  We study the Euler-Lagrange cohomology and explore the symplectic or
multisymplectic geometry and their preserving properties in classical mechanism
and classical field theory in Lagrangian and Hamiltonian formalism in each case
respectively. By virtue of the Euler-Lagrange cohomology that is nontrivial in
the configuration space, we show that the symplectic or multisymplectic
geometry and related preserving property can be established not only in the
solution space but also in the function space if and only if the relevant
closed Euler-Lagrange cohomological condition is satisfied in each case. We
also apply the cohomological approach directly to Hamiltonian-like ODEs and
Hamiltonian-like PDEs no matter whether there exist known Lagrangian and/or
Hamiltonian associated with them.
