Completely integrable Hamiltonian systems

This article is dedicated to one of the greatest mathematicians of our time: V.I. Arnold, who died suddenly Thursday, June 3, 2010 in France. Integrable hamiltonian systems are nonlinear ordinary differential equations described by a hamiltonian function and possessing sufficiently many independent constants of motion in involution. 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 purely differential geometric fact: a compact and connected n-dimensional manifold on which there exist n vector fields which commute and are independent at every point is diffeomorphic to an n-dimensional real torus and each vector field will define a linear flow there. We make a careful study of the connection with the concept of completely integrable systems and we apply the methods to several problems.