  The adjoint method introduced in [Eva] and [Tra] is used, to construct
analogs to the Aubry-Mather measures for non convex Hamiltonians. More
precisely, a general construction of probability measures, that in the convex
setting agree with Mather measures, is provided. These measures may fail to be
invariant under the Hamiltonian flow and a dissipation arises, which is
described by a positive semidefinite matrix of Borel measures. However, in the
important case of uniformly quasiconvex Hamiltonians the dissipation vanishes,
and as a consequence the invariance is guaranteed.
