Central configurations of three nested regular polyhedra for the spatial 3n-body problem

JOURNAL OF GEOMETRY AND PHYSICS

卷:59

Article

Corbera, M.¹  Llibre, J.² 

[1] Univ Vic, Dept Tecnol Digitals & Informac, Barcelona 08500, Spain

[2] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Spain

关键词: 3n-body problem;    Spatial central configurations;    Nested regular polyhedra;   

DOI  :  10.1016/j.geomphys.2008.11.012

来源: Elsevier

【 摘 要 】

Three regular polyhedra are called nested if they have the same number of vertices n, the same center and the positions of the vertices of the inner polyhedron r(i), the ones of the medium polyhedron R-i and the ones of the outer polyhedron R-i satisfy the relation R-i = rho r(i) and R-i = R r(i) for some scale factors R > rho > 1 and for all i = 1, ... , n. We consider 3n masses located at the vertices of three nested regular polyhedra. We assume that the masses of the inner polyhedron are equal to m(1), the masses of the medium one are equal to m(2), and the masses of the outer one are equal to m(3). We prove that if the ratios of the masses m(2)/m(1) and m(3)/m(1) and the scale factors rho and R satisfy two convenient relations, then this configuration is central for the 3n-body problem. Moreover there is some numerical evidence that, first, fixed two values of the ratios m(2)/m(1) and m(3)/m(1), the 3n-body problem has a unique central configuration of this type; and second that the number of nested regular polyhedra with the same number of vertices forming a central configuration for convenient masses and sizes is arbitrary. (C) 2008 Elsevier B.V. All rights reserved.

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