Two connected bodies in a central gravitational field

The dynamics of two bodies connected by a hinge joint, and moving in a plane under the action of a central gravitational force field is analyzed. Each body is modeled as a rigid massless link with a point mass at one end; their other ends are connected together by a hinge joint. The equations of motion of the connected bodies include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom give rise to a complex dynamical system. Relative equilibria, corresponding to circular orbits of fixed radius, are obtained from these equations of motion. The free dynamics has a symmetry due to the cyclic coordinate representing the true anomaly. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. We carry out stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations from these relative equilibria, to confirm their stability properties. These numerical simulations also suggest the use of shape change to alter the overall orientation and orbit of the assembly.