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...