  Elegant integration schemes of second and fourth order for simulations of
rigid body systems are presented which treat translational and rotational
motion on the same footing. This is made possible by a recent implementation of
the exact solution of free rigid body motion. The two schemes are
time-reversible, symplectic, and exactly respect conservation principles for
both the total linear and angular momentum vectors. Simulations of simple test
systems show that the second order scheme is stable and conserves all constants
of the motion to high precision. Furthermore, the schemes are demonstrated to
be more accurate and efficient than existing methods, except for high
densities, in which case the second order scheme performs at least as well,
showing their general applicability. Finally, it is demonstrated that the
fourth order scheme is more efficient than the second order scheme provided the
time step is smaller than a system-dependent threshold value.
