Three natural mechanical systems on Stiefel varieties

We consider integrable generalizations of the spherical pendulum system to the Stiefel variety $V(n,r)=SO(n)\mathrm{/}SO(n-r)$ 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 $G^{+}(n,r)=SO(n)\mathrm{/}SO(r)\times SO(n-r)$.