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