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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book illustrates the powerful interplay between topological, algebraic and complex analytical methods, within the field of integrable systems, by addressing several theoretical and practical aspects. Contemporary integrability results, discovered in the last few decades, are used within different areas of mathematics and physics. Integrable Systems incorporates numerous concrete examples and exercises, and covers a wealth of essential material, using a concise yet instructive approach. This book is intended for a broad audience, ranging from mathematicians and physicists to students pursuing graduate, Masters or further degrees in mathematics and mathematical physics. It also serves as an excellent guide to more advanced and detailed reading in this fundamental area of both classical and contemporary mathematics.
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Seitenzahl: 550
Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Dedication
Title Page
Copyright
Preface
1 Symplectic Manifolds
1.1. Introduction
1.2. Symplectic vector spaces
1.3. Symplectic manifolds
1.4. Vectors fields and flows
1.5. The Darboux theorem
1.6. Poisson brackets and Hamiltonian systems
1.7. Examples
1.8. Coadjoint orbits and their symplectic structures
1.9. Application to the group
SO
(
n
)
1.10. Exercises
2 Hamilton–Jacobi Theory
2.1. Euler–Lagrange equation
2.2. Legendre transformation
2.3. Hamilton’s canonical equations
2.4. Canonical transformations
2.5. Hamilton–Jacobi equation
2.6. Applications
2.7. Exercises
3 Integrable Systems
3.1. Hamiltonian systems and Arnold–Liouville theorem
3.2. Rotation of a rigid body about a fixed point
3.3. Motion of a solid through ideal fluid
3.4. Yang–Mills field with gauge group
SU
(2)
3.5. Appendix (geodesic flow and Euler–Arnold equations)
3.6. Exercises
4 Spectral Methods for Solving Integrable Systems
4.1. Lax equations and spectral curves
4.2. Integrable systems and Kac–Moody Lie algebras
4.3. Geodesic flow on
SO
(
n
)
4.4. The Euler problem of a rigid body
4.5. The Manakov geodesic flow on the group
SO
(4)
4.6. Jacobi geodesic flow on an ellipsoid and Neumann problem
4.7. The Lagrange top
4.8. Quartic potential, Garnier system
4.9. The coupled nonlinear Schrödinger equations
4.10. The Yang–Mills equations
4.11. The Kowalewski top
4.12. The Goryachev–Chaplygin top
4.13. Periodic infinite band matrix
4.14. Exercises
5 The Spectrum of Jacobi Matrices and Algebraic Curves
5.1. Jacobi matrices and algebraic curves
5.2. Difference operators
5.3. Continued fraction, orthogonal polynomials and Abelian integrals
5.4. Exercises
6 Griffiths Linearization Flows on Jacobians
6.1. Spectral curves
6.2. Cohomological deformation theory
6.3. Mittag–Leffler problem
6.4. Linearizing flows
6.5. The Toda lattice
6.6. The Lagrange top
6.7. Nahm’s equations
6.8. The
n
-dimensional rigid body
6.9. Exercises
7 Algebraically Integrable Systems
7.1. Meromorphic solutions
7.2. Algebraic complete integrability
7.3. The Liouville–Arnold–Adler–van Moerbeke theorem
7.4. The Euler problem of a rigid body
7.5. The Kowalewski top
7.6. The Hénon–Heiles system
7.7. The Manakov geodesic flow on the group
SO
(4)
7.8. Geodesic flow on
SO
(4) with a quartic invariant
7.9. The geodesic flow on
SO
(
n
) for a left invariant metric
7.10. The periodic five-particle Kac–van Moerbeke lattice
7.11. Generalized periodic Toda systems
7.12. The Gross–Neveu system
7.13. The Kolossof potential
7.14. Exercises
8 Generalized Algebraic Completely Integrable Systems
8.1. Generalities
8.2. The RDG potential and a five-dimensional system
8.3. The Hénon–Heiles problem and a five-dimensional system
8.4. The Goryachev–Chaplygin top and a seven-dimensional system
8.5. The Lagrange top
8.6. Exercises
9 The Korteweg–de Vries Equation
9.1. Historical aspects and introduction
9.2. Stationary Schrödinger and integral Gelfand–Levitan equations
9.3. The inverse scattering method
9.4. Exercises
10 KP–KdV Hierarchy and Pseudo-differential Operators
10.1. Pseudo-differential operators and symplectic structures
10.2. KdV equation, Heisenberg and Virasoro algebras
10.3. KP hierarchy and vertex operators
10.4. Exercises
References
Index
End User License Agreement
Guide
Cover
Table of Contents
Dedication
Title Page
Copyright
Preface
Begin Reading
References
Index
End User License Agreement
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To my Professor Pierre van Moerbeke
Integrable Systems
Ahmed Lesfari
First published 2022 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
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UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2022
The rights of Ahmed Lesfari to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2022932445
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-827-6
Preface
This book is intended for a wide readership of mathematicians and physicists: students pursuing graduate, masters and higher degrees in mathematics and mathematical physics. It is devoted to some geometric and topological aspects of the theory of integrable systems and the presentation is clear and well-organized, with many examples and problems provided throughout the text. Integrable Hamiltonian systems are nonlinear ordinary differential equations that are described by a Hamiltonian function and possess sufficiently many independent constants of motion in involution. The problem of finding and integrating Hamiltonian systems has attracted a considerable amount of attention in recent decades. Besides the fact that many integrable systems have been the subject of powerful and beautiful theories of mathematics, another motivation for their study is the concepts of integrability that are applied to an increasing number of physical systems, biological phenomena, population dynamics and chemical rate equations, to mention but a few applications. However, it still seems hopeless to describe, or even to recognize with any facility, the Hamiltonian systems which are integrable, even though they are exceptional.
Chapter 1 is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some interesting properties of one-parameter groups of diffeomorphisms or of flow, Lie derivative, interior product or Cartan’s formula, as well as the study of a central theorem of symplectic geometry, namely, Darboux’s theorem. We also show how to determine explicitly symplectic structures on adjoint and coadjoint orbits of a Lie group, with particular attention given to the group SO(n).
Chapter 2 deals with the study of some notions concerning the Hamilton–Jacobi theory in the calculus of variations. We will establish the Euler–Lagrange differential equations, Hamilton’s canonical equations and the Hamilton–Jacobi partial differential equation and explain how it is widely used in practice to solve some problems. As an application, we will study the geodesics, the harmonic oscillator, the Kepler problem and the simple pendulum.
In Chapter 3, we study the Arnold–Liouville theorem: the regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following differential geometric fact; a compact and connected n-dimensional manifold on which there exist n vector fields that commute and are independent at every point is diffeomorphic to an n-dimensional real torus, and there is a transformation to so-called action-angle variables, mapping the flow into a straight line motion on that torus. We give a proof as direct as possible of the Arnold–Liouville theorem and we make a careful study of its connection with the concept of completely integrable systems. Many problems are studied in detail: the rotation of a rigid body about a fixed point, the motion of a solid in an ideal fluid and the Yang–Mills field with gauge group SU(2).
In Chapter 4, we give a detailed study of the integrable systems that can be written as Lax equations with a spectral parameter. Such equations have no a priori Hamiltonian content. However, through the Adler–Kostant–Symes (AKS) construction, we can produce Hamiltonian systems on coadjoint orbits in the dual space to a Lie algebra whose equations of motion take the Lax form. We outline an algebraic-geometric interpretation of the flows of these systems, which are shown to describe linear motion on a complex torus. The relationship between spectral theory and completely integrable systems is a fundamental aspect of the modern theory of integrable systems. This chapter surveys a number of classical and recent results and our purpose here is to sketch a motivated overview of this interesting subject. We present a Lie algebra theoretical schema leading to integrable systems based on the Kostant–Kirillov coadjoint action. Many problems on Kostant–Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac–Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the AKS theorem, and the van Moerbeke–Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety (or some subabelian variety of it) of the spectral curve. The complex flows generated by the constants of the motion are straight line motions on these varieties. This chapter describes a version of the general scheme, and shows in detail how several important classes of examples fit into the general framework. Several examples of integrable systems of relevance in mathematical physics are carefully discussed: geodesic flow on SO(n), the Euler problem of a rigid body, Manakov geodesic flow on the group SO(4), Jacobi geodesic flow on an ellipsoid, the Neumann problem, the Lagrange top, a quartic potential or Garnier system, coupled nonlinear Schrödinger equations, Yang–Mills equations, the Kowalewski spinning top, the Goryachev–Chaplygin top and the periodic infinite band matrix.
The aim of Chapter 5 is to describe some connections between spectral theory in infinite dimensional Lie algebras, deformation theory and algebraic curves. We study infinite continued fractions, isospectral deformation of periodic Jacobi matrices, general difference operators, Cauchy–Stieltjes transforms and Abelian integrals from an algebraic geometrical point of view. These results can be used to obtain insight into integrable systems.
In Chapter 6, we present in detail the Griffiths’ approach and his cohomological interpretation of the linearization test for solving integrable systems without reference to Kac–Moody algebras. His method is based on the observation that the tangent space to any deformation lies in a suitable cohomology group and on algebraic curves, higher cohomology can always be eliminated using duality theory. We explain how results from deformation theory and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. These conditions are cohomological and the Lax equations turn out to have a natural cohomological interpretation. Several nonlinear problems in mathematical physics illustrate these results: the Toda lattice, Nahm’s equations and the n-dimensional rigid body.
In Chapter 7, the notion of algebraically completely integrable Hamiltonian systems in the Adler–van Moerbeke sense is explained, and techniques to find and solve such systems are presented. These are integrable systems whose trajectories are straight line motions on Abelian varieties (complex algebraic tori). We make, via the Kowalewski–Painlevé analysis, a study of the level manifolds of the systems, which are described explicitly as being affine part of Abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of Abelian integrals. We describe an explicit embedding of these Abelian varieties that complete the generic invariant surfaces into projective spaces. Many problems are studied in detail: the Euler problem of a rigid body, the Kowalewski top, the Hénon–Heiles system, Manakov geodesic flow on the group SO(4), geodesic flow on SO(4) with a quartic invariant, geodesic flow on SO(n) for a left invariant metric, the periodic five-particle Kac–van Moerbeke lattice, generalized periodic Toda systems, the Gross–Neveu system and the Kolossof potential.
In Chapter 8, we discuss the study of generalized algebraic completely integrable systems. There are many examples of differential equations that have the weak Painlevé property that all movable singularities of the general solution have only a finite number of branches, and some interesting integrable systems appear as coverings of algebraic completely integrable systems. The invariant varieties are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. These systems are Liouville integrable and by the Arnold–Liouville theorem, the compact connected manifolds invariant by the real flows are tori, the real parts of complex affine coverings of Abelian varieties. Most of these systems of differential equations possess solutions that are Laurent series of t1/n (t being complex time) and whose coefficients depend rationally on certain algebraic parameters. We discuss some interesting examples: Ramani–Dorizzi– Grammaticos (RDG) potential, the Hénon–Heiles system, the Goryachev–Chaplygin top, a seven-dimensional system and the Lagrange top.
Chapter 9 covers the stationary Schrödinger equation, the integral Gelfand–Levitan equation and the inverse scattering method used to solve exactly the Korteweg–de Vries (KdV) equation. The latter is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena.
In Chapter 10, we study some generalities on the algebra of infinite order differential operators. The algebras of Virasoro, Heisenberg and nonlinear evolution equations such as the KdV, Boussinesq and Kadomtsev–Petviashvili (KP) equations play a crucial role in this study. We make a careful study of some connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato–Date–Jimbo–Miwa–Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations and the Gelfand–Dickey flows, the Heisenberg and Virasoro algebras are given. The study of the KP and KdV hierarchies, the use of tau functions related to infinite dimensional Grassmannians, Fay identities, vertex operators and the Hirota’s bilinear formalism led to obtaining remarkable properties concerning these algebras such as, for example, the existence of an infinite family of first integrals functionally independent and in involution.
It is well known that when studying integrable systems, elliptic functions and integrals, compact Riemann surfaces or algebraic curves, Abelian surfaces (as well as the basic techniques to study two-dimensional algebraic completely integrable systems) play a crucial role. These facts, which may be well known to the algebraic reader, can be found, for example, in Adler and van Moerbeke (2004); Fay (1973); Griffiths and Harris (1978); Lesfari (2015b) and Vanhaecke (2001).
I would like to thank and am grateful to P. van Moerbeke and L. Haine, from whom I learned much of this subject through conversations and remarks. I would also like to thank the editors for their interest, seriousness and professionalism. Finally my thanks go to my wife and our children for much encouragement and undeniable support, who helped bring this book into being.
Ahmed LESFARI
September 2021
