Three natural mechanical systems on Stiefel varieties

Abstract

We consider integrable generalizations of the spherical pendulum system to the Stiefel variety V(n, r) = SO(n)/SO(n r) for a certain metric. For the case of V(n, 2) an alternative integrable model of the pendulum is presented. We also describe a system on the Stiefel variety with a fourth-degree potential. The latter has invariant relations on T*V(n, r) which provide the complete integrability of the flow reduced on the oriented Grassmannian variety G +(n, r) = SO(n)/SO(r) × SO(n r).