  We introduce a novel technique for constructing higher-order variational
integrators for Hamiltonian systems of ODEs. In particular, we are concerned
with generating globally smooth approximations to solutions of a Hamiltonian
system. Our construction of the discrete Lagrangian adopts Hermite
interpolation polynomials and the Euler-Maclaurin quadrature formula, and
involves applying collocation to the Euler-Lagrange equation and its
prolongation. Considerable attention is devoted to the order analysis of the
resulting variational integrators in terms of approximation properties of the
Hermite polynomials and quadrature errors. A performance comparison is
presented on a selection of these integrators.
