American Institute of Mathematical Sciences

Advanced
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 - A, 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.   Google Scholar

[2]

N. Faycal, On the classification of pyramidal central configurations,, Proc. Amer. Math. Soc., 124 (1996), 249.  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.  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.   Google Scholar

[5]

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

[8]

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

[15]

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

[16]

S. Zhang and Q. Zhou, Double pyramidal central configurations,, Phys. Lett. A, 281 (2001), 240.  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.   Google Scholar

[2]

N. Faycal, On the classification of pyramidal central configurations,, Proc. Amer. Math. Soc., 124 (1996), 249.  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.  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.   Google Scholar

[5]

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

[8]

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

[15]

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

[16]

S. Zhang and Q. Zhou, Double pyramidal central configurations,, Phys. Lett. A, 281 (2001), 240.  doi: 10.1016/S0375-9601(01)00140-2.  Google Scholar

[1]

Martha Alvarez, Joaquin Delgado, Jaume Llibre. On the spatial central configurations of the 5--body problem and their bifurcations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 505-518. doi: 10.3934/dcdss.2008.1.505
[2]

Eduardo Piña. Computing collinear 4-Body Problem central configurations with given masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1215-1230. doi: 10.3934/dcds.2013.33.1215
[3]

Allyson Oliveira, Hildeberto Cabral. On stacked central configurations of the planar coorbital satellites problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3715-3732. doi: 10.3934/dcds.2012.32.3715
[4]

Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589
[5]

Elbaz I. Abouelmagd, Juan L. G. Guirao, Aatef Hobiny, Faris Alzahrani. Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1047-1054. doi: 10.3934/dcdss.2015.8.1047
[6]

Martin Swaczyna, Petr Volný. Uniform motions in central fields. Journal of Geometric Mechanics, 2017, 9 (1) : 91-130. doi: 10.3934/jgm.2017004
[7]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[8]

Joan-Josep Climent, Francisco J. García, Verónica Requena. On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables. Advances in Mathematics of Communications, 2008, 2 (4) : 421-431. doi: 10.3934/amc.2008.2.421
[9]

Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137
[10]

Keisuke Hakuta, Hisayoshi Sato, Tsuyoshi Takagi. On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$. Advances in Mathematics of Communications, 2016, 10 (2) : 221-228. doi: 10.3934/amc.2016002
[11]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
[12]

Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561
[13]

R. Ryham, Chun Liu, Zhi-Qiang Wang. On electro-kinetic fluids: One dimensional configurations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 357-371. doi: 10.3934/dcdsb.2006.6.357
[14]

Klara Stokes, Maria Bras-Amorós. Associating a numerical semigroup to the triangle-free configurations. Advances in Mathematics of Communications, 2011, 5 (2) : 351-371. doi: 10.3934/amc.2011.5.351
[15]

Irina Berezovik, Carlos García-Azpeitia, Wieslaw Krawcewicz. Symmetries of nonlinear vibrations in tetrahedral molecular configurations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2473-2491. doi: 10.3934/dcdsb.2018261
[16]

Mao Chen, Xiangyang Tang, Zhizhong Zeng, Sanya Liu. An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle. Journal of Industrial & Management Optimization, 2020, 16 (1) : 495-510. doi: 10.3934/jimo.2018164
[17]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[18]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
[19]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[20]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences