Geometry of the isotropic oscillator driven by the conformal mode

TL;DR

This paper introduces a modified Eisenhart lift to geometrize the isotropic oscillator influenced by a conformal mode, extending the geometric approach to more complex dynamical systems.

Contribution

It proposes a novel modification of the Eisenhart lift to describe the isotropic oscillator with a conformal mode in arbitrary dimensions.

Findings

New geometric formulation of the isotropic oscillator

Extension of Eisenhart lift to conformal mode systems

Potential applications in theoretical physics and geometry

Abstract

Geometrization of a Lagrangian conservative system typically amounts to reformulating its equations of motion as the geodesic equations in a properly chosen curved spacetime. The conventional methods include the Jacobi metric and the Eisenhart lift. In this work, a modification of the Eisenhart lift is proposed which describes the isotropic oscillator in arbitrary dimension driven by the one-dimensional conformal mode.