On Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law

  • Martha Alvarez-Ramírez
  • Alan Almeida Santos
  • Claudio Vidal


We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which goes from the kite containing an equilateral triangle up to the square shape. After, by putting a fifth mass at the center, we feature the planar cc of five bodies as a tensor of corange two see, “Albouy and Chenciner (Invent Math 131:151–184, 1998)” and we prove that cc is stacked see, “Hampton (Nonlinearity 18:2299–2304, 2005b)” in a such way that the center of mass of the four bodies should be the center of the circle. We emphasize that our approach includes not only the Newtonian force law, but the homogeneous ones with exponent \(a\le -1\).


\(N\)-body problem Central configurations Co-circular configurations Homogeneous force law 



The second author acknowledges the support of Unión Matemática de América Latina y el Caribe—UMALCA to enable the visit to the Department of Mathematics at the Universidad del Bío Bío, Concepción, Chile. The first author was partially supported by SEP-CONACyT Grant Number SEP-2004-C01-47768. The second author was partially supported by UMALCA.


  1. 1.
    Albouy, A.: Symétrie des configurations centrales de quatre corps. C. R. Acad. Sci. Paris 320, 217–220 (1995a)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Albouy, A.: The symmetric central configurations of four equal masses. Contemp. Math. 198, 131–135. ISSN 0271–4132 (1995b)Google Scholar
  3. 3.
    Albouy, A.: Recherches sur le problème des \(n\) corps. Notes Scientifiques et Techniques du Bureau des Longitudes S058. Notes Institut de Mècanique Cèleste et Calcul des Èphèmèrides, Paris (1997)Google Scholar
  4. 4.
    Albouy, A.: On a paper of Moeckel on central configurations. Regul. Chaot. Dyn. 8(2), 33–42 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Albouy, A., Chenciner, A.: Le problème des n corps et les distances mutuelles. Invent. math. 131, 151–184. ISSN 0020–9910 (1998)Google Scholar
  6. 6.
    Albouy, A., Fu, Y., Sun, S.: Symmetry of planar four-body convex central configurations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464(2093), 1355–1365 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Albouy, A., Kaloshin, V.: Finiteness of central configurations of five bodies in the plane. Ann. Math. 176(1), 535–588 (2012). doi: 10.4007/annals.2012.176.1.10 MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cors, J., Roberts, G.: Cyclic central configurations in the four-body problem. Preprint (2010)Google Scholar
  9. 9.
    Euler, L.: De motu rectilineo trium corporum se mutuo attahentium. Novi Commun. Acad. Sci. Imp. Petrop. 11, 144–151 (1767)Google Scholar
  10. 10.
    Hampton, M.: Co-circular central configurations in the four-body problem. In: Equadiff: International Conference on Differential Equations. World Scientific Publishing Co. Pte, Ltd, pp. 993–998 (2003). ISBN 981-256-169-2 (2005a)Google Scholar
  11. 11.
    Hampton, M.: Stacked central configurations: new examples in the planar five-body problem. Nonlinearity 18, 2299–2304 (2005b). doi: 10.1088/0951-7715/18/5/021 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hampton, M., Jensen, A.: Finiteness of spatial central configurations in the five-body problem. Celest. Mech. Dyn. Astron. 109(4), 321–332 (2011). doi: 10.1007/s10569-010-9328-9 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hampton, M., Moeckel, R.: Finiteness of relative equilibria of the four-body problem. Invent. Math. 163, 289–312. ISSN 0020–9910 (2006)Google Scholar
  14. 14.
    Lagrange, J.: Essai sur le problème des trois corps. In: Euvres, vol. 6. Gauthier-Villars, Paris (1772)Google Scholar
  15. 15.
    Santos, A., Vidal, C.: Symmetry of the restricted 4+1 body problem with equal masses. Regul. Chaot. Dyn. 12(1), 27–38. ISSN 1560–3547 (2007)Google Scholar
  16. 16.
    Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton (1941)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Martha Alvarez-Ramírez
    • 1
  • Alan Almeida Santos
    • 2
  • Claudio Vidal
    • 3
  1. 1.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IztapalapaMexicoMexico
  2. 2.Departamento de Matemática, Campus Professor Alberto CarvalhoUniversidade Federal de SergipeItabaianaBrazil
  3. 3.Departamento de Matemática, Facultad de CienciasUniversidad del Bío BíoConcepcionChile

Personalised recommendations