Saari's Conjecture for the Collinear $n$-Body Problem

TL;DR

This paper proves Saari's conjecture for collinear solutions in the Newtonian n-body problem, showing that constant moment of inertia solutions are relative equilibria, and characterizes solutions with non-zero angular momentum as homographic motions with central configurations.

Contribution

It establishes Saari's conjecture for collinear configurations and characterizes solutions with non-zero angular momentum for homogeneous potentials.

Findings

Constant moment of inertia solutions are relative equilibria.

Collinear solutions with non-zero angular momentum are homographic motions.

Results hold for any potential involving mutual distances.

Abstract

In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.