Eisenhart lift of 2-dimensional mechanics

Allan P. Fordy, Anton Galajinsky

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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
Volume79
Issue number4
DOIs
Publication statusPublished - 1 Apr 2019

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Mechanics
symmetry
Degrees of freedom (mechanics)
Equations of motion
Tensors
classical mechanics
Dynamical systems
Einstein equations
dynamical systems
equations of motion
degrees of freedom
signatures
tensors
scalars
energy

ASJC Scopus subject areas

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

Cite this

Eisenhart lift of 2-dimensional mechanics. / Fordy, Allan P.; Galajinsky, Anton.

In: European Physical Journal C, Vol. 79, No. 4, 301, 01.04.2019.

Research output: Contribution to journalArticle

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