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SMF/AMS Texts and Monographs
2008; 149 pp; softcover
List Price: US$60
Member Price: US$48
Order Code: SMFAMS/15
Differential Equations and Quantum Groups: Andrey A. Bolibrukh Memorial Volume - Daniel Bertrand, Benjamin Enriquez, Claude Mitschi, Claude Sabbah and Reinhard Schafke
Hamiltonian systems began as a mathematical approach to the study of mechanical systems. As the theory developed, it became clear that the systems that had a sufficient number of conserved quantities enjoyed certain remarkable properties. These are the completely integrable systems. In time, a rich interplay arose between integrable systems and other areas of mathematics, particularly topology, geometry, and group theory.
This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations.
Audin has included many examples and exercises. Most of the exercises build on the material in the text. None of the important proofs have been relegated to the exercises. Many of the examples are classical, rather than abstract.
This book would be suitable for a graduate course in Hamiltonian systems.
Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.
Graduate students interested in Hamiltonian and integrable systems.
From a review of the French edition:
"The book is addressed to graduate students without previous exposure to these topics ... this is a refreshing attempt at giving a bird's eye view of disparate techniques that enter the geometric/differential nature of integrability of certain Hamiltonian systems. ... The book is intended to be readable by a non-expert; ... Several examples conclude each chapter, a good feature as they are workable and instructive ..."
-- Mathematical Reviews
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