The result of Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multi-dimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov-type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincaré-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian.

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A geometric criterion for the existence of chaotic trajectories of a reversible Hamiltonian system with the configuration space T_N, with N a compact manifold, is given. The main result is a variational version of the theorem of D. V. Turayev and L. P. Shilnikov (Dokl. Akad. Nauk SSSR 304, 1989, 811 814) on the symbolic representation of trajectories of Hamiltonian systems possessing several homoclinics to a saddle equilibrium.1998 Academic Press

We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions.

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