Three natural mechanical systems on Stiefel varieties

TL;DR

This paper explores integrable mechanical systems on Stiefel varieties, generalizing the spherical pendulum, and introduces new models with complete integrability on related Grassmannian spaces.

Contribution

It presents new integrable models on Stiefel varieties, including an alternative pendulum model and a system with a four-degree potential, expanding the understanding of integrable systems on these manifolds.

Findings

New integrable models on Stiefel varieties for specific metrics

Complete integrability of flows on Grassmannian varieties

Invariant relations enabling integrability of complex systems

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 $T^*V(n,r)$ which provide the complete integrability of the flow reduced on the oriented Grassmannian variety $G^+(n,r)=SO(n)/SO(r)\times SO(n-r)$.