  Numerical lattice gauge theory computations to generate gauge field
configurations including the effects of dynamical fermions are usually carried
out using algorithms that require the molecular dynamics evolution of gauge
fields using symplectic integrators. Sophisticated integrators are in common
use but are hard to optimise, and force-gradient integrators show promise
especially for large lattice volumes. We explain why symplectic integrators
lead to very efficient Monte Carlo algorithms because they exactly conserve a
shadow Hamiltonian. The shadow Hamiltonian may be expanded in terms of Poisson
brackets, and can be used to optimize the integrators. We show how this may be
done for gauge theories by extending the formulation of Hamiltonian mechanics
on Lie groups to include Poisson brackets and shadows, and by giving a general
method for the practical computation of forces, force-gradients, and Poisson
brackets for gauge theories.
