Eisenhart Lift of $2$--Dimensional Mechanics

Understand

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.