High order variational integrators: A polynomial approach

Abstract

We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the structural properties of these systems, like the symplectic form, the evolution of the momentum maps or the energy behaviour. Also they are easily applicable to optimal control problems based on mechanical systems as proposed in Ober-Blöbaum et al. (ESAIM Control Optim Calc Var 17(2):322-352, 2011). Following the same approach, we develop a family of variational integrators to which we refer as symplectic Galerkin schemes in contrast to symplectic partitioned Runge-Kutta. These two families of integrators are, in principle and by construction, different one from the other. Furthermore, the symplectic Galerkin family can as easily be applied to optimal control problems, for which Campos et al. (Higher order variational time discretization of optimal control problems. In: Proceedings of the 20th international symposium on mathematical theory of networks and systems, Melbourne, 2012) is a particular case.