Abstract
Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m 1,..., m n we denote by N(m 1, ..., m n, k) the number of central configurations' of the n-body problem in ℝk modulus dilatations and rotations. If m n 1,..., m n, k) is finite, then we give a bound of N(m 1,..., m n, k) which only depends of n and k.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buck, G.: 1991, ‘On clustering in central configurations’, Proc. Amer. Math. Soc., in print.
Buck, G.: 1989, ‘The collinear central configuration of n equal masses’ preprint.
Cedó, F. and Llibre, J.: 1989, ‘Symmetric central configurations of the spatial nbody problem’ J. of Geometry and Physics, 6, 367–394.
Dziobek, 0.: 1900, ‘Uber einen merkwurdigen fall des vielkorperproblems’, Astron. Nach., 152, 32–46.
Elmabsout, B.: 1988, ‘Sur l'existence de certaines configurations d'equilibre relatif dans le problem des N corps’, Celest. Mech., 41, 131–151.
Hagihara, Y.: 1970, Celestial Mechanics, Vol. 1, MIT press, Cambridge.
Mac-Millan, W.D. and Bartky, W.: 1932, ‘Permanent configurations in the problem of four bodies’, Trans. Amer. Math. Soc., 34, 838–875.
Meyer, K.R.: 1987, ‘Bifurcation of a central configuration’ Celest. Mech., 40, 273–282.
Meyer, K.R. and Schmidt, D.: 1988a, ‘Bifurcation of relative equilibria in the four and five body problem’, Ergod. Th. and Dyn. Sys. 8*, 215–225.
Meyer, K.R. and Schmidt, D.: 1988b, ‘Bifurcations of relative equilibria in the n-body and Kirchhoff problems’, Siam J. Math. Anal., 19, 1295–1313.
Moeckel, R.: 1985, lsRelative equilibria of the four-body problem’, Erg. Th. and Dyn. Sys., 5, 417–435.
Moeckel, R.: 1989, ‘On central configurations’, preprint.
Moulton, F.R.: 1910, ‘The straight line solutions of the problem of N bodies’, Ann. of Math., 12, 1–17.
Pacella, F.: 1987, ‘Central configurations of the n-body problem via equivariant Morse theory’, Archive for Rat. Mech. and Anal., 97, 59–74.
Palmore, J.: 1973, ‘Classifying relative equilibria I’, Bull. Amer. Math. Soc., 79, 904–908.
Palmore, J.: 1975a, ‘Classifying relative equilibria II’, Bull. Amer. Math. Soc., 81, 489–491.
Palmore, J.: 1975b, ‘Classifying relative equilibria III’, Letters in Math. Phys., 1, 71–73.
Perko, L.M. and Walter, E.L.: 1985, ‘Regular polygon solutions of the n-body problem’, Proc. Amer. Mat. Soc., 94, 301–309.
Saari, D.: 1980, ‘On the role and properties of central configurations’, Celest. Mech., 21, 9–20.
Schmidt, D.S.: 1988, ‘Central configurations in ℝ2 and ℝ3, Contemporary Mathematics, 81, 59–76.
Shub, M.: 1970, ‘Appendix to Smale's paper : Diagonals and relative equilibria’, in Manifolds-Amsterdam 1970, Springer Lecture Notes in Math., 197, 199–201.
Simo, C.: 1977, ‘Relative equilibrium solutions in the four body problem’, Celest. Mech., 18, 165–184.
Smale, S.: 1970a, ‘Topology and mechanics II: The planar n-body problem’, Inventiones math., 11, 45–64.
Smale, S.: 1970b, ‘Problems on the nature of relative equilibria in Celestial Mechanics’, in Manifolds-Amsterdam 1970, Springer Lecture Notes in Math., 197, 194–198. Wintner, A.: 1941, The Analytical Foundations of Celestial Mechanics, Princeton Univ. Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Llibre, J. On the number of central configurations in the N-body problem. Celestial Mech Dyn Astr 50, 89–96 (1990). https://doi.org/10.1007/BF00048988
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00048988