Applications of Lie Groups to Differential Equations

Springer Science & Business Media, Jan 21, 2000 - Language Arts & Disciplines - 513 pages
Symmetry methods have long been recognized to be of great importance for the study of the differential equations arising in mathematics, physics, engineering, and many other disciplines. The purpose of this book is to provide a solid introduction to thos

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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
 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

 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
 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 Copyright