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Some counterexamples to a generalized Saari's conjecture


Author: Gareth E. Roberts
Journal: Trans. Amer. Math. Soc. 358 (2006), 251-265
MSC (2000): Primary 70F10, 70F15; Secondary 37J45
DOI: https://doi.org/10.1090/S0002-9947-05-03697-4
Published electronically: January 21, 2005
MathSciNet review: 2171232
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Abstract: For the Newtonian $n$-body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.


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Additional Information

Gareth E. Roberts
Affiliation: Department of Mathematics and Computer Science, 1 College Street, College of the Holy Cross, Worcester, Massachusetts 01610
Email: groberts@radius.holycross.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03697-4
Keywords: Saari's conjecture, $n$-body problems, relative equilibria, Hamiltonian systems
Received by editor(s): September 12, 2003
Received by editor(s) in revised form: February 9, 2004
Published electronically: January 21, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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