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

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

2009-09-27

arxiv.org/abs/0909.4909v1

Trans. Amer. Math. Soc. 357 (2005), 4215-4223

Physics

Mathematical Physics

Scientific paper

10.1090/S0002-9947-04-03606-2

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.

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