Applications of Lie Groups to Differential Equations

Preface to First Edition Preface to Second Edition Acknowledgments Introduction Notes to the Reader CHAPTER 1 Introduction to Lie Groups 1.1. Manifolds Change of Coordinates Maps Between Manifolds The Maximal Rank Condition Submanifolds Regular 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. Lie Algebras One-Parameter Subgroups Subalgebras The Exponential Map Lie Algebras of Local Lie Groups Structure Constants Commutator Tables Infinitesimal Group Actions 1.5. Differential Forms Pull-Back and Change of Coordinates Interior Products The Differential The de Rham Complex Lie Derivatives Homotopy Operators Integration and Stokes'Theorem Notes Exercises CHAPTER 2 Symmetry Groups of Differential 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 Differential Equations Prolongation of Group Actions Invariance of Differential Eguations 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 Differential Equations Notes Exercises CHAPTER 3 Group-Invariant Solutions 3.1. Construction of Group-Invariant Solutions 3.2. Examples of Group-Invariant Solutions 3.3. Classification of Group-Invariant Solutions The Adjoint Representation Classification of Subgroups and Subalgebras Classification of Group-Invariant Solutions 3.4. Quotient Manifolds Dimensional Analysis 3.5. Group-Invariant Prolongations and Reduction Extended Jet Bundles Differential Equations Group Actions The Invariant Jet Space Connection with the Quotient Manifold The Reduced Equation Local Coordinates Notes Exercises CHAPTER 4 Symmetry Groups and Conservation Laws 4.1. The Calculus of Variations The Variational Derivative Null Lagrangians and Divergences Invariance of the Euler Operator 4.2. Variational Symmetries Infinitesimal Criterion of Invariance Symmetries of the Euler-Lagrange Equations Reduction of Order 4.3. Conservation Laws Trivial Conservation Laws Characteristics of Conservation Laws 4.4. Noether's Theorem Divergence Symmetries Notes Exercises CHAPTER 5 Generalized Symmetries 5.1. Generalized Symmetries of Differential Equations Differential Functions Generalized Vector Fields Evolutionary Vector Fields Equivalence and Trivial Symmetries Computation of Generalized Symmetries Group Transformations Symmetries and Prolongations The Lie Bracket Evolution Equations 5.2. Recursion Operators,Master Symmetries and Formal Symmetries Fr¨¦chet Derivatives Lie Derivatives of Differential Operators Criteria for Recursion Operators The Korteweg-de Vries Equation Master Symmetries Pseudo-differential Operators Formal Symmetries 5.3. Generalized Symmetries and Conservation Laws Adjoints of Differential Operators Characteristics of Conservation Laws Variational Symmetries Group Transformations Noether's Theorem Self-adjoint Linear Systems Action of Symmetries on Conservation Laws Abnormal Systems and Noether's Second Theorem Formal Symmetries and Conservation Laws 5.4. The Variational Complex The D-Complex Vertical Forms Total Derivatives of Vertical Forms Functionals and Functional Forms The Variational Differential Higher Euler Operators The Total Homotopy Operator Notes Exercises CHAPTER 6 Finite-Dimensional Hamiltonian Systems 6.1. Poisson Brackets Hamiltonian Vector Fields The Structure Functions The Lie-Poisson Structure 6.2. Symplectic Structures and Foliations The Correspondence Between One-Forms and Vector Fie Rank of a Poisson Structure Symplectic Manifolds Maps Between Poisson Manifolds Poisson Submanifolds Darboux'Theorem The Co-adjoint Representation 6.3. Symmetries, First Integrals and Reduction of Order First Integrals Hamiltonian Symmetry Groups Reduction of Order in Hamiltonian Systems Reduction Using Multi-parameter Groups Hamiltonian Transformation Groups The Momentum Map Notes Exercises CHAPTER 7 Hamiltonian Methods for Evolution Equations 7.1. Poisson Brackets The Jacobi Identity Functional Minlti-vectors 7.2. Symmetries and Conservation Laws Distinguished Functionals Lie Brackets Conservation Laws 7.3. Bi-Hamiltonian Systems Recursion Operators Notes Exercises References Symbol Index Author Index Subject Index