Jacobi-Maupertius metric and Kepler equation
Sumanto Chanda, G.W. Gibbons, Partha Guha
TL;DR
This paper explores the application of the Jacobi-Eisenhart lift, Jacobi metric, and Maupertius transformation to the Kepler system, analyzing various geometric and conformal approaches to understand its integrability and symmetries.
Contribution
It introduces new methods of applying the Jacobi lift and related transformations to Kepler systems, including conformal, contact, and spacetime symmetry perspectives.
Findings
Different geometric descriptions of Kepler system are developed.
Connections between Kepler problem and oscillator via Bohlin transformation are established.
Insights into integrability and symmetry properties of Kepler system are provided.
Abstract
This article studies the application of the Jacobi-Eisenhart lift, Jacobi metric and Maupertius transformation to the Kepler system. We start by reviewing fundamentals and the Jacobi metric. Then we study various ways to apply the lift to Kepler related systems: first as conformal description and Bohlin transformation of Hooke's oscillator, second in contact geometry, third in Houri's transformation (Houri, Liouville integrability of Hamiltonian systems and spacetime symmetry, http://www.geocities.jp/football\_physicien/publication.html), coupled with Milnor's construction (Milnor, The American Mathematical Monthly 90 (1983) 353-365) with eccentric anomaly.
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