Planar central configurations with five different positive masses

Abstract

We study planar five-body central configurations with different positive masses. From Laura–Andoyer equations, we recover the variational formulation in terms of the potential energy and the total inertia moment, including constraints on the distances. We use coordinates to prove the permanence of the central configuration by Newton equations of motion. The masses appear in our coordinates, forming a rigid orthocentric simplex in four dimensions, with the masses at the vertices. The linear dimensions are determined by a distance. We use different angles, with one angle measuring the rotation around the center of mass. The rest of the coordinates are angles that are constants of motion for central configurations. One angle measures the ratio of the two principal moments of inertia. The rest of the coordinates determine the rotation around the center of mass of the rigid simplex of masses with respect to the plane and to the direction of projection from four to two dimensions. We transform the Laura–Andoyer equations into different expressions in 10 and 5 dimensions, including the properties of the areas formed by five point bodies in the plane. Furthermore, we presented three numerical computed configurations of planar five-body central configurations, one convex and two concave, which give numerical confirmation of our equations with a precision better than 10−12.