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 four-degree potential. The latter has invariant relations on $T^{\ast }V(n,r)$ which provide the complete integrability of the flow reduced on the oriented Grassmannian variety $G^{+}(n,r)=SO(n)/SO(r)\times SO(n-r)$.

14 pages

References in corpus (3)

• A variational problem on Stiefel manifolds
• Geodesic Flows and Neumann Systems on Stiefel Varieties. Geometry and Integrability
• On the Cartan Model of the Canonical Vector Bundles over Grassmannians

Cited by in corpus (1)

• Contact magnetic geodesic and sub-Riemannian flows on $V_{n,2}$ and integrable cases of a heavy rigid body with a gyrostat