Consecutive collision orbits in the restricted three-body problem above the first critical energy value

TL;DR

This paper investigates the existence of symmetric collision orbits in the restricted three-body problem at energy levels just above the first critical value, using Floer homology and contact geometry.

Contribution

It establishes the existence of symmetric collision orbits and shows that, generically, there are at least two such orbits with consecutive collisions, advancing understanding of orbital dynamics.

Findings

Existence of symmetric collision orbits above the first critical energy.

No periodic symmetric collision orbit with an odd number of collisions for generic parameters.

At least two symmetric consecutive collision orbits exist under generic conditions.

Abstract

In this paper, we study the planar circular restricted three-body problem for energy levels slightly above the first critical value. We first observe that the energy hypersurfaces in the Birkhoff regularization corresponding to these energy levels are of contact type. Then, using a version of Rabinowitz Floer homology, we establish the existence of either a periodic symmetric collision orbit or infinitely many symmetric consecutive collision orbits. Furthermore, by an analytic continuation argument, for generic mass ratios and energy levels, we prove that there is no periodic symmetric collision orbit with odd number of collisions. This in turn implies the existence of at least two symmetric consecutive collision orbits.