Geometry of Curves with Fractional Derivatives in Lorentz Plane

Abstract

In this paper, the geometry of curves is discussed based on the Caputo fractional derivative \rin the Lorentz plane. Firstly, the tangent vector of a spacelike plane curve is defined in terms of the \rfractional derivative. Then, by considering a spacelike curve in the Lorentz plane, the arc length and \rfractional ordered frame of this curve are obtained. Later, the curvature and Frenet-Serret formulas are \rfound for this fractional ordered frame. Finally, the relation between the fractional curvature and \rclassical curvature of a spacelike plane curve is obtained. In the last part of the study, considering the \rtimelike plane curve in the Lorentz plane, new results are obtained with the method in the previous \rsection.