Superconvergence of Galerkin variational integrators

We study the order of convergence of Galerkin variational integrators for ordinary differential equations. Galerkin variational integrators approximate a variational (Lagrangian) problem by restricting the space of curves to the set of polynomials of degree at most $s$ and approximating the action integral using a quadrature rule. We show that, if the quadrature rule is sufficiently accurate, the order of the integrators thus obtained is $2s$.