  We study the time-periodic version of Evans approach to weak KAM theory.
Evans minimization problem is equivalent to a first oder mean field game
system. For the mechanical Hamiltonian we prove the existence of smooth
solutions. We introduce the corresponding effective Lagrangian and Hamiltonian
and prove that they are smooth. We also consider the limiting behavior of the
effective Lagrangian and Hamiltonian, Mather measures and minimizers.
