Equivalence, Invariants, and Symmetry

Peter J. Olver

Cambridge University Press, 1995

Table of Contents

• Preface v
• Acknowledgments ix
• Introduction 1

• Chapter 1 Geometric Foundations 7
  • Manifolds 7
  • Functions 10
  • Submanifolds 13
  • Vector Fields 17
  • Lie Brackets 21
  • The Differential 22
  • Differential Forms 23
  • Equivalence of Differential Forms 29

• Chapter 2 Lie Groups 32
  • Transformation Groups 35
  • Invariant Subsets and Equations 39
  • Canonical Forms 42
  • Invariant Functions 44
  • Lie Algebras 48
  • Structure Constants 51
  • The Exponential Map 52
  • Subgroups and Subalgebras 53
  • Infinitesimal Group Actions 55
  • Classification of Group Actions 58
  • Infinitesimal Invariance 61
  • Invariant Vector Fields 65
  • Lie Derivatives and Invariant Differential Forms 68
  • The Maurer--Cartan Forms 71

• Chapter 3 Representation Theory 75
  • Representations 75
  • Representations on Function Spaces 78
  • Multiplier Representations 81
  • Infinitesimal Multipliers 85
  • Relative Invariants 91
  • Classical Invariant Theory 95

• Chapter 4 Jets and Contact Transformations 105
  • Transformations and Functions 106
  • Invariant Functions 109
  • Jets and Prolongations 111
  • Total Derivatives 115
  • Prolongation of Vector Fields 117
  • Contact Forms 121
  • Contact Transformations 125
  • Infinitesimal Contact Transformations 129
  • Classification of Groups of Contact Transformations 134

• Chapter 5 Differential Invariants 136
  • Differential Invariants 136
  • Dimensional Considerations 139
  • Infinitesimal Methods 141
  • Stabilization and Effectiveness 143
  • Invariant Differential Operators 146
  • Invariant Differential Forms 153
  • Several Dependent Variables 157
  • Several Independent Variables 164

• Chapter 6 Symmetries of Differential Equations 175
  • Symmetry Groups and Differential Equations 175
  • Infinitesimal Methods 178
  • Integration of Ordinary Differential Equations 187
  • Characterization of Invariant Differential Equations 191
  • Lie Determinants 199
  • Symmetry Classification of Ordinary Differential Equations 202
  • A Proof of Finite Dimensionality 206
  • Linearization of Partial Differential Equations 209
  • Differential Operators 211
  • Applications to the Geometry of Curves 218

• Chapter 7 Symmetries of Variational Problems 221
  • The Calculus of Variations 222
  • Equivalence of Functionals 227
  • Invariance of the Euler--Lagrange Equations 230
  • Symmetries of Variational Problems 235
  • Invariant Variational Problems 238
  • Symmetry Classification of Variational Problems 240
  • First Integrals 242
  • The Cartan Form 244
  • Invariant Contact Forms and Evolution Equations 246

• Chapter 8 Equivalence of Coframes 252
  • Frames and Coframes 252
  • The Structure Functions 256
  • Derived Invariants 259
  • Classifying Functions 261
  • The Classifying Manifolds 266
  • Symmetries of a Coframe 274
  • Remarks and Extensions 276

• Chapter 9 Formulation of Equivalence Problems 280
  • Equivalence Problems Using Differential Forms 280
  • Coframes and Structure Groups 287
  • Normalization 291
  • Overdetermined Equivalence Problems 297

• Chapter 10 Cartan's Equivalence Method 304
  • The Structure Equations 304
  • Absorption and Normalization 307
  • Equivalence Problems for Differential Operators 310
  • Fiber-preserving Equivalence of Scalar Lagrangians 321
  • An Inductive Approach to Equivalence Problems 327
  • Lagrangian Equivalence under Point Transformations 328
  • Applications to Classical Invariant Theory 333
  • Second Order Variational Problems 337
  • Multi-dimensional Lagrangians 342

• Chapter 11 Involution 347
  • Cartan's Test 350
  • The Transitive Case 355
  • Divergence Equivalence of First Order Lagrangians 357
  • The Intrinsic Method 358
  • Contact Transformations 361
  • Darboux' Theorem 364
  • The Intransitive Case 366
  • Equivalence of Nonclosed Two-Forms 367

• Chapter 12 Prolongation of Equivalence Problems 372
  • The Determinate Case 373
  • Equivalence of Surfaces 377
  • Conformal Equivalence of Surfaces 385
  • Equivalence of Riemannian Manifolds 386
  • The Indeterminate Case 394
  • Second Order Ordinary Differential Equations 397

• Chapter 13 Differential Systems 409
  • Differential Systems and Ideals 409
  • Equivalence of Differential Systems 415
  • Vector Field Systems 416

• Chapter 14 Frobenius' Theorem 421
  • Vector Field Systems 421
  • Differential Systems 427
  • Characteristics and Normal Forms 428
  • The Technique of the Graph 431
  • Global Equivalence 440

• Chapter 15 The Cartan--K\"ahler Existence Theorem 447
  • The Cauchy--Kovalevskaya Existence Theorem 447
  • Necessary Conditions 449
  • Sufficient Conditions 455
  • Applications to Equivalence Problems 460
  • Involutivity and Transversality 465

• Tables 472
• References 477
• Subject Index 490
• Author Index 499
• Symbol Index 504