Contact Geometry of the Restricted Three-Body Problem on $\mathbb{S}^2$

TL;DR

This paper investigates the contact geometry of energy hypersurfaces in the symmetric restricted three-body problem on the sphere, revealing their contact type and contactomorphic structures, and establishing the existence of periodic orbits.

Contribution

It characterizes the contact structures of energy hypersurfaces in a spherical three-body problem and applies Taubes' Weinstein conjecture to prove periodic orbit existence.

Findings

Energy hypersurfaces are of contact type below and slightly above the first critical value.

These hypersurfaces are contactomorphic to real projective 3-space or its connected sum.

Periodic orbits exist in all considered energy regimes.

Abstract

We study the contact geometry of the connected components of the energy hypersurface, in the symmetric restricted 3-body problem on $\mathbb{S}^2$, for a specific type of motion of the primaries. In particular, we show that these components are of contact type for all energies below the first critical value and slightly above it. We prove that these components, suitably compactified using a Moser-type regularization are contactomorphic to $R\mathbb{P}^3$ with its unique tight contact structure or to the connected sum of two copies of it, depending on the value of the energy. We exploit Taubes' solution of the Weinstein conjecture in dimension three, to infer the existence of periodic orbits in all these cases.