  We consider a nearly integrable, non-isochronous, a-priori unstable
Hamiltonian system 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
proved via classical perturbation theory.
