Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system

Abstract

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincar\'e second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu\to 0$, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system.