  Some problems on variations are raised for classical discrete mechanics and
field theory and the difference variational approach with variable step-length
is proposed motivated by Lee's approach to discrete mechanics and the
difference discrete variational principle for difference discrete mechanics and
field theory on regular lattice. Based upon Hamilton's principle for the
vertical variations and double operation of vertical exterior differential on
action, it is shown that for both continuous and variable step-length
difference cases there exists the nontrivial Euler-Lagrange cohomology as well
as the necessary and sufficient condition for symplectic/multi-symplectic
structure preserving properties is the relevant Euler-Lagrange 1-form is closed
in both continuous and difference classical mechanics and field theory. While
the horizontal variations give rise to the relevant identities or relations of
the Euler-Lagrange equation and conservation law of the energy/energy-momentum
tensor for continuous or discrete systems. The total variations are also
discussed. Especially, for those discrete cases the variable step-length of the
difference is determined by the relation between the Euler-Lagrange equation
and conservation law of the energy/energy-momentum tensor. In addition, this
approach together with difference version of the Euler-Lagrange cohomology can
be applied not only to discrete Lagrangian formalism but also to the
Hamiltonian formalism for difference mechanics and field theory.
