Narrow Results

In this study, we obtain the special type of magnetic trajectories associated with a magnetic field ${{𝒱}}_b$ defined on a 3D Riemannian manifold. We investigate a new representation of binormal spherical indicatrices of magnetic curves. Thus, we study $B_b$-magnetic curves terms of inextensible flows. Furthermore, we give some new characterizations of curvatures in terms of some partial differential equations. Finally, we examine some geometrical and physical features of the moving charged particle corresponding to the $B_b$-magnetic curves. Namely, we compute uniformity of the $B_b$-magnetic curves.

We find the equations of the magnetic curves corresponding to the magnetic fields defined by the Killing vector fields on the 3-dimensional Heisenberg manifold.

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere $S^2$. In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame $\{T_e,N_e,B_e\}$ of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for $T_e,N_e,B_e$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces.

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples.