Partially rigid motions in the planar three-body problem

TL;DR

This paper investigates partially rigid motions in the planar three-body problem and proves that such motions cannot exist unless they are fully rigid, i.e., relative equilibria.

Contribution

It establishes that in the planar three-body problem with equal masses, partial rigidity implies full rigidity, ruling out partially rigid motions.

Findings

Partially rigid motions are impossible in the planar three-body problem with equal masses.

Any motion with even one constant mutual distance must be a relative equilibrium.

The result constrains the types of motions possible in this classical problem.

Abstract

A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid motions, where some but not all of the distances are constant. For the planar three-body problem with equal masses, we show that partially rigid motions are impossible -- if even one of the three mutual distances is constant, the motion must be a relative equilibrium.