Convex Four-Body Central Configurations with Some Equal Masses


We prove that there is an unique convex noncollinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such a central configuration possesses a symmetry line and it is a kite-shaped quadrilateral. We also show that there is exactly one convex noncollinear central configuration when the opposite masses are equal. Such a central configuration also possesses a symmetry line and it is a rhombus.

This is a preview of subscription content, access via your institution.


  1. 1.

    Albouy A. (1995). Symétrie des configurations centrales de quatre corps. C.R. Acad. Sci Paris. 320: 217–220

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Albouy A. (1996). The Symmetric Central Configurations of Four Equal Masses. Contemp Math. V 198: 131–135

    MathSciNet  Google Scholar 

  3. 3.

    Albouy A. (2003). On a Paper of Moeckel on Central Configurations. Reg. Chaotic Dynamics. 8: 133–142

    MATH  Article  ADS  MathSciNet  Google Scholar 

  4. 4.

    Albouy, A.: Mutual Distances in Celestial Mechanics. preprint (2004)

  5. 5.

    Albouy A. and Chenciner A. (1988). Le probléme des n corps et les distances mutuells. Invent. Mat. 131: 151–184

    Article  ADS  MathSciNet  Google Scholar 

  6. 6.

    Albouy, A., Fu, Y.:Euler Configurations and Quasi-Polynomial Systems. preprint (2006)

  7. 7.

    Chazy J. (1918). Sur Certaines trajectoires du probléme des n corps. Bull. Astron. 35: 321–389

    Google Scholar 

  8. 8.

    Hampton M. and Moeckel R. (2006). Finiteness of Relative Equilibria of the Four-Body Problem. Invent.math. 163: 289–312

    MATH  Article  ADS  MathSciNet  Google Scholar 

  9. 9.

    Leandro E.S.G. (2003). Finitness and Bifurcations of some Symmetrical Classes of Central Configurations. Arch. Rational Mech Anal. 167: 147–177

    MATH  Article  ADS  MathSciNet  Google Scholar 

  10. 10.

    Long Y. and Sun S. (2002). Four-Body Central Configurations with some Equal Masses. Arch. Rational Mech Anal. 162: 24–44

    Article  ADS  MathSciNet  Google Scholar 

  11. 11.

    Moulton F.R. (1910). The Straight Line Solutions of the Problem of n-bodies. Ann. Math. 12: 1–17

    Article  MathSciNet  Google Scholar 

  12. 12.

    Smale S. (1970). Topology and Mechanics II. Inv. Math. 11: 45–64

    MATH  Article  ADS  MathSciNet  Google Scholar 

  13. 13.

    Smale S. (1998). Mathematical Problems for the Next Century. Math. Intell. 20: 7–15

    MATH  Article  MathSciNet  Google Scholar 

  14. 14.

    Tien, F.: Recursion Formulas of Central Configurations, Thesis, University of Minnesota 1993

  15. 15.

    Uspensky J.V. (1948). Theory of Equations. McGraw Hill, New York

    Google Scholar 

  16. 16.

    Wintner, A.:The Analytical Foundations of Celestial Mechanics. Princeton Math. Series 5. Princeton University Press, Princeton NJ, 1941

  17. 17.

    Xia Z. (1991). Central Configurations with Many Small Masses. J. Diff. Equations. 91: 168–179

    MATH  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Ernesto Perez-Chavela.

Additional information

Communicated by P. Rabinowitz

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Perez-Chavela, E., Santoprete, M. Convex Four-Body Central Configurations with Some Equal Masses. Arch Rational Mech Anal 185, 481–494 (2007).

Download citation


  • Relative Equilibrium
  • Body Problem
  • Equal Mass
  • Symmetry Line
  • Central Configuration