Higher Order Variational Integrators: a polynomial approach

TL;DR

This paper introduces a new family of high-order variational integrators called symplectic Galerkin schemes, which preserve geometric properties of mechanical systems and are suitable for optimal control applications.

Contribution

It develops symplectic Galerkin integrators as a novel alternative to existing methods, expanding the toolkit for structure-preserving numerical integration.

Findings

Symplectic Galerkin schemes are different from symplectic partitioned Runge-Kutta methods.

These schemes preserve symplectic structure and are applicable to optimal control.

The approach generalizes existing variational integrators for mechanical systems.

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-Bl\"obaum et al. [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 in optimal control problems, for which Campos et al. [2012b] is a particular case.