  We present a combination of tools which allows for investigation of the
coupled orbital and rotational dynamics of two rigid bodies with nearly
arbitrary shape and mass distribution, under the influence of their mutual
gravitational potential. Methods for calculating that mutual potential and
resulting forces and moments for a polyhedral body representation are simple
and efficient. Discrete equations of motion, referred to as the Lie Group
Variational Integrator (LGVI), preserve the structure of the configuration
space, SE(3), as well as the geometric features represented by the total energy
and the total angular momentum. The synthesis of these approaches allows us to
simulate the full two body problem accurately and efficiently. Simulation
results are given for two octahedral rigid bodies for comparison with other
integration methods and to show the qualities of the results thus obtained. A
significant improvement is seen over other integration methods while correctly
capturing the interesting effects of strong orbit and attitude dynamics
coupling, in multiple scenarios.
