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

Florin Diacu, Ernesto Perez-Chavela, Manuele Santoprete

Published 2009-09-27Version 1

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.