经典力学与天体力学中的数学问题

图书标准信息

• 作者 阿诺德（Arnold.L.I） 著
• 出版社 科学出版社
• 出版时间 2009-01
• 版次 1
• ISBN 9787030235077
• 定价 96.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 518页
• 字数 653千字
• 正文语种 英语

【内容简介】

　　Thisworkdescribesthefundamentalprinciples,problems,andmethodsofclassicalmechanics.Themainattentionisdevotedtothemathematicalsideofthesubject.Theauthorshaveendeavoredtogiveanexpositionstressingtheworkingapparatusofclassicalmechanics.Thebookissignificantlyexpandedcomparedtothepreviousedition.Theauthorshaveaddedtwochaptersonthevariationalprinciplesandmethodsofclassicalmechanicsaswellasontensorinvariantsofequationsofdynamics.Moreover,variousothersectionshavebeenrevised,addedorexpanded.Themainpurposeofthebookistoacquaintthereaderwithclassicalmechanicsasawhole,inbothitsclassicalanditscontemporaryaspects.Thebookaddressesallmathematicians,physicistsandengineers.

【目录】

1BasicPrinciplesofClassicalMechanics
1.1NewtonianMechanics
1.1.1Space，Time，Motion
1.1.2Newton-LaplacePrincipleofDeterminacy
1.1.3PrincipleofRelativity
1.1.4PrincipleofRelativityandForcesofInertia
1.1.5BasicDynamicalQuantities.ConservationLaws
1.2LagrangianMechanics
1.2.1PreliminaryRemarks
1.2.2VariationsandExtremals
1.2.3LagrangesEquations
1.2.4PoincaresEquations
1.2.5MotionwithConstraints
1.3HamiltonianMechanics
1.3.1SymplecticStructuresandHamiltonsEquations
1.3.2GeneratingFunctions
1.3.3SymplecticStructureoftheCotangentBundle
1.3.4TheProblemofnPointVortices
1.3.5ActioninthePhaseSpace
1.3.6IntegralInvariant
1.3.7ApplicationstoDynamicsofIdealFluid
1.4VakonomicMechanics
1.4.1LagrangesProblem
1.4.2VakonomicMechanics
1.4.3PrincipleofDeterminacy
1.4.4HamiltonsEquationsinRedundantCoordinates
1.5HamiltonianFormalismwithConstraints
1.5.1DiracsProblem
1.5.2Duality
1.6RealizationofConstraints
1.6.1VariousMethodsofRealizationofConstraints
1.6.2HolonomicConstraints
1.6.3AnisotropicFriction
1.6.4AdjointMasses
1.6.5AdjointMassesandAnisotropicFriction
1.6.6SmallMasses

2Then-BodyProblem
2.1TheTwo-BodyProblem
2.1.1Orbits
2.1.2Anomalies
2.1.3CollisionsandRegularization
2.1.4GeometryofKeplersProblem
2.2CollisionsandRegularization
2.2.1NecessaryConditionforStability
2.2.2SimultaneousCollisions
2.2.3BinaryCollisions
2.2.4SingularitiesofSolutionsofthen-BodyProblem
2.3ParticularSolutions
2.3.1CentralConfigurations
2.3.2HomographicSolutions
2.3.3EffectivePotentialandRelativeEquilibria
2.3.4PeriodicSolutionsintheCaseofBodiescfEqualMasses
2.4FinalMotionsintheThree-BodyProblem
2.4.1ClassificationoftheFinalMotionsAccordingtoChazy.
2.4.2SymmetryofthePastandFuture
2.5RestrictedThree-BodyProblem
2.5.1EquationsofMotion.TheJacobiIntegral
2.5.2RelativeEquilibriaandHillRegions
2.5.3HillsProblem
2.6ErgodicTheoremsofCelestialMechanics
2.6.1StabilityintheSenseofPoisson
2.6.2ProbabilityofCapture
2.7DynamicsinSpacesofConstantCurvature
2.7.1GeneralizedBertrandProblem
2.7.2KeplersLaws
2.7.3CelestialMechanicsinSpacesofConstantCurvature
2.7.4PotentialTheoryinSpacesofConstantCurvature

3SymmetryGroupsandOrderReduction.
3.1SymmetriesandLinearIntegrals
3.1.1NSthersTheorem
3.1.2SymmetriesinNon-HolonomicMechanics
3.1.3SymmetriesinVakonomicMechanics
3.1.4SymmetriesinHamiltonianMechanics
3.2ReductionofSystemswithSymmetries
3.2.1OrderReduction(LagrangianAspect)
3.2.2OrderReduction(HamiltonianAspect)
3.2.3Examples：FreeRotationofaRigidBodyandtheThreeBodyProblem
3.3RelativeEquilibriaandBifurcationofIntegralManifolds
3.3.1RelativeEquilibriaandEffectivePotential
3.3.2IntegralManifolds，RegionsofPossibleMotion，andBifurcationSets
3.3.3TheBifurcationSetinthePlanarThree-BodyProblem
3.3.4BifurcationSetsandIntegralManifoldsintheProblemofRotationofaHeavyRigidBodywithaFixedPoint

4VariationalPrinciplesandMethods
4.1GeometryofRegionsofPossibleMotion
4.1.1PrincipleofStationaryAbbreviatedAction
4.1.2GeometryofaNeighbourhoodoftheBoundary
4.1.3RiemannianGeometryofRegionsofPossibleMotionwithBoundary
4.2PeriodicTrajectoriesofNaturalMechanicalSystems
4.2.1RotationsandLibrations
4.2.2LibrationsinNon-Simply-ConnectedRegionsofPossibleMotion
4.2.3LibrationsinSimplyConnectedDomainsandSeifertsConjecture
4.2.4PeriodicOscillationsofaMulti-LinkPendulum
4.3PeriodicTrajectoriesofNon-ReversibleSystems
4.3.1SystemswithGyroscopicForcesandMultivaluedFunctionals
4.3.2ApplicationsoftheGeneralizedPoincareGeometricTheorem
4.4AsymptoticSolutions.ApplicationtotheTheoryofStabilityofMotion
4.4.1ExistenceofAsymptoticMotions
4.4.2ActionFunctioninaNeighbourhoodofanUnstableEquilibriumPosition
4.4.3InstabilityTheorem
4.4.4Multi-LinkPendulumwithOscillatingPointofSuspension
4.4.5HomoclinicMotionsClosetoChainsofHomoclinicMotions

5IntegrableSystemsandIntegrationMethods
5.1BriefSurveyofVariousApproachestoIntegrabilityofHamiltonianSystems
5.1.1Quadratures
5.1.2CompleteIntegrability
5.1.3NormalForms
5.2CompletelyIntegrableSystems
5.2.1Action-AngleVariables
5.2.2Non-CommutativeSetsofIntegrals
5.2.3ExamplesofCompletelyIntegrableSystems
5.3SomeMethodsofIntegrationofHamiltonianSystems
5.3.1MethodofSeparationofVariables
5.3.2MethodofL-APairs
5.4IntegrableNon-HolonomicSystems
5.4.1DifferentialEquationswithInvariantMeasure
5.4.2SomeSolvedProblemsofNon-HolonomicMechanics.

6PerturbationTheoryforIntegrableSystems
6.1AveragingofPerturbations
6.1.1AveragingPrinciple
6.1.2ProcedureforEliminatingFastVariables.Non-ResonantCase
6.1.3ProcedureforEliminatingFastVariables.Resonantase
6.1.4AveraginginSingle-FrequencySystems
6.1.5AveraginginSystemswithConstantFrequencies
6.1.6AveraginginNon-ResonantDomains
6.1.7EffectofaSingleResonance
6.1.8AveraginginTwo-FrequencySystems
6.1.9AveraginginMulti-FrequencySystems
6.1.10AveragingatSeparatrixCrossing
6.2AveraginginHamiltonianSystems
6.2.1ApplicationoftheAveragingPrinciple
6.2.2ProceduresforEliminatingFastVariables
6.3KAMTheory
6.3.1UnperturbedMotion.Non-DegeneracyConditions
6.3.2InvariantToriofthePerturbedSystem
6.3.3SystemswithTwoDegreesofFreedom
6.3.4DiffusionofSlowVariablesinMultidimensionalSystemsanditsExponentialEstimate
6.3.5DiffusionwithoutExponentiallySmallEffects
6.3.6VariantsoftheTheoremonInvariantTori
6.3.7KAMTheoryforLower-DimensionalTori
6.3.8VariationalPrincipleforInvariantTori.Cantori
6.3.9ApplicationsofKAMTheory
6.4AdiabaticInvariants
6.4.1AdiabaticInvarianceoftheActionVariableinSingle-FrequencySystems
……
7Non-IntegrableSystems
8TheoryofSmallOscillations
9TensorInvariantsofEquationsofDynamics
RecommendedReading
Bibliography
IndexofNames
SubjectIndex