Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system

TL;DR

This paper extends Shilnikov's Lemma to Hamiltonian systems with a nondegenerate critical manifold, demonstrating the existence of trajectories shadowing homoclinic chains and applying it to the three-body problem with small masses.

Contribution

It introduces an analog of Shilnikov Lemma for symplectic critical manifolds in Hamiltonian systems and proves the existence of shadowing trajectories for small energy levels.

Findings

Trajectories shadowing chains of homoclinic orbits are characterized as extremals.

Existence of such trajectories is rigorously proved for small energy levels.

Application to the three-body problem with small masses and collisions.

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.