Equivalence between Type I Liouville dynamical systems in the plane and the sphere

TL;DR

This paper demonstrates a unified approach connecting Type I Liouville separable Hamiltonian systems on the sphere and the plane using gnomonic projections, revealing their equivalence and classical relationships.

Contribution

It introduces a method to transform and relate spherical and planar Liouville Type I systems via gnomonic projections, unifying their descriptions.

Findings

Gnomonic projection maps spherical systems to planar systems and vice versa.

Several spherical and planar separable systems are explicitly connected.

The approach provides a classical framework for understanding these systems' equivalence.

Abstract

Separable Hamiltonian systems either in sphero-conical coordinates on a $S^2$ sphere or in elliptic coordinates on a ${\mathbb R}^2$ plane are described in an unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with an spherical configuration space to its Liouville Type I partner where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context.