Qualitative Theory of Dynamical Systems

, Volume 8, Issue 2, pp 357–370 | Cite as

Finiteness in the Planar Restricted Four-Body Problem

  • Julianne L. Kulevich
  • Gareth E. Roberts
  • Christopher J. Smith
Article

Abstract

Using BKK theory, we show that the number of equilibria (central configurations) in the planar, circular, restricted four-body problem is finite for any choice of masses. Moreover, the number of such points is bounded above by 196.

Keywords

Celestial mechanics Restricted four-body problem Central configurations BKK theory 

Mathematics Subject Classification (2000)

Primary 70F10 70F15 Secondary 37N05 14M25 

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References

  1. 1.
    Arenstorff R.F.: Central configurations of four bodies with one inferior mass. Celest. Mech. 28, 9–15 (1982)CrossRefGoogle Scholar
  2. 2.
    Bernstein D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)MATHCrossRefGoogle Scholar
  3. 3.
    Bruder, F.: Saaris Vermutung für das restringierte Dreikörperproblem und für eine Form des restringierten Vierkörperproblems. Masters Thesis, University of Hamburg, Hamburg, Germany (2008)Google Scholar
  4. 4.
    Cox D., Little J., O’Shea D.: Using algebraic geometry, 2nd edn. Springer, New York (2005)MATHGoogle Scholar
  5. 5.
    Hampton M., Moeckel R.: Finiteness of relative equilibria of the four-body problem. Invent. Math. 163(2), 289–312 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lagrange, J.L.: Essai sur le problème des trois corps. Œvres 6 (1772), Gauthier-Villars, Paris, pp. 272–292Google Scholar
  7. 7.
    Leandro E.S.G.: On the central configurations of the planar restricted four-body problem. J. Differ. Equ. 226(1), 323–351 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lindow M.: Ein Spezialfall des Vierkörperproblems. Astron. Nachr. 216(21), 389–408 (1922)CrossRefGoogle Scholar
  9. 9.
    Maple, version 11.01 (2007), Maplesoft, Waterloo Maple Inc.Google Scholar
  10. 10.
    MATLAB, version 7.2.0.294 (2006), The MathWorks, Inc.Google Scholar
  11. 11.
    Moeckel R.: A computer-assisted proof of Saari’s conjecture for the planar three-body problem. Trans. Amer. Math. Soc. 357(8), 3105–3117 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Palmore J.I.: Collinear relative equilibria of the planar n-body problem. Celest. Mech. 28, 17–24 (1982)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pedersen, P.: Librationspunkte im restringierten Vierkörperproblem. Dan. Mat.-Fys. Medd 21(6), 80 pp (1944)Google Scholar
  14. 14.
    Qhull, The Geometry Center, University of Minnesota (2002). http://www.qhull.org
  15. 15.
    Roberts G.E.: A continuum of relative equilibria in the five-body problem. Phys. D 127, 141–145 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Roberts G.E., Melanson L.: Saari’s conjecture for the restricted three-body problem. Celest. Mech. Dyn. Astr. 97, 211–223 (2007)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    SAGE Mathematical Software, Version 3.1.1 (2008). http://www.sagemath.org
  18. 18.
    Simó C.: Relative equilibrium solutions in the four body problem. Celest. Mech. 18(2), 165–184 (1978)MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser/Springer Basel AG 2010

Authors and Affiliations

  • Julianne L. Kulevich
    • 1
  • Gareth E. Roberts
    • 1
  • Christopher J. Smith
    • 2
  1. 1.Department of Mathematics and Computer ScienceCollege of the Holy CrossWorcesterUSA
  2. 2.MilfordUSA

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