Some remarks on symmetric periodic orbits in the restricted three-body problem

TL;DR

This paper explores symmetric periodic orbits in the restricted three-body problem by computing Lagrangian Rabinowitz Floer homology, revealing new insights into the structure and existence of symmetric orbits.

Contribution

It introduces the computation of Lagrangian Rabinowitz Floer homology for the regularized PCRTBP and links it to symmetric periodic orbits, advancing understanding of symmetric dynamics.

Findings

Homology computation related to symmetric periodic orbits

Connection between homology and symmetric orbit existence

Insights into the structure of energy hypersurfaces

Abstract

The planar circular restricted three body problem (PCRTBP) is symmetric with respect to the line of masses and there is a corresponding anti-symplectic involution on the cotangent bundle of the 2-sphere in the regularized PCRTBP. Recently it was shown that each bounded component of an energy hypersurface with low energy for the regularized PCRTBP is fiberwise starshaped. This enable us to define a Lagrangian Rabinowitz Floer homology which is related to periodic orbits symmetric for the anti-symplectic involution in the regularized PCRTBP and hence to symmetric periodic orbits in the unregularized problem. In this paper we compute of this homology and discuss about symmetric periodic orbits.