Equivalence, invariants, and symmetry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1995
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 525 S. |
| ISBN: | 0521478111 |
Staff View
MARC
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| 245 | 1 | 0 | |a Equivalence, invariants, and symmetry |c Peter J. Olver |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1995 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface xi
Acknowledgments xv
Introduction 1
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
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 62
Invariant Vector Fields 65
Lie Derivatives and Invariant Differential Forms 68
The Maurer Cartan Forms 71
3. Representation Theory 75
Representations 75
Representations on Function Spaces 78
Multiplier Representations 81
Infinitesimal Multipliers 85
viii Contents
Relative Invariants 91
Classical Invariant Theory 95
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
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
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
7. Symmetries of Variational Problems 221
The Calculus of Variations 222
Equivalence of Functional 227
Invariance of the Euler Lagrange Equations 230
Symmetries of Variational Problems 235
Contents ix
Invariant Variational Problems 238
Symmetry Classification of Variational Problems 240
First Integrals 242
The Cartan Form 244
Invariant Contact Forms and Evolution Equations 246
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
9. Formulation of Equivalence Problems 280
Equivalence Problems Using Differential Forms 280
Coframes and Structure Groups 287
Normalization 291
Overdetermined Equivalence Problems 297
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
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
x Contents
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
13. Differential Systems 409
Differential Systems and Ideals 409
Equivalence of Differential Systems 415
Vector Field Systems 416
14. Probenius Theorem 421
Vector Field Systems 421
Differential Systems 427
Characteristics and Normal Forms 428
The Technique of the Graph 431
Global Equivalence 440
15. The Cartan Kahler 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
Symbol Index 490
Author Index 499
Subject Index 504
|
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| id | DE-604.BV010480590 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:53:13Z |
| institution | BVB |
| isbn | 0521478111 |
| language | English |
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| oclc_num | 31206565 |
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| physical | XVI, 525 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Olver, Peter J. 1952- Verfasser (DE-588)11113501X aut Equivalence, invariants, and symmetry Peter J. Olver 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1995 XVI, 525 S. txt rdacontent n rdamedia nc rdacarrier Géométrie différentielle Géométrie différentielle ram Invariants Invariants ram Symétrie (Physique) Symétrie (Physique) ram Geometry, Differential Symmetry (Physics) Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Invariante (DE-588)4128781-2 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Invariante (DE-588)4128781-2 s Symmetrie (DE-588)4058724-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006983921&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Olver, Peter J. 1952- Equivalence, invariants, and symmetry Géométrie différentielle Géométrie différentielle ram Invariants Invariants ram Symétrie (Physique) Symétrie (Physique) ram Geometry, Differential Symmetry (Physics) Differentialgeometrie (DE-588)4012248-7 gnd Symmetrie (DE-588)4058724-1 gnd Invariante (DE-588)4128781-2 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4058724-1 (DE-588)4128781-2 |
| title | Equivalence, invariants, and symmetry |
| title_auth | Equivalence, invariants, and symmetry |
| title_exact_search | Equivalence, invariants, and symmetry |
| title_full | Equivalence, invariants, and symmetry Peter J. Olver |
| title_fullStr | Equivalence, invariants, and symmetry Peter J. Olver |
| title_full_unstemmed | Equivalence, invariants, and symmetry Peter J. Olver |
| title_short | Equivalence, invariants, and symmetry |
| title_sort | equivalence invariants and symmetry |
| topic | Géométrie différentielle Géométrie différentielle ram Invariants Invariants ram Symétrie (Physique) Symétrie (Physique) ram Geometry, Differential Symmetry (Physics) Differentialgeometrie (DE-588)4012248-7 gnd Symmetrie (DE-588)4058724-1 gnd Invariante (DE-588)4128781-2 gnd |
| topic_facet | Géométrie différentielle Invariants Symétrie (Physique) Geometry, Differential Symmetry (Physics) Differentialgeometrie Symmetrie Invariante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006983921&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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