Generalization of the Lagrange equilateral-triangle solution and the Euler collinear solution to nongravitational forces in the three-body problem

In the gravitational three-body problem there are special configurations that allow exact solutions. They are the equilateral-triangle and collinear configurations. In this paper I show that exact solutions for these special configurations and others as well exist for forces other than gravitational. The interactions include "scalar" forces or those derived from a world scalar potential as well as Coulomb or "vector" forces. In the formalism used for this classical treatment the aspects emphasized are those which can be carried over to quantum mechanics. An application to an atomic system yields a reasonable estimate of the energy levels for helium-like atoms.