The Zhukovskiy-Volterra gyroscope: geometry and reduction

Abstract

We describe a symplectic reduction of the Zhukovskiy integrable case of motion of a rigid body about a fixed point with a gyroscope in $\mathbb R^n$ to an integrable magnetic flow on a sphere $S^{n-1}$. We also present Zhukovskiy's geometric interpretation of motion of the Euler top with a gyroscope, aka the Zhukovskiy--Volterra gyroscope, and perform a symplectic reduction of this system to an integrable magnetic flow on a $2$-dimensional sphere.