Research Article Motion of Charged Spinning Particles in a Unified Field M. I. Wanas 1,2 and Mona M. Kamal 2,3 1 Astronomy Department, Faculty of Science, Cairo University, Egypt 2 Egyptian Relativity Group (ERG), Cairo, Egypt 3 Mathematics Department, Faculty of Girls, Ain Shams University, Egypt Correspondence should be addressed to M. I. Wanas; wanas@scu.eg Received 15 May 2021; Accepted 11 October 2021; Published 25 November 2021 Academic Editor: Shi Hai Dong Copyright © 2021 M. I. Wanas and Mona M. Kamal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3 . Using a geometry wider than Riemannian one, the parameterized absolute parallelism (PAP) geometry, we derived a new curve containing two parameters. In the context of the geometrization philosophy, this new curve can be used as a trajectory of charged spinning test particle in any unified field theory constructed in the PAP space. We show that imposing certain conditions on the two parameters, the new curve can be reduced to a geodesic curve giving the motion of a scalar test particle or/and a modified geodesic giving the motion of neutral spinning test particle in a gravitational field. The new method used for derivation, the Bazanski method, shows a new feature in the new curve equation. This feature is that the equation contains the electromagnetic potential term together with the Lorentz term. We show the importance of this feature in physical applications. 1. Introduction According to the geometrization philosophy, the curve in a certain geometry represents the equation of motion of a the- ory which constructed in this geometry. Together with the field equations of any theory, we need the equation of motion which characterizes the theory used. In general relativity, geodesic curve is considered as an equation of motion of a scalar test particle moving in a gravitational field. Geodesic equation can be derived using the Lagrangian (cf. [1]): L 1 = def : g μν _ x μ _ x ν , ð1Þ where g μν is the metric tensor and _ x μ ð= def : dx μ /dsÞ is the unit tangent vector to the curve. Euler-Lagrange equation is given by the following (cf. [2]): d ds ∂L 1 ∂ _ x γ − ∂L 1 ∂x γ = 0, ð2Þ such that s is the scalar parameter varying along the curve. Using Lagrangian (Equation (1)) and Equation (2), we get the following: € x α + α μν () _ x μ _ x ν = 0, ð3Þ where α μν () is the coefficient of Levi-Civita linear connec- tion which is defined as follows: α μν () = def : 1 2 g ασ g μσ,ν + g νσ,μ − g μν,σ   : ð4Þ Equation (3) is the curve equation of the Riemannian geometry. The Lagrangian used for deriving the equation of motion of a charged particle moving in the presence of electromag- netic field is defined by the following (cf. [3]): L 2 = def : g μν V μ + βA μ ð ÞV ν , ð5Þ where A μ is a vector field and β is a conversion parameter Hindawi Advances in High Energy Physics Volume 2021, Article ID 4970469, 6 pages https://doi.org/10.1155/2021/4970469