Elliptical magnetic trajectories of a point on the elliptical 2 -sphere

Elliptical trajectories play numerous significant roles in mathematics and physics. The focus of this work is to analyze the trajectories of a point on the ellipsoid Sa1,a2,a32 while it is under the influence of a killing vector field K. For this purpose, we introduce the generalized Darboux frame and the variational vector fields of Sa1,a2,a32. We were able to get the trajectory equations of the charged particle on the ellipsoid by utilizing the elliptical structure and elliptic inner and cross product. We adopted a clearer and more comprehensible strategy in this way. With the aid of a mathematical program, we were able to display the paths we had determined based on the particle's velocity vector's position in the magnetic field. These trajectories are spiral, helical, and elliptical.