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

TL;DR

This paper introduces the Eisenhart lift as an accessible didactical tool to connect Hamiltonian dynamics with modern geometric concepts like curved spaces and symmetries, aiding graduate education and research.

Contribution

It demonstrates how the Eisenhart lift can be derived from basic principles and used to illustrate advanced geometric ideas in physics, bridging pedagogy and current research.

Findings

Eisenhart lift provides a geometric interpretation of Hamiltonian trajectories.

It serves as an educational tool for teaching modern geometry in physics.

The method has applications in dynamical systems and non-relativistic holography.

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.