Systèmes dynamiques algébriquement complètement intégrables et géométrie

In this paper I present the basic ideas and properties of the complex algebraic completely integrable dynamical systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several examples of algebraic completely integrable systems : Kowalewski’s top, geodesic flow on SO(4), Hénon-Heiles system, Garnier potential, two coupled nonlinear Schrödinger equations and Yang-Mills system.