Reduction and unfolding: the Kepler problem

TL;DR

This paper systematically relates the Kepler problem to the isotropic harmonic oscillator using Lagrangian formalism, providing a tangent bundle version of the Kustaanheimo-Stiefel map, which offers a new geometric perspective.

Contribution

It introduces a tangent bundle approach to connect the Kepler problem with the harmonic oscillator, extending previous methods in a Lagrangian framework.

Findings

Established a systematic relation between Kepler and harmonic oscillator problems.

Developed a tangent bundle version of the Kustaanheimo-Stiefel map.

Enhanced geometric understanding of classical mechanics problems.

Abstract

In this paper we show, in a systematic way, how to relate the Kepler problem to the isotropic harmonic oscillator. Unlike previous approaches, our constructions are carried over in the Lagrangian formalism dealing with second order vector fields. We therefore provide a tangent bundle version of the Kustaahneimo-Stiefel map.