https://doi.org/10.1140/epjc/s10052-019-6812-6
Regular Article - Theoretical Physics
Eisenhart lift of 2-dimensional mechanics
1
School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK
2
School of Physics, Tomsk Polytechnic University, 634050, Tomsk, Russia
* e-mail: galajin@tpu.ru
Received:
30
January
2019
Accepted:
25
March
2019
Published online:
3
April
2019
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 
-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.
© The Author(s), 2019
