Action minimizing methods in Hamiltonian dynamics : an introduction to Aubry Mather theory Tonelli Lagrangians and Hamitonians on compact manifoldsFrom KAM theory to Aubry Mather theory Action minimizing invariant measures for Tonelli Lagrangians Action minimizing curves for Tonelli Lagragians The Hamilton Jacobi equation and weak KAM theory Appendices On the existence of invariant Lagrangian graphs Schwartzman asymptotic cycle and dynamics.