Stability of motions near resonances in quasi-integrable Hamiltonian systems

Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.Partially supported by Ministere della Pubblica Istruzione.Partially supported by Grant N.S.F. DMS 85-03333 and by Ministero della Pubblica Istruzione.