Discrete variation, Euler-Lagrange cohomology and symplectic, multisymplectic structures

We introduce the discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object. By virtue of this variational principle, we get the difference discrete Euler-Lagrange equations for the difference discrete classical mechanics and classical field theory. We also explore the difference discrete versions for the Euler-Lagrange cohomology and apply to the symplectic or multisymplectic geometry and preserving property in discrete mechanics and field theory. 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 equations but also in the function space in each case if and only if the relevant closed Euler-Lagrange cohomological conditions are satisfied.