The Eisenhart Lift and Hamiltonian Systems

TL;DR

This paper explores the Eisenhart lift's application to Hamiltonian systems, revealing new geometric insights, stability analysis tools, and conformal classes of ODEs, with extensions to complex ODEs and dual Riemannian metrics.

Contribution

It extends the Eisenhart lift framework to complex ODEs and dual Riemannian metrics, providing new stability criteria and boundary results for Hamiltonian systems.

Findings

Curvature and conjugate points aid in stability analysis.

Existence of conformal classes of ODEs from lightlike geodesics.

A boundary value theorem and conserved quantity for lifted ODE families.

Abstract

It is well known in general relativity that trajectories of Hamiltonian systems lift to geodesics of pp-wave spacetimes, an example of a more general phenomenon known as the "Eisenhart lift." We review and expand upon the benefits of this correspondence for dynamical systems theory. One benefit is the use of curvature and conjugate points to study the stability of Hamiltonian systems. Another benefit is that this lift unfolds a Hamiltonian system into a family of ODEs akin to a moduli space. One such family arises from the conformal invariance of lightlike geodesics, by which any Hamiltonian system unfolds into a "conformal class" of non-diffeomorphic ODEs with solutions in common. By utilizing higher-index versions of pp-waves, a similar lift and conformal class are shown to exist for certain second-order complex ODEs. Another such family occurs by lifting to a Riemannian metric that is dual to a pp-wave, a process that in certain cases yields a "square root" for the Hamiltonian. We prove a two-point boundary result for the family of ODEs arising from this lift, as well as the existence of a constant of the motion generalizing conservation of energy.