Trajectories in Logarithmic Potentials

TL;DR

This paper analyzes the motion of particles in logarithmic potentials, specifically focusing on electron trajectories in cylindrical capacitors, providing series solutions and characterizing the resulting orbits.

Contribution

It introduces a series expansion method for solving equations of motion in logarithmic potentials and characterizes the resulting trajectories, including rosette and circular orbits.

Findings

Series solutions for trajectories are derived and shown to converge.

Rosette orbits with period 4π/3 are identified.

Circular paths are also characterized.

Abstract

Trajectories in logarithmic potentials are investigated by taking as example the motion of an electron within a cylindrical capacitor. The solution of the equation of motion in plane polar coordinates, (r,{\phi}) is attained by forming a series expansion of r and of 1/r as a function of {\phi}. The terms of the series contain polynomials, the recurrence relation of which is given, together with some further characteristics. By the comparison-theorem of infinite series, the convergence of the solution is demonstraded. The simplest trajectories in logarithmic potentials are represented by rosette type orbits with a period of 4{\pi}/3, and by circular paths.