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Given a smooth Hamiltonian H, we study the existence of measures invariant under the Hamiltonian flow, when H is non-convex. To simplify the treatment, we restrict ourselves to the periodic setting. Due to the lack of convexity, it is not possible to pass to the Lagrangian formulation, and to use the classical variational approach. Instead, we apply the "Adjoint method" recently developed by Evans. This is a joint work with Diogo Gomes (IST, Lisbon) and Hung V. Tran (University of California, Berkeley).
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