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.
Highlights
It would not be an exaggeration to say that, since the discovery of general relativity, geometry and theoretical physics go parallel
An embedding of the isotropic oscillator driven by the one-dimensional conformal mode into the geodesics of the Eisenhart-like metric is discussed in Sect
As compared to the conventional Eisenhart prescription, in which coordinates parametrizing the spacetime are associated with degrees of freedom of the original dynamical system, the conformal mode ρ(t) enters the metric (29) as a specific scale factor whose time evolution is governed by the Einstein equations
Summary
It would not be an exaggeration to say that, since the discovery of general relativity, geometry and theoretical physics go parallel. Geometrization of a Lagrangian conservative system, whose kinetic term involves a positive definite metric, amounts to reformulating its equations of motion as the geodesic equations in a properly chosen curved spacetime or embedding them into the geodesics of a larger theory such that the time evolution of the extra degrees of freedom is unambiguously fixed, provided the dynamics of the original model is known. The Jacobi metric [1] and the Eisenhart lift [2,3] represent the conventional tools of that kind (for a recent application to time-dependent systems see [4]). An embedding of the isotropic oscillator driven by the one-dimensional conformal mode into the geodesics of the Eisenhart-like metric is discussed in Sect. A set of vector fields is found which all together form the Newton–Hooke algebra under the commutator The requirement that they be the Killing vector fields of the metric in Sect. An alternative parametrization of the coset space for the l-conformal Newton–Hooke algebra was recently discussed in [9]
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