On averaging of motion in two-scale Hamiltonian systems with locally periodic potentials

We review some recent work on averaging of motion in periodic potentials perturbed by slowly varying ones. We show that for a "critical" scaling, provided the trajectories are local minimizers of the action at an appropriate "mesoscale", the solutions converge to those described by the effective Lagrangians and effective Hamiltonians. We discuss a relations of the microscopic part to "temperature", and some other related issues.