  In this work we derive and analyze variational integrators of higher order
for the structure-preserving simulation of mechanical systems. The construction
is based on a space of polynomials together with Gauss and Lobatto quadrature
rules to approximate the relevant integrals in the variational principle. The
use of higher order schemes increases the accuracy of the discrete solution and
thereby decrease the computational cost while the preservation properties of
the scheme are still guaranteed. The order of convergence of the resulting
variational integrators are investigated numerically and it is discussed which
combination of space of polynomials and quadrature rules provide optimal
convergence rates. For particular integrators the order can be increased
compared to the Galerkin variational integrators previously introduced in
Marsden & West 2001. Furthermore, linear stability properties, time
reversibility, structure-preserving properties as well as efficiency for the
constructed variational integrators are investigated and demonstrated by
numerical examples.
