Celestial Mechanics and Dynamical Astronomy

, Volume 109, Issue 1, pp 27–43 | Cite as

On the central configurations of the planar 1 + 3 body problem

  • Montserrat Corbera
  • Josep Maria CorsEmail author
  • Jaume Llibre
Original Article


We consider the Newtonian four-body problem in the plane with a dominat mass M. We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision arithmetics, we provide evidence that the number of central configurations varies from five to seven.


1 + 3 body problem Central configurations Coorbital satellites 


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  1. Albouy A., Fu Y.: Relative equilibria of four identical satellites. Proc. R. Soc. A 465, 2633–2645 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. Casasayas J., Llibre J., Nunes A.: Central configurations of the planar 1 + n body problem. Celest. Mech. Dyn. Astron. 60, 273–288 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. Cors J.M., Llibre J., Ollé M.: Central configurations of the planar coorbital satellite problem. Celest. Mech. Dyn. Astron. 89, 319–342 (2004)zbMATHCrossRefADSGoogle Scholar
  4. Hall, G.R.: Central configurations in the planar 1 + n body problem, Preprint (1988)Google Scholar
  5. Lang S.: Algebra, 3rd edn. Addison–Wesley, Reading (1993)zbMATHGoogle Scholar
  6. Lee T.-L., Santoprete M.: Central configurations of the five-body problem with equal masses. Celest. Mech. Dyn. Astron. 104, 369–381 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. Maxwell, J.C.: On the stability of the motion of Saturn’s rings. In: Maxwell on Saturn’s Rings, pp 69–158. MIT Press (1859)Google Scholar
  8. Moeckel R.: Linear stability of relative equilibria with a dominant mass. J. Dyn. Diff. Equ. 6, 37–51 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. Olver P.: Classical invariant Theory, London Mathematical Society Student Texts, vol. 44. Cambridge University Press, New York (1999)Google Scholar
  10. Renner S., Sicardy B.: Stationary configurations for co–orbital satellites with small arbitary masses. Celest. Mech. Dyn. Astron. 88, 397–414 (2004)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. Roberts G.: Linear stability of the n + 1-gon relative equilibrium. Hamiltonian Systems and Celestial Mechanics (HAMSYS-98). World Sci. Monogr. Ser. Math. 6, 303–330 (2000)CrossRefADSGoogle Scholar
  12. Scheeres D.J., Vinh N.X.: Linear stability of a self-graviting rings. Celest. Mech. Dyn. Astron. 51, 83–103 (1991)zbMATHCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Montserrat Corbera
    • 1
  • Josep Maria Cors
    • 2
    Email author
  • Jaume Llibre
    • 3
  1. 1.Departament de Tecnologies Digitals i de la InformacióUniversitat de VicVic, BarcelonaSpain
  2. 2.Matemàtica Aplicada IIIUniversitat Politècnica de CatalunyaManresa, BarcelonaSpain
  3. 3.Departament de MatemàtiquesUniversitat Autònoma de Barcelona, BellaterraBarcelonaSpain

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