On Backlund transformation and motion of null Cartan curves

Abstract

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]