Applications of Lie Groups to Differential Equations (Google eBook)

Front Cover
Springer Science & Business Media, Feb 1, 2000 - Language Arts & Disciplines - 513 pages
2 Reviews
Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented so that graduate students and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.
  

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Contents

Introduction to Lie Groups
1
11 Manifolds
2
Change of Coordinates
6
Maps Between Manifolds
7
Submanifolds
8
Regular Submanifolds
11
Curves and Connectedness
12
12 Lie Groups
13
The Variational Derivative
244
Null Lagrangians and Divergences
247
Invariance of the Euler Operator
249
42 Variational Symmetries
252
Infinitesimal Criterion of Invariance
253
Symmetries of the EulerLagrange Equations
255
Reduction of Order
257
43 Conservation Laws
261

Lie Subgroups
17
Local Lie Groups
18
Local Transformation Groups
20
Orbits
22
13 Vector Fields
24
Flows
27
Action on Functions
30
Differentials
32
Lie Brackets
33
Tangent Spaces and Vector Fields on Submanifolds
37
Frobenius Theorem
38
14 Lie Algebras
42
OneParameter Subgroups
44
Subalgebras
46
The Exponential Map
48
Structure Constants
50
Infinitesimal Group Actions
51
15 Differential Forms
53
PullBack and Change of Coordinates
56
The Differential
57
The de Rham Complex
58
Lie Derivatives
60
Homotopy Operators
63
Integration and Stokes Theorem
65
Notes
67
Exercise
69
Symmetry Groups of Differential Equations
75
21 Symmetries of Algebraic Equations
76
Invariant Functions
77
Infinitesimal Invariance
79
Local Invariance
83
Invariants and Functional Dependence
84
Methods for Constructing Invariants
87
22 Groups and Differential Equations
90
23 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 Theorem
110
Properties of Prolonged Vector Fields
115
24 Calculation of Symmetry Groups
116
25 Integration of Ordinary Differential Equations
130
First Order Equations
131
Higher Order Equations
137
Differential Invariants
139
Multiparameter Symmetry Groups
145
Solvable Groups
151
Systems of Ordinary Differential Equations
154
26 Nondegeneracy Conditions for Differential Equations
157
Invariance Criteria
161
The CauchyKovalevskaya Theorem
162
Characteristics
163
Normal Systems
166
Notes
172
Exercises
176
GroupInvariant Solutions
183
31 Construction of GroupInvariant Solutions
185
32 Examples of GroupInvariant Solutions
190
33 Classification of GroupInvariant Solutions
199
Classification of Subgroups and Subalgebras
203
Classification of GroupInvariant Solutions
207
34 Quotient Manifolds
209
Dimensional Analysis
214
35 GroupInvariant 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
Exercise
238
Symmetry Groups and Conservation Laws
242
41 The Calculus of Variations
243
Trivial Conservation Laws
264
Characteristics of Conservation Laws
266
44 Noethers Theorem
272
Divergence Symmetries
278
Notes
281
Exercise
283
Generalized Symmetries
286
51 Generalized Symmetries of Differential Equations
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
52 Recursion Operators Master Symmetries and Formal Symmetries
304
Frechet Derivatives
307
Lie Derivatives of Differential Operators
308
Criteria for Recursion Operators
310
The Kortewegde Vries Equation
312
Master Symmetries
315
Pseudodifferential Operators
318
Formal Symmetries
322
53 Generalized Symmetries and Conservation Laws
328
Characteristics of Conservation Laws
330
Variational Symmetries
331
Group Transformations
333
Noethers Theorem
334
Selfadjoint Linear Systems
336
Action of Symmetries on Conservation Laws
341
Abnormal Systems and Noethers Second Theorem
342
Formal Symmetries and Conservation Laws
346
54 The Variational Complex
350
The DComplex
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
Exercise
379
FiniteDimensional Hamiltonian Systems
389
61 Poisson Brackets
390
Hamiltonian Vector Fields
392
The Structure Functions
393
The LiePoisson Structure
396
62 Symplectic Structures and Foliations
398
Rank of a Poisson Structure
399
Symplectic Manifolds
400
Maps Between Poisson Manifolds
401
Poisson Submanifolds
402
Darboux Theorem
404
The Coadjoint Representation
406
63 Symmetries First Integrals and Reduction of Order
408
Hamiltonian Symmetry Groups
409
Reduction of Order in Hamiltonian Systems
412
Reduction Using Multiparameter Groups
416
Hamiltonian Transformation Groups
418
The Momentum Map
420
Notes
427
EXercise
428
Hamiltonian Methods for Evolution Equations
433
71 Poisson Brackets
434
The Jacobi Identity
436
Functional Multi vectors
439
72 Symmetries and Conservation Laws
446
Conservation Laws
447
73 BiHamiltonian Systems
452
Recursion Operators
458
Notes
461
Exercise
463
References
467
Symbol Index
489
Author Index
497
Subject Index
501
Copyright

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Page xviii - ... discovery that techniques designed to solve particular unrelated types of ODEs, such as separable, homogeneous and exact equations, were in fact all special cases of a general form of integration procedure based on the invariance of the differential equation under a continuous group of symmetries. Roughly speaking a symmetry group of a system of differential equations is a group that transforms solutions of the system to other solutions. Once the symmetry group has been identified a number of...
Page xix - Y* are said to be equivalent if one can be transformed into the other by a transformation of coordinates in ^n(&).

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