Darboux inversions of the Kepler problem

TL;DR

This paper explores Darboux inversions of the Kepler problem on constant curvature surfaces, explaining their property of having all orbits periodic, and connects this to classical inverse transformations in differential geometry.

Contribution

It provides a detailed description of Darboux inverses of the Kepler problem, clarifying their geometric and dynamical properties and historical context.

Findings

Darboux inverses of Kepler problem have all orbits periodic on open sets.

These inverses are related to conformal maps and classical differential geometry.

The paper explains the geometric reason behind the periodicity of orbits.

Abstract

While extending a famous problem asked and solved by Bertrand in 1873, Darboux found in 1877 a family of abstract surfaces of revolution, each endowed with a force function, with the striking property that all the orbits are periodic on open sets of the phase space. We give a description of this family which explains why they have this property: they are the Darboux inverses of the Kepler problem on constant curvature surfaces. What we call the Darboux inverse was briefly introduced by Darboux in 1889 as an alternative approach to the conformal maps that Goursat had just described.