v.(1909), No. 9. ActaMath, xxxvi.(1912), p.105. The essential features ofthework were originally publishedinActa Societatis Sclent. Fennicae in1906and 1909. ||Levi-Civita regularised thedifferential equations oftherestricted problemofthree bodies byanelementary transformation inActaMath. xxx. (1906), p.306;andinalater paper, Rend, d.Lincei, xxiv.(1915), p.61,heextended this totheproblemofthree bodies inaplane. IfThe variables canbeexpandedinascending powers of(ti-1)&,wheretirepresents the instant ofcollision :theorbits hare cuspsatthepointofimpact. 412Integration byTrigonometric Series[CH.xvi isonlyonecase ofexception, namely when allthree bodies collide simul taneously:butthiscanhappen onlyinavery special typeofmotion, inwhich alltheconstants ofangular momentum arezerotogether*. Disregardingthis case oftriple collision, Sundman introduced anew independentvariable wdefined bytheequation where r ,r1}rzdenote thethree mutual distances, and Iisthelowerbound already mentioned. The coordinates ofthebodies, andthetime, arethen holomorphicfunctions ofwwithin aband offinite breadth 2Ointhew-plane, boundedbytwolinesparalleltotherealaxisandoneither sideofit.There exists acontinuous one-to-onecorrespondence between therealvalues oftand therealvalues ofw,sothatwhen tvaries from ooto+oo,wlikewise varies from ooto+ac . Lastly, SundmanappliedPoincare stransformation 2H,I+T , iv=log-- 7T81-T inorder totransform theband inthew-planeintoacircle ofradiusunityin theplaneofanewvariable r.Thecoordinates ofthethree bodies, andthe time, arenowholomorphicfunctions ofTeverywherewithin theunit circle in ther-plane:andthereforetheycanbeexpandedasconvergentseriesofpowers ofTforallrealvaluesofthetime,whether there arecollisions ornot :the case oftriplecollision alonebeing excepted. 182.Trigonometricseries. The series discussed intheprecedingarticles areallopentotheobjection thatthey givenoevident indication ofthenature ofthemotion ofthe systemafter thelapseofagreatinterval oftime :theyalsothrow nolighton thenumber andcharacter ofthedistincttypesofmotion which arepossible intheproblem:andtheactual execution oftheprocessesdescribed isattended withgreatdifficulties. Under these circumstances weareledtoinvestigate expansionsofanaltogether differenttype. Ifinthesolution oftheproblemofthesimple pendulum (44)weconsider theoscillatory typeofmotion, andreplacetheelliptic functionbyitsex pansionasatrigonometricseries,wehave 27T-g4.- (2s-l)^(t-t Q)-8~~ *This last facthadbeenknown toWeierstrass :cf.ActaMath. xxxv.p.55.Themotion is then inoneplane. tAsimpler equation available inthe restricted problem ofthree bodies wasgiven by G-.Armellini, Comptes Rendus, CLVIII.(1914), p.253. JCf.Whittaker andWatson, ModernAnalysis, 226. 181-183] Integration byTrigonometricSeries 413 where 6denotes theinclination ofthependulumtothevertical attime t;Kand tmayberegardedasthetwoarbitraryconstants ofthesolution, and /j,isadefinite constant, whileqdenotes e~vKIK ,whereKisthecomplete elliptic integral complementarytoK.Thisexpansion,eachterm ofwhich isatrigonometricfunction oft,isvalid foralltime. Moreover, when the constantqisnotlarge,the firstfewterms oftheseriesgiveacloseapproxi mation tothemotion for allvalues of t.Thecirculatory typeofmotion ofthependulum maybesimilarly expressed byatrigonometric series ofthe samegeneralcharacter. Turning now toCelestial Mechanics, wefindthat series oftrigonometric terms havelongbeenrecognisedasthemostconvenient method ofexpressing thecoordinates ofthemembers ofthesolarsystem;these series areofthe type 2tfn,,n 2,...,n fccos(nl6l+n.262-\-...+nkk), where thesummation istaken overpositive andnegative integer values of %1}nz,...,nk,and6risoftheform\rt+er;thequantities a,X,and ebeing constants. Delaunay* shewed in1860 thatthecoordinates ofthemoon can beexpressedinthisway;Newcombf in1874 obtained asimilar result for thecoordinates oftheplanets, and several later writersJhavedesigned processesforthesolution ofthegeneral Problem ofThree Bodies inthis form;theseprocessesarealsoapplicabletootherdynamical systems whose equationsofmotion areofacertaintype resemblingthose oftheProblem ofThree Bodies. Inthefollowingarticles weshallgiveamethod which isapplicabletoalldynamical systems andleads tosolutions intheform of trigonometricseries :themethod consistsessentially,aswillbeseen, inthe repeated applicationofcontact-transformations, whichultimately reduce the problemtotheequilibrium -problem. 183. Removaloftermsofthefirst degree from theenergy function. Consider then adynamical system, whoseequations ofmotion are dqr_dH dpr__dff dt~dp r dt~~dfr1,2, ...,w), where theenergyfunctionHdoesnotinvolve thetime texplicitly. Thealgebraicsolution ofthe2nsimultaneousequations 8/=0,^=(r-1,*.,,)dpr oqr willfurnish ingeneral oneormore setsofvalues(a,,az,...,an,bl}62,...,6n) forthevariables(q,, q.2&gt;...,qn,plt...,p n);andeach ofthese setsofvalues *Theorie dumouvement delalune.Paris, 1860. fSmithsonianContributions, 1874. Je.g.Lindstedt, Tisserand, andPoincare. Whittaker, Proc. Land. Math. Soc.xxxiv.(1902), p.206. 414 Integration byTrigonometricSeries[OH.xvi willcorrespondtoaform ofequilibriumor(iftheaboveequationsarethose ofareducedsystem) steadymotion ofthesystem. Letanyoneofthese setsofvalues (al,a2,...,an,b1}b2,...,bn)beselected; weshallshewhow tofindexpansionswhichrepresentthesolution ofthe problemwhen themotion isofatypeterminated bythisform ofequilibrium orsteadymotion. Thus ifthesystemconsidered were thesimple pendulum, andtheform ofequilibriumchosen were that inwhich thependulum hangs verticallydownwards atrest, ouraimwould betofind series which would representthesolution ofthependulum problem when themotion isofthe oscillatory type. Take thennew variables(q,f ,q2,...,qn,pS,p2f ,...,p n\defined bythe equations qr=ar+qr,pr=br+pr (r=1,2, ...,n) ; theequationsofmotion become dqrdH dpr dH .a _* (f=I/ ni7. "i /) 7i O / V *&gt; "&gt; )"/&gt;dtdp,dtdqr and forsufficientlysmall values ofthenew variables thefunctionHcanbe expandedasamultiple powerseries intheform H=H +Hl+H.2+H,+ ..., whereHkdenotes terms homogeneousofthekihdegreeinthevariables (qi, q*,&gt;qn,pi, -,pn)- SinceHdoesnotcontain anyofthevariables, itmaybeomitted :andthe factthatthedifferential equationsaresatisfied when (&lt;//,qz,...,qn ,p,1 ,...,pn) arepermanentlyzerorequiresthatHlshould vanishidentically. Theexpansion ofHtherefore beginswith thetermsH2,which(suppressingtheaccents of thenew variables) maybewritten intheform H.2=^2(arrgr-+2or Sqrqs)+2bnqrp8+^(crrpS4-2cnprp,\ where ttrs=asr, crs=csr , butbnisnotnecessarily equaltobsr.IfthetermsHs,H4,...wereneglected incomparisonwithH2,theequationswould become those ofavibrational problem (Chapter VII). 184. Determination ofthenormal coordinates byacontact-transformation. Weshallnowapplyacontact-transformation tothesysteminorder to expressH2inasimplerform *,infact, toobtain variables whichcorrespond tonormal coordinates forsmall vibrations ofthesystem. *Inobtaining thetransformation ofthis article amethod isusedwhich wassuggested tothe author byDrBromwich, andwhich furnishes thetransformation more directly than themethod originally devised. 183,184] Integration byTrigonometricSeries 415 Consider thesetof2nequations (r=l,2,...,w) ~8aSr^dy rz^ 1*x *&gt; &gt;Xn &gt;y^&gt; y^)- J orsyr=arlx1+arzx.2+ ...+arnxn+\)ny^+...+brnyn\ J U.A j.7 l(r=l,2,...,n).sx.rolrxl+o.2rxz+...+onrxn+cnyl+ ...+crnyn) Onsolvingtheseequations, weobtain forsthedeterminantalequation which in84wasdenoted byf(s)=:weshallsuppose thatH2isapositive definite form, and (asin84)weshall denote theroots oftheequation by isi, is2,..., isn;thequantitiesslts.2,...,snare allreal,and for simplicity weshallsupposenotwoofthem tobeequal. Toeach rootthere willcorrespondasetofvalues fortheratios ofthe quantities (xl,xz,...,xn,ylt...,yn};letthesetwhichcorrespondtotheroot isrbedenoted by(y^, rx2,...,fXn,ry^,..., ryn),and letthesetwhich corre spondtotheroot isrbedenoted by(_,.#!, _,^ 2,...,-rXn,-ryi,...,-ryn)&gt;so thatwehave Multiplytheseequations bykxpand kyprespectively, addthem, andsumwith respecttop;wethus obtain theequation n is,. 2,\tjcpkyp ifKpryp)=-H(?%k), where 1\i -,/t/ )ll^***iIc^l&lt;~^**32jx- ]Jb"^2*~K^\T^^f"""^H \vQb] jfcVi r*Jt^- it1/i/~T~* sothatH(r,k)issymmetricallyrelated torand k. Interchangingrand k,wehave * isk2(kXpryprXpkyp)=H(r,k\ P=I n andtherefore(sr+sk)^(^ ryp^xpkyp)=0. P=I So,unless sr+skiszero,wehave n -^\.^^p kyj)^~kptjPs andconsequently H(r,k)iszero :ifsr+skiszero,wehave^p=-rXp, kyp= -,-!/]&gt;,andtherefore n 1Sr \rXpryp rXpryp)=" \f&gt; **/ p=l Integration byTrigonometricSeries[CH.xvi Ifnowwedefine newvariables((?/,q2,...,qn ,PI,,Pn)bytheequations n v{ ~fr 1 7";21&gt;V~T*1 6i - -i74 /*2nI"*//-A /* _-lit I i1 "i / (r=J.. . ....n). i . i . . , /1v Pr=tfr and if8andAdenote anytwoindependentmodes ofvariation, itisevident n n thaithecoefficient oftarAc*in2(Sqi&pt-bqiSpi)is2(M-kyi--&i fyi), 1=1 n which iszerowhen risnotequaltok.Thus2(Sqi&pi AqiSpi)contains no terms exceptsuch as(Sqr&p,.-Agv&p/),andthecoefficient ofthisterm is 2(i-ici-ryi-ri ryi)-Now hitherto theactual values ofrXi, ryihavenotbeen fixed, asonlytheir ratios aredetermined from theirequationsofdefinition; wemaytherefore choose their values sothat 2O*-ryi--M ryi)=1(r=i,2,...,n), 1=1 andthenweshallhave n n 2(8qi&pi &qi$pi)= 2(Sqr&p rAgvSp/), 1=1 r=l sothat(128)thetransformation from thevariables(ql}q2,...,qn,PI,.Pn) tothevariables(g/, q%,....qn ,pi, ...,pn)isacontact-transformation. Moreover, ifinH2wesubstitute for(qltq2,...,qn,plt...,p n)interms of (q\&gt; q*&gt;-&gt;qn,Pi,,Pn),weobtain H2=2H(r,-r)qrpr r=l n or H2=i2srqrpr. r=l Now applytothevariables (&lt;?/,q2,...,qn,PI, ...,pn)thecontact- transformation definedbytheequations dW &lt;n Ir 12 ^ ^rPr~dq r n! ipr"2 where W=2(pr"qr+i-iisrq,2 }, n which gives H2=i2(pr"2+sr2qr"2 ). r=l As allthetransformations concerned have been linear, weseethat H3,H4,...willbehomogeneous polynomialsofdegrees 3,4,...inthenew variables :and thus, omittingthe accents, wehave the result that the equations ofmotion ofthedynamical systemhave beenbroughttotheform dqr_m dp_r__dH dt~ dpr dt~ dqr 184,185] Integration byTrigonometric Series 417 where H=H2+H3+H4+..., inwhichHrisahomogeneous polynomial ofdegreerinthevariables, andin particular Itisclear that ifweneglectHS)H4,...incomparison withH2,and integratetheequations,thesolution obtained willbeidentical with that found in 84. 185.Transformationtothetrigonometric form ofH. Thesystemwillnowbefurther transformedbyapplyingtoitacontact- transformation from thevariables(qltq2,...,qn,p^,...,pn)tonew variables (&lt;?/,q*,,qn,Pi,,pn),definedbytheequations ,dW dW where W=S |qrarcsm r=l sothat pr=(2srqr^sinpr, qr= (&lt;2qr)?sr~$cospr, (r=1,2,...,). The differentialequations become dtdpr"~dt where H=s^+s2q,+...+snqn+H 3+H4+...; andnowHrdenotes anaggregateofterms which arehomogeneousof degree \rinthequantities qr ,andhomogeneousofdegreerinthe quantities cos_p/, ainp r. Since aproductofpowersofcospr,sinprcanbeexpressedasasum of sines andcosines ofanglesoftheform(n^pi+n2p.J+...+nnpn),where n1}n2,...,nnhaveintegerorzero values, itfollows thatHrcanbeexpressed asthesumofafinitenumber ofterms, each oftheform sin tf^ft1"1...qnmn(nlPl+n2p2+...+nnpn),L-Ub where m,+m2+ ...+mn=r, [nr^.2mr, andtherefore |nx+nz|+...+ r. ThefunctionHisthusexpressedintheform ,)H.....nin /* /* /olll .. where foreachtermwehave %+na |+...+nn |^2(m,-t-m2+...+mn), w.D. 418 Integration byTrigonometricSeries[CH.xvi andclearlytheseries isabsolutely convergentforallvalues ofp^,p2,...,pn , provided &lt;?/,q2,...,qndonotexceed certain limits ofmagnitude. From the absolute convergenceitfollows thattheorder oftheterms canberearranged inanyarbitrary way:weshallsuppose them soordered that alltheterms involvingthesame argument n^+...+nnpnfarecollectedtogether,so thatHtakes theform where thecoefficients aandbarefunctions of &lt;//,q2,...,qnandtheexpansion ofan,n,...,n norbniyn2,...,n ninpowersofqi,q 2,...,^ncontains noterms oforder lower than{ |^ |+n2+...+\nn\\;andwhere thesummations extend over allpositiveandnegative integerandzerovalues ofn1}n2,...,nn,except thecombination ?ix=nz=...=nn=0. Moreover, theexpansionofa0j0,...,o(whichwillbecalled thenon-periodic part ofH,therestoftheexpansion beingcalled theperiodic part) beginswith theterms MI+Ma++s n&lt;ln, and,wheng/,q2,...,q naresmall, these arethemostimportantterms inH, sincetheycontribute terms independentof &lt;?/,q2,...,qntothedifferential equations. Forconvenience weshall oftenspeakofg/,q2,...,qnas"small," inorder tohave adefinite idea oftherelative importanceoftheterms which occur. Itwillbeunderstood thatq^,q^,...,qnarenot,however, infinitesimal, and infactarenotrestricted atallinmagnitude exceptsofarasisrequiredto ensure theconvergenceofthevarious series which areused. Toavoid unnecessary complexity,weshallignoretheterms S^.n,, ...,sin(n^pi+...+nnpn} inH.astheyaretobetreated inthesamewayastheterms Sofn,,n 2,...,,,cos(n 1p1/+...+nnpn), andtheirpresence complicates,butdoesnotinanyimportant respect modify, thelaterdevelopments. Theform towhich theproblemhasnowbeenbrought maytherefore be stated asfollows (suppressingtheaccents inthevariables):Theequations of motion are aqrdH dpr=m (r=1;2)...,.),dtdpr dt dqr where H=u0(0.....+2an ,,...,cos(n lpl+n2p2+...+nnpn\ and thecoefficientsaarefunctions ofql,q2,...,q nonly ;moreover, theperiodic partofHissmall comparedwith thenon-periodic parta 0)0&gt;...,o/atermwhich hasforargument (n lp1+n2p2+...+nnpn)has itscoefficient a ni&gt;^, ...,atleast 185,186] Integration byTrigonometric Series 419 oforder ^{ \%,+ |n2\+...+ jnn\}inthesmallquantities qltq2,...,qn;and theexpansion ofa,o,...,obeginswith theterms(slqi+s2q2+...+snqn\ Itfollows from thisthatwhen thevariablesqltq2,..,,qnaresmallthey varyvery slowly,while thevariables p1}p2,...,pnvaryalmostproportionally tothetime. 186. Othertypes ofmotion which lead toequations ofthesameform. Theequationswhich havenowbeen obtained have beenshewn tobe applicable when themotion isofatypenotfarremoved from asteady motion oranequilibrium -configuration, e.g.theoscillatory motion ofthesimple pendulum,orthosetypesofmotion oftheProblem ofThree Bodies which have been studied in 171. Buttheseequations maybeshewn tobe applicablealsotomotion which isnotofthis character, and inparticularto motion such asthat oftheplanetsround thesun, orthemoon round the earth *. For lettheequationsofmotion oftheProblem ofThree Bodies betaken intheform obtained in160; and letthecontact-transformation which is definedbytheequationsdW Pr= beappliedtothissystem, where or-*~,r-~ (r=l,2,3,4)oqr dqr ,+1-- 77,--I- ^rdq. 2 2 Thenewvariables canbeinterpretedinthefollowing way. Suppose that at theinstant talltheforcesactingontheparticle //,cease, exceptaforce of magnitude m^m^q^ directed totheorigin;and letabethesemi-majoraxis and etheeccentricityoftheellipse described after thisinstant :then q,=it^m^a-e2? , q3= Further, ifthelower limits oftheintegralsaresuitably chosen, /?/+q3is thetrueanomalyof/*initsellipse, and-p3isthemeananomaly. The variablesqj,qt,p2,p^stand inacorrespondingrelation totheparticle ///. Theequationsofmotion nowtaketheform dqr_dH dp; dH W-fa"W"9ff(^=1,2,3,4); when theparticles w2and ra3aresupposedtobeofsmallmasscompared with Wj,andaredescribingorbits ofaplanetarycharacter aboutml}itisreadily found thatHcanbeexpandedinterms ofthenewvariables intheform H=ao,o,o,o+2a,li)W2i,l3Fn4cos(n^pf+n2p2+n3p3+n^ ), *Delaunay, Theorie delaLune;Tisserand, Annales deIObs.deParis, Memoires, xvm.(1885). 272 420 Integration byTrigonometricSeries[cu.xvi where thecoefficients aarefunctions of(qi, q,,qa,q4)only,thesummation extends overpositiveandnegative integerandzero values ofnltw2,ns,?i4, andthecoefficient a0)0)0)0ismuch themostimportant partofthe series. As thisexpansionofHisofthesame character asthatobtained in 185,it follows that themethodofsolution giveninthefollowingarticles isapplicable either tomotion oftheplanetary typeortomotion ofthetypestudied in171. 187. Removal ofaperiodictermfromH. We shallnowapplytothesystemanother contact-transformation, the effect ofwhich willbetheremoval ofoneoftheperiodicterms fromH;this willfurther accentuate thefeature already noted, namelythatthenon-periodic partofHismuch more importantthan theperiodic part*. Letoneoftheperiodicterms inHbeselected, say n,,n 2,...,nncos(ni.Pi+n?P2++nnpn)- WriteH=a , , ...,&lt;&gt;+,,...,.cos&lt;XPi+n*P*++n ^P^&gt;+R &gt; sothatRdenotes therestoftheperiodicterms ofH;whenwewish toput inevidence theargumentsofwhich&lt;*,,...,isafunction, weshall write it Applytothesystemthecontact-transformation defined bytheequations ,_dW =dW 12, where W=qfa+q2p2+...+qnpn+f(?i,?2/ , ,?n,0) weshallsupposethat/isafunction, asyetundetermined, ofthearguments indicated. Theproblemisnowexpressed bytheequations dt dpr"dtdqr where 8/ ) -f-:Hi~Q,...,qn+w-no/j )cos^ "T"R &gt; and6andRaresupposedtobeexpressedinterms ofthenew variables by means oftheequationsoftransformation df,"tf *Readers familiar with Celestial Mechanics willnotice theanalogy ofthismethod with that ofDelaunayslunar theory: theanalysisisdifferent fromDelaunay s,buttheidea isessentially thesame. 186,187] Integration byTrigonometricSeries Thefunction/is,asyet,undetermined andatourdisposal.Itwillbe chosen soastosatisfythecondition that 6shallidentically disappearfrom theexpression D,o,....o i+HI- ,...,qn+nn ,~\f "r\f\+n,, ...,(qi+MIg0,.,qn+nn^jcos sothat thisquantityisafunction of &lt;?/,9a,...,qnalone, say a&gt;o,,...,o(qi,qs, &gt;&lt;/ ) Then theequation 9/9 ( 3/ 9A(*, .....nn(qi+h figIn+nn^)cosB=a determines9/730interms of &lt;//, &lt;?2&gt; &gt;^n,- o,.....o&gt;andcos 8. Supposethat thesolution ofthisequationforf)f/d&isexpressedinthe form ofaseries ofcosines ofmultiplesof6(which canbedone, forinstance, bysuccessiveapproximation),sothat df^=c+2ckcoskO,ov k=i where c,Cj,c2,...areknown functions ofq^,q^,...,qn,a 0&gt;0...... Nowa0,0,.,.,0isasyetundetermined, and isatourdisposal. Imposethe condition that cistobezero; thisdetermines a,o,...,oasafunction of qi, q-2,&gt;qn ;and,onsubstitutingitsvalue intheseries fordf/dO,wehave dffi8= Ickcosk0, k=i where now c1}C2,c3,...areknown functions ofq^,q2,...,qn.Integrating thisequationwithrespectto6,and forourpurpose takingtheconstant of integrationtobezero,wehave /=1^sin&tf.t-i* Theequations denningthetransformation nowbecome %1dck.ia\ Pr=p-r+ 275-7Sinfvd k-lkd^(r-\2 n)^i,&,..., ii). qr=qr+nr2ckcoskd I k=--\ Multiplythe first setoftheseequations bynltnz,..., nnrespectively, and addthem :writing WiK+n2p2+...+nnpn= , 1 /V /I -V**-/"vie OClfwehave v=6+2,-I?h5^5+w2 ,+... Sm 422 Integration byTrigonometricSeries[OH.xvi Reversingthis series, wehave 6=&+Sdksinke\ where d1}d2,...areknown functions of &lt;//,q2,...,qn.Substitutingthis value ofdintheequationsoftransformation, theybecome sn COS where allthecoefficients rVk&gt;9kareknown functions of &lt;?/, g./,5#&gt;/ Now, before thetransformation, thefunction Rconsisted ofanaggregate ofterms ofthetype R=2aWi )OT2&gt;...&gt;TOBcos(w^PJ+...+mnpn); when thevalues which have been found for(q1}q2&gt;...,qn,p\, ,pn)are substituted inthisexpression,andtheseries isreduced byreplacing powers andproductsoftrigonometricfunctions of /&gt;/, p.?,...,pnbycosines ofsums ofmultiplesofp,,p,,...,pn ,itisclear thatRwillconsist ofanaggregate ofterms ofthetype R=2ami)Ml2)...&gt;Mncos(m^pi+m2p2+...+mnpn), where thecoefficients aareknown functions of(#/,q^, ...,qn}- Wethushave theresult(omittingtheaccents ofthenew variables) that afterthetransformationhasbeeneffected,thesystemisstillexpressed byaset ofequations oftheform dqr_dH dpr_dH - - ~~i/x,.j**/ dtdpr dtdqr whereH=a^o.-.o +Sam,, ,,,... &gt;Wncos(m lpl+m2p2+...+mnpn), andwhere thecoefficientsaareknown functions ofq1}q2,...,qn. Letusnowreview thewhole effect ofthetransformation. The differential equationsofmotion have thesamegeneralform asbefore; butfrom the equation Oo,o,...,o+ani,n2,...,n nCOS(n^+??2&gt;2+...+nnpn)=a,o,...,o weseethatoneterm hasbeen transferred from theperiodic partofHtoits non-periodic part:theperiodic partofHislessimportant,incomparison with thenon-periodic part,than itwasbefore thetransformation wasmade. 188. Removal offurther periodictermsfromH. Havingnowcompletedtheabsorptionofthisperiodicterm intothenon- periodic partofH,weproceedtoabsorb oneoftheperiodicterms ofthenew expansionofHintothenon-periodic part,byarepetitionofthesame process.Inthiswaywecancontinuallyenrich thenon-periodic partof 187-189] Integration byTrigonometricSeries 423 Hattheexpenseoftheperiodic part,andultimately,after anumber of applicationsofthetransformation, theperiodic partofHwillbecome so insignificantthat itmaybeneglected.Let(al,o2,...,an,/31;/32, &gt;/3n)be thevariables atwhich wearrive asaresult ofthefinal transformation :then theequationsofmotion are doir_dH dpr__dH ,_ ~di= d@r dT~~~^r where H,consisting onlyofitsnon-periodic part,isafunction of (1}a.2,...,an)only.Wehave therefore -0,*~/g*(~1, *,...,&gt;. which shews that thequantitiesaareconstants, andthequantities ftareof theform dH ft,.= fjirt+er,where ar=-^ (r=1,2,...,n);oar thequantities e,.arearbitrary constants, andthepartoffirindependentof (d,cr2,...,an)is-ST. 189. Reversion totheoriginalcoordinates. Havingnowsolved theequationsofmotion intheir final form, itremains onlytoexpresstheoriginalcoordinates ofthedynamical systeminterms of theultimate coordinates(a1,a2,...,an,/3l5..., (3n).Rememberingthatthe result ofperforming anynumber ofcontact-transformations insuccession is acontact-transformation, itiseasilyseen that thevariables(q1}q.2,...,qn, plt...,p n)used attheendof185canbeexpressedinterms of(1( 2,... ,n, /3] ,...,/8n)byequationsoftheform mnsn where thecoefficients aand 6arefunctions of(a1}a2,....an). From this itfollows that thevariables(q^,q2,...,qn,pi,--^Pn)of183, interms ofwhich theconfigurationofthedynamical systemwasoriginally expressed,areobtained intheform oftrigonometric series, proceedingin sinesandcosines ofsums ofmultiplesofthenangles @i,/32,..., {3n.These anglesarelinear functions ofthetime, oftheformfj,rt+er;thequantities 424Integration byTrigonometricSeries[CH.xvi erarenofthe2narbitraryconstants ofthesolution, while thequantities /j,rareoftheform Pr=-*.-+S ckl,*..., kni*a*2n*n , *1,K2,... thecoefficients cbeing independentoftheconstants ofintegration. The coefficients inthetrigonometricseries arefunctions ofthearbitraryconstants (i, 2, -,n)only. Theexpansionsthusobtainedrepresent afamily ofsolutions ofthedynamical system,thelimiting memberofthefamily beingtheposition ofequilibriumor steady motion which wasourstarting-point. Evidently also,byapplyingtheintegration-processof187 189tothe equationsofmotion found in186,weobtain asolution oftheProblemof Three Bodies, when themotion isoftheplanetary type,intermsoftrigono metric seriesofthekind abovespecified. Forthefurther developmentofthetheoryofthepresent chapter,inconnexion with theProblem ofThree Bodies, reference maybemade totreatises onCelestial Mechanics : inparticular, thesecond volume ofPoincare sNouvelles Methodes delaMecaniqueCeleste contains anaccount ofseveral methods ofderiving expansions, with adiscussion ofthe convergenceoftheseries obtained. Themost recent discussion ofthesubjectwillbefound inapaper bythepresentwriter OntheAdelphic Integral oftheDifferential Equations of Dynamics (Proc. Roy.Soc.Edin., Nov. 1916). MISCELLANEOUS EXAMPLES. 1.Let &lt;f&gt;denote anyfunction ofthevariablesq^,g2,...,qn,p1,...,pnofadynamical system whichpossessesanintegralofenergyH(qi,?2, &gt;?n&gt;Pi,&gt;p)=Constant; let%,a2,...,an,61}...,bnbethevalues ofqltq2,...,qn,pi, iPnrespectivelyatthe instant t=t;and let{/,g}denote thevalue ofthePoisson-bracket(/,g]when thequan titiesqi, &lt;?2, , &lt;lniP\i &gt;Pnoccurringinitarereplaced respectively by 1? 2&gt;&gt;a, bi,...,bn. Shew that 2.Shew thatthedynamical system whose equations ofmotion are dq_dH ^=_^ dt~ dp^t~ dq JWIW where H= %p&lt;+^- - , possessesafamilyofsolutions represented bytheexpansion (retaining onlyterms oforder lessthan a7 ) ,3a/2a\4 3a where p=-k+~t+f, andaand earearbitrary constants. INDEX OFAUTHORS QUOTED (Thenumbersrefertothepages.} Abdank-Abakanowicz, B.214 Albeggiani, M.L.337 Amontons, G.227 Appell,P.73,258,279 Armellini, G.81,412 Bennett, T.L.356 Bernoulli, Daniel 62,177,186 John 62,229 Bertrand,J.88,260, 320, 331, 332, 338, 349 Bessel,F.W.91 Bisconcini, G.411 Block, H.411 B6cher, M.183 Bohlin, K.340, 358,407 Boltzmann, L.41,278 Bonacini, C.408 Bonnet, 0.94 Bour, E.356 Brell, H.259 Bromwich, T.J.414 deBrun, F.166 Brims, H.357,358 Burgatti,P.69,166,325 Burnside, W. 3,5 Cailler, C.89 Cassie, W.R.118 Cauchy,A.L.4,124, 264,316 Cayley, A.9,12,115 Cerruti, V.330,331 Charlier, C.V.L.395 Chasles, M.4 Chretien, H.90 Christoflfel,E.B.39 Cigala, A.R.396 Clairaut, A.C.78Clebsch, A.311 Conway, A.W.26 Cotes, R.83 Culverwell, E.P.251 Curtis, A.H.97,113 Dainelli, V.96,112,114 DAlembert, J.leR.177,230 Dall Acqua,F.A.316 Darboux, G.80,109, 261, 333,396 Darwin, SirG.H.407 Dautheville,S.320 Davaux, E.400 Delaunay,C.413, 419,420 Derriman, W.H.118 Donkin, W.F.264 Dumas, G.166 Elliott, E.B.210 amEnde, H.112 Euler,L.2,8,9,41,72,93,97,100, 117, 124, 127, 144, 177,248 Ferrers, X.M.215 Flye Sainte-Marie, C.174 Forster, W.262 Ford, L.R.12 Forsyth,A.R.358 Fouret, G.130 Frank, P.408 Galilei, G.62,72,99,177 deGasparis, A.356 Gauss, C.F.9,255 Gautier, A.339 Gebbia, M.174 Glaisher,J.W.L.80 Gorjatscheff,D.166 426 Goursat, E.262,336 Grant, R.339 Green, G.38 Grinwis,C.H.89 Grossi, P.330 Hadamard, J.69,403 Halphen, G.5,106 Hamel, G.41Legendre, A.M.81 Lehmann-Filhes, R.316 Leibnitz, G.W.von35 Leitinger, R,256 Levi-Civita, T.90,325, 343, 385, 396,411 Levy, M.330,331 Liapounoff, A.M.400 Lie, S.275, 290, 295, 301, 322,343 Linders, F.J.393 Hamilton, SirW.R.3,9,55,79,246, 264, Lindstedt, A.413 288,290,315, 316, 323,409 Hazzidakis, J.N.102 Helmholtz, H.von 45,55,247,305 Hertz, H.255 Heun, K.37 Hill, G.W.407 Hiltebeitel, A.M.99 Hirsch, A.45,287 Holder, 0.249 Hoppe,R.130 Husson, E.166 Huygens,C.62,72,99,117Liouville,J.67,281,323 R.167 Lipschitz,R.256 Longley, W.R.393 Lovett,E.0.339, 393,395 MacMillan, W.D.85 Marcolongo,R.166 Mathieu, E.301 Maupertuis,P.L.N.de248 Mayer,A.45 Mehmke, R.77 Monge,G.264,316 Jacobi, C.G.J.104,144, 174,251,276, 281, Moulton, F.R.106, 390, 391, 393,395 287, 295, 307,316, 341, 342, 349,354 Jordan, C.179 Joukovsky, N.Ill Kasner, E.409 Kelvin, Lord (W.Thomson) 261 Kepler,J.60,90 Kerkhoven-Wythoff, A.G.222 Klein,F.12,193,207 Kobb, G.109 Koenigs, G.1,88,275 Kolosoff, G.167 Korkine, A.337 Korteweg,D.398,400 Kotter, F.166 Kowalevski, N.166 S.164Muth,P.183 Nanson, E.J.183 Neumann, C.116, 215,239 Newcomb, S.413 Newton,Sir I.27,29,30,47,48,59,62,77, 78,82,83,86,90,103,229 Nicomedi, R.112 Nobile, V.82 Novikoff,P.M.407 Oekinghaus,E.91 Ogura, K.409 Olsson,0.166 Ostrogradsky, M.264,265 Painleve,P.70,227, 262, 379, 385, 389, 411 Lagrange,J.L.34,38,41,50,62,91,94, Pascal, E.214 97,104, 156, 177, 183,248, 264, 298,316, Pavanini, G.391 322,340,393 Laisant,C.A.113 Lamb,II.203,305 Lambert,J.H.91 Lame, G.104 Larmor,SirJ.278 Laurent, H.337 P.A.198 Lazzarino, 0.166Pennacchietti, G.338 Pfaff,J.F.264, 296, 307,316 diPirro, G.335 Poincare, H.203, 267, 286, 343, 354, 380, 385, 387, 400, 403, 407, 410, 413,424 Poinsot, L.2,152 Poisson,S.D.163,230, 264, 281, 299, 320, 407 Puiseux, V.106 Index ofAuthors Quoted 427 Quanjel,J.316 Radau, R.348 Rayleigh, Lord230,261 Resal, H.115 Rodrigues, O.3,9 Routh, E.J.55 Rueb, A.S.144 Salkowski, E.109 Scheffler, H.255 Schenkl, E.256 Schoute,P.H.85 Schouten, G.408 Segner,J.A.124 Siacci,F.21,24,154, 174,230,325 Signorini, A.388 Sommerfeld, A.193 Stackel,P.109, 166,335 Stader,J.F.81 Stekloff, V.166 Stokes, SirG.G.271 Stromgren,E.395 Sturm,J.C.F.396 Suchar,I.80Sundman, K.F.411,412 Sylvester,J.J.184 Tait, P.G.409 Taylor, Brook 177 Tchapligine,S.A.166,167 Thomson, W.,seeKelvin, Lord Tisserand, F.419 Tissot, A.104 Tonelli, L.388 Vierkandt, A.215 vonVieth, J.23 Voss, A.249 Wallis, J.48,234 Wassmuth, A.256 Weber, W.45 Weierstrass, K.183, 197,412 Whewell, W.78 Whittaker, E.T.64,339,343, 388,393,407, 413,424 Woronetz, P.221,343 Wren, SirC.48,234 INDEX OFTERMS EMPLOYED (Thenumbersrefertothepages, where theterm occursforthefirsttime inthebook orisdefined.} Absolute integral-invariants, 271 Acceleration, 14 ActioAgentis,30 Action, Least, 248 Action andReaction, Lawof,29 Adjoint systems, 287 Admit,to(atransformation), 319 Angles, Eulerian, 9 Angular Momentum, Integral of,59 Anomaly, true, eccentric, andmean, 89 Aphelion, 86 Apocentre,85 Appellsequations, 258 Apse, 86 Attractive regionsofafield offorce, 403 Axes ofinertia, principal, 124 Axis ofrotation, instantaneous, 2 Azimuth,19 Bernoulli sprinciple, 186 Bertrand stheorem ondetermination of forces, 331 impulses, 260 Bilinear covariant, 297 Boltzmann-Larmor representation ofthe LastMultiplier, 278 Bonnet stheorem, 94 Bracket-expressions, Lagrange s,298 Poissons,299 Brims theorem, 358 Canonical form ofequationsofmotion, 264 Cayley-Klein parameters,12 Central forces, 77 Centre ofrotation, instantaneous, 3 Centrifugal forces, 41 Characteristicexponents, 400 function, 289Chasles theorem, 4 ChristofFel ssymbol, 39 Classicalintegrals, 358 Coefficient offriction, 227 ofstability, 396 CollinearLagrangesparticles, 392 Collision, orbitsof,391 Collisions, 234 Components ofmomentum, 48 ofavector, 14 Conjugate point, 252 Conservation ofangular momentum, 59 ofenergy, 62 ofmomentum, 59 Conservativeforces, 38 Constraint, 255 Contact-transformation, 290,293 homogeneous, 301 infinitesimal, 292, 302 Continuation, analytic, 410 Coordinates ofadynamical system, 32 elliptic, 97 ignorableorcyclic, 54 normal orprincipal,181 quasi-, 42 Cotesspirals, 83 Covariant, bilinear, 297 ,, deformation-, 111 Curvature, Least, 255 Cyclic coordinates, 54 Definite quadratic form, 36 Deformation-covariant, 111,404 Degreesoffreedom, 34 Density, 117 Differential form, 296 parameters,111 Index ofTerms Employed429 Displacement,1 possible, 33 Dissipation-function,231 Dissipative systems,226 Distance, mean, 87 Divisors, elementary,183 Eccentric anomaly, 89 Ejection,orbitsof,391 Elementary divisors, 183 Elimination ofthenodes, 341 Ellipsoidofgyration,124 ofinertia, 124 momenta!,124 Elliptic coordinates, 97 Energy, integralof.62 Kinetic, 35 Potential, 38 Equations, Appell s,258 first Pfaffs system of,307 Hamiltons,ofmotion, 263 partial differential, 315 Jacobis,342 Lagrangian, 37 inquasi-coordi nates, 43 withundetermined multipliers, 213 variational, 268 Equidistant Lagrangesparticles, 393 Equilibrium-configuration,177 Equilibrium-problem,315 Equimomental bodies, 117 Eulerian angles,9 Exponents, characteristic, 400 Extended point-transformations, 293 External forces, 32,37 Field offorce, 30 conservative, 38 parallel, 93 First PfaflPssystem, 307 Fixity, 26 Fixtures, sudden, 169 Flux ofavector, 14 Focus, kinetic, 252 Force, 29 Forces, central, 77 centrifugal,41 external andmolecular, 32,37 Form, differential, 296 Freedom, degrees of,34Frictional systems, 227 Function, dissipation-, 231 Hamilton scharacteristic, 289 principal, 317 Hamiltonian, 265 Jacobis,342 Function-group,322 Gaussprinciple, 255 Geodesies, 254 Gravity, 27 Group, function-, 322 Group-property, 293 Gyration, ellipsoid of,124 radiusof,118 Gyroscopic terms, 195 Hadamard stheorem, 406 Halphenstheorem,5 Hamilton spartialdifferentialequation, 315 ,, principle,246 theorem, 79 Hamiltonian form ofequationsofmotion, 264 function, 265 Herpolhode,154 Hertz sprinciple,255 Holonomic, 33 Homogeneous contact-transformations, 301 Homography,12 Ignorable coordinates, 54 Ignorationofcoordinates, 56 Impact, 234 Impulse, 49 Impulsive motion. 48 Index ofstability, 398 Inelastic bodies, 234 Inertia, ellipsoid of,124 moment aridproduct of,117 principalaxesof,124 Infinitesimal contact-transformations, 292, 302 Initial motions, 45 Instability, 186, 193, 203,396 Instantaneous axis ofrotation,2 centre ofrotation, 3 Integralofadynamical system, 53 ofangular momentum, 59 classical, 358 ofenergy, 62 Jacobian, 354 Index ofTerms Employed Integralofmomentum, 58 ofasystem ofequations, 53 Integral-invariants, 268 absolute and relative, 271 Invariable lineandplane, 144,346 Invariant relations, 326 Invariants, integral, 268 Inverse ofatransformation, 293 Involution, 322 Isoperimetrical systems, 267 Jacobi sequation,342 function, 342 Jacobianintegral, 354 Joukovskystheorem, 107 Kinematics,1 Kineticenergy,35 focus, 252 ,, potential,38 Kineto-statics, 37 Koenigs andLiestheorem, 275 Kortewegstheorem, 397 Kowalevski stop,164 Lagrangesbracket-expressions, 298 equations, 34 with undetermined multipliers, 213 ofimpulsive motion, 50 forquasi-coordinates, 43 threeparticles, 393 Lagrangian function, 39 Lambert stheorem, 91 Larmor-Boltzmannrepresentationofthe LastMultiplier, 278 LastMultiplier, 277 Law ofAction andReaction, 29 Newtonian, 86 Least Action, 248 Curvature orConstraint, 255 Levi-Civita stheorem, 325 Levystheorem, 330 LieandKoenigs theorem, 275 Liouville stype, systems of,67 Localised vectors, 15 Mass, 28 Mathieu transformations, 301 Mean anomaly, 89Meandistance, 87 ,,motion, 88 Meridianplane, 18 Molecularforces, 32 Moment ofaforce, 30 ,, ofinertia, 117 Moinentalellipsoid, 124 Momentum, 48 angular, 59 correspondingtoacoordinate, 54 integral of,58 Motion, impulsive, 48 initial, 45 mean, 88 steady, 193 Multiplier, Last, 277 Naturalfamilyoforbits, 389 systems, 57 Newton stheorem onrevolving orbits, 83 Newtonianlaw,86 Node, 349 Nodes, elimination ofthe,341 Non-holonomic, 33 Normal coordinates, 181 vibrations, 186,195 Orbit, 78 periodic, 386 Order ofanintegral-invariant, 268 ,,ofasystem, 52 Oscillation, centreof,132 Parallel fields offorce, 93 Parameters, Cayley-Klein,11 differential, 111 Eulers,8 Particle, 27 Particles, Lagrange s,393 Pendulum, simple,72 spherical, 104 Perfect roughness,31 Pericentre, 85 Perihelion, 86 Perihelion-constant, 87 Period, 73 Periodic solutions ororbits, 386 time, 87 Pfaff sexpression, 296 ,,systemofequations, 307 Pitch ofascrew,5 Plane, invariable, 346 Index ofTerms Employed431 Planetoid, 353 Poincare snormal variables, 387 theorem, 380 Poinsot srepresentation,152 Point-transformation, 293 Poisson sbracket-expressions,299 stability, 407 theorem, 320 Polhode, 154 Possible displacement,33 Potential energy, 38 Kinetic, 38 ,, involvingthe velocities, 44 Principalaxes ofinertia, 124 coordinates, 181 ,, function, 317 moments ofinertia, 124 Principle, Hamiltons,246 ,,ofLeast Action, 248 ofLeast Curvature orConstraint, 255 ofRelativity, 26 ,,ofSuperpositionofVibrations, 186 Problem ofThree Bodies, 339 inaplane,351 restricted, 353 Product ofinertia, 116 Prolongationofmotion after acollision, 411 Quadratures, problemssolubleby,54 Quantitas Motus, 48 Quasi-coordinates,41 Quaternions, 9 Radius ofgyration, 118 Rayleighsdissipation-function, 230 Reaction, lawofAction and,29 Reciprocal theorem, Helmholtzs,304 Reciprocation, 291 Regularisatiori, 411 Relations, invariant, 326 Relativevelocity,14 integral-invariants, 271 Relativity, principle of,26 Repellent regionsoffield offorce, 403 Resistance ofair,229 Restrictedproblem ofthree bodies, 353 Resultant ofvectors, 14 Reversed forces, 47 motion, 305 Revolving orbits, 83 Rigid, 1,32Rodrigues andHamilton stheorem, 3 Rotation about aline,1 point,1 instantaneous axisof,2 centreof,3 Rough, 31 Screwdisplacement,5 Similarityindynamical systems, 47 Sleeping top,206 Smooth, 31 Spherical pendulum,104 top,159 Spirals, Cotes,83 Stabilityofequilibrium, 186 oforbits, 396,407 ofsteady motion, 193 coefficientof,396 indexof,398 secular, 203 Steady motion, 163,193 Sub-group,301 Sudden fixture, 169 Superpositionofvibrations, 186 Surface-density, 118 Suspension,centre of,132 Sylvesterstheorem, 183 Symbol,Christoffels,39 ofaninfinitesimaltransformation, 303 System, adjoint, 287 Systems, dissipative, 226 frictional, 227 involution-, 322 isoperimetrical, 267 Pfaffs,307 Thomson stheorem, 261 Three Bodies, Problemof,339 inaplane,351 restricted, 353 Time, 27 periodic, 87 Top, 155 Kowalevskis,164 sleeping, 206 spherical, 159 Trajectory, 78,245 Transformation, contact-, 290,293 Mathieu s,301 Poincares,410 point-, 293 Translation,1 432 Index ofTerms Employed Trojan groupofasteroids, 393 True anomaly,89 Two centres ofgravitation, 97 Type,Liouvilles,67 Unstable, 186, 193, 203,396 Variational equations, 268 Vector, localised, 15 Vectors, 13 Velocity, 14,33 angular,15 , relative, 14Velocity, correspondingtoacoordinate, 33 Vibrations aboutequilibrium, 177 steady motion, 193 normal, 186,195 ofdissipative systems, 232 ofnon-holonomicsystems,221 Virtual work, 264 VisMatrix, 29 VisViva, 35 Wave-fronts, 289 Weber slawofattraction, 45 Work, 30 CAMBRIDGE: PRINTED BYj.B.PEACE, M.A., ATTHEUNIVERSITY PRESS KClUKN Astronomy Mathematics/Statistics Computer Science Library TO* 100Evans Hall 642-3381 PERIOD 12 4-M Hr-i-Hi ALLBOOKSMAY BERECALLED AFTER 7DAYS DUEASSTAMPED BELOW NI*?^r GOT 1 1997 121996 APR121998 ... 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