“…Mather [51] has developed a theory in any dimension, closely related to the present article, of minimizing measures for a Tonelli's Lagrangian, that is, 1-periodic superlinear strictly convex Lagrangian L : T M × T 1 → R. In dimension 1, Moser [52] has proved the equivalence between two approaches, monotone twist map versus Tonelli's Lagrangian, showing that such a monotone twist map can always be seen as the time one map of some 1-periodic smooth Hamiltonian H : T * M ×T 1 → R. As noticed by Herman [38], there are interesting connections between configurations with minimal energy and Lagrangian tori invariant under symplectic diffeormorphisms of the cotangent bundle of the d-dimensional torus. Massart has generalized several results of the Aubry-Mather theory in [45,46,47].…”