# Eisenhart Lift of $2$--Dimensional Mechanics

## TL;DR

This paper explores the Eisenhart lift of 2D mechanics on curved backgrounds, deriving symmetries, constructing solutions to Einstein equations, and analyzing superintegrable models with hidden symmetries.

## Contribution

It provides a detailed study of the Eisenhart lift for 2D systems, including conformal symmetries, energy-momentum tensors, and conditions for physically viable metrics.

## Key findings

- Derived conformal symmetry and quadratic integral.
- Constructed energy--momentum tensor satisfying Einstein equations.
- Identified conditions under which Darboux--Koenigs metrics meet energy conditions.

## 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.