李群在微分方程中的应用（第2版）

PrefacetoFirstEdition
PrefacetoSecondEdition
Acknowledgments
Introduction
NotestotheReader
CHAPTER1
IntroductiontoLieGroups
1.1.Manifolds
ChangeofCoordinates
MapsBetweenManifolds
TheMaximalRankCondition
Submanifolds
RegularSubmanifolds
ImplicitSubmanifolds
CurvesandConnectedness
1.2.LieGroups
LieSubgroups
LocalLieGroups
LocalTransformationGroups
Orbits
1.3.VectorFields
Flows
ActiononFunctions
Differentials
LieBrackets
TangentSpacesandVectorsFieldsonSubmanifolds
Frobenius'Theorem
1.4.LieAlgebras
One-ParameterSubgroups
Subalgebras
TheExponentialMap
LieAlgebrasofLocalLieGroups
StructureConstants
CommutatorTables
InfinitesimalGroupActions
1.5.DifferentialForms
Pull-BackandChangeofCoordinates
InteriorProducts
TheDifferential
ThedeRhamComplex
LieDerivatives
HomotopyOperators
IntegrationandStokes'Theorem
Notes
Exercises
CHAPTER2
SymmetryGroupsofDifferentialEquations
2.1.SymmetriesofAlgebraicEquations
InvariantSubsets
InvariantFunctions
InfinitesimalInvariance
LocalInvariance
InvariantsandFunctionalDependence
MethodsforConstructingInvariants
2.2.GroupsandDifferentialEquations
2.3.Prolongation
SystemsofDifferentialEquations
ProlongationofGroupActions
InvarianceofDifferentialEquations
ProlongationofVectorFields
InfinitesimalInvariance
TheProlongationFormula
TotalDerivatives
TheGeneralProlongationFormula
PropertiesofProlongedVectorFields
CharacteristicsofSymmetries
2.4.CalculationofSymmetryGroups
2.5.IntegrationofOrdinaryDifferentialEquations
FirstOrderEquations
HigherOrderEquations
DifferentialInvariants
Multi-parameterSymmetryGroups
SolvableGroups
SystemsofOrdinaryDifferentialEquations
2.6.NondegeneracyConditionsforDifferentialEquations
LocalSolvability
InvarianceCriteria
TheCauchy-KovalevskayaTheorem
Characteristics
NormalSystems
ProlongationofDifferentialEquations
Notes
Exercises
CHAPTER3
Group-InvafiantSolutions
3.1.ConstructionofGroup-InvariantSolutions
3.2.ExamplesofGroup-InvariantSolutions
3.3.ClassificationofGroup-InvariantSolutions
TheAdjointRepresentation
ClassificationofSubgroupsandSubalgebras
ClassificationofGroup-InvariantSolutions
3.4.QuotientManifolds
DimensionalAnalysis
3.5.Group-InvariantProlongationsandReduction
ExtendedJetBundles
DifferentialEquations
GroupActions
TheInvariantJetSpace
ConnectionwiththeQuotientManifold
TheReducedEquation
LocalCoordinates
Notes
Exercises
CHAPTER4
SymmetryGroupsandConservationLaws
4.1.TheCalculusofVariations
TheVariationalDerivative
NullLagrangiansandDivergences
InvarianceoftheEulerOperator
4.2.VariationalSymmetries
InfinitesimalCriterionofInvariance
SymmetriesoftheEuler-LagrangeEquations
ReductionofOrder
4.3.ConservationLaws
TrivialConservationLaws
CharacteristicsofConservationLaws
4.4.Noether'sTheorem
DivergenceSymmetries
Notes
Exercises
CHAPTER5
GeneralizedSymmetries
5.1.GeneralizedSymmetriesofDifferentialEquations
DifferentialFunctions
GeneralizedVectorFields
EvolutionaryVectorFields
EquivalenceandTrivialSymmetries
ComputationofGeneralizedSymmetries
GroupTransformations
SymmetriesandProlongations
TheLieBracket
EvolutionEquations
5.2.RecursionOperators,MasterSymmetriesandFormalSymmetries
FrechetDerivatives
LieDerivativesofDifferentialOperators
CriteriaforRecursionOperators
TheKorteweg-deVriesEquation
MasterSymmetries
Pseudo-differentialOperators
FormalSymmetries
5.3.GeneralizedSymmetriesandConservationLaws
AdjointsofDifferentialOperators
CharacteristicsofConservationLaws
VariationalSymmetries
GroupTransformations
Noether'sTheorem
Self-adjointLinearSystems
ActionofSymmetriesonConservationLaws
AbnormalSystemsandNoether'sSecondTheorem
FormalSymmetriesandConservationLaws
5.4.TheVariationalComplex
TheD-Complex
VerticalForms
TotalDerivativesofVerticalForms
FunctionalsandFunctionalForms
TheVariationalDifferential
HigherEulerOperators
TheTotalHomotopyOperator
Notes
Exercises
CHAPTER6
Finite-DimensionalHamiltonianSystems
6.1.PoissonBrackets
HamiltonianVectorFields
TheStructureFunctions
TheLie-PoissonStructure
6.2.SymplecticStructuresandFoliations
TheCorrespondenceBetweenOne-FormsandVectorFields
RankofaPoissonStructure
SymplecticManifolds
MapsBetweenPoissonManifolds
PoissonSubmanifolds
Darboux'Theorem
TheCo-adjointRepresentation
6.3.Symmetries,FirstIntegralsandReductionofOrder
FirstIntegrals
HamiltonianSymmetryGroups
ReductionofOrderinHamiitonianSystems
ReductionUsingMulti-parameterGroups
HamiltonianTransformationGroups
TheMomentumMap
Notes
Exercises
CHAPTER7
HamiltonianMethodsforEvolutionEquations
7.1.PoissonBrackets
TheJacobiIdentity
FunctionalMulti-vectors
7.2.SymmetriesandConservationLaws
DistinguishedFunctionals
LieBrackets
ConservationLaws
7.3.Bi-HamiltonianSystems
RecursionOperators
Notes
Exercises
References
SymbolIndex
AuthorIndex
SubjectIndex