Using Mathematica software to solve ordinary differential equations and applying it to the graphical representation of trajectories

TL;DR

This paper demonstrates how to use Mathematica software for solving ordinary differential equations related to charged particle trajectories under electromagnetic fields, including exact, numerical solutions, graphical representations, and video simulations.

Contribution

It provides a detailed methodology for solving Lorentz force equations with Mathematica and visualizing particle trajectories, including dynamic simulations.

Findings

Exact and numerical solutions for Lorentz force equations

Graphical representations of particle trajectories

Video simulations of oscillating electric field effects

Abstract

We discuss the great importance of using mathematical software in solving problems in today's society. In particular, we show how to use Mathematica software to solve ordinary differential equations exactly and numerically. We also show how to represent these solutions graphically. We treat the particular case of a charged particle subject to an oscillating electric field in the xy plane and a constant magnetic field. We show how to construct the equations of motion, defined by the vectors position, velocity, electric and magnetic fields. We show how to solve these equations of Lorentz force, and graphically represent the possible trajectories. We end by showing how to build a video simulation for an oscillating electric field trajectory particle case.