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 Poincaré 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.

References in corpus (2)

• Normally hyperbolic invariant manifolds near strong double resonance
• Oscillatory singularities in Bianchi models with magnetic fields

Cited by in corpus (1)

• Degenerate billiards in celestial mechanics