An anisotropic geometrical approach for non-relativistic extended dynamics

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 ω), 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). (2010): 53C60, 53C80, 83C10. Mathematics Subject Classification

Optics in nonuniform media and Lagrange geometry

“…here the negative values of f correspond to a sign "-" in (14), while positive values of f are in the range of W defined by a sign "+". Taking into account W < 0, one finds that f < 1.…”

“…To this end, we transform the second equation in (9) to the variable f . Substituting the expressions ( 15), (14) in the second equation in (9) and taking into account that the derivative of V (2) over t can be represented as dV (2) dt = dV (2) df df dt = dV (2) df f ′ , one get the equation dV (2) df + 1 (f + 2)…”

“…where n = n(x) is interpreted as the local refractive index of the nonuniform medium (see [2], [7], [11]- [14]).…”

“…In the sequel, we will examine the special case of an anisotropic dynamical model, which is governed by the Lagrangian [13]:…”

“…Assuming that the refractive index n(x) is invariant with respect to Lorentz transformations, we conclude that the Lagrangian (5) is also invariant. A similar 3-dimensional Lagrangian was studied in [14], the corresponding non-relativistic Lagrangian being invariant with respect to the orthogonal group O(3).…”