An anisotropic geometrical approach for non-relativistic extended dynamics

TL;DR

This paper develops a specialized anisotropic Riemannian geometric framework inspired by optics in non-uniform media, deriving equations of motion and specific geodesic solutions influenced by the medium's refractive index.

Contribution

It introduces a novel geometrical approach using nonlinear connection and Cartan's linear connection for non-relativistic extended dynamics inspired by optical media.

Findings

Derived equations of motion for optical media with variable refractive index

Identified specific geodesic trajectories such as helices, circles, and straight lines

Demonstrated the influence of non-constant refractive index on geodesic paths

Abstract

In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and some particular solutions are given. For instance, we obtain as geodesic trajectories some circular helices (depending on an angular velocity \omega), certain circles situated in some planes (ones are parallel with xOy, and other ones are orthogonal on xOy), or some straight lines which are parallel with the axis Oz. All these geometrical geodesics are very specific because they are completely determined by the non-constant index of refraction n(x).