Constructing equivalence-preserving Dirac variational integrators with forces

Abstract

Abstract The dynamical motion of mechanical systems possesses underlying geometric structures and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single mechanical system, structure preservation can be achieved by adopting the variational integrator construction (Marsden, J. & West, M. (2001) Discrete mechanics and variational integrators. Acta Numer., 10, 357–514). This construction has been generalized to more complex systems involving forces or constraints as well as to the setting of Dirac mechanics (Leok, M. & Ohsawa, T. (2011) Variational and geometric structures of discrete Dirac measures. Found. Comput. Math., 11, 529–562). Forced Lagrange–Dirac systems are described by a Lagrangian and an external force pair, and two pairs of Lagrangians and external forces are said to be equivalent if they yield the same equations of motion. However, the variational discretization of a forced Lagrange–Dirac system discretizes the Lagrangian and forces separately, and will generally depend on the choice of representation. In this paper we derive a class of Dirac variational integrators with forces that yield well-defined numerical methods that are independent of the choice of representation. We present a numerical simulation to demonstrate how such equivalence-preserving discretizations avoid spurious solutions that otherwise arise from poorly chosen representations.