  We investigate the dynamics of quasi-periodic Hamiltonian systems from the
perspective of weak KAM theory. We obtain that the limit $u$, which obtained
from convergence of a sequence of functional minimizers, satisfies
Hamilton-Jacobi equations in a weak sense. This is the so-called weak KAM
solutions.
  Meanwhile, we also get a minimal measures $\mu$.Finally, we discuss the
existence and uniqueness of smooth solutions to a kind of divergence equation
by the continuation method, which is crucial to our approach.
