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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

X. Liu, On double pyramidal central configuration with parallelogram base, Xinan Shifan Daxue Xuebao Ziran Kexue Ban, 26 (2001), 521-525.  Google Scholar

[5]

X. Liu, Double pyramidal central configurations with a concave quadrilateral base, J. Chongqing Univ., 1 (2002), 67-69.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. Liu and X. Chen, Double pyramidal central configurations of 5-bodieswith arbitrary triangle base, Sichuan Daxue Xuebao, 40 (2003), 190-194.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. Ouyang, Z. Xie and S. Zhang, Pyramidal central configurations and perverse solutions, Electron. J. Differential Equations, 106 (2004), 1-9.  Google Scholar

[15]

D. Yang and S. Zhang, Necessary conditions for central configurations of six-body problems, Southeast Asian Bull. Math., 27 (2003), 739-747.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

X. Liu, On double pyramidal central configuration with parallelogram base, Xinan Shifan Daxue Xuebao Ziran Kexue Ban, 26 (2001), 521-525.  Google Scholar

[5]

X. Liu, Double pyramidal central configurations with a concave quadrilateral base, J. Chongqing Univ., 1 (2002), 67-69.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. Liu and X. Chen, Double pyramidal central configurations of 5-bodieswith arbitrary triangle base, Sichuan Daxue Xuebao, 40 (2003), 190-194.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. Ouyang, Z. Xie and S. Zhang, Pyramidal central configurations and perverse solutions, Electron. J. Differential Equations, 106 (2004), 1-9.  Google Scholar

[15]

D. Yang and S. Zhang, Necessary conditions for central configurations of six-body problems, Southeast Asian Bull. Math., 27 (2003), 739-747.  Google Scholar

[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.  Google Scholar

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