On the Schwarzschild-Type Polygonal ( n + 1 ) - Body Problem and on the Associated Restricted Problem

The planar symmetrical (n+1)-body problem in a Schwarzschild-type field is investigated. It is shown that, if n equal masses are initially situated at the vertices of a regular polygon centered in the (n+1)-th mass, and if the initial velocities form a vector field symmetrical with respect to the central mass, then the polygonal configuration is preserved all along the motion, but with variable side and with variable rotation around the center. The motion of every mass relative to the center is given by the solution of the Schwarzschild-type two-body problem. All possible behaviors of the polygonal solution are surveyed. The relative equilibria of the (n+1)-body problem are considered. A restricted problem is associated with them, for which the Jacobi integral is shown to exist. (Author)