Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra

Contents......Page 9
1 Introduction......Page 17
2.1 Definitions and Examples......Page 23
2.2 Computer Algebra Applications......Page 33
2.3 Exercises......Page 36
3.1 Definitions......Page 37
3.2 Concepts for Lie Groups......Page 40
3.3 Haar Measure......Page 44
3.4 Computer Algebra Applications......Page 48
3.5 Exercises......Page 49
4.1 Introduction......Page 51
4.2 Classical Groups......Page 52
4.2.1 Affine Groups......Page 61
4.2.2 Group Actions and Group Representations......Page 64
4.3 Computer Algebra Applications......Page 69
4.4 Exercises......Page 70
5.1 Introduction......Page 73
5.2 Infinitesimal Generators......Page 77
5.3 Multi-Parameter Lie Groups of Transformations......Page 82
5.4 Computer Algebra Applications......Page 86
5.5 Exercises......Page 89
6.1 Definition and Examples......Page 91
6.3 Lie Algebras and Fermi Operators......Page 94
6.4 Lie Algebras and Lie Groups......Page 95
6.5 Classical Lie Algebras......Page 100
6.6 Important Concepts......Page 104
6.7 Adjoint Representation......Page 109
6.8 Casimir Operators......Page 112
6.9 Cartan-Weyl Basis......Page 113
6.10 Computer Algebra Applications......Page 115
6.11 Exercises......Page 117
7.1 One-Dimensional Linear Wave Equation......Page 119
7.2 One-Dimensional Diffusion Equation......Page 123
7.3 Stationary Flow......Page 124
7.4 Gauge Transformation......Page 127
7.5 Computer Algebra Applications......Page 134
7.6 Exercises......Page 135
8.1 Vector Fields and Tangent Bundles......Page 137
8.2 Differential Forms and Cotangent Bundles......Page 139
8.3 Exterior Derivative......Page 143
8.4 Pull-Back Maps......Page 145
8.5 Interior Product or Contraction......Page 146
8.6 Riemannian Manifolds......Page 147
8.7 Hodge Star Operator......Page 150
8.8 Pseudospherical Surfaces......Page 154
8.9 Computer Algebra Applications......Page 155
8.10 Exercises......Page 158
9.2 Definitions......Page 161
9.3 Invariance and Conformal Invariance......Page 165
9.4 Computer Algebra Applications......Page 175
9.5 Exercises......Page 176
10.1.1 Introductory Example......Page 179
10.1.2 Local Jet Bundle Formalism......Page 182
10.1.3 Prolongation of Vector Fields......Page 186
10.1.4 Partial Differential Equations on Jet Bundles......Page 188
10.2 Invariance of Differential Equations......Page 191
10.3 Similarity Solutions......Page 200
10.4 Transforming Solutions to Solutions......Page 209
10.5 Direct Method......Page 211
10.6 Computer Algebra Application......Page 213
10.7 Exercises......Page 215
11.1 Definitions and Examples......Page 217
11.2 Invariant Solutions......Page 226
11.3 Computer Algebra Applications......Page 229
11.4 Exercises......Page 231
12.1 Lie Point Symmetry Vector Fields......Page 233
12.2 Lie-Backlund Vector Fields......Page 238
12.4 Exercises......Page 240
13.2 Listing......Page 241
13.3 Exercises......Page 260
14.1 Gateaux Derivative......Page 261
14.2 Definition and Examples......Page 269
14.3 Computer Algebra Applications......Page 273
14.4 Exercises......Page 274
15.1 Definitions......Page 275
15.2 Examples......Page 277
15.3 Computer Algebra Applications......Page 281
15.4 Exercises......Page 282
16.1 Definitions......Page 283
16.2 Examples......Page 285
16.3 Sato\'s Theory......Page 289
16.4 Computer Algebra Applications......Page 292
16.5 Exercises......Page 293
17.1 Basic Concepts......Page 295
17.2 Exterior Differential Systems......Page 301
17.3 Cartan Fundamental Form......Page 307
17.4 String Theory and Invariants......Page 320
17.5 Computer Algebra Applications......Page 323
17.6 Exercises......Page 326
18.1 Introduction......Page 327
18.2 Ordinary Differential Equations......Page 328
18.3 Invertible Point Transformation......Page 333
18.4 Partial Differential Equations......Page 338
18.5 Symmetries by Truncated Expansions......Page 341
18.6 Painleve Test and Recursion Operators......Page 344
18.7 Singular Manifold and Similarity Variables......Page 346
18.8.1 Hirota Technique......Page 350
18.8.2 Connection with Painleve Test......Page 356
18.9 Integrability Concepts......Page 365
18.10 Computer Algebra Applications......Page 366
18.11 Exercises......Page 367
19.1 Introduction......Page 369
19.2 Ziglin\'s Theorems......Page 370
19.3 Applications......Page 374
19.4 Computer Algebra Application......Page 380
19.5 Exercises......Page 381
20.1 Introduction......Page 383
20.2 Covariant Exterior Derivative......Page 386
20.3 Yang-Mills Equations......Page 387
20.4 Maurer-Cartan Equation......Page 391
20.5 Application......Page 392
20.6 Yang-Mills Equation and Chaos......Page 394
20.7 Self-Dual Yang-Mills Equations......Page 396
20.8 Anti-Self Dual Yang-Mills Equation......Page 400
20.9 Geometry of SU(n)......Page 402
20.10 Computer Algebra Applications......Page 405
20.11 Exercises......Page 407
21.1 Embedding and Bose Operators......Page 409
21.2 Examples......Page 412
21.3 Embedding and Bose Field Operators......Page 414
21.4 Bose Operators and Maps......Page 417
21.5 Computer Algebra Applications......Page 420
21.6 Exercises......Page 421
22.1 Discrete Dynamical Systems......Page 423
22.2 Logistic Map......Page 425
22.3 Discrete Painleve Equations......Page 428
22.4 Computer Algebra Applications......Page 429
22.5 Exercises......Page 430
23.1 Computer Algebra Packages......Page 431
23.2 Programs for Lie Symmetries......Page 432
23.3.1 Expression Tree......Page 436
23.3.2 Examples......Page 439
23.4 Examples......Page 444
A.1 Definitions......Page 449
A.2 Examples of Differentiable Manifolds......Page 451
A.3 Exercises......Page 457
Bibliography......Page 459
Index......Page 469