Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on thefunction space of the infinite set and construct a homotopy operator to prove the analogue of the Poincarè lemma forthe difference complex. Then the horizontal and vertical complexesare introduced with the total differential map and vertical exterior derivative. As the application of thedifferential calculus, we derive the schemes with the conservationof symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents.Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.