From Brake to Syzygy

TL;DR

This paper investigates brake orbits in the planar three-body problem, analyzing the flow-induced Poincaré map to understand syzygy configurations, and provides variational characterizations and examples of periodic brake orbits.

Contribution

It introduces a continuous Poincaré map for brake orbits, extends it to collision orbits, and offers variational characterizations and explicit examples of periodic solutions.

Findings

Defined a flow-induced Poincaré map for brake orbits

Extended the map to Lagrange triple collision orbits

Identified simple periodic brake orbits

Abstract

In the planar three-body problem, we study solutions with zero initial velocity (brake orbits). Following such a solution until the three masses become collinear (syzygy), we obtain a continuous, flow-induced Poincar\'e map. We study the image of the map in the set of collinear configurations and define a continuous extension to the Lagrange triple collision orbit. In addition we provide a variational characterization of some of the resulting brake-to-syzygy orbits and find simple examples of periodic brake orbits.