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Celestial Mechanics and Dynamical Astronomy

, Volume 60, Issue 2, pp 273–288 | Cite as

Central configurations of the planar 1+N body problem

  • Josefina Casasayas
  • Jaume Llibre
  • Ana Nunes
Article

Abstract

In this paper, we give a new derivation of the equations for the central configurations of the 1+n body problem. In the case of equal masses, we show that forn large enough there exists only one solution. Our lower bound forn improves by several orders of magnitude the one previously found by Hall.

Key words

Central configurations N-body restricted problem 

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References

  1. 1.
    Casasayas, J. and Llibre, J.: 1982, ‘Invariant manifolds associated to homoclinic orbits in then-body problem’,Indiana Univ. J. Math. 31, 463–470.Google Scholar
  2. 2.
    Hall, G.R.: 1988, ‘Central configurations in the Planar 1+n body problem’, preprint.Google Scholar
  3. 3.
    Hagihara, Y.: 1970,Celestial Mechanics, Vol. 1, MIT Press, Cambridge.Google Scholar
  4. 4.
    Llibre, J.: 1991, ‘On the number of central configurations in theN-body problem’,Celest. Mech. Dyn. Astron. 50, 89–96.Google Scholar
  5. 5.
    Llibre, J.: 1977, ‘Posiciones de equilibrio relativo del problema de 4 cuerpos’,Pub. Sec. Mat. U. A. B. 3, 73–86.Google Scholar
  6. 6.
    Llibre, J. and Simó, C.: 1981, ‘Characterization of transversal homothetic solutions in theN-body problem’,Archive for Rat. Mech. and Anal. 77, 189–198.Google Scholar
  7. 7.
    Maxwell, J.C.: 1859,On the Stability of Motion of Saturn's Rings, Macmillan & Co., London.Google Scholar
  8. 8.
    Moeckel, R.: 1985, ‘Relative equilibria of the four-body problem’,Ergod. Th. and Dyn. Sys. 5, 417–435.Google Scholar
  9. 9.
    Moeckel, R.: 1991, ‘Linear stability of relative equilibria with a dominant mass’, communicated to the International Symposium on Hamiltonian Systems and Celestial Mechanics, Guanajuato.Google Scholar
  10. 10.
    Pacella, F.: 1987, ‘Central configurations of then-body problem via equivariant Morse theory’,Archive for Rat. Mech. and Anal. 97, 59–74.Google Scholar
  11. 11.
    Palmore, J.I.: 1973, ‘Classifying relative equilibria, I’,Bull. Amer. Math. Soc. 79, 904–908.Google Scholar
  12. 12.
    Palmore, J.I.: 1975, ‘Classifying relative equilibria, II’,Bull. Amer. Math. Soc. 81, 489–491.Google Scholar
  13. 13.
    Palmore, J.I.: 1976, ‘Measure of degenerate relative equilibria, I’,Annal. Math. 104, 421–429.Google Scholar
  14. 14.
    Saari, D.G.: 1980, ‘On the role and properties ofn-body central configurations’,Celest. Mech. 25, 9–20.Google Scholar
  15. 15.
    Simó, C.: 1978, ‘Relative equilibrium solutions in the four body problem’,Celest. Mech. 18, 165–184.Google Scholar
  16. 16.
    Smale, S.: 1970, ‘Topology and Mechanics, II’,Inventiones Math. 11, 45–64.Google Scholar
  17. 17.
    Smale, S.: 1971, ‘Problems on the nature of relative equilibria in celestial mechanics’,Springer Lecture Notes Maths. 197, 194–198.Google Scholar
  18. 18.
    Wintner, A.: 1941,The Analytical Foundations of Celestial Mechanics, Princeton Univ. Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Josefina Casasayas
    • 1
  • Jaume Llibre
    • 2
  • Ana Nunes
    • 3
  1. 1.Department de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  2. 2.Department de MatemàtiquesUniversitat Autònoma de Barcelona, BellaterraBarcelonaSpain
  3. 3.Departmento de FísicaUniversidade de LisboaLisboaPortugal

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