Applications of Lie Groups to
Differential Equations
i
Second Edition
Щ Springer
Table of Contents
Preface to First Edition v Preface to Second Edition vii
Acknowledgments ix Introduction xvii Notes to the Reader xxv CHAPTER 1
Introduction to Lie Groups 1
1.1. Manifolds 2 Change of Coordinates 6
Maps Between Manifolds 7 The Maximal Rank Condition 7
Submanifolds 8 Regular Submanifolds 11
Implicit Submanifolds 11 Curves and Connectedness 12
1.2. Lie Groups 13 Lie Subgroups 17 Local Lie Groups 18 Local Transformation Groups 20
Orbits 22 1.3. Vector Fields 24
Flows 27 Action on Functions 30
Differentials 32 Lie Brackets 33 Tangent Spaces and Vectors Fields on Submanifolds 37
Frobenius' Theorem 38
XI
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 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
Table of Contents xiii
2.6. Nondegeneracy Conditions for Differential Equations 157
Local Solvability 157 Invariance Criteria 161 The Cauchy-Kovalevskaya Theorem 162
Characteristics 163 Normal Systems 166 Prolongation of Differential Equations 166
Notes 172 Exercises 176 CHAPTER 3
Group-Invariant Solutions 183 3.1. Construction of Group-Invariant Solutions 185
3.2. Examples of Group-Invariant Solutions 190 3.3. Classification of Group-Invariant Solutions 199
The Adjoint Representation 199 Classification of Subgroups and Subalgebras 203
Classification of Group-Invariant Solutions 207
3.4. Quotient Manifolds 209 Dimensional Analysis 214 3.5. Group-Invariant Prolongations and Reduction 217
Extended Jet Bundles 218 Differential Equations 222 Group Actions 223 The Invariant Jet Space 224
Connection with the Quotient Manifold 225
The Reduced Equation 227 Local Coordinates 228
Notes 235 Exercises 238 CHAPTER 4
Symmetry Groups and Conservation Laws 242
4.1. The Calculus of Variations 243 The Variational Derivative 244 Null Lagrangians and Divergences 247 Invariance of the Euler Operator 249
4.2. Variational Symmetries 252 Infinitesimal Criterion of Invariance 253
Symmetries of the Euler-Lagrange Equations 255
Reduction of Order 257 4.3. Conservation Laws 261
Trivial Conservation Laws 264 Characteristics of Conservation Laws 266
4.4. Noether's Theorem 272 Divergence Symmetries 278
Notes 281 Exercises 283
CHAPTER 5
Generalized Symmetries 286 5.1. Generalized Symmetries of Differential Equations 288
Differential Functions 288 Generalized Vector Fields 289 Evolutionary Vector Fields 291 Equivalence and Trivial Symmetries 292
Computation of Generalized Symmetries 293
Group Transformations 297 Symmetries and Prolongations 300
The Lie Bracket 301 Evolution Equations 303 5.2. Recursion Operators, Master Symmetries and Formal Symmetries 304
Frechet Derivatives 307 Lie Derivatives of Differential Operators 308
Criteria for Recursion Operators 310 The Korteweg-de Vries Equation 312
Master Symmetries 315 Pseudo-differential Operators 318
Formal Symmetries 322 5.3. Generalized Symmetries and Conservation Laws 328
Adjoints of Differential Operators 328 Characteristics of Conservation Laws 330
Variational Symmetries 331 Group Transformations 333 Noether's Theorem 334 Self-adjoint Linear Systems 336 Action of Symmetries on Conservation Laws 341
Abnormal Systems and Noether's Second Theorem 342 Formal Symmetries and Conservation Laws 346
5.4. The Variational Complex 350 The D-Complex 351 Vertical Forms 353 Total Derivatives of Vertical Forms 355
Functional and Functional Forms 356 The Variational Differential 361 Higher Euler Operators 365 The Total Homotopy Operator 368
Notes 374 Exercises 379 CHAPTER 6
Finite-Dimensional Hamiltonian Systems 389
6.1. Poisson Brackets 390 Hamiltonian Vector Fields 392
The Structure Functions 393 The Lie-Poisson Structure 396
Table of Contents xv
6.2. Symplectic Structures and Foliations 398 The Correspondence Between One-Forms and Vector Fields 398
Rank of a Poisson Structure 399 Symplectic Manifolds 400 Maps Between Poisson Manifolds 401
Poisson Submanifolds 402 Darboux' Theorem 404 The Co-adjoint Representation 406
6.3. Symmetries, First Integrals and Reduction of Order 408
First Integrals 408 Hamiltonian Symmetry Groups 409
Reduction of Order in Hamiltonian Systems 412 Reduction Using Multi-parameter Groups 416 Hamiltonian Transformation Groups 418
The Momentum Map 420
Notes 427 Exercises 428 CHAPTER 7
Hamiltonian Methods for Evolution Equations 433
7.1. Poisson Brackets 434 The Jacobi Identity 436 Functional Mnlti-vectors 439 7.2. Symmetries and Conservation Laws 446
Distinguished Functionals 446
Lie Brackets 446 Conservation Laws 447 7.3. Bi-Hamiltonian Systems 452
Recursion Operators 458
Notes 461 Exercises 463 References 467 Symbol Index 489 Author Index 497 Subject Index 501
