  We study constrained Hamiltonian systems by utilizing general forms of time
discretization. We show that for explicit discretizations, the requirement of
preserving the canonical Poisson bracket under discrete evolution imposes
strong conditions on both allowable discretizations and Hamiltonians. These
conditions permit time discretizations for a limited class of Hamiltonians,
which does not include homogeneous cosmological models. We also present two
general classes of implicit discretizations which preserve Poisson brackets for
any Hamiltonian. Both types of discretizations generically do not preserve
first class constraint algebras. Using this observation, we show that time
discretization provides a complicated time gauge fixing for quantum gravity
models, which may be compared with the alternative procedure of gauge fixing
before discretization.
