The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics

Abstract

This work originates from part of a final year undergraduate research project on the Eisenhart lift for Hamiltonian systems. The Eisenhart lift is a procedure to describe trajectories of a classical natural Hamiltonian system as geodesics in an enlarged space. We point out that it can be easily obtained from basic principles of Hamiltonian dynamics, and as such it represents a useful didactical way to introduce graduate students to several modern concepts of geometry applied to physics: curved spaces, both Riemannian and Lorentzian, conformal transformations, geometrisation of interactions and extra dimensions, geometrisation of dynamical symmetries. For all these concepts the Eisenhart lift can be used as a theoretical tool that provides easily achievable examples, with the added benefit of also being a topic of current research with several applications, among which the study of dynamical systems and non-relativistic holography.

Jacobi-Maupertuis-Eisenhart metric and geodesic flows

“…One way to deal with time-dependent systems is the Eisenhart-Duval lift. The Eisenhart-Duval lift, developed by L.P. Eisenhart [15] and rediscovered by C. Duval [16], with applications demonstrated in [23,24] embeds non-relativistic theories into Lorentzian geometry. It is one example of a method for geometrizing interactions, where a classical system in n dimensions is shown to be dynamically equal to a Lorentzian n+2 spacetime.…”

Integrability of geodesic motions in curved manifolds through nonlocal conserved charges

“…A similar correspondence can be claimed for regular (non-parametrization invariant) mechanical systems up to a transformation in time with the use of the Jacobi metric [2,8,20,21]. An alternative way of a geometrization of classical regular mechanical problems is supplemented by the Eisenhart-Duval lift [22][23][24][25] which was initially introduced by Eisenhart and later rediscovered in a more physical context by Duval and collaborators; for a recent treatment of two dimensional problems see [26]. Thus, we see that the motion of a relativistic particle in a curved manifold can be associated with several different problems.…”

Finite measure for the initial conditions of inflation

“…The Eisenhart lift [38][39][40] is a formalism in classical mechanics that may be used to write any system subject to a conservative force as an equivalent free system moving on a higher-dimensional curved manifold. This technique has been recently extended to scalar field theories [41], and can therefore be readily applied to the theory of inflation.…”

“…Clearly, a new outlook on the problem is required. In this paper, we will examine the Eisenhart lift [38][39][40][41] as a possible approach to the measure problem. The Eisenhart-lift formalism allows any scalar field theory to be transformed ("lifted") into a purely geometric system without altering its classical dynamics.…”