Integrability of the magnetic geodesic flow on the sphere with a constant 2-form

TL;DR

This paper proves that magnetic geodesic flows on spheres with a specific constant 2-form are Liouville integrable, confirming a recent conjecture and identifying quadratic and linear integrals in momenta.

Contribution

It establishes the Liouville integrability of magnetic geodesic flows on spheres with a constant 2-form, confirming a conjecture and characterizing the integrals involved.

Findings

Magnetic geodesic flow on spheres with a constant 2-form is Liouville integrable.

The integrals of motion are quadratic and linear in momenta.

The result confirms a recent conjecture by Dragovic et al.

Abstract

We prove a recent conjecture of Dragovic et al arXiv2504.20515 stating that the magnetic geodesic flow on the standard sphere $S^n\subset \mathbb R^{n+1}$ whose magnetic 2-form is the restriction of a constant 2-form from $\mathbb{R}^{n+1}$ is Liouville integrable. The integrals are quadratic and linear in momenta.