  We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian
systems with a (trigonometric polynomial) $O(\mu)$-perturbation which does not
preserve the unperturbed tori. We prove the existence of Arnold diffusion with
diffusion time $ T_d = O((1/ \mu) \log (1/ \mu))$ by a variational method which
does not require the existence of ``transition chains of tori'' provided by KAM
theory. We also prove that our estimate of the diffusion time $T_d $ is optimal
as a consequence of a general stability result derived from classical
perturbation theory.
