  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 $2s$.
