Error analysis of forced discrete mechanical systems

The purpose of this paper is to perform an error analysis the variational integrators mechanical systems subject external forcing. Essentially, we prove that when a discretization contact order $r$ Lagrangian and force are used, integrator has same order. Our performed first for discrete forced defined over $TQ$, where study existence flows, construction properties exact flows (variational integrators) in terms original systems. Then use those results derive corresponding analogous $Q\times Q$.