ON THE STABILITY OF THE PERTURBED CENTRAL MOTION PROBLEM: A QUASICONVEXITY AND A NEKHOROSHEV TYPE RESULT

The aim of this thesis is the study of the dynamics of small perturbations of the spatial central motion problem. Our main result consists in proving that if the central potential is analytic, then, except for the Harmonic and the Keplerian case, the unperturbed system written in action angle variables is quasiconvex. Thus, when it is perturbed, one can apply a Nekhoroshev type theorem ensuring the stability over exponentially long times of the modulus of the angular momentum and of the energy of the unperturbed system. Being a \emph{superintegrable} system, namely, a system which admits a number of independent integrals of motion larger than the number of degrees of freedom, the version of Nekhoroshev theorem provided here is the one for superintegrable systems. We also give a complete proof (à la Lochak) of this result.