  Following the discrete embedding formalism, we give a new derivation of the
mid-point variational integrators as developed by J.M. Wendlandt and J.E.
Marsden by defining an adapted order two discrete differential and integral
calculus. This allows us to obtain a clearer correspondence between the
discrete and continuous case. We also discuss the corresponding definition of a
discrete Hamiltonian system. A complete comparaison with the results of J.M.
Wendlandt and J.E. Marsden is provided.
