  In this paper we provide a variational derivation of the Euler-Poincar\'e
equations for systems subjected to external forces using an adaptation of the
techniques introduced by Galley and others. Moreover, we study in detail the
underlying geometry which is related to the notion of Poisson groupoid.
Finally, we apply the previous construction to the formal derivation of the
variational error for numerical integrators of forced Euler-Poincar\'e
equations and the application of this theory to the derivation of geometric
integrators for forced systems.
