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.

We develop a discrete analogue of the Hamilton-Jacobi theory in the framework
of the discrete Hamiltonian mechanics. We first reinterpret the discrete
Hamilton-Jacobi equation derived by Elnatanov and Schiff in the language of
discrete mechanics. The resulting discrete Hamilton-Jacobi equation is discrete
only in time, and is shown to recover the Hamilton-Jacobi equation in the
continuous-time limit. The correspondence between discrete and continuous
Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi's
solution to the Hamilton-Jacobi equation.

This paper formulates an optimal control problem for a system of rigid bodies
that are connected by ball joints and immersed in an irrotational and
incompressible fluid. The rigid bodies can translate and rotate in
three-dimensional space, and each joint has three rotational degrees of
freedom. We assume that internal control moments are applied at each joint. We
present a computational procedure for numerically solving this optimal control
problem, based on a geometric numerical integrator referred to as a Lie group
variational integrator.