Equivalent Integrable Metrics on the Sphere with Quartic Invariants

TL;DR

This paper explores transformations linking classical geodesic flows on the sphere to new integrable systems with quartic invariants, introducing potentials to generate novel integrable models.

Contribution

It introduces a method to construct new integrable systems on the sphere with quartic invariants via canonical transformations and potential additions.

Findings

Established canonical transformations between geodesic flows

Constructed new integrable systems with quartic invariants

Extended known models by adding potentials

Abstract

We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.