Central Configurations of Four Bodies with Gravitational Charges of Both Signs

Abstract

The Newtonian Four-Body Problem with positive and negative gravitational charges is studied from a physical point of view that considers inertial masses that are always positive and positive or negative charges in the gravitational interaction forces. The difference with respect to the differential equation of motion used in several mathematical papers is stressed. Formulation of the modified Dziobek equations is developed. We prove that four masses with one charge of different sign lead to a concave central configuration with the charge of different sign in the convex hull of the other three. For two charges of one sign and two charges of the opposite sign, we prove that the central configuration is convex with the two charges of the same sign on each diagonal. An algorithm used for computing the six distances between particles of a planar Four-Body central configuration, presented previously for positive gravitational masses, is adapted to compute physical central configurations with charges of different sign.

Libration Points Inside a Spherical Cavity of a Uniformly Rotating Gravitating Ball

“…Remark 7. The mechanics of systems with negative masses (either gravitational, or inertial) is the subject of active study not only in celestial mechanics (see, e. g., [22][23][24][25]), but also in theoretical physics (see, e. g., [26]). Objects with cavities are, in our opinion, good examples of systems that often exhibit somewhat unexpected dynamical effects (see, e. g., [27]).…”