Geometry and solutions of the planar problem of two centers of gravitation

Abstract

The planar problem of two centers of gravitation was studied by Euler, who found a second ``momentum--like'' integral and thus the problem turned out to be completely integrable. We present some effective solutions of the motion of the free particle under the influence of the two centers. These solutions are expressed by elliptic theta functions. We also classify all types of such motions from topological point of view. There exist exactly 16 types of motions. Ten of them are unbounded and six are bounded.