Applications of Lie Groups to Differential Equations

by Olver, Peter J.; Axler, S.; Gehring, F. W.; Ribet, K. A.

• ISBN13: 9780387950006
• ISBN10: 0387950001
• Edition: 2nd
• Format: Paperback
• Copyright: 2000-01-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Preface of First Edition	v
Preface to Second Edition	vii
Acknowledgments	ix
Introduction	xvii
Notes to the Reader	xxv
Introduction to Lie Groups	1	(74)
Manifolds	2	(11)
Change of Coordinates	6	(1)
Maps Between Manifolds	7	(1)
The Maximal Rank Condition	7	(1)
Submanifolds	8	(3)
Regular Submanifolds	11	(1)
Implicit Submanifolds	11	(1)
Curves and Connectedness	12	(1)
Lie Groups	13	(11)
Lie Subgroups	17	(1)
Local Lie Groups	18	(2)
Local Transformation Groups	20	(2)
Orbits	22	(2)
Vector Fields	24	(18)
Flows	27	(3)
Action on Functions	30	(2)
Differentials	32	(1)
Lie Brackets	33	(4)
Tangent Spaces and Vectors Fields on Submanifolds	37	(1)
Frobenius' Theorem	38	(4)
Lie Algebras	42	(11)
One-Parameter Subgroups	44	(2)
Subalgebras	46	(2)
The Exponential Map	48	(1)
Lie Algebras of Local Lie Groups	48	(2)
Structure Constants	50	(1)
Commutator Tables	50	(1)
Infinitesimal Group Actions	51	(2)
Differential Forms	53	(22)
Pull-Back and Change of Coordinates	56	(1)
Interior Products	56	(1)
The Differential	57	(1)
The de Rham Complex	58	(2)
Lie Derivatives	60	(3)
Homotopy Operators	63	(2)
Integration and Stokes' Theorem	65	(2)
Notes	67	(2)
Exercises	69	(6)
Symmetry Groups of Differential Equations	75	(108)
Symmetries of Algebraic Equations	76	(14)
Invariant Subsets	76	(1)
Invariant Functions	77	(2)
Infinitesimal Invariance	79	(4)
Local Invariance	83	(1)
Invariants and Functional Dependence	84	(3)
Methods for Constructing Invariants	87	(3)
Groups and Differential Equations	90	(4)
Prolongation	94	(22)
Systems of Differential Equations	96	(2)
Prolongation of Group Actions	98	(2)
Invariance of Differential Equations	100	(1)
Prolongation of Vector Fields	101	(2)
Infinitesimal Invariance	103	(2)
The Prolongation Formula	105	(3)
Total Derivatives	108	(2)
The General Prolongation Formula	110	(5)
Properties of Prolonged Vector Fields	115	(1)
Characteristics of Symmetries	115	(1)
Calculation of Symmetry Groups	116	(14)
Integration of Ordinary Differential Equations	130	(27)
First Order Equations	131	(6)
Higher Order Equations	137	(2)
Differential Invariants	139	(6)
Multi-parameter Symmetry Groups	145	(6)
Solvable Groups	151	(3)
Systems of Ordinary Differential Equations	154	(3)
Nondegeneracy Conditions for Differential Equations	157	(26)
Local Solvability	157	(4)
Invariance Criteria	161	(1)
The Cauchy-Kovalevskaya Theorem	162	(1)
Characteristics	163	(3)
Normal Systems	166	(1)
Prolongation of Differential Equations	166	(6)
Notes	172	(4)
Exercises	176	(7)
Group-Invariant Solutions	183	(59)
Construction of Group-Invariant Solutions	185	(5)
Examples of Group-Invariant Solutions	190	(9)
Classification of Group-Invariant Solutions	199	(10)
The Adjoint Representation	199	(4)
Classification of Subgroups and Subalgebras	203	(4)
Classification of Group-Invariant Solutions	207	(2)
Quotient Manifolds	209	(8)
Dimensional Analysis	214	(3)
Group-Invariant Prolongations and Reduction	217	(25)
Extended Jet Bundles	218	(4)
Differential Equations	222	(1)
Group Actions	223	(1)
The Invariant Jet Space	224	(1)
Connection with the Quotient Manifold	225	(2)
The Reduced Equation	227	(1)
Local Coordinates	228	(7)
Notes	235	(3)
Exercises	238	(4)
Symmetry Groups and Conservation Laws	242	(44)
The Calculus of Variations	243	(9)
The Variational Derivative	244	(3)
Null Lagrangians and Divergences	247	(2)
Invariance of the Euler Operator	249	(3)
Variational Symmetries	252	(9)
Infinitesimal Criterion of Invariance	253	(2)
Symmetries of the Euler--Lagrange Equations	255	(2)
Reduction of Order	257	(4)
Conservation Laws	261	(11)
Trivial Conservation Laws	264	(2)
Characteristics of Conservation Laws	266	(6)
Noether's Theorem	272	(14)
Divergence Symmetries	278	(3)
Notes	281	(2)
Exercises	283	(3)
Generalized Symmetries	286	(103)
Generalized Symmetries of Differential Equations	288	(16)
Differential Functions	288	(1)
Generalized Vector Fields	289	(2)
Evolutionary Vector Fields	291	(1)
Equivalence and Trivial Symmetries	292	(1)
Computation of Generalized Symmetries	293	(4)
Group Transformations	297	(3)
Symmetries and Prolongations	300	(1)
The Lie Bracket	301	(2)
Evolution Equations	303	(1)
Recursion Operators, Master Symmetries and Formal Symmetries	304	(24)
Frechet Derivatives	307	(1)
Lie Derivatives of Differential Operators	308	(2)
Criteria for Recursion Operators	310	(2)
The Korteweg-de Vries Equation	312	(3)
Master Symmetries	315	(3)
Pseudo-differential Operators	318	(4)
Formal Symmetries	322	(6)
Generalized Symmetries and Conservation Laws	328	(22)
Adjoints of Differential Operators	328	(2)
Characteristics of Conservation Laws	330	(1)
Variational Symmetries	331	(2)
Group Transformations	333	(1)
Noether's Theorem	334	(2)
Self-adjoint Linear Systems	336	(5)
Action of Symmetries on Conservation Laws	341	(1)
Abnormal Systems and Noether's Second Theorem	342	(4)
Formal Symmetries and Conservation Laws	346	(4)
The Variational Complex	350	(39)
The D-Complex	351	(2)
Vertical Forms	353	(2)
Total Derivatives of Vertical Forms	355	(1)
Functionals and Functional Forms	356	(5)
The Variational Differential	361	(4)
Higher Euler Operators	365	(3)
The Total Homotopy Operator	368	(6)
Notes	374	(5)
Exercises	379	(10)
Finite-Dimensional Hamiltonian Systems	389	(44)
Poisson Brackets	390	(8)
Hamiltonian Vector Fields	392	(1)
The Structure Functions	393	(3)
The Lie--Poisson Structure	396	(2)
Symplectic Structures and Foliations	398	(10)
The Correspondence Between One-Forms and Vector Fields	398	(1)
Rank of a Poisson Structure	399	(1)
Symplectic Manifolds	400	(1)
Maps Between Poisson Manifolds	401	(1)
Poisson Submanifolds	402	(2)
Darboux' Theorem	404	(2)
The Co-adjoint Representation	406	(2)
Symmetries, First Integrals and Reduction of Order	408	(25)
First Integrals	408	(1)
Hamiltonian Symmetry Groups	409	(3)
Reduction of Order in Hamiltonian Systems	412	(4)
Reduction Using Multi-parameter Groups	416	(2)
Hamiltonian Transformation Groups	418	(2)
The Momentum Map	420	(7)
Notes	427	(1)
Exercises	428	(5)
Hamiltonian Methods for Evolution Equations	433	(34)
Poisson Brackets	434	(12)
The Jacobi Identity	436	(3)
Functional Multi-vectors	439	(7)
Symmetries and Conservation Laws	446	(6)
Distinguished Functionals	446	(1)
Lie Brackets	446	(1)
Conservation Laws	447	(5)
Bi-Hamiltonian Systems	452	(15)
Recursion Operators	458	(3)
Notes	461	(2)
Exercises	463	(4)
References	467	(22)
Symbol Index	489	(8)
Author Index	497	(4)
Subject Index	501