Eisenhart lift of 2-dimensional mechanics

Allan P. Fordy, Anton Galajinsky

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Article number301
JournalEuropean Physical Journal C
Issue number4
Publication statusPublished - 1 Apr 2019

ASJC Scopus subject areas

  • Engineering (miscellaneous)
  • Physics and Astronomy (miscellaneous)

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