Effective Hamiltonians and Averaging for Hamiltonian Dynamics I

Abstract:

This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE (partial differential equation) methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell” or “corrector” PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study of periodic homogenization for Hamilton-Jacobi equations, formally induces a canonical change of variables, in terms of which the dynamics are trivial. We investigate to what extent this observation can be made rigorous in the case that the Hamiltonian is strictly convex in the momenta, given that the relevant PDE does not usually in fact admit a smooth solution.