GEOMETRIC PERSPECTIVE OF BERRY'S PHASE ACCORDING TO ALTERNATIVE ORTHOGONAL MODIFIED FRAME

It is very important to determine the Frenet vector fields to obtain the characterizations of curves in 3-dimensional Euclidean space. However, it may not be possible to define Frenet vector fields at some points of the curve, or the Frenet frame may not be sufficient to determine the curve's characterization. New frames can be defined as alternatives to the Frenet frame in these situations. Lots of work has been done for these cases such as Bishop frame, {d_2,C,W}- frame, orthogonal frame, etc. In this study, we produce the magnetic curves related to an alternative modified frame that can be used in cases where the Frenet frame and, {d_2,C,W}- frame cannot be defined at some points. It is known that the trajectories of a charged particle in a magnetic field are called magnetic curves. The adapted frame vectors defined on the curve are critical when examining these trajectories. In this context, the new alternative orthogonal {N,C,W} frame described in this article will be beneficial as it is defined at every point of the magnetic curve. Thus, we obtain the characterization of the magnetic and electromagnetic curves associated with this frame. We found that magnetic and electromagnetic curves are circular helices or constant precession slant helices. Also, we showed that the electric field parallel transported with the Fermi Walker parallel transportation rule along a curve with the {N,C,W} frame. Furthermore, we provide some examples and visualize them using the Mathematica and Maple programs.