On some collinear configurations in the planar three-body problem

TL;DR

This paper investigates collinear configurations in the planar three-body problem, proving that all negative energy solutions with zero angular momentum lead to syzygies, and extends previous results with new bounds using elementary Sturm-Liouville theory.

Contribution

It provides an independent proof of Montgomery's result and broadens the scope to include negative energy cases with new bounds, using elementary Sturm-Liouville theory.

Findings

All negative energy solutions with zero angular momentum lead to collinear configurations.

Extended bounds for negative energy solutions in the three-body problem.

Reinforced the connection between velocity alignments and syzygies in the three-body problem.

Abstract

In this paper, we further investigate the planar Newtonian three-body problem with a focus on collinear configurations, where either the three bodies or their velocities are aligned. We provide an independent proof of Montgomery's result, stating that apart from the Lagrange solution, all negative energy solutions to the zero angular momentum case result in syzygies, i.e., collinear configurations of positions. The concept of generalised syzygies, inclusive of velocity alignments, was previously explored by the author for bounded solutions. In this study, we broaden our scope to encompass negative energy cases and provide new bounds. Our methodology builds upon the elementary Sturm-Liouville theory and the Wintner-Conley "linear" form of the three-body problem, as previously explored in the works of Albouy and Chenciner.