  Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and
super-linear). In \cite{Vi,MVZ}, Mather's $\alpha$ function is recovered from
the homogenization of symplectic capacities. This allows the authors to extend
the Mather functional to non convex cases. This article shows that the relation
between invariant measures and the subdifferential of Mather's functional
(which is the foundational statement of Mather) is preserved in the non convex
case.
  We give applications in the context of the classical KAM theory to the
existence of invariant measures with large rotation vector after the possible
disappearance of some KAM tori.
