Central Configurations for Newtonian N + 2 p + 1 -Body Problems

Abstract

We show the existence of spatial central configurations for the $N+2p+1$-body problems. In the $N+2p+1$-body problems, N bodies are at the vertices of a regular N-gon T ; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T; the $N+2p+1$th is located at the the center of T. The masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal.