TL;DR
This paper explores a new variant of the Eisenhart lift inspired by the Bohlin transformation, embedding dynamical systems into higher-dimensional spacetime to find conformally flat metrics with special symmetries.
Contribution
It introduces a novel Eisenhart lift variant that constructs conformally flat metrics with higher rank Killing tensors, expanding geometric methods in dynamical systems.
Findings
Constructed new conformally flat metrics with higher rank Killing tensors.
Linked Bohlin transformation to Eisenhart lift for dynamical systems.
Provided geometric insights into integrable systems.
Abstract
Inspired by the Bohlin transformation relating the planar harmonic oscillator to the Kepler problem, a variant of the Eisenhart lift is studied, in which a Lagrangian conservative dynamical system with d degrees of freedom is embedded into timelike geodesics of a conformally flat metric on a (d+2)-dimensional space-time of the Lorentzian signature. The uplift is used to construct novel examples of conformally flat metrics admitting higher rank Killing tensors.
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