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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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Saari’s homographic conjecture of the three-body problem
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by Florin Diacu, Toshiaki Fujiwara, Ernesto Pérez-Chavela and Manuele Santoprete PDF
Trans. Amer. Math. Soc. 360 (2008), 6447-6473 Request permission


Saari’s homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $n\ge 3$.
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Additional Information
  • Florin Diacu
  • Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada
  • Email:
  • Toshiaki Fujiwara
  • Affiliation: College of Liberal Arts and Sciences, Kitasato University, 1-15-1 Kitasato, Sagamihara, Kanagawa 228-8555, Japan
  • Email:
  • Ernesto Pérez-Chavela
  • Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
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  • Manuele Santoprete
  • Affiliation: Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada
  • Email:
  • Received by editor(s): November 27, 2006
  • Published electronically: May 29, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 6447-6473
  • MSC (2000): Primary 70F10, 70H05
  • DOI:
  • MathSciNet review: 2434294