eBook ISBN:  9781470400569 
Product Code:  MEMO/100/479.E 
List Price:  $31.00 
MAA Member Price:  $27.90 
AMS Member Price:  $18.60 
eBook ISBN:  9781470400569 
Product Code:  MEMO/100/479.E 
List Price:  $31.00 
MAA Member Price:  $27.90 
AMS Member Price:  $18.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 100; 1992; 101 ppMSC: Primary 15; 34; 70
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 systems—such as the Neumann system, periodic Toda, geodesic flow on an ellipsoid, etc.—can also be analyzed by this method.
ReadershipGraduate students and researchers in integrable systems.

Table of Contents

Chapters

1. Introduction

2. The discrete EulerArnold equation (I)

3. The discrete EulerArnold equation (II)

4. Billiards in an elliptical region

5. Loop groups and rank 2 extensions

Appendix. Classical $R$matrix theory


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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 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 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.

Chapters

1. Introduction

2. The discrete EulerArnold equation (I)

3. The discrete EulerArnold equation (II)

4. Billiards in an elliptical region

5. Loop groups and rank 2 extensions

Appendix. Classical $R$matrix theory