Transactions of the American Mathematical Society

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

Author(s): Richard Moeckel
Journal: Trans. Amer. Math. Soc. 357 (2005), 3105-3117.
MSC (2000): Primary 70F10, 70F15, 37N05
Posted: May 10, 2004
Retrieve article in: PDF DVI PostScript

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.

1.: A. Albouy and A. Chenciner, Le problème des n corps et les distances mutuelles, Inv. Math., 131 (1998) 151-184. MR 98m:70017
2.: D. N. Bernstein, The number of roots of a system of equations, Fun. Anal. Appl., 9 (1975) 183-185. MR 55:8034
3.: T. Christof and A. Loebel. PORTA: Polyhedron Representation Transformation Algorithm, Version 1.3.2. http://www.iwr.uni-heidelberg.de/ 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, $\OE$ uvres, 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.: D. Saari, On bounded solutions of the -body problem, in G.E.O. Giacaglia, ed., Periodic Orbits, Stability and Resonances, D.Reidel, Dordrecht (1970). MR 42:8753
11.: I. R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space, Springer-Verlag, Berlin, Heidelberg, New York (1994). MR 95m:14001
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.: J. Waldvogel, Symmetric and regularized coordinates on the plane triple collision manifold, Cel. Mech., 28 (1982) 69-82. MR 83m:70021
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 4.1.5.0, Wolfram Research, Inc.
16.: G. Zeigler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York (1995). MR 96a:52011

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 70F10, 70F15, 37N05

Richard Moeckel
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: rick@math.umn.edu

DOI: 10.1090/S0002-9947-04-03527-5
PII: S 0002-9947(04)03527-5
Keywords: Celestial mechanics, three-body problem, computational algebra
Received by editor(s): September 11, 2003
Posted: May 10, 2004
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS