Magnetic flows on spheres

Abstract

We consider motion of a material point placed in a constant homogeneous magnetic field in $\mathbb{R}^n$ and restricted to the sphere $S^{n-1}$. While there is an obvious integrability of the magnetic system in $$\mathbb{R}^n$$, the integrability of the system restricted to the sphere $S^{n-1}$ is highly non-trivial. We prove complete integrability of the obtained restricted magnetic systems for $n\le 6$.