Applications of Lie Groups to Differential Equations

Peter J. Olver

Springer Science & Business Media, 1993 - Language Arts & Disciplines - 513 pages

Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented so that graduate students and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.

Contents

• Introduction to Lie Groups: 1
• 11 Manifolds: 2
• Change of Coordinates: 6
• Maps Between Manifolds: 7
• Submanifolds: 8
• Regular Submanifolds: 11
• Curves and Connectedness: 12
• 12 Lie Groups: 13
• Lie Subgroups: 17
• Local Lie Groups: 18
• Local Transformation Groups: 20
• Orbits: 22
• 13 Vector Fields: 24
• Flows: 27
• Action on Functions: 30
• Differentials: 32
• Lie Brackets: 33
• Tangent Spaces and Vector Fields on Submanifolds: 37
• Frobenius Theorem: 38
• 14 Lie Algebras: 42
• OneParameter Subgroups: 44
• Subalgebras: 46
• The Exponential Map: 48
• Structure Constants: 50
• Infinitesimal Group Actions: 51
• 15 Differential Forms: 53
• PullBack and Change of Coordinates: 56
• The Differential: 57
• The de Rham Complex: 58
• Lie Derivatives: 60
• Homotopy Operators: 63
• Integration and Stokes Theorem: 65
• Notes: 67
• Exercise: 69
• Symmetry Groups of Differential Equations: 75
• 21 Symmetries of Algebraic Equations: 76
• Invariant Functions: 77
• Infinitesimal Invariance: 79
• Local Invariance: 83
• Invariants and Functional Dependence: 84
• Methods for Constructing Invariants: 87
• 22 Groups and Differential Equations: 90
• 23 Prolongation: 94
• Systems of Differential Equations: 96
• Prolongation of Group Actions: 98
• Invariance of Differential Equations: 100
• Prolongation of Vector Fields: 101
• Infinitesimal Invariance: 103
• The Prolongation Formula: 105
• Total Derivatives: 108
• The General Prolongation Formula Theorem: 110
• Properties of Prolonged Vector Fields: 115
• 24 Calculation of Symmetry Groups: 116
• 25 Integration of Ordinary Differential Equations: 130
• First Order Equations: 131
• Higher Order Equations: 137
• Differential Invariants: 139
• Multiparameter Symmetry Groups: 145
• Solvable Groups: 151
• Systems of Ordinary Differential Equations: 154
• 26 Nondegeneracy Conditions for Differential Equations: 157
• Invariance Criteria: 161
• The CauchyKovalevskaya Theorem: 162
• Characteristics: 163
• Normal Systems: 166
• Notes: 172
• Exercises: 176
• GroupInvariant Solutions: 183
• 31 Construction of GroupInvariant Solutions: 185
• 32 Examples of GroupInvariant Solutions: 190
• 33 Classification of GroupInvariant Solutions: 199
• Classification of Subgroups and Subalgebras: 203
• Classification of GroupInvariant Solutions: 207
• 34 Quotient Manifolds: 209
• Dimensional Analysis: 214
• 35 GroupInvariant 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
• Exercise: 238
• Symmetry Groups and Conservation Laws: 242
• 41 The Calculus of Variations: 243
• The Variational Derivative: 244
• Null Lagrangians and Divergences: 247
• Invariance of the Euler Operator: 249
• 42 Variational Symmetries: 252
• Infinitesimal Criterion of Invariance: 253
• Symmetries of the EulerLagrange Equations: 255
• Reduction of Order: 257
• 43 Conservation Laws: 261
• Trivial Conservation Laws: 264
• Characteristics of Conservation Laws: 266
• 44 Noethers Theorem: 272
• Divergence Symmetries: 278
• Notes: 281
• Exercise: 283
• Generalized Symmetries: 286
• 51 Generalized Symmetries of Differential Equations: 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
• 52 Recursion Operators Master Symmetries and Formal Symmetries: 304
• Frechet Derivatives: 307
• Lie Derivatives of Differential Operators: 308
• Criteria for Recursion Operators: 310
• The Kortewegde Vries Equation: 312
• Master Symmetries: 315
• Pseudodifferential Operators: 318
• Formal Symmetries: 322
• 53 Generalized Symmetries and Conservation Laws: 328
• Characteristics of Conservation Laws: 330
• Variational Symmetries: 331
• Group Transformations: 333
• Noethers Theorem: 334
• Selfadjoint Linear Systems: 336
• Action of Symmetries on Conservation Laws: 341
• Abnormal Systems and Noethers Second Theorem: 342
• Formal Symmetries and Conservation Laws: 346
• 54 The Variational Complex: 350
• The DComplex: 351
• Vertical Forms: 353
• Total Derivatives of Vertical Forms: 355
• Functionals and Functional Forms: 356
• The Variational Differential: 361
• Higher Euler Operators: 365
• The Total Homotopy Operator: 368
• Notes: 374
• Exercise: 379
• FiniteDimensional Hamiltonian Systems: 389
• 61 Poisson Brackets: 390
• Hamiltonian Vector Fields: 392
• The Structure Functions: 393
• The LiePoisson Structure: 396
• 62 Symplectic Structures and Foliations: 398
• Rank of a Poisson Structure: 399
• Symplectic Manifolds: 400
• Maps Between Poisson Manifolds: 401
• Poisson Submanifolds: 402
• Darboux Theorem: 404
• The Coadjoint Representation: 406
• 63 Symmetries First Integrals and Reduction of Order: 408
• Hamiltonian Symmetry Groups: 409
• Reduction of Order in Hamiltonian Systems: 412
• Reduction Using Multiparameter Groups: 416
• Hamiltonian Transformation Groups: 418
• The Momentum Map: 420
• Notes: 427
• EXercise: 428
• Hamiltonian Methods for Evolution Equations: 433
• 71 Poisson Brackets: 434
• The Jacobi Identity: 436
• Functional Multi vectors: 439
• 72 Symmetries and Conservation Laws: 446
• Conservation Laws: 447
• 73 BiHamiltonian Systems: 452
• Recursion Operators: 458
• Notes: 461
• Exercise: 463
• References: 467
• Symbol Index: 489
• Author Index: 497
• Subject Index: 501

Bibliographic information

• Title: Applications of Lie Groups to Differential Equations; Volume 107 of Graduate Texts in Mathematics, ISSN 0072-5285
• Author: Peter J. Olver
• Edition: illustrated, reprint
• Publisher: Springer Science & Business Media, 1993
• ISBN: 0387950001, 9780387950006
• Length: 513 pages
• Subjects: Language Arts & Disciplines / Library & Information Science / General; Mathematics / Algebra / Abstract; Mathematics / Algebra / Linear; Mathematics / Calculus; Mathematics / Differential Equations / General; Mathematics / Functional Analysis; Mathematics / Group Theory; Mathematics / Mathematical Analysis; Mathematics / Optimization; Science / System Theory