The pyramidal configurations of N + 1 bodies cannot rotate

For the (N + 1)-body problem, we assume that N bodies are at the vertices of a unit regular polygon and the (N + 1)st body is along the vertical line normal to the plane formed by the former N bodies. If N bodies rotate at the unit circle and the (N + 1)st body oscillates along the vertical line of the plane formed by the former N bodies and passing through the geometrical center, then we prove that the (N + 1)st body must locate at the geometrical center of unit regular polygon.  2003 Elsevier Inc. All rights reserved.