Eisenhart lift of Koopman-von Neumann mechanics

TL;DR

This paper geometrizes Koopman-von Neumann classical mechanics using Eisenhart lift, revealing transformations between fundamental systems like harmonic oscillator, linear potential, and free particle within this geometric framework.

Contribution

It introduces a geometric approach to KvN mechanics via Eisenhart lift, connecting classical and relativistic physics and identifying transformations among key classical systems.

Findings

Geometrization of KvN mechanics using Eisenhart lift

Identification of transformations between classical systems in KvN framework

Enhanced understanding of classical mechanics structure through geometry

Abstract

The Eisenhart lift establishes a fascinating connection between non-relativistic and relativistic physics, providing a space-time geometric understanding of non-relativistic Newtonian mechanics. What is still little known, however, is the fact that there is a Hilbert space representation of classical mechanics (also called Koopman-von Neumann mechanics) that attempts to give classical mechanics the same mathematical structure that quantum mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN) mechanics using the Eisenhart toolkit. We then use a geometric view of KvN mechanics to find transformations that relate the harmonic oscillator, linear potential, and free particle in the context of KvN mechanics.