Null lifts and projective dynamics

TL;DR

This paper introduces a geometric framework using projective geometry to analyze Hamiltonian systems, revealing dualities and transformations, and discusses potential quantum extensions.

Contribution

It develops a novel approach to Hamiltonian systems via null lifts and projective structures, connecting classical and quantum dynamics.

Findings

Identifies Hamiltonian systems with projective conics

Reveals dualities like coupling-constant metamorphosis

Proposes extension to quantum Eisenhart-Duval lifts

Abstract

We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart-Duval lifts.