# Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions

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*Percy Alec Deift; Carlos Tomei; Luen-Chau Li*

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.

#### Table of Contents

# Table of Contents

## Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions

- Table of Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. The discrete Euler-Arnold equation (I) 1221 free
- Chapter 3. The discrete Euler-Arnold equation (II) 4251
- Chapter 4. Billiards in an elliptical region 5968
- Chapter 5. Loop groups and rank 2 extensions 8392
- Appendix. Classical R-matrix theory 93102
- Bibliography 99108