出版时间:1999-11 出版社:世界图书出版公司 作者:P.J.Olver 页数:513
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内容概要
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.
书籍目录
Preface to First EditionPreface to Second EditionAcknowledgmentsIntroductionNotes to the ReaderCHAPTER 1 Introduction to Lie Groups 1.1 Manifolds Change ofCoordinates Maps Between Manifolds The Maximal Rank Condition Submanifolds Reaular Submanifolds Implicit Submanifolds Curves and Connectedness 1.2 Lie Groups Lie Subgroups Local Lie Groups Local Transformation Groups Orbits 1.3. Vector Fields Flows Action on Functions Differentials Lie Brackets Tangent Spaces and Vectors Fields on Submanifolds Frobenius' Theorem 1.4 LieAlgebras One-Parameter Subgroups Subalgebras The Exponential Map Lie Algebras of Local Lie Groups Structure Constants Commutator Tables Infinitesimal Group Actions 1.5. Differential Forms PuIl-Back and Change ofCoordinates Interior Products The Differential The de Rham Complex Lie Derivatives Homotopy Operators Integration and Stokes' Theorem Notes ExercisesCHAPTER 2 Symmetry Groups of DifTerential Equations 2.1 Symmetries of Algebraic Equations Invariant Subsets Invariant Functions Infinitesimal Invariance Local Invariance Invariants and Functional Dependence Methods for Constructing Invariants 2.2. Groups and Differential Equations 2.3. Prolongation Systems of DifTerential Equations Prolongation ofGroup Actions Invariance of Differential Equations Prolongation of Vector Fields Infinitesimal Invariance The Prolongation Formula Total Derivatives The General Prolongation Formula Properties of Prolonged Vector Fields Characteristics of Symmetries 2.4. Calculation of Symmetry Groups 2.5. Integration of Ordinary Differential Equations First Order Equations Higher Order Equations Differential Invariants Multi-parameter Symmetry Groups Solvable Groups Systems of Ordinary Differential Equations 2.6. Nondegeneracy Conditions for Differential Equations Local Solvability Invariance Criteria The Cauchy-Kovalevskaya Theorem Characteristics Normal Systems Prolongation of DifTerential Equations Notes ExercisesCHAPTER 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 Index
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