  We analyze the canonical treatment of classical constrained mechanical
systems formulated with a discrete time. We prove that under very general
conditions, it is possible to introduce nonsingular canonical transformations
that preserve the constraint surface and the Poisson or Dirac bracket
structure. The conditions for the preservation of the constraints are more
stringent than in the continuous case and as a consequence some of the
continuum constraints become second class upon discretization and need to be
solved by fixing their associated Lagrange multipliers. The gauge invariance of
the discrete theory is encoded in a set of arbitrary functions that appear in
the generating function of the evolution equations. The resulting scheme is
general enough to accommodate the treatment of field theories on the lattice.
This paper attempts to clarify and put on sounder footing a discretization
technique that has already been used to treat a variety of systems, including
Yang--Mills theories, BF-theory and general relativity on the lattice.
