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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Some counterexamples to a generalized Saari’s conjecture
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by Gareth E. Roberts PDF
Trans. Amer. Math. Soc. 358 (2006), 251-265 Request permission


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:
  • Received by editor(s): September 12, 2003
  • Received by editor(s) in revised form: February 9, 2004
  • Published electronically: January 21, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 251-265
  • MSC (2000): Primary 70F10, 70F15; Secondary 37J45
  • DOI:
  • MathSciNet review: 2171232