Integrable Systems

7Algebraically Integrable Systems

This chapter presents an excellent introduction to the problems, techniques and results of algebraic complete integrability. We will mainly focus on algebraic integrability in the sense of Adler–van Moerbeke, where the fibers of the momentum map are affine parts of complex algebraic tori (Abelian varieties). It is well known that most of the problems of classical mechanics are of this form. Many important problems will be studied: Euler and Kowalewski tops, the Hénon–Heiles system, geodesic flow on SO(n), the Kac–van Moerbeke lattice, generalized periodic Toda systems, the Gross–Neveu system, the Kolossof potential, as well as other systems.

7.1. Meromorphic solutions

Consider the system of nonlinear differential equations

[7.1]

where f₁, ..., f_n are functions of n + 1 complex variables t, z₁, ..., z_n and which apply a domain of ℂⁿ⁺¹ into ℂ. The Cauchy problem is the search for a solution (z₁(t), ..., z_n(t)) in a neighborhood of a point t₀, passing through the given point , that is, satisfying the initial conditions . The system [7.1] can be written in a vector form in , by putting z = (z₁, ..., z_n) and f = (f₁, ..., f_n). In this case, the Cauchy ...