Using the theory of J. N. Mather [Math. Z. 207, No. 2, 169–207 (1991; Zbl 0716.58020)] the author describes some dynamical properties of trajectories of the Euler-Lagrange flow associated to the Lagrangian \(L\) which are contained in the support of a probability \(\mu\) which have a given rotation vector and minimizes the action function \(A_L(\mu)=\int L\,d\mu\). It is shown that the minimal action function \(\beta(h)=\min\{\int L\,d\mu:\rho(\mu)=h\}\), has radial derivative, (where \(\rho(\mu)\in H_1(M,\mathbb R)\) is the homology vector of the invariant probability \(\mu\)) and that there exists a one-to-one correspondence between minimizing measures for Lagrangians associated to mechanical systems and minimizing measures for the geodesic problem on a fixed energy level.