KAM Theory and Celestial Mechanics

Abstract

Kolmogorov–Arnold–Moser (KAM) theory deals with the construction of quasi–periodic trajectories in nearly–integrable Hamiltonian systems and it was motivated by classical problems in Celestial Mechanics such as the n– body problem. Notwithstanding the formidable bulk of results, ideas and techniques produced by the founders of the modern theory of dynamical systems, most notably by H. Poincare and G.D. Birkhoff, the fundamental question about the persistence under small perturbations of invariant tori of an integrable Hamiltonian system remained completely open until 1954. In that year A.N Kolmogorov stated what is now usually referred to as the KAM Theorem (in the real–analytic setting) and gave a precise outline of its proof presenting a striking new and powerful method to overcome the so–called small divisor problem (resonances in Hamiltonian dynamics produce, in the