# Back to epicycles - relativistic Coulomb systems in velocity space

## TL;DR

This paper explores relativistic Coulomb systems in velocity space, revealing that their trajectories are epicyclic in hyperbolic velocity space, which simplifies analysis and offers new insights into their orbital behavior.

## Contribution

It introduces the concept of epicyclic orbits in relativistic velocity space and demonstrates the linearity of the velocity equation for $1/r$ interactions, simplifying the study of such systems.

## Key findings

- Relativistic Coulomb orbits in velocity space are epicyclic.
- Velocity space for relativistic systems is hyperbolic ($H^3$).
- Orbits can be understood through their properties in velocity space.

## Abstract

The study of relativistic Coulomb systems in velocity space is prompted by the fact that the study of Newtonian Kepler/Coulomb systems in velocity space, although less familiar than the analytic solutions in ordinary space, provides a much simpler (also more elegant) method. The simplicity and elegance of the velocity-space method derives from the linearity of the velocity equation, which is the unique feature of $1/r$ interactions for Newtonian and relativistic systems alike. The various types of possible trajectories are presented, their properties deduced from the orbits in velocity space, accompanied with illustrations. In particular, it is found that the orbits traversed in the relativistic velocity space (which is hyperbolic ($H^3$) rather than Euclidean) are epicyclic -- circles whose centres also rotate -- thus the title.