# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1908.02751