On symplectic and multisymplectic structures and their discrete versions in Lagrangian formalism

We introduce the Euler-Lagrange cohomology to study the symplectic and mult isymplectic structures and their preserving properties in finite and infini te dimensional Lagrangian systems respectively We also explore their cei ta in difference discrete counterparts in the relevant regularly discretized f inite and infinite dimensional Lagrangian systems by means of the differenc e discrete variational principle with the difference being regarded as an e ntire geometric object and the noncommutative differential calculus on regu lar lattice. In order to show that in all these cases the symplectic and mu ltisymplectic preserving properties do not necessarily depend on the releva nt Euler-Lagrange equations, the Euler-Lagrange cohomological concepts and content in the configuration space are employed.