Geometric integrators for higher-order mechanics on Lie groups

This paper develops a structure-preserving numerical integration scheme for a class of higher-order mechanical systems. The dynamics of these systems are governed by invariant variational principles defined on higher-order tangent bundles of Lie groups. The variational principles admit Lagrangians that depend on acceleration, for example. The symmetry reduction method used in the Hamilton--Pontryagin approach for developing variational integrators of first-order mechanics is extended here to higher order. The paper discusses the general approach and then focuses on the primary example of Riemannian cubics. Higher-order variational integrators are developed both for the discrete-time integration of the initial value problem and for a particular type of trajectory-planning problem. The solution of the discrete trajectory-planning problem for higher-order interpolation among points on the sphere illustrates the approach.