Abstract and Applied Analysis
Volume 2012 (2012), Article ID 948356, 19 pages
http://dx.doi.org/10.1155/2012/948356
Research Article

New Classes of Spatial Central Configurations for N + N + 2-Body Problem

Liu Xuefei, Zhang Chuntao, Luo Jianmei, and Zhang Gan

Department of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100, China

Received 9 December 2011; Accepted 16 February 2012

Academic Editor: Wing-Sum Cheung

Copyright © 2012 Liu Xuefei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Under arbitrary masses, in this paper, we discuss the existence of new families of spatial central configurations for the N + N + 2-body problem, 𝑁 ≥ 2 . We study some necessary conditions and sufficient conditions for a families of spatial double pyramidical central configurations (d.p.c.c.), where 2N bodies are at the vertices of a nested regular N-gons Γ 1 ∪ Γ 2 , and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains Γ 1 ∪ Γ 2 and passes through the geometric center of Γ 1 ∪ Γ 2 . We prove that if the bodies are in a d.p.c.c., then the masses on each N-gon are equal, and the other two are also equal. And also we prove the existence and uniqueness of the central configurations for any given ratios of masses.