A jet symplectic algorithm for Euler-Lagrange systems is studied.It is shown that the discrete Euler-Lagrange (DEL) equation,which was given by the second author in 1998,has fundamental geometric structures that preserve along solutions obtained directly from the variatonal principle.It is shown that these difference schemes are jet symplectic and the Nother's theorem exists by which we give the definition of a discrete version,the momentum map.A numerical example in jet symplectic difference scheme is given.A comparison with other discretization schemes was made.

A method preserving structures of the Hamiltonian systems is considered. On the basis of the jet symplectic difference scheme for canonical Hamiltonians the jet symplectic difference scheme for Hamiltonian systems in general symplectic structure with variable coefficientsic is defined. According to the general approach of the generating function method for the symplectic difference schemes a relation between the general symplectic structure and the generating functions is found. The jet symplectic difference schemes for classical Hamiltonian systems are constructed in terms of Hamilton-Jacobi equation.