Variational integrators using forced discrete Hamiltonian systems

Abstract

We study discrete Hamiltonian systems defined on cotangent bundles that are subjected to external forces, whose trajectories are determined by a discrete variational principle. We analyze the evolution of the canonical symplectic structure and, when a Lie group of symmetries is present, the corresponding evolution of the associated momenta. Given a continuous forced Hamiltonian system, we construct an exact discrete analogue whose order-$r$ approximations yield trajectories that approximate the continuous ones with accuracy of at least order $r$. We also give two methods to build approximate discrete systems. Combining these, we obtain a variational integrator: first approximate the exact discrete system and then solve the resulting algebraic equations of motion.