Applications of Lie Groups to Differential Equations

Peter J. Olver (auth.)9780387940076, 0387940073, 3540940073

Symmetry methods have long been recognized to be of great importance for the study of the differential equations arising in mathematics, physics, engineering, and many other disciplines. The purpose of this book is to provide a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice, including determination of symmetry groups, integration of orginary differential equations, construction of group-invariant solutions to partial differential equations, symmetries and conservation laws, generalized symmetries, and symmetry methods in Hamiltonian systems. The computational methods are presented so that grauate students and researchers in other fields 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. This second edition contains a new section on formal symmetries and the calculus of pseudo-differential operators, simpler proofs of some theorems, new exercises, and a substantially updated bibiography.

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Table of contents :
Front Matter….Pages i-xxvi
Introduction to Lie Groups….Pages 1-76
Symmetry Groups of Differential Equations….Pages 77-185
Group-Invariant Solutions….Pages 186-245
Symmetry Groups and Conservation Laws….Pages 246-291
Generalized Symmetries….Pages 292-377
Finite-Dimensional Hamiltonian Systems….Pages 378-422
Hamiltonian Methods for Evolution Equations….Pages 423-474
Back Matter….Pages 457-500