Periodic Orbits and Binary Collisions in the Classical Coulomb Three-Body Problem

TL;DR

This paper develops a method to identify periodic orbits in the classical Coulomb three-body problem, specifically for helium-like systems, aiding semiclassical quantization efforts.

Contribution

It introduces a novel procedure using symbolic dynamics to find and verify periodic orbits in the two-dimensional Coulomb three-body problem.

Findings

Predicted sequences of periodic orbits are confirmed numerically.

The method provides a foundation for discovering remaining orbits for semiclassical applications.

Results advance understanding of classical dynamics in Coulomb systems.

Abstract

In the helium case of the classical Coulomb three-body problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. A sequence of periodic orbits are predicted and are actually found numerically. The results obtained here will be a cornerstone for finding the remaining periodic orbits, which needed for semiclassical applications such as periodic orbit quantization.