Jacobi dynamics and the N-body problem with variable masses

An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered.