Arnold's diffusion and formally gradient dynamics of the action functional

In a celebrated 1964 paper Arnold constructed an example of a
Hamiltonian system arbitrarily close to integrable, but with a "wandering"
orbit visiting different regions of the phase space. Arnold's conjecture
that this is typical (generic) for Hamiltonian systems is, despite many
partial results and announcements, still open. For example, even for the
classical Arnold's example, published results seem to cover only the case of
very small parameter values. We show existence of Arnold's diffusion in the
Arnold's example for all parameter values by analyzing formally gradient
dynamics of the action functional as an extended gradient system.