Rigid Body Dynamics

123 Pages·2001·4.214 MB·English

by  Borisov A.V., Mamaev I.S.

UDK515.1,531.01
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PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
OurbooksyouareinterestedinyoucaneasilybuyusingtheservicesofourInternet-
shop.Theregistrationinthisshopwillallowyou Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
tobuybookscheaperthananywhereelse;
(cid:15) tosubscribeforregularreceptionofmessagesaboutnewbooks; RigidBodyDynamicsCreators . . . . . . . . . . . . . . . . . . . . . 19
(cid:15) thefastestbuyingofbooksbeforetheybecomeavailableinthebookstores. Euler, Leonard (19). Lagrange, Joseph Louis (19). Poinsot, Louis (20).
(cid:15) Kirchhoff, Gustav Robert (20). Clebsch, Rudolph Fridrich Alfred (21).
Joukovskiy,NikolayEgorovitch(21). Kowalevskaya,SophiaVasilievna(22).
Poincare(cid:1), Henri Jules (23). Lyapunov, Alexander Mikhailovitch (24).
Steklov, Vladimir Andreyevitch (24). Chaplygin, Sergei Alexeyevitch (24).
Kozlov,ValeriyVasilievitch(24).
BorisovA.V.,MamaevI.S.
Rigid Body Dynamics. (cid:22) Izhevsk: NIC (cid:16)Regular & Chaotic Dynamics(cid:17), 2001, Chapter1. RigidBodyMotionEquations andtheir Integration . . . 26
384p. 1. PoissonBracketsandHamiltonianFormalism. . . . . . . . . . . 26
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1.PoissonManifolds . . . . . . . . . . . . . . . . . . . . . . . 26
The book discusses main forms of equations of motion of a rigid body, including
Poisson brackets and theirproperties (26). Nondegenerate bracket. Sym-
themotioninpotential(cid:28)elds,in(cid:29)uid(Kirchhoff’sequations),andmotionofarigidbody
plecticstructure.(29). Symplecticfoliation. Darbouxtheoremgeneraliza-
withcavities(cid:28)lledwith(cid:29)uid.Thebookcontainsconditionsoftheorderreductionofthese
tion.(30).
equations,andexistenceofcyclicvariables.Itcollectsalmostallintegrablecasespresently
2.TheLie(cid:21)Poisson Bracket . . . . . . . . . . . . . . . . . . . 31
known, and methods of their explicit integration. For the purpose of investigation, the
computertechniques,allowingvividrepresentationofthemotionpicture,arewidelyused.
2. Poincare(cid:1) andPoincare(cid:1)(cid:21)ChetayevEquations . . . . . . . . . . . 32
Themajorityofresultspresentedinthebookbelongstotheauthors.
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1.Poincare(cid:1) Equations . . . . . . . . . . . . . . . . . . . . . . . 32
For students and graduate students of mechanical, mathematical and physical
2.Poincare(cid:1)(cid:21)ChetayevEquations . . . . . . . . . . . . . . . . . 34
departments of universities, mathematical physicists, and specialists on dynamical
systems. Historicalcomment(35).
3.EquationsonLieGroups . . . . . . . . . . . . . . . . . . . . 35
4.Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ISBN5-93972-055-2
3. VariousSystemsofVariablesinRigidBodyDynamics . . . . . . 37
Thebookwaspublishedwiththe(cid:28)nancialsupportof x 1.TheEulerAngles . . . . . . . . . . . . . . . . . . . . . . . . 38
theUdmurtianStateUniversity 2.The Euler Variables. Momentum Components and Direction
Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.Rodrigue(cid:21)HamiltonQuaternionParameters . . . . . . . . . . 40
c NIC(cid:16)Regular&ChaoticDynamics(cid:17),2001
(cid:13) 4.Andoyaer(cid:21)DepritVariables . . . . . . . . . . . . . . . . . . 43
5.Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
http://rcd.ru
4 Contents Contents 5
4. EquationsofMotioninVariousForms . . . . . . . . . . . . . . 46 Chapter2. Euler(cid:21)Poisson Equations andTheirGeneralizations . . . 82
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1.EquationsofMotionofaRigidBodywithaFixedPoint . . . . 46 1. TheEuler(cid:21)PoissonEquationsandIntegrableCases . . . . . . . . 82
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Euler(cid:21)Poincare(cid:1) equations on group SO(3) (46). Equations of motion in 1.ARigid Bodywith aFixedPoint . . . . . . . . . . . . . . . . 82
termsofangularvelocitiesandquaternions(47). Kineticenergyofarigid 2.Kirchhoff’sAnalogyforanElasticThread . . . . . . . . . . . 84
bodywitha(cid:28)xedpoint(48). 3.IntegrableCases . . . . . . . . . . . . . . . . . . . . . . . . 85
2.HamiltonianFormofEquationsofMotionforVariousSystemsof 4.AbsoluteMotion . . . . . . . . . . . . . . . . . . . . . . . . 88
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 FixedpointsonthePoissonsphere(88). PeriodicsolutionsonthePoisson
Equationsofmotioninthealgebraicform(49). Quaternionrepresentation sphere(89). Quasiperiodic(double-frequency)paths(89). Regularpreces-
ofequationsofmotion(51). Canonicalequations intheEuleranglesand sions(89). Absolutemotion: integrableandnonintegrable cases(89).
Andoyaer(cid:21)Depritvariables(51). 2. TheEulerCase . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.ThePoincare(cid:1) Cross-SectionandChaoticMotions . . . . . . . 53 x
1.TheGeometricInterpretationbyPoinsot . . . . . . . . . . . . 92
2.ExplicitIntegrationandBifurcationalAnalysis . . . . . . . . . 93
5. EquationsofRigidBodyMotioninEuclidianSpace . . . . . . . 56
Motionintheabsolutespace(94). Aherpolhode(95).
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1.LagrangeFormalismandPoincare(cid:1) EquationsonaGroupE(3) . 56 3.Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.KineticEnergyofaRigidBodyinR3 . . . . . . . . . . . . . 58
3.The Hamiltonian Form of Equations of Motion of a Rigid Body
3. TheLagrangeCase . . . . . . . . . . . . . . . . . . . . . . . . 98
inR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 x
The reduction to one degree of freedom (98). Complete system dynam-
ics(99).
6. ExamplesandRelatedProblemStatements . . . . . . . . . . . . 60 1.BifurcationalPatternandGeometricalAnalysis ofMotion . . . 100
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1.TheMotionofaRigidBodywithaFixedPointintheSuperposi- 2.VariousReducedSystems(withrespectto andto ’) . . . . . 101
tionofPermanentUniformForceFields . . . . . . . . . . . . 60 3.ConjugatePoissonStructures . . . . . . . . . . . . . . . . . . 102
2.AFreeRigidBodyinaQuadraticPotential . . . . . . . . . . 61 4.HistoricalComments . . . . . . . . . . . . . . . . . . . . . . 103
3.The Motionof a Bodywith a FixedPoint in a RotatingFrame of
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4. TheKowalevskayaCase . . . . . . . . . . . . . . . . . . . . . 108
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AgyroscopeandFoucault’spendulum(63). AsatelliteattheEarthcircular 1.ExplicitIntegration. TheKowalevskayaVariables . . . . . . . 108
orbit(64). 2.ABifurcationalPatternandtheAppelrotClasses . . . . . . . . 109
4.RelativeMotionofaRigidBodywithaFixedPoint . . . . . . 64 I. The Delauney solution [70] (111). II. (114). III. (115). IV. (117).
5.RigidBodyMotiononaSmoothPlane . . . . . . . . . . . . . 65 Bobylev(cid:21)Steklovsolution(117).
6.AGyroscopeinaGimbal . . . . . . . . . . . . . . . . . . . 66 3.PhasePortraitandVisualizationoftheMostRemarkableSolutions119
Historicalcomment(67). Phase portrait at c = 0 (119). Phase portrait at c = 1:15 (120). The
7.RigidBodyMotioninaPerfectIncompressibleFluid(Kirchhoff’s Delauney solution (121). TheBobylev(cid:21)Steklovsolution(121). Unstable
Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 periodicsolutionsandseparatrices(124).
8.AHeavyBodyFall inFluid,theChaplyginEquations . . . . . 69 4.HistoricalComments . . . . . . . . . . . . . . . . . . . . . . 125
Comments(70). The Kowalevskaya method (125). The Kowalevskaya case, its analysis
7. TheoremsaboutIntegrabilityandTechniquesofIntegration. . . . 70 andgeneralizations(127).
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1.HamiltonianSystems. TheLiouville(cid:21)ArnoldTheorem . . . . . 71 5. TheGoryachev(cid:21)ChaplyginCase . . . . . . . . . . . . . . . . . 128
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2.TheLastMultiplierTheory. TheEuler(cid:21)JacobiTheorem . . . . 73 1.ExplicitIntegration . . . . . . . . . . . . . . . . . . . . . . . 128
3.SeparationofVariables. TheHamilton(cid:21)JacobiMethod . . . . 75 2.ABifurcationalPatternandaPhasePortrait . . . . . . . . . . 129
Geodesic(cid:29)owonanellipsoid(theJacobiproblem)[183](76). Thesystem 3.VisualizingtheMostRemarkableSolutions . . . . . . . . . . 130
withaquadraticpotentialonasphere(theNeumannproblem)[251](78). The Goryachev solution [65] (133). Stable and unstable periodic solu-
Comments(79). tions(136).
6 Contents Contents 7
6. ParticularSolutions . . . . . . . . . . . . . . . . . . . . . . . . 137 2.IntegrableCases . . . . . . . . . . . . . . . . . . . . . . . . 180
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1.TheHessSolution . . . . . . . . . . . . . . . . . . . . . . . 137 3.TheCase ofAxialSymmetry(ofH.Poincare(cid:1)) . . . . . . . . . 181
2.TheStaudePermanentRotations . . . . . . . . . . . . . . . . 139 4.TheSchottky(cid:21)ManakovCase . . . . . . . . . . . . . . . . . 181
3.TheGrioliRegularPrecessions . . . . . . . . . . . . . . . . . 142 5.Steklov’sCase . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.TheBobylev(cid:21)SteklovSolution(1896) . . . . . . . . . . . . . 145 6.TheIntegrableCasewithFourthDegreeIntegral(M.Adler,P.van
Thestabilityofparticularsolutions(145). Moerbeke) . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7. EquationsofMotionofaHeavyGyrostat . . . . . . . . . . . . . 146 7.ParticularCases at(M;p)=0 . . . . . . . . . . . . . . . . . 190
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1.AGyrostat . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 The (cid:28)rstcaseof Bogoyavlenskiy (190). The second caseof Bogoyavlen-
2.TheJoukovskiy(cid:21)VolterraCase . . . . . . . . . . . . . . . . . 148 skiy(190).
The separation of variables for the Joukovskiy(cid:21)Volterra case (151). The 8.TheHess CaseGeneralization . . . . . . . . . . . . . . . . . 191
explicitsolutionbyV.Volterra(152). 9.IntegrableGeneralizationswithLinearTermsin Hamiltonian. . 191
3.TheExplicitIntegrationofOtherCases . . . . . . . . . . . . 153 TheanalogueoftheRubanovskiycaseonso(4)(191). Thegeneralization
ofthe(cid:28)rstcaseofBogoyavlenskiy(192).
8. ConnectedSystemsofRigidBodies,aRotator . . . . . . . . . . 153 11. TheRemarkableBoundaryCase of thePoincare(cid:1)(cid:21)JoukovskiyEqua-
x Aconnectedsystemoftwotops(153). Thebodywitharotator(155). Com- x tions. TheCountableFamilyofFirst Integrals . . . . . . . . . . 193
ments(157). TheLiouvilleequations(157). 12. A RigidBodyin anArbitraryPotentialField . . . . . . . . . . . 200
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1.GeneralizedEuler(cid:21)PoissonEquations . . . . . . . . . . . . . 201
Chapter3. RelatedProblemsofRigidBodyDynamics . . . . . . . . 159
The Euler case (202). The generalized Lagrange case (202). The gen-
9. Kirchhoff’sEquations. . . . . . . . . . . . . . . . . . . . . . . 159 eralized Kowalevskaya case (202). The generalization of the Delauney
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1.EquationsofMotionandPhysicalInterpretations. . . . . . . . 159 case (204). The generalized spherical top (204). The Hess case ana-
Rigid body dynamics in (cid:29)uids (159). Brun’s problem (160). Grioli’s logue(205).
problem (161). Neumann’s system [251] (161). Jacobi’s problem about 2.TheBrunSystem . . . . . . . . . . . . . . . . . . . . . . . . 206
geodesicsonanellipsoid[183](162). TheLaxrepresentationand(cid:28)rstintegrals([21,31])(206). Thedynamical
2.IntegrableCases . . . . . . . . . . . . . . . . . . . . . . . . 163 symmetrycase(209). TheBrunprobleminasingle(cid:28)eld(210).
Comments(165). 3.QuaternionEuler(cid:21)PoissonEquations. . . . . . . . . . . . . . 210
3.TheCaseofAxialSymmetry . . . . . . . . . . . . . . . . . . 166 Aspherical top(a =a =a )(212). (cid:16)TheKowalevskaya case(cid:17)(212).
1 2 3
4.Clebsch’scase . . . . . . . . . . . . . . . . . . . . . . . . . 166 (cid:16)TheGoryachev(cid:21)Chaplygincase(cid:17)(213).
5.TheSteklov(cid:21)LyapunovFamily . . . . . . . . . . . . . . . . 167
Chapter4. CyclicIntegralsand OrderReduction . . . . . . . . . . . 214
Comments(169).
6.Chaplygin’scase(I) . . . . . . . . . . . . . . . . . . . . . . 170 1. LinearIntegralsin RigidBodyDynamics . . . . . . . . . . . . . 214
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7.Chaplygin’sCase (II) . . . . . . . . . . . . . . . . . . . . . . 171 1.Classical AreaIntegralN3 =(M; (cid:13))=c=const . . . . . . 217
8.IntegrableGeneralizationswithLinearTermsinaHamiltonian . 171 2.IntegralN3 M3 =(M; (cid:13)) M3 =c=const . . . . . . . . 219
(cid:0) (cid:0)
3.IntegralM =c=const(theLagrangianintegral) . . . . . . . 220
Theequations ofmotionof amulticonnected body (172). Rubanovskiy’s 3
4.LiftingofIntegrableSystems . . . . . . . . . . . . . . . . . . 221
generalizationofSteklov(cid:21)Lyapunov integrablefamily(172). TheGener-
alizationofChaplygin’sCase(I)(173). GeneralizationoftheYehia(cid:21)Kowalevskayafamily(222). Thegeneralized
Goryachev(cid:21)Chaplyginfamily(224).
10. Poincare(cid:1)(cid:21)JoukovskiyEquations . . . . . . . . . . . . . . . . . 174
x 2. DynamicalSymmetryandLagrange’sIntegral . . . . . . . . . . 225
1.EquationsofMotionandTheirPhysicalInterpretation . . . . . 174
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1.An Explicit Quadrature of the Generalized Lagrange Case. The
Poisson’s structureand equations ofmotion (174). Poincare(cid:1)(cid:21)Joukovskiy
ConditionsofIntegralExistence . . . . . . . . . . . . . . . . 225
equations (175). Historical comments (176). Dynamics of a rigid body
with a cavity containing (cid:29)uid (177). Rigid body dynamics in (cid:0)4: a four 2.AToponaSmoothPlaneinaGravityField . . . . . . . . . . 228
dimensionalEulertop(178). Arigidbodyincurvedspace(179). Arigid Comments(229).
bodyinS3 in(cid:29)uid(179). Asystemofinteractingspins(179). 3.AGyroscopeinaGimbalinanAxiallySymmetricField . . . . 230
8 Contents Contents 9
4.TheAxialSymmetryCase inChapligin’sEquations . . . . . . 230 2.˛Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:229) æº(cid:243)(cid:247)(cid:224)(cid:255)ˆ(cid:238)(cid:240)(cid:255)(cid:247)(cid:229)(cid:226)(cid:224)(cid:21)(cid:215)(cid:224)(cid:239)ºßªŁ(cid:237)(cid:224) . . . . . . . . . . 294
Comments(231). 3.(cid:209)º(cid:243)(cid:247)(cid:224)Ø ˆ(cid:238)(cid:240)(cid:255)(cid:247)(cid:229)(cid:226)(cid:224) . . . . . . . . . . . . . . . . . . . . . . . 296
5.TheAnalogybetweentheLagrangeTopandtheLeggetteSystem231 (cid:159)8. —(cid:224)(cid:231)(cid:228)(cid:229)º(cid:229)(cid:237)Ł(cid:229) (cid:239)(cid:229)(cid:240)(cid:229)(cid:236)(cid:229)(cid:237)(cid:237)ßı . . . . . . . . . . . . . . . . . . . . . 297
3. TheHessCase: Geometry,CyclicVariable,andExplicitIntegration 233 1.—(cid:224)(cid:231)(cid:228)(cid:229)º(cid:255)(cid:254)øŁ(cid:229) (cid:239)(cid:240)(cid:229)(cid:238)Æ(cid:240)(cid:224)(cid:231)(cid:238)(cid:226)(cid:224)(cid:237)Ł(cid:255)(cid:226) Ł(cid:237)(cid:242)(cid:229)ª(cid:240)Ł(cid:240)(cid:243)(cid:229)(cid:236)ßı(cid:231)(cid:224)(cid:228)(cid:224)(cid:247)(cid:224)ı(cid:228)Ł(cid:237)(cid:224)-
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1.APotentialSystemonAlgebrae(3). ACyclicCoordinate . . . 233 (cid:236)ŁŒŁ (cid:242)(cid:226)(cid:229)(cid:240)(cid:228)(cid:238)ª(cid:238)(cid:242)(cid:229)º(cid:224) . . . . . . . . . . . . . . . . . . . . . . 297
2.TheClassical HessCase . . . . . . . . . . . . . . . . . . . . 235 (cid:209)Łæ(cid:242)(cid:229)(cid:236)(cid:224) ˘(cid:243)Œ(cid:238)(cid:226)æŒ(cid:238)ª(cid:238)(cid:21)´(cid:238)º(cid:252)(cid:242)(cid:229)(cid:240)(cid:240)(cid:224) (298). (cid:209)º(cid:243)(cid:247)(cid:224)Ø ˚(cid:238)(cid:226)(cid:224)º(cid:229)(cid:226)æŒ(cid:238)Ø (300).
Historicalcomment(238). ˇ(cid:240)(cid:229)(cid:238)Æ(cid:240)(cid:224)(cid:231)(cid:238)(cid:226)(cid:224)(cid:237)Ł(cid:229) (cid:213)(cid:224)Ø(cid:237)(cid:229)(cid:21)(cid:213)(cid:238)(cid:240)(cid:238)(cid:231)(cid:238)(cid:226)(cid:224) (cid:228)º(cid:255) æŁæ(cid:242)(cid:229)(cid:236)ß ˚(cid:238)(cid:226)(cid:224)º(cid:229)(cid:226)æŒ(cid:238)Ø (304).
4. TheHessCase Generalizations . . . . . . . . . . . . . . . . . . 240 (cid:192)(cid:237)(cid:224)º(cid:238)ªŁ(cid:255) ˚(cid:238)º(cid:238)æ(cid:238)(cid:226)(cid:224) Ł (cid:229)(cid:229) (cid:238)Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:255) (306). ¨æ(cid:242)(cid:238)(cid:240)Ł(cid:247)(cid:229)æŒŁØ Œ(cid:238)(cid:236)(cid:236)(cid:229)(cid:237)(cid:242)(cid:224)-
x Linearandquadraticpotentials(241). TheKnownintegrablecases(243). A (cid:240)ŁØ(308).(cid:209)º(cid:243)(cid:247)(cid:224)Ø(cid:215)(cid:224)(cid:239)ºßªŁ(cid:237)(cid:224)(I)(308).(cid:209)Łæ(cid:242)(cid:229)(cid:236)(cid:224)`(cid:238)ª(cid:238)(cid:255)(cid:226)º(cid:229)(cid:237)æŒ(cid:238)ª(cid:238)(309).
rigid body on a smooth plane (244). A gyroscope in a gimbal (246). The 2.ˇ(cid:229)(cid:240)(cid:229)(cid:236)(cid:229)(cid:237)(cid:237)ß(cid:229)(cid:190)(cid:228)(cid:229)Øæ(cid:242)(cid:226)Ł(cid:229)¿ Ł (cid:240)(cid:224)(cid:231)(cid:228)(cid:229)º(cid:255)(cid:254)øŁ(cid:229)(cid:239)(cid:229)(cid:240)(cid:229)(cid:236)(cid:229)(cid:237)(cid:237)ß(cid:229) . . . . 310
HessintegralintheChaplyginequations(246). (cid:159)9. ˜(cid:226)(cid:238)(cid:255)Œ(cid:238)(cid:224)æŁ(cid:236)(cid:239)(cid:242)(cid:238)(cid:242)Ł(cid:247)(cid:229)æŒŁ(cid:229) (cid:228)(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:255) (cid:228)º(cid:255)Ł(cid:237)(cid:242)(cid:229)ª(cid:240)Ł(cid:240)(cid:243)(cid:229)(cid:236)ßıæŁæ(cid:242)(cid:229)(cid:236) 313
(cid:209)º(cid:243)(cid:247)(cid:224)Ø (cid:221)غ(cid:229)(cid:240)(cid:224) (314). (cid:209)º(cid:243)(cid:247)(cid:224)Ø ¸(cid:224)ª(cid:240)(cid:224)(cid:237)(cid:230)(cid:224) (315). (cid:209)º(cid:243)(cid:247)(cid:224)Ø ˘(cid:243)Œ(cid:238)(cid:226)æŒ(cid:238)ª(cid:238)(cid:21)
ˆ¸(cid:192)´(cid:192)5. SpecialProblemsofRigidBodyDynamics . . . . . . . . . 248 ´(cid:238)º(cid:252)(cid:242)(cid:229)(cid:240)(cid:240)(cid:224)(317).˚(cid:238)(cid:236)(cid:236)(cid:229)(cid:237)(cid:242)(cid:224)(cid:240)ŁŁ(317).
(cid:159)1. (cid:210)(cid:226)(cid:229)(cid:240)(cid:228)(cid:238)(cid:229)(cid:242)(cid:229)º(cid:238)(cid:226)æ(cid:238)(cid:239)(cid:240)(cid:238)(cid:242)Ł(cid:226)º(cid:255)(cid:254)ø(cid:229)Øæ(cid:255) æ(cid:240)(cid:229)(cid:228)(cid:229) . . . . . . . . . . . . 248 (cid:159)10.˜Ł(cid:237)(cid:224)(cid:236)ŁŒ(cid:224)(cid:226)(cid:238)º(cid:247)Œ(cid:224) Ł(cid:236)(cid:224)(cid:242)(cid:229)(cid:240)Ł(cid:224)º(cid:252)(cid:237)(cid:238)Ø (cid:242)(cid:238)(cid:247)ŒŁ(cid:237)(cid:224)æ(cid:244)(cid:229)(cid:240)(cid:229)Ł(cid:253)ººŁ(cid:239)æ(cid:238)Ł(cid:228)(cid:229) 318
(cid:209)Łæ(cid:242)(cid:229)(cid:236)(cid:224)¸(cid:238)(cid:240)(cid:229)(cid:237)(cid:246)(cid:224)(252).(cid:190)˜Ł(cid:224)ª(cid:238)(cid:237)(cid:224)º(cid:252)(cid:237)(cid:224)(cid:255) (cid:228)ŁææŁ(cid:239)(cid:224)(cid:246)Ł(cid:255)¿(253). ˚(cid:238)(cid:236)(cid:236)(cid:229)(cid:237)(cid:242)(cid:224)- 1.˜(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:229) (cid:242)(cid:238)(cid:247)ŒŁ (cid:239)(cid:238) æ(cid:244)(cid:229)(cid:240)(cid:229) Ł (cid:253)ººŁ(cid:239)æ(cid:238)Ł(cid:228)(cid:243) (n = 2;3). (cid:192)(cid:237)(cid:224)º(cid:238)ªŁ(cid:255)
(cid:240)ŁØ(254).(cid:216)(cid:224)(cid:240)(cid:238)(cid:226)(cid:238)Ø(cid:226)(cid:238)º(cid:247)(cid:238)Œæ(cid:238)æº(cid:238)(cid:230)(cid:237)(cid:238)Ø(cid:228)ŁææŁ(cid:239)(cid:224)(cid:246)Ł(cid:229)Ø(254). æ(cid:228)Ł(cid:237)(cid:224)(cid:236)ŁŒ(cid:238)Ø (cid:226)(cid:238)º(cid:247)Œ(cid:224) . . . . . . . . . . . . . . . . . . . . . . 318
(cid:159)2. ´ß(cid:226)(cid:238)(cid:228) (cid:243)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)ŁØ ˚Ł(cid:240)ıª(cid:238)(cid:244)(cid:224), ˇ(cid:243)(cid:224)(cid:237)Œ(cid:224)(cid:240)(cid:229)(cid:21)˘(cid:243)Œ(cid:238)(cid:226)æŒ(cid:238)ª(cid:238) Ł (cid:247)(cid:229)(cid:242)ß(cid:240)(cid:229)ı- ˜(cid:226)(cid:243)(cid:236)(cid:229)(cid:240)(cid:237)ßØ(cid:253)ººŁ(cid:239)æ(cid:238)Ł(cid:228)Łæ(cid:244)(cid:229)(cid:240)(cid:224)(E2;S2)(319).(cid:210)(cid:240)(cid:229)ı(cid:236)(cid:229)(cid:240)(cid:237)ßØ(cid:253)ººŁ(cid:239)æ(cid:238)Ł(cid:228)
(cid:236)(cid:229)(cid:240)(cid:237)(cid:238)ª(cid:238)(cid:226)(cid:238)º(cid:247)Œ(cid:224) . . . . . . . . . . . . . . . . . . . . . . . . . 255 Łæ(cid:244)(cid:229)(cid:240)(cid:224)(E3,S3)(320).
1.˜(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:229) (cid:242)(cid:226)(cid:229)(cid:240)(cid:228)(cid:238)ª(cid:238)(cid:242)(cid:229)º(cid:224)(cid:226)Ł(cid:228)(cid:229)(cid:224)º(cid:252)(cid:237)(cid:238)Ø (cid:237)(cid:229)æ(cid:230)Ł(cid:236)(cid:224)(cid:229)(cid:236)(cid:238)Ø(cid:230)Ł(cid:228)Œ(cid:238)æ(cid:242)Ł 255 2.ˆ(cid:224)(cid:240)(cid:236)(cid:238)(cid:237)Ł(cid:247)(cid:229)æŒŁØ (cid:238)æ(cid:246)Łºº(cid:255)(cid:242)(cid:238)(cid:240) (cid:237)(cid:224) S2;S3. ˛Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:229) (cid:231)(cid:224)(cid:228)(cid:224)(cid:247) ˝(cid:229)Ø-
(cid:211)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)Ł(cid:255) ˚Ł(cid:240)ıª(cid:238)(cid:244)(cid:224) (255). (cid:211)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)Ł(cid:255) (cid:228)(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:255) (cid:228)º(cid:255) (cid:236)(cid:237)(cid:238)ª(cid:238)æ(cid:226)(cid:255)(cid:231)(cid:237)(cid:238)ª(cid:238) (cid:236)(cid:224)(cid:237)(cid:224) Ł (cid:223)Œ(cid:238)ÆŁ . . . . . . . . . . . . . . . . . . . . . . . . . 322
(cid:242)(cid:229)º(cid:224)(260). ˆ(cid:243)Œ(cid:238)(cid:226)æŒŁ(cid:229) (cid:246)(cid:229)(cid:237)(cid:242)(cid:240)ß (cid:237)(cid:224) æ(cid:244)(cid:229)(cid:240)(cid:229) (322). ˛Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:229) (cid:231)(cid:224)(cid:228)(cid:224)(cid:247)Ł ˝(cid:229)Ø(cid:236)(cid:224)(cid:237)(cid:224) (cid:237)(cid:224)
2.(cid:211)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)Ł(cid:255)ˇ(cid:243)(cid:224)(cid:237)Œ(cid:224)(cid:240)(cid:229)(cid:21)˘(cid:243)Œ(cid:238)(cid:226)æŒ(cid:238)ª(cid:238) . . . . . . . . . . . . . . 263 S2 (324). ˛Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:229) (cid:231)(cid:224)(cid:228)(cid:224)(cid:247)Ł (cid:223)Œ(cid:238)ÆŁ (cid:237)(cid:224) E2 (325). ˛Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:229) æŁæ(cid:242)(cid:229)-
3.˜(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:229) (cid:242)(cid:226)(cid:229)(cid:240)(cid:228)(cid:238)ª(cid:238) (cid:242)(cid:229)º(cid:224) c ªŁ(cid:240)(cid:238)æ(cid:242)(cid:224)(cid:242)(cid:238)(cid:236) (cid:226) ŁæŒ(cid:240)Ł(cid:226)º(cid:229)(cid:237)(cid:237)(cid:238)(cid:236) (cid:239)(cid:240)(cid:238)- (cid:236)ß˝(cid:229)Ø(cid:236)(cid:224)(cid:237)(cid:224)(cid:237)(cid:224)S3 (325).
æ(cid:242)(cid:240)(cid:224)(cid:237)æ(cid:242)(cid:226)(cid:229).(cid:209)(cid:242)(cid:224)(cid:246)Ł(cid:238)(cid:237)(cid:224)(cid:240)(cid:237)ß(cid:229) (cid:228)(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:255) . . . . . . . . . . . . . 267 3.˙(cid:224)(cid:228)(cid:224)(cid:247)(cid:224)n ª(cid:243)Œ(cid:238)(cid:226)æŒŁı(cid:246)(cid:229)(cid:237)(cid:242)(cid:240)(cid:238)(cid:226) (cid:237)(cid:224)æ(cid:244)(cid:229)(cid:240)(cid:229) . . . . . . . . . . . . 327
(cid:209)(cid:226)(cid:238)Æ(cid:238)(cid:228)(cid:237)(cid:238)(cid:229)(cid:228)(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:229)(cid:242)(cid:229)º(cid:224)(cid:226)S3(268).˜(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:229)æ(cid:226)(cid:255)(cid:231)ŒŁ(cid:228)(cid:226)(cid:243)ı(cid:242)(cid:229)º.(cid:211)(cid:240)(cid:224)(cid:226)- 4.(cid:209)Łæ(cid:242)(cid:229)(cid:236)(cid:224) ˆ(cid:224)(cid:244)(cid:244)(cid:229) . . . . . . . . . . . . . . . . . . . . . . . . 328
(cid:237)(cid:238)(cid:226)(cid:229)ł(cid:229)(cid:237)(cid:237)ßØ ªŁ(cid:240)(cid:238)æ(cid:242)(cid:224)(cid:242) (270). (cid:211)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)Ł(cid:255) ˚Ł(cid:240)ıª(cid:238)(cid:244)(cid:224) (cid:237)(cid:224) S3; L3 (272). (cid:159)11.˝(cid:229)Æ(cid:229)æ(cid:237)(cid:224)(cid:255)(cid:236)(cid:229)ı(cid:224)(cid:237)ŁŒ(cid:224)(cid:237)(cid:224)(cid:228)(cid:226)(cid:243)(cid:236)(cid:229)(cid:240)(cid:237)(cid:238)ØŁ(cid:242)(cid:240)(cid:229)ı(cid:236)(cid:229)(cid:240)(cid:237)(cid:238)Øæ(cid:244)(cid:229)(cid:240)(cid:224)ı . . . . 329
(cid:209)(cid:226)(cid:238)Æ(cid:238)(cid:228)(cid:237)(cid:238)(cid:229)(cid:242)(cid:226)(cid:229)(cid:240)(cid:228)(cid:238)(cid:229)(cid:242)(cid:229)º(cid:238)(cid:226)(cid:239)(cid:240)(cid:238)æ(cid:242)(cid:240)(cid:224)(cid:237)æ(cid:242)(cid:226)(cid:229)¸(cid:238)Æ(cid:224)(cid:247)(cid:229)(cid:226)æŒ(cid:238)ª(cid:238)(272).˚(cid:238)(cid:236)(cid:236)(cid:229)(cid:237)- 1.˙(cid:224)(cid:228)(cid:224)(cid:247)(cid:224)˚(cid:229)(cid:239)º(cid:229)(cid:240)(cid:224) . . . . . . . . . . . . . . . . . . . . . . . . 329
(cid:242)(cid:224)(cid:240)ŁØ(273). 2.(cid:221)غ(cid:229)(cid:240)(cid:238)(cid:226)(cid:224)(cid:231)(cid:224)(cid:228)(cid:224)(cid:247)(cid:224)(cid:228)(cid:226)(cid:243)ı(cid:246)(cid:229)(cid:237)(cid:242)(cid:240)(cid:238)(cid:226) . . . . . . . . . . . . . . . . 331
(cid:159)3. (cid:192)ºª(cid:229)Æ(cid:240)(cid:224)e(4)Ł(cid:229)(cid:229)(cid:238)(cid:240)ÆŁ(cid:242)ß . . . . . . . . . . . . . . . . . . . . 274 —(cid:224)(cid:231)(cid:228)(cid:229)º(cid:229)(cid:237)Ł(cid:229) (cid:239)(cid:229)(cid:240)(cid:229)(cid:236)(cid:229)(cid:237)(cid:237)ßı (332). ˇ(cid:229)(cid:240)(cid:226)ß(cid:229) Ł(cid:237)(cid:242)(cid:229)ª(cid:240)(cid:224)ºß (333). ˜(cid:238)Æ(cid:224)(cid:226)º(cid:229)(cid:237)Ł(cid:229)
(cid:159)4. ˝(cid:238)(cid:226)(cid:224)(cid:255)L A-(cid:239)(cid:224)(cid:240)(cid:224)(cid:238)Æ(cid:238)Æø(cid:229)(cid:237)(cid:237)(cid:238)ª(cid:238)(cid:226)(cid:238)º(cid:247)Œ(cid:224)ˆ(cid:238)(cid:240)(cid:255)(cid:247)(cid:229)(cid:226)(cid:224)(cid:21)(cid:215)(cid:224)(cid:239)ºßªŁ(cid:237)(cid:224) 277 ª(cid:243)Œ(cid:238)(cid:226)æŒŁı(cid:246)(cid:229)(cid:237)(cid:242)(cid:240)(cid:238)(cid:226)(334).
(cid:0) 3.˜(cid:226)Ł(cid:230)(cid:229)(cid:237)Ł(cid:229) (cid:231)(cid:224)(cid:240)(cid:255)(cid:230)(cid:229)(cid:237)(cid:237)(cid:238)Ø (cid:247)(cid:224)æ(cid:242)Ł(cid:246)ß (cid:226) (cid:239)(cid:238)º(cid:229) (cid:236)(cid:224)ª(cid:237)Ł(cid:242)(cid:237)(cid:238)ª(cid:238) (cid:236)(cid:238)(cid:237)(cid:238)(cid:239)(cid:238)º(cid:255)
(cid:159)5. ˜Ł(cid:237)(cid:224)(cid:236)ŁŒ(cid:224)(cid:244)(cid:229)(cid:240)(cid:240)(cid:238)(cid:236)(cid:224)ª(cid:237)(cid:229)(cid:242)ŁŒ(cid:224)(cid:226)(cid:236)(cid:224)ª(cid:237)Ł(cid:242)(cid:237)(cid:238)(cid:236) (cid:239)(cid:238)º(cid:229) . . . . . . . . . 281
Ł Œ(cid:243)º(cid:238)(cid:237)(cid:238)(cid:226)æŒ(cid:238)ª(cid:238)(cid:246)(cid:229)(cid:237)(cid:242)(cid:240)(cid:224)(cid:237)(cid:224) (cid:242)(cid:240)(cid:229)ı(cid:236)(cid:229)(cid:240)(cid:237)(cid:238)Øæ(cid:244)(cid:229)(cid:240)(cid:229) . . . . . . . . . 335
(cid:159)6. (cid:211)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)Ł(cid:229) ¸(cid:224)(cid:237)(cid:228)(cid:224)(cid:243)(cid:21)¸Ł(cid:244)łŁ(cid:246)(cid:224), (cid:228)ŁæŒ(cid:240)(cid:229)(cid:242)(cid:237)ß(cid:229) æŁæ(cid:242)(cid:229)(cid:236)ß Ł (cid:231)(cid:224)(cid:228)(cid:224)(cid:247)(cid:224)
ˇº(cid:238)æŒ(cid:238)(cid:229)(cid:239)(cid:240)(cid:238)æ(cid:242)(cid:240)(cid:224)(cid:237)æ(cid:242)(cid:226)(cid:238)(336).¨æŒ(cid:240)Ł(cid:226)º(cid:229)(cid:237)(cid:237)(cid:238)(cid:229)(cid:239)(cid:240)(cid:238)æ(cid:242)(cid:240)(cid:224)(cid:237)æ(cid:242)(cid:226)(cid:238)(337).
˝(cid:229)Ø(cid:236)(cid:224)(cid:237)(cid:224) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
(cid:159)12.˝(cid:238)(cid:226)ßØ Ł(cid:237)(cid:242)(cid:229)ª(cid:240)(cid:224)º (cid:247)(cid:229)(cid:242)(cid:226)(cid:229)(cid:240)(cid:242)(cid:238)Ø æ(cid:242)(cid:229)(cid:239)(cid:229)(cid:237)Ł (cid:243)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)ŁØ ˚Ł(cid:240)ıª(cid:238)(cid:244)(cid:224) Ł ˇ(cid:243)-
1.(cid:211)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)Ł(cid:229)¸(cid:224)(cid:237)(cid:228)(cid:224)(cid:243)(cid:21)¸Ł(cid:244)łŁ(cid:246)(cid:224) . . . . . . . . . . . . . . . . 284
(cid:224)(cid:237)Œ(cid:224)(cid:240)(cid:229)(cid:21)˘(cid:243)Œ(cid:238)(cid:226)æŒ(cid:238)ª(cid:238) . . . . . . . . . . . . . . . . . . . . . . . 338
2.(cid:192)(cid:237)Ł(cid:231)(cid:238)(cid:242)(cid:240)(cid:238)(cid:239)(cid:237)(cid:224)(cid:255)XYZ-(cid:236)(cid:238)(cid:228)(cid:229)º(cid:252)ˆ(cid:229)Ø(cid:231)(cid:229)(cid:237)Æ(cid:229)(cid:240)ª(cid:224) . . . . . . . . . . . 285
¨æ(cid:242)(cid:238)(cid:240)Ł(cid:247)(cid:229)æŒŁØŒ(cid:238)(cid:236)(cid:236)(cid:229)(cid:237)(cid:242)(cid:224)(cid:240)ŁØ(340).
(cid:204)(cid:237)(cid:238)ª(cid:238)(cid:236)(cid:229)(cid:240)(cid:237)ß(cid:229)(cid:238)Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:255)(287).
3.(cid:221)ººŁ(cid:239)æ(cid:238)Ł(cid:228)(cid:224)º(cid:252)(cid:237)ßØ ÆŁº(cid:252)(cid:255)(cid:240)(cid:228) Ł(cid:228)ŁæŒ(cid:240)(cid:229)(cid:242)(cid:237)ß(cid:229) (cid:226)(cid:238)º(cid:247)ŒŁ . . . . . . 288 ¸Ł(cid:242)(cid:229)(cid:240)(cid:224)(cid:242)(cid:243)(cid:240)(cid:224) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
˚(cid:238)(cid:236)(cid:236)(cid:229)(cid:237)(cid:242)(cid:224)(cid:240)ŁŁ(288).
(cid:159)7. —(cid:224)(cid:231)ºŁ(cid:247)(cid:237)ß(cid:229) (cid:238)Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:255) æº(cid:243)(cid:247)(cid:224)(cid:229)(cid:226) Ł(cid:237)(cid:242)(cid:229)ª(cid:240)Ł(cid:240)(cid:243)(cid:229)(cid:236)(cid:238)æ(cid:242)Ł (cid:243)(cid:240)(cid:224)(cid:226)(cid:237)(cid:229)(cid:237)ŁØ (cid:221)Ø-
º(cid:229)(cid:240)(cid:224)(cid:21)ˇ(cid:243)(cid:224)ææ(cid:238)(cid:237)(cid:224) . . . . . . . . . . . . . . . . . . . . . . . . . 289
1.˛Æ(cid:238)Æø(cid:229)(cid:237)Ł(cid:229) æº(cid:243)(cid:247)(cid:224)(cid:255)˚(cid:238)(cid:226)(cid:224)º(cid:229)(cid:226)æŒ(cid:238)Ø . . . . . . . . . . . . . . . 289
PREFACE 11
thekindofattentionthatwas arousedbythealreadyfoundandcitedhereones.
Classics tried to use them to understand motion, and they achieved temporary
PREFACE successes. In rigid body dynamics the enthusiasm about geometric interpreta-
tionsofmotion,tracingbacktoPoinsot,everynowandthenwasreplacedbythe
analytical investigations,the majorityof which, unfortunately,was unnecessary
neither for physicists, nor for engineers, and soon became comprehensible for
thenarrowspecialistsonly.
I.
We havestartedwritingthisbooktwoyearsagowiththeaimofcollecting In this book we, probably, somewhat ignored proofs and precise formula-
all the integrable cases, known in rigid body dynamics. We felt that such a tions. Wesimultaneouslyusedachievementsoftopology,analysisandcomputer
projectcouldberealizedratherfast,andthebookwassupposedtobepublished experiments to receive suf(cid:28)ciently complete understanding of motion. It’s not
in 2000 (cid:22) the year of 150-anniversaryof S.V.Kowalevskaya. We also wanted that easy to assert if we attainedour aim, but one thing can’tbe doubted: even
to present comprehensive information concerning the case and the method she theclassicalcases(likeLagrange,Kowalevskaya,Goryachev(cid:21)Chaplygincases)
haddiscovered. have experienced in this approach the second birth, they have best the frame-
However,ourplanswere graduallyextended,mainlybecauseofthe active workofdullcomputationsandbecomerathertangible. Probably,itshouldbethe
application of numerical experiments and computer visualization methods to- ambition of mechanics(cid:22) to present a certainalgorithm, by meansof whichwe
gether with analytical computations. In the end, we have developeda perfectly can look into the whole variety of motions and clearly imagine each particular
newviewpointtooneoftheclassical(cid:28)eldsofmechanics,theonewhichallows motionandits peculiarities.
generalizationencompassingthewholedynamics.
In the preface we give honor to computer dynamics whose development In this book we try to revive traditions of mathematical literature of Eu-
and application to dynamic problems of top theory the reader will be meeting ler times. According to Jacobi expression [183], (cid:16)Euler himself in spite of
throughthewholebook. Computerinvestigationsindynamics,orjustcomputer considering only particular cases, selects them so well that the general method
dynamics,is, inouropinion,aseparatescienti(cid:28)c(cid:28)eld,establishinggeneralreg- determinedlateraddstohisresultsonlylittleornothingatall(cid:17).
ularitiesofthemotionofrealphysicalsystemsbymeansofaseriesofnumerical
methods and techniques. Each of these methods has its own peculiar features Thus,ifweconsiderthelawsofnature,leadingtoacertainsystemofdiffer-
(stability and others) and possesses some internal parameters (like a pitch and entialequations,beestablished,thenforitsanalysisthecomputerandanalytical
precision). That’s why results of such an investigationare related to the reality methods turn out to be complementary. Here we’d like to emphasize the dif-
onlyindirectly. However,similarconclusionsmaybealsomadefortheordinary ference between our viewpointand the prevailingone that the (cid:16)real science(cid:17) is
analytical(orpurelymathematical)methodsthatdemandrigorousproofsateach analytical, and the computeris capable of givingonly illustrations to analytical
stepoftheprocess. Atthatalotofphysicallyevidentfactsmayleadtounsolv- methodsandimpulsionforstatementsofnewtheorems. That’scertainlytrue,as
ableproblems(theseareespeciallynumerousinnonlineardynamicsandmathe- well, butit’s onlya byproductofcomputerinvestigations. The latter havetheir
matical theoryofchaos). Here we aregoingto indicateonlyproblemswith the owninnerlogicandasystemofdescriptionsofphysicalphenomena. Systematic
ergodicityproof,entropy computation,small parameterestimates, KAM-theory development of computer investigations , revealing new areas of computer (or
applicability and so on. Nevertheless, the solution of these problems, will not (cid:16)virtual(cid:17))dynamics,is thematterofthenearestfuture.
at the least advance our understandingof the remarkable regularities which we
observe,followingtheevolutionofchaosinparticularsystems. Asahistoricalprospect,orratherasafunnything,illustratingtheexcessive
In this book a classical branch of rigid body dynamics, dealing with the belief in the power of logical method, note that Leibnitz and Descartes in their
search of possible integrable cases, (cid:28)nds its natural conclusion. It’s probable papers,beforedevelopingpropermathematicalmethods,(cid:16)proved(cid:17)theexistence
that other cases and integrals that may be found in the future, will never arise ofmotionandevenGod.
12 PREFACE PREFACE 13
II. We decided to neglect sections, concerning nonholonomic systems, and
also multidimensional generalizations of rigid body dynamics. They are rather
In addition to the idea of computer dynamics, in the book we tried to extensive,andwe’lltrytoexplainthemseparately.
show the most modern methods of Poisson dynamics and geometry, theories In the beginning of the book we gathered short historical accounts about
of Lie groups and algebras, which were only designated in our previous book the creators of rigid body dynamics. These assays allow to trace the evolution
Poisson Structures and Lie Algebras in Hamiltonian Mechanics, which, as we ofideas inthis (cid:28)eld,and,probably,to correctsomehistoricaldiscrepancies.
feel, was quite a success. Rigid body dynamics plays a special role in the
development of these methods. In a certain sense it represents a ground for
testingnewmathematicalmeans,andatthetimebeingit’sdif(cid:28)culttoappreciate
itssigni(cid:28)cance,especiallyforthedevelopmentofmanysectionsoftopologyand
nonlinearPoissonstructures,nonholonomicgeometry,theoryofsymmetriesand
tensorinvariants.
We can even assert that, similar to the way the understandingof profound
ideas of H.Poincare(cid:1),concerningthe nonintegrabilityof dynamicalsystems, be-
camepossibleduetotheanalysisofthreebodyproblem,resultsandtechniques
ofSophusLieenteredgeneralmathematicalculturebecauseoftheirapplication
to theory of tops, exemplifying the mechanical realization of the most natural
Lie groups and algebras. Besides, unlike celestial mechanics and theory of os-
cillations, rigidbodydynamics contains, onthe one hand,a series of nontrivial
integrable cases, and, on the other hand, on account of con(cid:28)gurational space
compactnessitis mostlypreferablefortheanalysisofchaoticmotions.
III.
While checking nearlyall modern and classical integrablecases , we used
the analytical computational system MAPLE. It happened so that some previ-
ously known results turned out to be not absolutely correct, and some others
wereconsiderablysimpli(cid:28)ed.
Computer visualization of motion and numerical integration were carried
out on the software complex (cid:16)Computer dynamics(cid:17) invented in the scien-
ti(cid:28)c-publishingcenter(cid:16)RegularandChaoticDynamics(cid:17).
Theproblemsofstabilityofparticularmotionsandthemajorityofapplied
and technical problems, whose thorough presentation requires a separate trea-
tise, were left beyond the book. Nevertheless, even a physicist or an engineer
mayunderstandfromthebookgeneralformalismofnotationofmaindynamical
equations , and key aspects of regular and chaotic behavior in rigid body dy-
namics. In this aspect, the book can be consideredas a referencebook, where,
nevertheless, we try to explain derivation of the main results, and sometimes
producecompleteproofs.
Introduction 15
Thestudyoftheseproblemsbecamepossibleduetothedevelopmentofgeneral
dynamicalformalismwhosesummitbecamethePoincare(cid:1)equations,allowingto
representrigidbodymotionequationsintermsofgroupvariables.
INTRODUCTION
Here we should also mention the progress of perfect (cid:29)uid hydrodynamics
and vortex theory whose foundationswere laid by H.Helmholtz. That was the
way to obtain equations for a vorticity vector, quite analogical to dynamical
equations of kinetic moment, and Poincare(cid:1) was the (cid:28)rst to study precession of
1. As an introductionwe are goingto present some short comments,con- the equator axis, using, as the Earth model, a rigid body (a mantle), having
cerningmainstagesofrigidbodydynamicsdevelopment. Integrablecaseswere cavities,(cid:28)lled withincompressiblevortex(cid:29)uid(core).
the (cid:28)rst to be studied. The most popular ones were found by Euler (1758)and
3. Asitwasalreadymentioned,intheclassicalperiodforvariousformsof
Lagrange(1788)atthestageofformationanddevelopmentofthemaindynam-
equationsitwasconsideredofprimeimportanceto(cid:28)ndsuchcases(thatcouldbe
ical principles. At this point the basic system, used for approbationsof various
(cid:28)xed by restrictions of parameters and initial conditions) of explicit solvability
mathematical methods duringnext centuries, was the system of Euler(cid:21)Poisson
of a problemin quadratures. In modernterminologythese are called integrable
equations,describingmotionofaheavyrigidbodyarounda(cid:28)xedpoint.
cases.
Substantially more dif(cid:28)cult case of integrability of Euler(cid:21)Poisson equa-
The integrable cases are usually connected with the names of their
tions was discoveredbe S.V.Kowalevskayain 1888. It has givenan impulsion
discoverers. Among them are famous Western mathematics and me-
to new investigations in the (cid:28)eld of integrable systems, This result was highly
chanics: G.Kirchhoff, A.Clebsch, P.Appell, F.Brun, V.Volterra. Great
appreciated by the French Academy of Sciences. In 1888, S.V.Kowalevskaya
achievements were made by Russian scientists A.M.Lyapunov, V.A.Steklov,
was awardedwiththeBaurdenPrizeforthememoironrotationofarigidbody
N.E.Joukovskiy, S.A.Chaplygin. In this respect rigid body dynamics may be
around a (cid:28)xed point. It should be mentioned that earlier the Academy of Sci-
consideredasa(cid:28)eld,(cid:28)lledwithinterestingintegrableproblems,constitutingthe
ences had announced about the competition on investigation of this problem
mostvaluablepossessionofmoderndynamics.
twice,butnobodyreceivedthePrize. Inspringof1889Kowalevskayawashon-
In the classical period, except for the (cid:28)nding of (cid:28)rst integrals, it was con-
ored with the Prize of the Swedish Royal Academy of Sciences for the second
sidered especially valuable to obtain explicit solutions in various classes of
memoirontheproblemofrigidbodyrotation.
functions, mainly, elliptical ones. Particular successes were achieved here by
The integrability of the Euler and Lagrange cases is stipulated by natu-
S.V.Kowalevskaya, V.Volterra, G.Halphen, and up to this very day their tech-
ral dynamical symmetries and preservation of the corresponding (cid:28)rst integrals.
niqueremainsunsurpassed.
S.V.Kowalevskayahasfoundhercaseofintegrability,startingfromnonevident
4. Inthe (cid:28)rst half oftwentiethcenturytheinterest tointegrable cases has,
analytical considerationsandusing theoryofalgebraicfunctions(whosepartic-
sotosay,decreased. Inmanyrespectsthatwasconnectedwithunderstandingby
ularcaseis ellipticfunctions),well developedatthetime. Sherequiredunique-
themajorityofmathematicianstheresults(obtainedbyH.Poincare(cid:1)),concerning
ness of the general solution on the complex plane of time, which in the future
nonintegrability of a typical Hamiltonian dynamical system [144]. In the con-
led to the beginningsof one of the most advancedmethods of dynamicsystem
sciousness of mathematicians this fact depreciated many results of classics and
analysis for integrability (cid:22) the Painleve(cid:21)Kowalevskaya test. As it is said, the
led to the development of new methods of perturbation theory: the averaging
Kowalevskayaintegraldoesn’thavenaturalsymmetryorigin;itssymmetriesare
principle,theKAM theoryandothers.
hidden, and the problem of motion description and explicit integration itself is
Inthegeneralcasethemainequationsofrigidbodydynamicsarealsonon-
essentiallymoredif(cid:28)cultinthiscase.
integrable. Thismeanstheyhavecomplicatedandunpredictablebehavior[144],
2. Startingfromthemiddleofnineteenthandinthebeginningoftwentieth whosestudyisasubjectofanew(cid:28)eldofinvestigationscalleddeterminatechaos.
century in rigid body dynamics there were found integrable cases for various Theeffectsofnonintegrabilityinrigidbodydynamics(cid:28)ndtheirsystematicstudy
statementsofproblemsonrigidbodymotion: motionofabodyin(cid:29)uid;motion inthetreatise byV.V.Kozlov[92].
of a body with cavities, (cid:28)lled with (cid:29)uid; gyrostats; nonholonomic problems. Thebook[92] is also important,because,unlikeunnaturalcravingofclas-
16 Introduction Introduction 17
sics for obtaining explicit solution, which allows to say but little about real integrals for a spin model, cited in the present day literature on quantum me-
motionofasystem,it involvesthequestion,concerningthequalitativeanalysis chanics(see,forexample,[259]),arebutsimpli(cid:28)edresults,obtainedbyclassics
of integrable dynamical systems, and by using examples of the Kowalevskaya (W.Frahm, F.Schottky) more than a hundred years ago. That is conditioned
andGoryachev(cid:21)Chaplygintopsthe authormakesgeneral conclusionsconcern- by the fact that many modern physicists who has gone far in the (cid:28)eld of their
ing the behavior of the line of nodes and proper rotation angles. The latter abstractandintricatetheories(likequantum(cid:28)eldtheory,gravitationtheory),ex-
resultswereobtainedbyapplyingLiouville(cid:21)ArnoldtheoremandWeyltheorem hibit poororientation in naturally originatedquestions,concerningdynamicsof
onuniformdistribution. anordinarytoytop.
5. The application of topological analysis methods to the integration 8. In a certain sense, even in the analysis of the integrable situation, for
of rigid body dynamics problems, namely the study of Liouville tori recon- whichthecompleteclassi(cid:28)cationofallsolutionsis,inprinciple,possible,acom-
structions under passing critical values, was for the (cid:28)rst time offered by puter has started a totally new era. If, earlier, in the investigation of integrable
M.P.Harlamov [170] and developed in topological invariant theory, created to systemsthereprevailedanalyticalmethods,makingitpossibletoobtainexplicit
classify integrable Hamiltonian systems with two degrees of freedom. Nearly quadraturesandgeometricinterpretations,whichinmanycaseslookedquitearti-
all known results, obtained by using this technique, are shown in the recently (cid:28)cial(forinstance,JoukovskiyinterpretationofKowalevskayatopmotion[76]),
published book [25]. The complex methods, generally leading to the similar the combination of ideas of topological analysis (bifurcational pattern), stabil-
results,areadvocatedinthebookbyM.Audin[134]. ity theory,phasesection methodanddirectcomputervisualizationofthe (cid:16)most
6. The increase of interest to rigid body dynamics integrable problems in remarkable(cid:17)solutionsiscapableofrepresentinganintegrablesituationandem-
1970(cid:21)1990,havingentailedthediscoveryofthe wholeseries ofnewintegrable phasizing the most characteristic features of motion. Such an investigation can
cases, is connected with the isospectral deformation method development(Lax provideaseriesofnewresults,evenforaseeminglyworn(cid:28)eld(forexample,for
representations, L A-pairs). As a rule, the majority of papers of that period the Kowalevskayatop, Goryachev(cid:21)Chaplygintop, Bobylev(cid:21)Steklovsolution).
(cid:0)
concernsmultidimensionalgeneralizationsofnaturalphysicalanaloguesalready The point is that these results are very dif(cid:28)cult to be detected in the cumber-
known. The development of this trend of researches is also associated with someanalyticalexpressions. Itseems tobepossibletoobtaintheproofofthese
the penetrationof ideas of Lie groupsand algebras, and the analysis of general facts analytically, as well, but only after their computer displaying. Here we
(nonlinear and degenerate) Poisson structures into dynamics. The present state should pay special attention to the analysis of motion in absolute space. Such
oftheseproblemsmaybefoundinourbook[31]. ananalysiswaspracticallynevercarriedout.
It should also be noted that it turned out to be possible to extend many Some curious motions, exhibited by integrable tops, perhaps, are capable
structuresoftheLiealgebraicapproachandqualitativeanalysismethodstonon- of evoking certain ideas concerning their practical application. Recall that, for
holonomicproblemsof rigid bodydynamics,where within last decades several example, the Kowalevskaya top (discovered more than a hundred years ago)
newintegrablesystemswereadded,aswell[52,36]. is still out of the application, just because nothing at all was known about its
7. During last decades there appeared some more trends, concerning top motion,inspiteofcompletesolutioninellipticfunctions.
dynamics. Oneappearedinquantummechanicsfromtheanalysisofsystemsof We also cite some unstable periodic solutions, generating a family of
interactingspinswithanisotropy(achainorXYZ-modelofHeisenberg). Here doubly-asymptotic motions, whose behavior is most complicated and even in
the classical model is a foundation for understanding dynamics at a quantum the presence of an additional integral looks chaotic. Under perturbation such
level,and,inacertainsense,itcanalsobeintegrableandchaotic. Thequantum solutionsarethe(cid:28)rsttobecomedestroyed,andnearthem,inphasespace,there
chaos is only starting to be investigated, but in short time these researches will arisewholedomains,fullof(cid:16)real(cid:17)chaoticpaths.
form a separate scienti(cid:28)c branch, where the essential place will be given to Computer researches make us (cid:16)revise(cid:17) many things and perceive the true
quantum descriptions of tops. First of all, that is because the top model is meaningofanalyticalinvestigations. Ifsomeanalyticalresults(cid:22)likeseparation
a basic model in quantum theory of angular momentum, applied in quantum ofvariables(cid:22) turnoutto bequiteusefulforstudyingbifurcationsandclassical
chemistryandmolecularspectroscopy. solutions, their further (cid:16)development(cid:17) up to obtaining explicit quadratures (in
It is also interesting to know that the conditions of integrability and the termsof(cid:18)-functions)ispracticallyofnouse. Theseresultsarecollected,forex-
18 Introduction
ample,inthebooks[61,72],buttheyareapplicableasexercisesondifferential
equations,ratherthandynamicalanalysismethods.
9. Asforthevalueofclassicalresultsinrigidbodydynamics,itwassome- RIGID BODY DYNAMICS CREATORS
what doubted already in the seventies of the last century (K.Magnus [119]).
The age of total belief in unlimited possibilities of computers generated the
opinion that these results are of no use and the suf(cid:28)ciently powerful com-
puter is capable of predicting motion at any interval of time with suf(cid:28)cient Here is a bit of information concerning the scientists, who obtained the
precision. However, the fact of the exponentially fast separation of paths main results, cited in the book. We meant to show their achievements in rigid
(connected with the instability in the whole domains of phase space) in typi- bodydynamicsonly,whilemanyofthemhavealsoreceivedwell-knownresults
cal dynamical systems, which are nonintegrable, made such a computation at in other (cid:28)elds of mathematics and mechanics. These brief sketches may be
rather large time intervals physically meaningless, as far as the initial con- useful for understandingthe evolutionof principal ideas and methods, andalso
ditions for particular (applied) systems are never known with absolute preci- foreliminationofsomehistoricaldiscrepancies.
sion. Allthesketchesarein chronologicalorder.
Itseemsthatonecanhopefornumericalmethodsonlyintheintegrablecase
Euler, Leonard (15.4.1707(cid:21)18.9.1783) (cid:22) a great mathematician and me-
wherethisseparationneverhappens. Nevertheless,itturnsoutthatconservative
chanic. He was born in Switzerland, but the substantial part of his life he has
systems preserve many elements of integrable dynamics even in the stochastic
spentinRussia(1727(cid:21)41,1766(cid:21)83). Eulerhascontributedtonearlyallbranches
case. Under small perturbation of an integrable problem there continue to ex-
ofmathematics,his workisdif(cid:28)culttobesurveyedandincludesmorethan865
ist non-degenerate periodic orbits, and the majority of conditionally-periodical
essays.
motionsdoesn’tbecomedestroyed(theKAMtheory).
In rigid body dynamics Euler has developed theory of
Under further increase of perturbation both periodic orbits, and invariant
moments of inertia and obtained the formula of veloc-
toriundergovariousbifurcations,havingsomecommonregularities. Theyde(cid:28)ne
ity distribution in a rigid body. In 1750 he obtained
thechangeofthewholestructureofaphase(cid:29)ow,combiningareaswithregular
the equations of motion in a (cid:28)xed frame of reference,
andchaoticbehavior,andprovidescenariosoftransitiontochaos. Inrigidbody
the ones which turned out to be of a little use in prac-
dynamicstheseinvestigations,whichareincidentallyimpossiblewithouthighly
tice. In the works of 1758(cid:21)1765 Euler, for the (cid:28)rst
precisecomputersimulation,werenotcarriedout. Inthepresentbookweshow
time,introducedamovingframeofreference,attached
only several examples of chaotic motion and hope that the nearest future will
tothebody,andobtainedtheEuler(cid:21)Poissonequations
bringalotofnewandinterestingresultsinthis(cid:28)eld.
in the (cid:28)nal form(the Poissoncontribution,re(cid:29)ectedin
the name, seems to consist in their systematic account
inhisfamouscourseonmechanics). Thesepapersalso
contain Euler angles, kinematic relations, named after
L.Euler
Euler,andanintegrablecaseintheabsenceofagravity
(cid:28)eld. As for the last case, Euler brings it up to quadratures and considers var-
ious particular solutions. In would be proper to mention the contribution Euler
made into applied sciences (cid:22) shipbuilding, artillery, turbine theory, strength of
materials.
Lagrange, Joseph Louis (25.1.1736(cid:21)10.4.1813) (cid:22) a great French math-
ematician, mechanic, and astronomer. In his famous treatise Analytical Me-
chanics (in 2 volumes), along with the general formalism of dynamics,
20 RigidBodyDynamicsCreators RigidBodyDynamicsCreators 21
he has shown equations of rigid body motion in an trodynamics. He has shown the analogy between the Euler(cid:21)Poisson equations
arbitrarypotentialforce(cid:28)eld, usingtheframeofrefer- and equations of an elastic curvebending. Followingthe idea of Thomsonand
ence, attached to the body,angularmomentumprojec- Tait,hereducedtheproblemofrigidbodymotioninperfect(cid:29)uid,tothesystem
tions and direction cosines (volume II). There he also ofordinarydifferentialequations. Hehasfoundanintegrablecase,characterized
mentionsan integrablecase, characterizedby the axial by the axial symmetry, shown its solution in elliptic functions and considered
symmetry,whichhereducedtoquadratures. Following variousparticularmotions.
his principle of avoiding drawings, Lagrange doesn’t
give geometrical study of motion, so an apex behav-
ior drawings, that were earlier included into nearly all
textbooks on mechanics, appeared for the (cid:28)rst time in
the paper by Poisson (1815) who has investigated this
J.L.Lagrange
problem as a totally new. Nevertheless, Poisson sys-
tematized notations that complicate understanding of
treatises byD’Alembert,Euler andLagrange,and consideredvarious particular
cases of motion (some textbooks refer to the Lagrange case as the Lagrange(cid:21)
Poisson case). In his turn, Lagrange has simpli(cid:28)ed the solution of the Euler
caseandhasprovidedthedirectproofofexistenceofthird-degreeequationreal
roots,de(cid:28)ningthepositionofprincipalaxes. We shouldnotethatLagrangehas
G.R.Kirchhoff A.Clebsch
also contributed into perturbation theory which enabled Jacobi to consider the
problemabouttheEulertopperturbationandobtainthesystemofcorresponding
(cid:16)osculating(cid:17)variables. Clebsch, Rudolph Fridrich Alfred (19.1.1833(cid:21)7.11.1872) (cid:22) a German
mathematician and mechanic. He has founded the journal Mathematishe An-
Poinsot,Louis(3.1.1777(cid:21)5.12.1859) (cid:22)aFrench
nalen which for sixty years was a leading mathematical journal. He was an
engineer, mechanic, and mathematician. He has given
expert in projective geometry and theory of invariants of algebraic forms. He
ageometricinterpretationoftheEulercase,introduced
offeredanew formofnotationforKirchhoff’sequationswhichis equivalentto
concepts of inertia ellipsoid, momentary axis of rota-
the transition from Lagrangian to Hamiltonian description. For these equations
tion,andthenotionsconnectedtoit(cid:22)polhodeandher-
hehasshownacase ofexistenceofan additionalquadraticintegral,which,asit
polhode(1851). Hehas showngeometricalanalysis of
turnedoutlater,isidenticalto integralsofBrunandTisserand.
stabilityofrigidbodyrotationaroundtheprincipalaxis
ofinertiaellipsoid. Poinsot,incontrasttoLagrange,in-
Joukovskiy,NikolayEgorovitch(17.1.1847(cid:21)17.3.1921) (cid:22)aRussianme-
sisted that in mechanics geometric methods should be
chanic, mathematician, and engineer, as V.I.Lenin has put it, (cid:16)the father of
preferredtoanalyticalones(cid:22)(cid:16)inallthesesolutionswe
Russian aviation(cid:17). In his master’s thesis (1885)he laid the foundationsof the-
canseeonlycomputationswithoutanyclearpictureof
ory of motion of a rigid body with cavities, completely (cid:28)lled with a perfect
the bodymotion(cid:17)[252]. Afterwards, the Poinsotideas
L.Poinsot incompressible (cid:29)uid. For multiconnected cavities he noticed the equivalence
weresupportedanddevelopedbyN.E.JoukovskiyandS.A.Chaplygin. Poinsot
of the obtained form of equation with the equation of motion of a rigid body
alsousedageometricalmethodforstudyingstatics(StaticsElements,1803).
with a (cid:29)y-wheel (cid:22) a gyrostat. He introduced correspondingdynamical charac-
Kirchhoff, Gustav Robert (12.3.1824(cid:21)17.10.1887) (cid:22) a German physicist and teristics and carried out their computations for cavities of various shapes. He
mechanic. InhisLecturesonMathematicalPhysics(1874(cid:21)94,v.1(cid:21)4)hehaslaid has indicated the case of integrability of a free gyrostat. The explicit solution
the foundationsof moderntheoryof elasticity, hydrodynamics,optics and elec- for this case was obtained by V.Volterra by means of elliptic functions (1899).