2019/01/10 by Allan P. Fordy, Anton Galajinsky · 1 citation
Physics and Astronomy · Mathematics · #nlin.SI #gr-qc #hep-th #math-ph #math.MP #physics.class-ph #msc:17B63 #msc:37J35 #msc:70G45 #msc:70H06
paper · pdf · doi:10.1140/epjc/s10052-019-6812-6
20 pages
arxiv created 2019/01/10 · arxiv updated 2019/05/01
The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy--momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux--Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented.