A Dynamical Approach to Position Vector of Timelike Curve by Vectorial Momentum, Torque and Tangential Dual Curve

Abstract

In this study, the position vector of a timelike curve $$\wp$$ ℘ is stated by a linear combination of its Serret Frenet frame with differentiable functions. The definition of tangential dual curve of the curve $$\wp$$ ℘ is stated by using these differentiable functions. Moreover, tangential torque curve of timelike curve $$\wp$$ ℘ is defined and investigated. New dynamically and physical results are stated depending on the torque of the timelike curve $$\wp$$ ℘ and the direction of the tangent vector component of the curve. Then, the position vector of a timelike W curve is again stated by differentiable functions. Therefore, solutions of differential equation of the position vector of timelike W curve with two different types depending on the values of curvature and torsion of timelike curve are obtained. By using the differentiable functions obtained as a result of these solutions, tangential dual and torque curve of the timelike W curve are obtained. Depending on the tangential dual and torque curve of the timelike W curve, results are given for two different cases separately.

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

“…Define the unit tangent vector T, the unit principal normal vector N, and the unit binormal vector B to the timelike curve α, and let {T, N, B} be the Frenet-Serret frame for the timelike curve α. The Frenet-Serret frame {T, N, B} in R 2,1 satisfies the following properties [27]:…”

“…Lemma 1. Let α(s) be the timelike curve in R 2,1 , then the Frenet-Serret equations are given by [27]:…”