  We study the difference discrete variational principle in the framework of
multi-parameter differential approach by regarding the forward difference as an
entire geometric object in view of noncomutative differential geometry. By
virtue of this variational principle, we get the difference discrete
Euler-Lagrange equations and canonical ones for the difference discrete
versions of the classical mechanics and classical field theory. We also explore
the difference discrete versions for the Euler-Lagrange cohomology and apply
them to the symplectic or multisymplectic geometry and their preserving
properties in both Lagrangian and Hamiltonian formalism. In terms of the
difference discrete Euler-Lagrange cohomological concepts, we show that the
symplectic or multisymplectic geometry and their difference discrete structure
preserving properties can always be established not only in the solution spaces
of the discrete Euler-Lagrange/canonical equations derived by the difference
discrete variational principle but also in the function space in each case if
and only if the relevant closed Euler-Lagrange cohomological conditions are
satisfied. We also apply the difference discrete variational principle and
cohomological approach directly to the symplectic and multisymplectic
algorithms.
