Abstract by Jacob Murri
Collinear Equilibria in a Simplified 4-Body Problem
The N-Body problem examines the movement of bodies in space under their mutual gravitational attraction. We consider a specific case of the 4-body problem which includes two primary masses and two small masses moving in the potential induced by the primaries. This problem warrants study because it forms a bridge between the restricted 3-body problem and the general 4-body problem (of which it is a special case). Our goal is to demonstrate the similarities and differences between these problems. We begin by consider collinear equilibrium solutions (for which the four masses remain on the same line in rotating coordinates). We show analytically that each of the 4-body problem’s twelve collinear equilibria persist in the simplified problem by restricting to one radial coordinate for each primary and considering twelve distinct regions produced on this 2-dimensional plane. Furthermore, we begin to address stability using the eigenvalues of the Hessian matrix for the Hamiltonian system.