Invariant Sets of Hamiltonian Systems and Variational Methods

We study the problem on the existence of homoclinic trajectories to Mather minimizing invariant sets (multidimensional generalization of Aubry-Mather sets) of positive definite time-periodic Hamiltonian systems [19]. These sets are supports of invariant probability measures in the Lagrangian L. For natural systems with L = ǁυǁ2/2 — V(x), the minimizing set is Γ = {V = h}, h = max V, and for time-periodic systems with reversible L the minimizing sets consist of brake orbits of minimal action. For natural Hamiltonian systems, the existence of homoclinics to Γ was proved in [3] using the Maupertuis-Jacobi functional $$ \smallint \sqrt {h - V(x)} \parallel dx\parallel ,$$ and for reversible time-periodic systems in [4] using Hamilton’s functional (see also [5], [16]). For nonreversible systems (for example, natural systems with gyroscopic forces), in general there are no Mather sets of simple structure. We extend the above existence results to arbitrary minimizing sets replacing homoclinic trajectories by semihomoclinic ones in Birkhoff’s sense [2]. A similar problem was studied in [20].