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

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, $N\ge 2$. 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 ${\Gamma }_1\cup {\Gamma }_2$, and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains ${\Gamma }_1\cup {\Gamma }_2$ and passes through the geometric center of ${\Gamma }_1\cup {\Gamma }_2$. 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.