On the stability problem for nearly{integrable Hamiltonian systems

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 diiusion") 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. 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 diiusion") 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 split