Galerkin Lie-group variational integrators based on unit quaternion interpolation

Abstract

Lie-group variational integrators of arbitrary order are developed using the Galerkin method, based on unit quaternion interpolation. To our knowledge, quaternions have not been used before for this purpose, though they allow a very simple and efficient way to perform the interpolation. The resulting integrators are symplectic and structure preserving, in the sense that certain symmetries in the Lagrangian of the mechanical system are carried over to the discrete setting, which leads to the preservation of the corresponding momentum maps. The integrators furthermore exhibit a very good long time energy behavior, i.e. energy is neither dissipated nor gained artificially. At the same time, the Lie-group structure is preserved by carefully defining the variations, the interpolation method and by solving the non-linear system of equations directly on the manifold, rather than constraining it in a surrounding space using Lagrange multipliers. As a consequence, we are able to show that Lie-group variational integrators based on the special orthogonal group, are equivalent to the variational integrators for constrained systems using the discrete null-space method employed e.g. in DMOCC (discrete mechanics and optimal control of constrained systems). We show new numerical results on the convergence rates, which are substantially higher than the known theoretical bounds, and on the relation between accuracy and computational cost.