Applications of Lie groups to differential equations:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York [u.a.]
Springer
2000
|
| Ausgabe: | 2. ed., 1. softcover printing |
| Schriftenreihe: | Graduate texts in mathematics
107 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016074725&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXVIII, 513 S. graph. Darst. |
Internformat
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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., 1. softcover printing | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a XXVIII, 513 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 107 | |
| 650 | 7 | |a Equations différentielles |2 ram | |
| 650 | 4 | |a Groupes symétriques | |
| 650 | 4 | |a Lie, Groupes de | |
| 650 | 7 | |a Lie, Groupes de |2 ram | |
| 650 | 4 | |a Équations différentielles | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Lie groups | |
| 650 | 0 | 7 | |a Differentialgleichung |0 (DE-588)4012249-9 |2 gnd |9 rswk-swf |
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| 689 | 0 | 2 | |a Physik |0 (DE-588)4045956-1 |D s |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016074725 | |
Datensatz im Suchindex
| _version_ | 1819327868334768128 |
|---|---|
| adam_text | 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
g
Regular Submanifolds
11
Implicit Submanifolds
11
Curves and Connectedness
12
1.2.
Lie Groups
13
Lie Subgroups
17
Locai
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
xii Table of Contents
1.4.
Lie Algebras 42
One-Parameter Subgroups ^
Subalgebras ^
The Exponential Map 48
Lie Algebras of Local Lie Groups 48
Structure Constants 50
Commutator Tables 50
Infinitesimal Group Actions 51
1.5.
Differential Forms
^3
Pull-Back and Change of Coordinates 56
Interior Products ^
The Differential 57
The
de Rham
Complex 5^
Lie Derivatives ™
Homotopy Operators
63
Integration and Stokes Theorem 6$
Notes 67
Exercises 69
CHAPTER
2
Symmetry Groups of Differential Equations
75
2.1.
Symmetries of Algebraic Equations 76
Invariant Subsets 7^
Invariant Functions 77
Infinitesimal
Invariance
Local
Invariance
^^
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
139
Multi-parameter Symmetry Groups
145
Solvable Groups
151
Systems of Ordinary Differential Equations
154
Table of
Contents xiii
2.6.
Nondegeneracy Conditions for Differential Equations
157
Local Solvability
157
In variance 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.
Noethcr 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
Frfechet 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
Author Index
497
Subject Index
501
|
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| author | Olver, Peter J. 1952- |
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| ctrlnum | (OCoLC)45110002 (DE-599)BVBBV022869595 |
| dewey-full | 515.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.35 |
| dewey-search | 515.35 |
| dewey-sort | 3515.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed., 1. softcover printing |
| format | Book |
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| id | DE-604.BV022869595 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:05:02Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016074725 |
| oclc_num | 45110002 |
| open_access_boolean | |
| owner | DE-706 DE-355 DE-BY-UBR DE-188 |
| owner_facet | DE-706 DE-355 DE-BY-UBR DE-188 |
| physical | XXVIII, 513 S. graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spellingShingle | Olver, Peter J. 1952- Applications of Lie groups to differential equations Graduate texts in mathematics Equations différentielles ram Groupes symétriques Lie, Groupes de Lie, Groupes de ram Équations différentielles Differential equations Lie groups Differentialgleichung (DE-588)4012249-9 gnd Physik (DE-588)4045956-1 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4045956-1 (DE-588)4035695-4 |
| 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 | Equations différentielles ram Groupes symétriques Lie, Groupes de Lie, Groupes de ram Équations différentielles Differential equations Lie groups Differentialgleichung (DE-588)4012249-9 gnd Physik (DE-588)4045956-1 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Equations différentielles Groupes symétriques Lie, Groupes de Équations différentielles Differential equations Lie groups Differentialgleichung Physik Lie-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016074725&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT olverpeterj applicationsofliegroupstodifferentialequations |