Total Collision in a Four-Body Problem with Jacobi Potential

Abstract

We study the dynamics of a spatial four-body problem where the bodies maintain a rhombus-shape configuration at all times: two of the bodies of equal mass move in the horizontal plane symmetrically with respect to the origin while another pair of bodies of equal mass move symmetrically opposed along the vertical axis. The bodies interact via the Jacobi potential, an attractive binary potential of the form $-1/x^2$, where x is the distance between the particles. We use appropriate transformations to blow up total collision into a manifold pasted onto the phase space for all levels of energy. We find that the topology of the total collision manifold changes as the angular momentum varies, and also that the dynamics on the collision manifold changes as a function of angular momentum and the mass ratio. We also give a qualitative description of the global flow of the problem for negative energy. This description utilizes knowledge concerning the flow on and near the collision manifold, the presence of an additional integral of motion, and takes advantage of the time-reversing symmetry inherent in the system of equations.