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

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.