Eisenhart lift for higher derivative systems

TL;DR

This paper extends the Eisenhart lift, a geometric method for second-order systems, to higher derivative models using Ostrogradsky's Hamiltonian, with the Pais-Uhlenbeck oscillator as an example.

Contribution

It generalizes the Eisenhart lift to higher derivative systems, providing a geometric framework for a specific class of potentials.

Findings

Eisenhart lift can be extended to certain higher derivative models.

A geometric description is feasible for a particular class of potentials.

The Pais-Uhlenbeck oscillator exemplifies the generalized scheme.

Abstract

The Eisenhart lift provides an elegant geometric description of a dynamical system of second order in terms of null geodesics of the Brinkmann-type metric. In this work, we attempt to generalize the Eisenhart method so as to encompass higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian. A consistent geometric description seems feasible only for a particular class of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.