Degenerate billiards in celestial mechanics

TL;DR

This paper studies degenerate billiards in celestial mechanics, proving the existence of trajectories that shadow degenerate billiard paths, which are limits of systems with singularities, motivated by second species solutions of Poincaré.

Contribution

It establishes the existence of shadowing trajectories in degenerate billiard systems arising from celestial mechanics singularities.

Findings

Existence of trajectories shadowing degenerate billiards.

Degenerate billiards as limits of singular celestial systems.

Application to second species solutions of Poincaré.

Abstract

In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincar\'e.