The Two Fixed Centers Problem Revisited

Abstract

The study of the motion of a point mass, moving in the gravitational field of two fixed attracting centers, is a problem first posed by Euler in the 18th century, as an intermediate step towards the solution of the famous three-body problem. Euler himself, in a series of three papers (Euler 1766a, 1766b and 1767), was able to integrate the equations of motion for the two-dimensional (2-D) case, i.e. the case where the point mass moves on a plane containing the two attracting centers. Almost a century later Jacobi (1842) showed that the corresponding potential of the full 3-D case is separable in prolate spheroidal coordinates. Another century later Erikson and Hill (1949) found, in explicit form, the third integral of motion of the full three-dimensional (3-D) case (besides the total energy and the angular momentum about the axis passing through the two centers). Since then the problem has been considered as a non-exciting example of a separable potential and it is included, as such, in many textbooks of Theoretical Mechanics.