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Bibliografická citace

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BK
2nd ed.
New York : Springer, c1993
xxviii, 513 s. : il. ; 24 cm

ISBN 0-387-95000-1 (dotisk ; brož.)
Graduate texts in mathematics ; 107
Obsahuje bibliografii na s. 467-488 a rejstříky
Popsáno podle brožovaného dotisku z roku 2000
000239653
Table of Contents // Preface to First Edition v // Preface to Second Edition vii // Acknowledgments ix // Introduction xvii // Notes to the Reader xxv // CHAPTER i // 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 // Table of Contents // xii // 1.4. Lie Algebras 42 // One-Parameter Subgroups 44 // Subalgebras 45 // The Exponential Map 4g // Lie Algebras of Local Lie Groups 48 // Structure Constants 50 // Commutator Tables 50 // Infinitesimal Group Actions 51 // 1.5. Differential Forms 53 // Pull-Back and Change of Coordinates 56 // Interior Products 56 // The Differential 57 // The de Rham Complex 58 // Lie Derivatives 60 // Homotopy Operators 63 // Integration and Stokes’ Theorem 65 // Notes 67 // Exercises 69 // CHAPTER 2 // Symmetry Groups of Differential Equations 75 // 2.1. Symmetries of Algebraic Equations 76 // Invariant Subsets 76 // Invariant Functions 77 // Infinitesimal Invariance 79 // Local Invariance 83 // Invariants and Functional Dependence 84 // Methods for Constructing Invariants 87
// 2.2. Groups and Differential Equations 90 // 2.3. 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 110 // Properties of Prolonged Vector Fields 115 // Characteristics of Symmetries 115 // 2.4. Calculation of Symmetry Groups 116 // 2.5. Integration of Ordinary Differential Equations 130 // First Order Equations 131 // Higher Order Equations 137 // Differential Invariants I39 // Multi-parameter Symmetry Groups 145 // Solvable Groups 151 // Systems of Ordinary Differential Equations 154 // - ir.e of Contents // xiii // 1: 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 // THAPTER 3 // Group-Invariant Solutions 183 // 3.1 Construction of Group-Invariant Solutions 185 // 3-1 Examples of Group-Invariant Solutions 190 // 33. Gassification of Group-Invariant Solutions 199 // The Adjoint Representation 199 // Classification of Subgroups and Subalgebras 203 // Classification of Group-Invariant Solutions 207 // 5 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 // xiv // Table of Contents // 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 // Fréchet 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 33? // 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 35O // The D-Complex 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 // 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 // - : e 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 Midti-vectors 439 // 72. 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

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