Applications of Lie groups to differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1993
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
107 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVIII, 513 S. graph. Darst. |
| ISBN: | 0387940073 3540940073 |
Staff View
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| 100 | 1 | |a Olver, Peter J. |d 1952- |e Verfasser |0 (DE-588)11113501X |4 aut | |
| 245 | 1 | 0 | |a Applications of Lie groups to differential equations |c Peter J. Olver |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1993 | |
| 300 | |a XXVIII, 513 S. |b graph. Darst. | ||
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| 650 | 4 | |a Lie, Groupes de | |
| 650 | 7 | |a Lie-groepen |2 gtt | |
| 650 | 4 | |a Équations différentielles | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Lie groups | |
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Record in the Search Index
| _version_ | 1823026246793035776 |
|---|---|
| adam_text |
PETER
J.
OLVER
APPLICATIONS OF
LIE
GROUPS TO
DIFFERENTIAL
EQUATIONS
SECOND
EDITION
SPRINGER-VERLAG
NEW
YORK BERLIN HEIDELBERG LONDON PARIS
TOKYO
HONG KONG BARCELONA BUDAPEST
TABLE OF CONTENTS
PREFACE TO FIRST EDITION V
PREFACE TO SECOND EDITION VII
ACKNOWLEDGMENTS IX
INTRODUCTION XVII
NOTES TO THE READER XXV
CHAPTER 1
INTRODUCTION TO LIE GROUPS 1
1.1. MANIFOLDS 2
CHANGE OF COORDINATES 6
MAPS BETWEEN MANIFOLDS 7
THE MAXIMAL RANK CONDITION 7
SUBMANIFOLDS 8
REGULAER 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
XLL
1.4. LIE ALGEBRAS
ONE-PARAMETER
SUBGROUPS
SUBALGEBRAS
THE
EXPONENTIAL MAP
LIE
ALGEBRAS OF LOCAL LIE GROUPS
STRUCTURE
CONSTANTS
COMMUTATOR
TABLES
INFINITESIMAL
GROUP ACTIONS
1.5.
DIFFERENTIAL FORMS
PULL-BACK AND CHANGE OF COORDINATES
INTERIOR
PRODUCTS
THE
DIFFERENTIAL
THE DE RHAM COMPLEX
LIE
DERIVATIVES
HOMOTOPY
OPERATORS
INTEGRATION
AND STOKES' THEOREM
NOTES
EXERCISES
CHAPTER 2
SYMMETRY
GROUPS OF DIFFERENTIAL EQUATIONS
2.1.
SYMMETRIES OF ALGEBRAIC EQUATIONS
INVARIANT
SUBSETS
INVARIANT
FUNCTIONS
INFINITESIMAL
INVARIANCE
LOCAL
INVARIANCE
INVARIANTS
AND FUNCTIONAL DEPENDENCE
METHODS FOR CONSTRUCTING INVARIANTS
2.2.
GROUPS AND DIFFERENTIAL EQUATIONS
2.3.
PROLONGATION
SYSTEMS
OF DIFFERENTIAL EQUATIONS
PROLONGATION OF GROUP ACTIONS
INVARIANCE
OF DIFFERENTIAL EQUATIONS
PROLONGATION
OF VECTOR FIELDS
INFINITESIMAL
INVARIANCE
THE
PROLONGATION FORMULA
TOTAL
DERIVATIVES
THE
GENERAL PROLONGATION FORMULA
PROPERTIES OF PROLONGED VECTOR FIELDS
CHARACTERISTICS
OF SYMMETRIES
2.4.
CALCULATION OF SYMMETRY GROUPS
2.5.
INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS
FIRST ORDER EQUATIONS
HIGHER
ORDER EQUATIONS
DIFFERENTIAL
INVARIANTS
MULTI-PARAMETER SYMMETRY GROUPS
SOLVABLE
GROUPS
SYSTEMS
OF ORDINARY DIFFERENTIAL EQUATIONS
TABLE OF CONTENTS
XLLL
2.6.
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
CHAPTER 3
GROUP-INVARIANT
SOLUTIONS 183
3.1.
CONSTRUCTION OF GROUP-INVARIANT SOLUTIONS 185
3.2. EXAMPLES OF GROUP-INVARIANT SOLUTIONS 190
3.3.
CLASSIFICATION OF GROUP-INVARIANT SOLUTIONS 199
THE ADJOINT REPRESENTATION 199
CLASSIFICATION OF SUBGROUPS AND SUBALGEBRAS 203
CLASSIFICATION OF GROUP-INVARIANT SOLUTIONS 207
3.4. 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
FRECHET
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 330
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 350
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
TABLE 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 MULTI-VECTORS 439
7.2. 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
AUTHORINDEX 497
SUBJECT INDEX 501 |
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| id | DE-604.BV008357517 |
| illustrated | Illustrated |
| indexdate | 2025-02-03T09:01:49Z |
| institution | BVB |
| isbn | 0387940073 3540940073 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005518647 |
| oclc_num | 722035314 |
| open_access_boolean | |
| owner | DE-384 DE-91G DE-BY-TUM DE-634 DE-83 DE-188 |
| owner_facet | DE-384 DE-91G DE-BY-TUM DE-634 DE-83 DE-188 |
| physical | XXVIII, 513 S. graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Olver, Peter J. 1952- Verfasser (DE-588)11113501X aut Applications of Lie groups to differential equations Peter J. Olver 2. ed. New York [u.a.] Springer 1993 XXVIII, 513 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 107 Differentiaalvergelijkingen gtt Equacoes diferenciais larpcal Espacios dimensionales Groupes symétriques Grupos de lie larpcal Lie, Groupes de Lie-groepen gtt Équations différentielles Differential equations Lie groups Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s Differentialgleichung (DE-588)4012249-9 s Physik (DE-588)4045956-1 s 1\p DE-604 Graduate texts in mathematics 107 (DE-604)BV000000067 107 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005518647&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Olver, Peter J. 1952- Applications of Lie groups to differential equations Graduate texts in mathematics Differentiaalvergelijkingen gtt Equacoes diferenciais larpcal Espacios dimensionales Groupes symétriques Grupos de lie larpcal Lie, Groupes de Lie-groepen gtt Équations différentielles Differential equations Lie groups Lie-Gruppe (DE-588)4035695-4 gnd Differentialgleichung (DE-588)4012249-9 gnd Physik (DE-588)4045956-1 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4012249-9 (DE-588)4045956-1 |
| title | Applications of Lie groups to differential equations |
| title_auth | Applications of Lie groups to differential equations |
| title_exact_search | Applications of Lie groups to differential equations |
| title_full | Applications of Lie groups to differential equations Peter J. Olver |
| title_fullStr | Applications of Lie groups to differential equations Peter J. Olver |
| title_full_unstemmed | Applications of Lie groups to differential equations Peter J. Olver |
| title_short | Applications of Lie groups to differential equations |
| title_sort | applications of lie groups to differential equations |
| topic | Differentiaalvergelijkingen gtt Equacoes diferenciais larpcal Espacios dimensionales Groupes symétriques Grupos de lie larpcal Lie, Groupes de Lie-groepen gtt Équations différentielles Differential equations Lie groups Lie-Gruppe (DE-588)4035695-4 gnd Differentialgleichung (DE-588)4012249-9 gnd Physik (DE-588)4045956-1 gnd |
| topic_facet | Differentiaalvergelijkingen Equacoes diferenciais Espacios dimensionales Groupes symétriques Grupos de lie Lie, Groupes de Lie-groepen Équations différentielles Differential equations Lie groups Lie-Gruppe Differentialgleichung Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005518647&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT olverpeterj applicationsofliegroupstodifferentialequations |