Partially rigid motions in the n-body problem

TL;DR

This paper investigates the existence of partially rigid motions in the n-body problem, specifically proving that hinged solutions do not exist for n=3 or 4, regardless of the space dimension.

Contribution

It establishes the non-existence of hinged solutions in the 3- and 4-body problems, extending understanding of rigid motion constraints.

Findings

Hinged solutions do not exist for n=3.

Hinged solutions do not exist for n=4.

Results are independent of ambient space dimension.

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. In particular, a {\em hinged} solution is one such that exactly one mutual distance varies. The goal of this paper is to show that hinged solutions don't exist when n=3 or n=4. For n=3 this means that if 2 of the 3 distances are constant so is the third and for n=4, if 5 of the 6 distances are constant, so is the sixth. These results hold independent of the dimension d of the ambient space.