  We present a new class of high-order variational integrators on Lie groups.
We show that these integrators are symplectic, momentum preserving, and can be
constructed to be of arbitrarily high-order, or can be made to converge
geometrically. Furthermore, these methods are stable and accurate for very
large time steps. We demonstrate the construction of one such variational
integrator for the rigid body, and discuss how this construction could be
generalized to other related Lie group problems. We close with several
numerical examples which demonstrate our claims, and discuss further extensions
of our work.
