Eisenhart Lift of $2$--Dimensional Mechanics

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.

20 pages

References in corpus (4)

• Superintegrable Systems in Darboux spaces
• Conformal Killing Tensors and covariant Hamiltonian Dynamics
• Ricci-flat spacetimes admitting higher rank Killing tensors
• Explicit metrics for a class of two-dimensional superintegrable systems

Cited by in corpus (2)

• Schwarzian mechanics via nonlinear realizations
• Integrability of geodesic motions in curved manifolds through non-local conserved charges