Central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the-3 body problem

The solutions of the 2-body problem (also called the Kepler problem) has been completely solved, but the solutions for the n-body for n>2, is still an open problem. For the Newtonian n-body problem the simplest possible motions are such that the confi guration formed by the n-bodies is constant up to rotations and scaling, such motions are called the homographic solutions of the n-body problem, and are the unique known explicit solutions of the n-body problem when n>2. Only some special confi gurations of particles are allowed in the homographic solutions of the n-body problem, called by Wintner [1] central confi gurations. Also, central confi gurations are of utmost importance when studying bifurcations of the hypersurfaces of constant energy and angular momentum, for more details see Meyer [2] and Smale [3]. These last years some central confi gurations have been used for different missions of the spacecrafts in the solar system, see for instance [4,5].