On some collinear configurations in the planar 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 (Montgomery 2007 Ergod. Theory Dyn. Syst. 27 311–40), stating that apart from the Lagrange’s 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 Tsygvintsev (2023 C. R. Acad. Sci., Paris 361 331–5). 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 (1997 Invent. Math. 131 151–84); Albouy (2004 Mutual Distances in Celestial Mechanics (Lectures at Nankai Institute)); Chenciner (2013 Acta Math. Vietnam. 38 165–86).