  In this paper, we present a new variational integrator for problems in
Lagrangian mechanics. Using techniques from Galerkin variational integrators,
we construct a scheme for numerical integration that converges geometrically,
and is symplectic and momentum preserving. Furthermore, we prove that under
appropriate assumptions, variational integrators constructed using Galerkin
techniques will yield numerical methods that are in a certain sense optimal,
converging at the same rate as the best possible approximation in a certain
function space. We further prove that certain geometric invariants also
converge at an optimal rate, and that the error associated with these geometric
invariants is independent of the number of steps taken. We close with several
numerical examples that demonstrate the predicted rates of convergence.
