  Most classical mechanical systems are based on dynamical variables whose
values are real numbers. Energy conservation is then guaranteed if the
dynamical equations are phrased in terms of a Hamiltonian function, which then
leads to differential equations in the time variable. If these real dynamical
variables are instead replaced by integers, and also the time variable is
restricted to integers, it appears to be hard to enforce energy conservation
unless one can also derive a Hamiltonian formalism for that case. We here show
how the Hamiltonian formalism works here, and how it may yield the usual
Hamilton equations in the continuum limit. The question was motivated by the
author's investigations of special quantum systems that allow for a
deterministic interpretation. The 'discrete Hamiltonian formalism' appears to
shed new light on these approaches.
