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A computer-assisted proof of Saari's conjecture for the planar three-body problem

Author: Richard Moeckel
Journal: Trans. Amer. Math. Soc. 357 (2005), 3105-3117
MSC (2000): Primary 70F10, 70F15, 37N05
Published electronically: May 10, 2004
MathSciNet review: 2135737
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Abstract | References | Similar Articles | Additional Information

Abstract: The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.

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  • 1. Alain Albouy and Alain Chenciner, Le problème des 𝑛 corps et les distances mutuelles, Invent. Math. 131 (1998), no. 1, 151–184 (French). MR 1489897,
  • 2. D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1–4 (Russian). MR 0435072
  • 3. T. Christof and A. Loebel. PORTA: Polyhedron Representation Transformation Algorithm, Version 1.3.2. iwr/comopt/soft/PORTA/readme.html
  • 4. A. G. Khovansky Newton polyhedra and toric varieties, Fun. Anal. Appl., 11 (1977) 289-296.
  • 5. A. G. Kushnirenko, Newton polytopes and the Bézout theorem, Fun. Anal. Appl., 10 (1976) 233-235.
  • 6. J.L. Lagrange, Essai sur le problème des trois corps, \OEuvres, vol. 6.
  • 7. J. Llibre and E. Piña, Saari's conjecture holds for the planar 3-body problem, preprint (2002).
  • 8. C. McCord, Saari's conjecture for the planar three-body problem with equal masses, preprint (2002).
  • 9. F. D. Murnaghan, A symmetric reduction of the planar three-body problem, Amer. J. Math, 58 (1936) 829-832.
  • 10. Periodic orbits, stability and resonances, Proceedings of a Symposium conducted by the University of São Paulo, the Technical Institute of Aeronautics of São José dos Campos, and the National Observatory of Rio de Janeiro, at the University of São Paulo, São Paulo, Brasil, 4-12 September, vol. 1969, D. Reidel Publishing Co., Dordrecht, 1970. MR 0273877
  • 11. Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
  • 12. E. R. van Kampen and A. Wintner, On a symmetric reduction of the problem of three bodies, Am. J. Math, 59 (1937) 153-166.
  • 13. Jörg Waldvogel, Symmetric and regularized coordinates on the plane triple collision manifold, Celestial Mech. 28 (1982), no. 1-2, 69–82. MR 682838,
  • 14. A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series 5, Princeton University Press, Princeton, NJ (1941). MR 3:215b
  • 15. S. Wolfram, Mathematica, version, Wolfram Research, Inc.
  • 16. Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028

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Additional Information

Richard Moeckel
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Keywords: Celestial mechanics, three-body problem, computational algebra
Received by editor(s): September 11, 2003
Published electronically: May 10, 2004
Article copyright: © Copyright 2004 American Mathematical Society