An important problem in mechanics is to determine physical trajectories satisfying given boundary conditions (for example trajectories which are periodic or connect two fixed point). For Hamiltonian systems these trajectories bear a special meaning: they are the stationary points of the action functional. In particular, we will concentrate on stationary points that globally minimize the action and see that they build interesting invariant sets (so-called Aubry-Mather Sets) for the dynamics. In the first part of the seminar, we will look at a toy model of this theory given by monotone twist maps, minimal geodesics on the two-torus and the Frenkel-Kontorova model following a paper by Bangert. In the second part, we will deal with the general picture and investigate its relationship with Weak KAM Theory, which studies weak solutions of the Hamilton-Jacobi equation.

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