  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.
