Geometric Integrators for Mechanical Systems on Lie Groups

Retraction and discretization maps form the seed for many numerical integrators, and hence provide a general framework for discretization methods on manifolds. This approach has been extended to carry out discretizations on both the tangent and cotangent bundle leading to structure preserving integrators for mechanical systems. We explore the particular case when the configuration space happens to be a Lie group and the mechanical system exhibits certain symmetries. This case is especially interesting since it appears, for instance, on the equations of the rigid body, heavy top and ideal fluids as some special cases. In such a scenario, the discretization framework simplifies owing to the symmetries and the fact that Lie groups along with their tangent and cotangent bundles are parallelizable. The geometric integrator thus obtained can be used to discretize the Lie-Poisson-type equations that govern the motion of many mechanical systems, and more importantly, easily extend to systems with forces and optimal control problems where the configuration space is a Lie group.