Difference equations for Hamiltonian systems are derived from a discrete variational principle. The difference equations completely determine piecewise-linear, continuous trajectories which exactly conserve the Hamiltonian function at the midpoints of each linear segment. A generating function exists for transformations between the vertices of the trajectories. Existence and uniqueness results are present as well as simulation results for a simple pendulum and an inverse square law system.
Shibberu, Yosi, "Time-Discretization of Hamiltonian Dynamical Systems" (1993). Mathematical Sciences Technical Reports (MSTR). 81.