Finiteness of central configurations of five bodies in the plane

Abstract

We prove there are finitely many isometry classes of planar central configurations (also called relative equilibria) in the Newtonian 5-body problem, except perhaps if the 5-tuple of positive masses belongs to a given codimension 2 subvariety of the mass space.

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  • [Win] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton, NJ: Princeton Univ. Press, 1941, vol. 5.
    @book {Win, MRKEY = {0005824},
      AUTHOR = {Wintner, Aurel},
      TITLE = {The {A}nalytical {F}oundations of {C}elestial {M}echanics},
      SERIES = {Princeton Math. Ser.},
      VOLUME={5},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1941},
      PAGES = {xii+448},
      MRCLASS = {85.0X},
      MRNUMBER = {0005824},
      MRREVIEWER = {E. J. Moulton},
      ZBLNUMBER = {0026.02302},
      }
  • [Xia] Go to document Z. Xia, "Central configurations with many small masses," J. Differential Equations, vol. 91, iss. 1, pp. 168-179, 1991.
    @article {Xia, MRKEY = {1106123},
      AUTHOR = {Xia, Zhihong},
      TITLE = {Central configurations with many small masses},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {91},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {168--179},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {70F10 (70F15)},
      MRNUMBER = {1106123},
      MRREVIEWER = {Alexander Burov},
      DOI = {10.1016/0022-0396(91)90137-X},
      ZBLNUMBER = {0724.70015},
      }

Authors

Alain Albouy

CNRS-UMR8028
Observatoire de Paris
77, avenue Denfert-Rochereau
75014 Paris
France

Vadim Kaloshin

Department of Mathematics
University of Maryland
College Park, MD 20742-4015