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Transactions of the American Mathematical Society

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Saari's conjecture for the collinear $n$-body problem

Authors: Florin Diacu, Ernesto Pérez-Chavela and Manuele Santoprete
Journal: Trans. Amer. Math. Soc. 357 (2005), 4215-4223
MSC (2000): Primary 70F10; Secondary 70F07
Published electronically: November 4, 2004
MathSciNet review: 2159707
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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.

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Additional Information

Florin Diacu
Affiliation: Pacific Institute for the Mathematical Sciences and Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045 STN CSC, Victoria, British Columbia, Canada V8W 3P4

Ernesto Pérez-Chavela
Affiliation: Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Apdo. 55534, México, D.F., México

Manuele Santoprete
Affiliation: Department of Mathematics, University of California, Irvine, 294 Multipurpose Science & Technology Building, Irvine, California 92697

Received by editor(s): September 26, 2003
Received by editor(s) in revised form: December 18, 2003
Published electronically: November 4, 2004
Article copyright: © Copyright 2004 American Mathematical Society