A nonholonomic approach to isoparallel problems and some applications

Abstract

Isoparallel problems are a class of optimal control problems on principal fibre bundles endowed with a connection and a Riemannian metric on the base space. These problems consist of finding the shortest curve on the base among those with a given parallel transport operator. It has been shown that when the structure group of the principal bundle admits a bi-invariant metric, the normal solutions are precisely the projections of the geodesics (relative to an appropriate Riemannian metric) on the bundle. In this work we obtain a generalization of this result that holds true for any structure group, by transforming the isoparallel problem into a nonholonomic problem of a generalized type. The latter reduces to the geodesic problem if the structure group has a bi-invariant metric. We illustrate the theory with an application to the optimal control of an elastic rolling ball (the plate–ball system), relating some aspects of this problem to the dynamics of a simple pendulum. Finally, we indicate how the study of locomotion of microorganisms can benefit from this approach. This work shows how optimal control and generalized nonholonomic mechanics are related within the context of Lagrangian reduction.