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Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions
Percy Deift, Luen-Chau Li, and Carlos Tomei

Memoirs of the American Mathematical Society
1992; 101 pp; softcover
Volume: 100
ISBN-10: 0-8218-2540-2
ISBN-13: 978-0-8218-2540-2
List Price: US$31
Individual Members: US$18.60
Institutional Members: US$24.80
Order Code: MEMO/100/479
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The theory of classical \(R\)-matrices provides a unified approach to the understanding of most, if not all, known integrable systems. This work, which is suitable as a graduate textbook in the modern theory of integrable systems, presents an exposition of \(R\)-matrix theory by means of examples, some old, some new. In particular, the authors construct continuous versions of a variety of discrete systems of the type introduced recently by Moser and Vesclov. In the framework the authors establish, these discrete systems appear as time-one maps of integrable Hamiltonian flows on co-adjoint orbits of appropriate loop groups, which are in turn constructed from more primitive loop groups by means of classical \(R\)-matrix theory. Examples include the discrete Euler-Arnold top and the billiard ball problem in an elliptical region in \(n\) dimensions. Earlier results of Moser on rank 2 extensions of a fixed matrix can be incorporated into this framework, which implies in particular that many well-known integrable systems--such as the Neumann system, periodic Toda, geodesic flow on an ellipsoid, etc.--can also be analyzed by this method.


Graduate students and researchers in integrable systems.

Table of Contents

  • The discrete Euler-Arnold equation (I)
  • The discrete Euler-Arnold equation (II)
  • Billiards in an elliptical region
  • Loop groups and rank \(2\) extensions
  • Appendix: Classical \(R\)-matrix theory
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