Memoirs of the American Mathematical Society 1992; 101 pp; softcover Volume: 100 ISBN10: 0821825402 ISBN13: 9780821825402 List Price: US$29 Individual Members: US$17.40 Institutional Members: US$23.20 Order Code: MEMO/100/479
 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 timeone maps of integrable Hamiltonian flows on coadjoint 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 EulerArnold 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 wellknown integrable systemssuch as the Neumann system, periodic Toda, geodesic flow on an ellipsoid, etc.can also be analyzed by this method. Readership Graduate students and researchers in integrable systems. Table of Contents  The discrete EulerArnold equation (I)
 The discrete EulerArnold equation (II)
 Billiards in an elliptical region
 Loop groups and rank \(2\) extensions
 Appendix: Classical \(R\)matrix theory
