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Generic finiteness for Dziobek configurations


Author: Richard Moeckel
Journal: Trans. Amer. Math. Soc. 353 (2001), 4673-4686
MSC (1991): Primary 70F10, 70F15, 37N05
DOI: https://doi.org/10.1090/S0002-9947-01-02828-8
Published electronically: April 24, 2001
MathSciNet review: 1851188
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Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this paper is to show that for almost all choices of $n$ masses, $m_i$, there are only finitely many central configurations of the Newtonian $n$-body problem for which the bodies span a space of dimension $n-2$ (such a central configuration is called a Dziobek configuration). The result applies in particular to two-dimensional configurations of four bodies and three-dimensional configurations of five bodies.


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  • 1. Alain Albouy, Symétrie des configurations centrales de quatre corps, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 2, 217–220 (French, with English and French summaries). MR 1320359
  • 2. A. Albouy, Recherches sur le problème des $n$ corps, Notes scientifiques et techniques du Bureau des Longitudes, Paris, (1997) 78.
  • 3. O. Dziobek, Über einen merkwürdigen Fall des Vielkörperproblems, Astron. Nach. 152 (1900) 33-46.
  • 4. Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558
  • 5. R. P. Kuz′mina, An upper bound for the number of central configurations in the plane 𝑛-body problem, Dokl. Akad. Nauk SSSR 234 (1977), no. 5, 1016–1019 (Russian). MR 0494263
  • 6. J.L. Lagrange, Ouvres, vol 6, 272.
  • 7. P. S. Laplace, Sur quelques points du système du monde, Mémoires de l'Académie Royale des Sciences de Paris (1789) article XXIII ou Oeuvres Complètes, vol 11, 553.
  • 8. R. Lehmann-Filhés, Ueber zwei Fälle des Vielkörpersprblems, Astron. Nach. 127 (1891) 137-143.
  • 9. W. D. MacMillan & W. Bartky, Permanent configurations in the problem of four bodies, Trans. Amer. Math. Soc. 34 (1932) 838-875.
  • 10. J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280. MR 0161339, https://doi.org/10.1090/S0002-9939-1964-0161339-9
  • 11. Richard Moeckel, Relative equilibria of the four-body problem, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 417–435. MR 805839, https://doi.org/10.1017/S0143385700003047
  • 12. F.R. Moulton, The straight line solutions of the problem of N bodies, in Ann. of Math. 2-12 (1910) 1-17.
  • 13. 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
  • 14. René Thom, Sur l’homologie des variétés algébriques réelles, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 255–265 (French). MR 0200942
  • 15. W. L. Williams, Permanent configurations in the problem of five bodies, Trans. Amer. Math. Soc. 44 (1938) 563-579.
  • 16. A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series 5, Princeton University Press, Princeton, NJ (1941). MR 3:215b

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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-01-02828-8
Keywords: Celestial mechanics, central configurations, $n$-body problem
Received by editor(s): December 29, 2000
Published electronically: April 24, 2001
Article copyright: © Copyright 2001 American Mathematical Society