Variational integrators for forced Birkhoffian systems

In this letter, we first extend the Pfaff-Birkhoff principle to include forces and obtain the Pfaff-Birkhoff-D'Alembert principle consequently. Well then motions of forced Birkhoffian systems coincide with extremals of the modified principle. Subsequently, compared with the continuous case, the discrete forced Birkhoffian equations are constructed by discretizing the Pfaff-Birkhoff-D'Alembert principle. Considered as algorithms, the discrete equations have good numerical behavior in terms of getting the correct amounts by which the energy changes over the integration process, demonstrated by the given example.