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

Weak KAM Theory in Quasi-periodic Hamiltonian Systems