内容简介

This book is devoted to explaining a wide range of applications of continuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems.1

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目录

Preface to First EditionPreface to Second EditionAcknowledgmentsIntroductionNotes to the ReaderCHAPTER 1 Introduction to Lie Groups1.1 ManifoldsChange ofCoordinatesMaps Between ManifoldsThe Maximal Rank ConditionSubmanifoldsReaular SubmanifoldsImplicit SubmanifoldsCurves and Connectedness1.2 Lie GroupsLie SubgroupsLocal Lie GroupsLocal Transformation GroupsOrbits1.3. Vector FieldsFlowsAction on FunctionsDifferentialsLie BracketsTangent Spaces and Vectors Fields on SubmanifoldsFrobenius' Theorem1.4 LieAlgebrasOne-Parameter SubgroupsSubalgebrasThe Exponential MapLie Algebras of Local Lie GroupsStructure ConstantsCommutator TablesInfinitesimal Group Actions1.5. Differential FormsPuIl-Back and Change ofCoordinatesInterior ProductsThe DifferentialThe de Rham ComplexLie DerivativesHomotopy OperatorsIntegration and Stokes' TheoremNotesExercisesCHAPTER 2 Symmetry Groups of DifTerential Equations2.1 Symmetries of Algebraic EquationsInvariant SubsetsInvariant FunctionsInfinitesimal InvarianceLocal InvarianceInvariants and Functional DependenceMethods for Constructing Invariants2.2. Groups and Differential Equations2.3. ProlongationSystems of DifTerential EquationsProlongation ofGroup ActionsInvariance of Differential EquationsProlongation of Vector FieldsInfinitesimal InvarianceThe Prolongation FormulaTotal DerivativesThe General Prolongation FormulaProperties of Prolonged Vector FieldsCharacteristics of Symmetries2.4. Calculation of Symmetry Groups2.5. Integration of Ordinary Differential EquationsFirst Order EquationsHigher Order EquationsDifferential InvariantsMulti-parameter Symmetry GroupsSolvable GroupsSystems of Ordinary Differential Equations2.6. Nondegeneracy Conditions for Differential EquationsLocal SolvabilityInvariance CriteriaThe Cauchy-Kovalevskaya TheoremCharacteristicsNormal SystemsProlongation of DifTerential EquationsNotesExercisesCHAPTER 3 Group-Invariant Solutions……CHAPTER 4 Symmetry Groups and Conservation LawsCHAPTER 5 Generalized SymmetriesCHAPTER 6 Finite-Dimensional Hamiltonian SystemsCHAPTER 7 Hamiltonian Methods for Evolution EquationsReferencesSymbol IndexAuthor IndexSubject Index1

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