A nonholonomic approach to isoholonomic problems and some applications

HERNÁN CENDRA; SEBASTIÁN J. FERRARO

DYNAMICAL SYSTEMS

Taylor & Francis

Año: 2006 vol. 21 p. 409 - 437

1468-9367

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.