On the Stability Problem for Nearly-Integrable Hamiltonian Systems

Abstract

The problem of stability of the action variables in nearly-integrable (real-analytic) Hamiltonian systems is considered. Several results (fully described in [CG2]) are discussed; in particular: (i) a generalization of Arnold’s method ([A]) allowing to prove instability (i.e. drift of action variables by an amount of order 1, often called “Arnold’s diffusion”) for general perturbations of “a-priori unstable integrable systems” (i.e. systems for which the integrable structure carries separatrices); (ii) Examples of perturbations of “a-priori stable sytems” (i.e. systems whose integrable part can be completely described by regular action-angle variables) exhibiting instability. In such examples, inspired by the “D’Alembert problem” in Celestial Mechanics (treated, in full details, in [CG2]), the splitting of the asymptotic manifolds is not exponentially small in the perturbation parameter.