Fermi–Walker conformable connection and the evolution of the conformable magnetically driven particles

We conduct a thorough exploration of the magnetic flows and physical dynamics exhibited by point charged particles in three-dimensional ordinary space. Our methodology is grounded in the application of conformable methods. Initially, we refine the general treatment to derive velocity and acceleration vector fields for point particles traversing conformable curves under the influence of distinct force fields. Subsequently, our inquiry extends into the realm of magnetism, where we investigate various types of dynamical magnetic curves that emerge from characterizing the movement of charged particles along conformable curves. In addition to this, we introduce the Fermi–Walker conformable derivative and provide an alternative representation for a unit-speed conformable curve by leveraging the Fermi–Walker connection. Throughout the entirety of the paper, we establish a meaningful correlation between the consistent motion of conformable charged particles and the trajectories dictated by specific external forces within the \(\alpha \) $\alpha$ -Frenet–Serret frame. The application of fractional calculus plays a pivotal role in our investigations, a facet we meticulously delve into in the paper. This work significantly contributes to an enhanced comprehension of magnetic phenomena and dynamics, offering valuable insights into the behavior of charged particles within conformable frameworks.