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

Mid-point embedding of Hamiltonian systems and variational integrators