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Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions
 
Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions
eBook ISBN:  978-1-4704-0056-9
Product Code:  MEMO/100/479.E
List Price: $31.00
MAA Member Price: $27.90
AMS Member Price: $18.60
Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions
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Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions
eBook ISBN:  978-1-4704-0056-9
Product Code:  MEMO/100/479.E
List Price: $31.00
MAA Member Price: $27.90
AMS Member Price: $18.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1001992; 101 pp
    MSC: 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 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.

    Readership

    Graduate students and researchers in integrable systems.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. The discrete Euler-Arnold equation (I)
    • 3. The discrete Euler-Arnold equation (II)
    • 4. Billiards in an elliptical region
    • 5. Loop groups and rank 2 extensions
    • Appendix. Classical $R$-matrix theory
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1001992; 101 pp
MSC: 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 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.

Readership

Graduate students and researchers in integrable systems.

  • Chapters
  • 1. Introduction
  • 2. The discrete Euler-Arnold equation (I)
  • 3. The discrete Euler-Arnold equation (II)
  • 4. Billiards in an elliptical region
  • 5. Loop groups and rank 2 extensions
  • Appendix. Classical $R$-matrix theory
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.