An AP structure with Finslerian Flavor: Path equations

Abstract

The Bazanski approach for deriving paths is applied to Finsler geometry. The approach is generalized and applied to a new developed geometry called "Absolute parallelism with a Finslerian Flavor" (FAP). A sets of path equations is derived for the FAP. This is the horizontal (h) set. A striking feature appears in this set, that is: the coefficient of torsion term, in the set, jumps by a step of one-half from one equation to the other. This is tempting to believe that the h-set admits some quantum features. Comparisons with the corresponding sets in other geometries are given. Conditions to reduce the set of path equations obtained, to well known path equations in some geometries are summarized in a schematic diagram.

Spinning Equations for Objects of Some Classes in Finslerian Geometry

“…Owing to this analogy, we reach to regard the behavior of the spin connection in non-Riemannian geometries as the untested effect of nonlinear connection in these types of geometries. Such a result will be extended to examine its role in torsion and curvature, having a Finslaerian flavor [23] on the spinning motion and the effect of their corresponding spinning jerk in our future work.…”

Spinning and Spinning Deviation Equations for Special Types of Gauge Theories of Gravity

“…This approach was applied for obtaining path equations for some geometries admitting non vanishing curvature and torsion simultaneously. [26][27][28] Due to the resultant equations (2.35),(2.36), we have figured out that torsion is explicitly mentioned in spin deviation equations ,even if one puts P µ = mU µ ,. Such a result is in favor of Hehl's argument falsifying the measurement of torsion from identifying the spin tensor, for a spinning object in an orbit.…”

Motion of Poly-vectors in Fractal Clifford Spaces

“…) Yet, this type of torsion (15) can related to the notation of torsion as mentioned by Hammond [8] as commutation of the potential associated by a prescribed scalar field φ [8]…”

“…We suggest the equivalent Bazanski Lagrangian [14][15][16][17][18][19] for deriving the equations for spinning and spinning poly-vectors of a Fractal Clifford Space (FCS) to be…”