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Article Contents

# On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base

• In this paper we prove the existence of central configurations of the $n+2$--body problem where $n$ equal masses are located at the vertices of a regular $n$--gon and the remaining $2$ masses, which are not necessarily equal, are located on the straight line orthogonal to the plane containing the $n$--gon passing through its center. Here this kind of central configurations is called bi--pyramidal central configurations. In particular, we prove that if the masses $m_{n+1}$ and $m_{n+2}$ and their positions satisfy convenient relations, then the configuration is central. We give explicitly those relations.
Mathematics Subject Classification: Primary: 70F10; Secondary: 70F15.

 Citation:

•  [1] F. Cedó and J. Llibre, Symmetric central configurations of the spatial $n$-body problem, J. of Geometry and Physics, 6 (1989), 367-394. [2] N. Faycal, On the classification of pyramidal central configurations, Proc. Amer. Math. Soc., 124 (1996), 249-258.doi: 10.1090/S0002-9939-96-03135-8. [3] E. S. G. Leandro, Finiteness and bifurcations of some symmetrical classes of central configurations, Arch. Ration. Mech. Anal., 167 (2003), 147-177.doi: 10.1007/s00205-002-0241-6. [4] X. Liu, On double pyramidal central configuration with parallelogram base, Xinan Shifan Daxue Xuebao Ziran Kexue Ban, 26 (2001), 521-525. [5] X. Liu, Double pyramidal central configurations with a concave quadrilateral base, J. Chongqing Univ., 1 (2002), 67-69. [6] X. Liu, A class of double pyramidal central configurations of 7-body with concave pentagon base, Xinan ShifanDaxue Xuebao Ziran Kexue Ban, 27 (2002), 494-496. [7] X. Liu, Existence and uniqueness for a class of double pyramidal central configurations with a concave pentagonal base, J. Chongqing Univ., 2 (2003), 28-30. [8] X. Liu and X. Chen, Double pyramidal central configurations of 5-bodieswith arbitrary triangle base, Sichuan Daxue Xuebao, 40 (2003), 190-194. [9] X. Liu, Existence and uniqueness of a class of double pyramidal central configurations in six-body problems, J. Chongqing Univ., 3 (2004), 97-101. [10] X. Liu, X. Du and T. Feng, Existence and uniqueness for a class of nine-bodies central configurations, J. Chongqing Univ., 5 (2006), 53-56. [11] L. F. Mello and A. C. Fernandes, New spatial central configurations in the $5$-body problem, An. Acad. Bras. Ciênc., 83 (2011), 763-774.doi: 10.1590/S0001-37652011005000023. [12] L. F. Mello and A. C. Fernandes, New classes of spatial central configurations for the $n+3$ body problem, Nonlinear Anal. Real World Appl., 12 (2011), 723-730.doi: 10.1016/j.nonrwa.2010.07.013. [13] R. Moeckel and C. Simó, Bifurcation of spatial central configurations from planar ones, SIAM J. Math. Anal., 26 (1995), 978-998.doi: 10.1137/S0036141093248414. [14] T. Ouyang, Z. Xie and S. Zhang, Pyramidal central configurations and perverse solutions, Electron. J. Differential Equations, 106 (2004), 1-9. [15] D. Yang and S. Zhang, Necessary conditions for central configurations of six-body problems, Southeast Asian Bull. Math., 27 (2003), 739-747. [16] S. Zhang and Q. Zhou, Double pyramidal central configurations, Phys. Lett. A, 281 (2001), 240-248.doi: 10.1016/S0375-9601(01)00140-2.