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

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March 2013, 33(3): 1049-1060. doi: 10.3934/dcds.2013.33.1049

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

Montserrat Corbera ^1, and Jaume Llibre ^2,

1.	Departament de Tecnologies Digitals i de la Informació, Universitat de Vic, C/. Laura, 13, 08500 Vic, Barcelona, Catalonia, Spain
2.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Received April 2011 Revised December 2011 Published October 2012

Abstract
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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.

Keywords: $n+2$-body problem, bi-pyramidal central configurations., Spatial central configurations.

Mathematics Subject Classification: Primary: 70F10; Secondary: 70F1.

Citation: Montserrat Corbera, Jaume Llibre. On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1049-1060. doi: 10.3934/dcds.2013.33.1049

References:

[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.

show all references

References:

[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.

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