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 |
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 |
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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 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.
ReadershipGraduate students and researchers in integrable systems.
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Table of Contents
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Chapters
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1. Introduction
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2. The discrete Euler-Arnold equation (I)
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3. The discrete Euler-Arnold equation (II)
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4. Billiards in an elliptical region
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5. Loop groups and rank 2 extensions
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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 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.
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