We are interested in defining new energy functionals and solving them by using the variational approach method. That is, we aim to define a new class of elastic curves in the three-dimensional ordinary space. We further improve an alternative method to find critical points of the bending energy functionals acting on a class of magnetic curves. Then, we classify these critical curves as elastic magnetic curves of the Frenet-Serret vector family. Moreover, we investigate the dynamics of the elastic magnetic curves of ferromagnetic and superparamagnetic models and discuss their numerical and analytical analysis.

We firstly discuss the unchanged quasi direction motion (UDQM) with biharmonicity condition in the Heisenberg space. We define the energy of velocity magnetic particles and some Lorentz fields. Also, we construct the new relationship between the Fermi-Walker parallel transportation and the unchanged quasi direction motion in the Heisenberg space. In other words, we obtain the applied geometric characterization for the unchanged quasi direction motion of biharmonic velocity magnetic particles in the Heisenberg space. This concept also boosts to discover some physical and geometrical characterizations belonging to the particle such as the magnetic motion, the electrical energy functional, the torque, and the Poynting vector. Finally, we obtain electrical energy with respect to its electric field and energy flux density in the radial direction.

In this study, we firstly introduce a different type of directional Fermi-Walker transportations along with vortex lines of a non-vanishing vector field in three-dimensional space. Thus we conclude that geometric quantities, which are used to characterize vortex lines, are also associated with the geometric phase and angular velocity vector (Darboux vector) of the system. Then we present directional magnetic vortex lines by computing the Lorentz force. Hence, we reach a remarkable relation between directional magnetic vortex lines and angular velocity vector of vortex lines with a non-rotating frame. We later determine the directional electric vortex lines by considering the electromagnetic force equation. We finally investigate the conditions of being uniform for magnetic fields of directional magnetic vortex lines and we improve a remarkable approach to find the electromagnetic curvature, which contains many geometrical features belonging to directional electric vortex line.

In this paper, we mainly focus on the theory of evolution of wave polarization in the normal direction of the curved path, which is assumed to be the trajectory of the propagated light beam. The polarization state of the wave is described by the unit complex transverse field component by eliminating the longitudinal field component, which reduces the dimension of the problem. A Coriolis term is also effectively used to describe the relationship between the geometric phase and the parallel transport law of the wave polarization vector of the evolving light beam in the normal direction of the curved path. We further present a unified geometric interpretation of the binormal evolution of the wave polarization vector in the normal direction of the curved path via the nonlinear Schrodinger equation of repulsive type. Finally, we can sum up these discussions by investigating the analytic solutions of the nonlinear Schrodinger equation of repulsive type, which represents binormal evolution of the polarization vector in the normal direction of the curved path trajectory, for some special cases by using the traveling wave hypothesis approach.