Relativistic motion in a Coulomb field

Live Script exploring motion of a relativistic particle in an inverse-square law force field.

It is well known that, in non-relativistic mechanics, only the harmonic oscillator and inverse square force law potentials give rise to closed (elliptical) orbits while perturbations to these potentials result in precession of the orbital axes. This educational Live Script solves the relativistic form of Newton's 2nd Law for the motion of a particle in Coulomb force field and illustrates precession and other unusual behavior.

The script reproduces and enhances trajectory plots from the Master's Thesis by Angelo Di Fiore [2], which are based on the analytic solutions presented by T.H. Boyer [1]. The script visualizes six distinct cases of relativistic orbits, including scattering, bound, and spiral trajectories. For each case, the analytic time-dependence is also computed and validated using a numerical integration of the relativistic equations of motion.

The script also compares the relativistic and non-relativistic trajectories for a beam of particles for both attractive and repulsive Coulomb potentials. In the non-relativistic case, no trajectory enters a halo region behind the scattering center, a region bounded by a quadratic curve. In the relativistic case, the halo region shrinks considerably.

This script may interest students and instructors in physics and is suitable for a course in intermediate level mechanics. It contains 'Try this' suggestions, coding 'Challenges", hyperlinks, and references for further exploration. Additional educational Live Scripts by the author are here.