Elliptical trajectories of a point on the elliptical 2-sphere

TL;DR

This paper investigates the trajectories of a point on an ellipsoid influenced by Killing vector fields, introducing a generalized Darboux frame and variational methods to analyze magnetic and helical paths, with visualizations.

Contribution

It introduces a generalized Darboux frame and variational approach to analyze magnetic trajectories and helices on ellipsoids under Killing vector fields, providing new geometric insights.

Findings

Derived Killing equations in terms of Darboux frame invariants.

Identified and characterized magnetic curves satisfying Lorentz force equations.

Visualized specific trajectories on ellipsoids using Mathematica.

Abstract

The focus of this work is to analyze the trajectories of a point on the ellipsoid $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$ 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 $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. Then, we determine the Killing equations in terms of the Darboux frame invariants along an ellipsoidal curve. The Killing equations make it possible for us to interpret the magnetic trajectory of a point on the ellipsoid $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. Then, we determine two special trajectories using the variational method. The first one is magnetic curves that are the trajectories produced by the Killing magnetic field $K$ are satisfied the following Lorentz force equation $F_{L} (t)=K\times _{E}t=\nabla _{T}t$, where $\times _{E}$ is elliptical cross product and $\nabla $ is the Levi-Civita connection of the ellipsoid $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. The second one is generalized magnetic helices that are trajectories described by the trajectory of a point on a great ellipse of the ellipsoid rolling without slipping on a fixed ellipse of the ellipsoid using the elliptical motion on the $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. Furthermore, we give various examples and visualized them with the program Mathematica.