Generic Finiteness for Dziobek Configurations
HTML articles powered by AMS MathViewer
- by Richard Moeckel
- Trans. Amer. Math. Soc. 353 (2001), 4673-4686
- DOI: https://doi.org/10.1090/S0002-9947-01-02828-8
- Published electronically: April 24, 2001
- PDF | Request permission
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.References
- 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
- A. Albouy, Recherches sur le problème des $n$ corps, Notes scientifiques et techniques du Bureau des Longitudes, Paris, (1997) 78.
- O. Dziobek, Über einen merkwürdigen Fall des Vielkörperproblems, Astron. Nach. 152 (1900) 33–46.
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
- J.L. Lagrange, Ouvres, vol 6, 272.
- 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.
- R. Lehmann-Filhés, Ueber zwei Fälle des Vielkörpersprblems, Astron. Nach. 127 (1891) 137–143.
- W. D. MacMillan & W. Bartky, Permanent configurations in the problem of four bodies, Trans. Amer. Math. Soc. 34 (1932) 838–875.
- J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280. MR 161339, DOI 10.1090/S0002-9939-1964-0161339-9
- Richard Moeckel, Relative equilibria of the four-body problem, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 417–435. MR 805839, DOI 10.1017/S0143385700003047
- F.R. Moulton, The straight line solutions of the problem of N bodies, in Ann. of Math. 2-12 (1910) 1–17.
- 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
- 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
- W. L. Williams, Permanent configurations in the problem of five bodies, Trans. Amer. Math. Soc. 44 (1938) 563–579.
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Bibliographic Information
- Richard Moeckel
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: rick@math.umn.edu
- Received by editor(s): December 29, 2000
- Published electronically: April 24, 2001
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1851188