ON MINIMIZING MEASURES OF THE ACTION OF AUTONOMOUS LAGRANGIANS

We apply J Mather's theory on minimizing measures to the case of positive definite autonomous Lagrangians L : TM --> R. We show that the minimal action function beta(h) = min{integral Ld mu\rho(mu) = h}, has radial derivative, rho(mu) is an element of H-1(M,R) is the homology vector of the invariant probability mu. As one consequence, for any supporting domain S of beta the closure of the union of the support of any minimizing measures with homology in S is contained in a fixed energy level. It is also shown that there is a one-to-one correspondence between minimizing measures for Lagrangians associated to mechanical systems (kinetic minus potential energy) and minimizing measures for the geodesic problem on a fixed energy level.