Syzygies in the two center problem

TL;DR

This paper provides a symbolic dynamics framework for Euler's two fixed centers problem, linking Sturmian sequences to periodic orbits and collisions, offering a complete description of the system's dynamics.

Contribution

It introduces a symbolic dynamics approach using Sturmian sequences to characterize all types of orbits in the two center problem, including collision and non-collision trajectories.

Findings

Periodic Sturmian sequences correspond to periodic orbits.

Finite Sturmian sequences represent collision-collision orbits.

Complete symbolic description of the system's dynamics.

Abstract

We give a complete symbolic dynamics description of the dynamics of Euler's problem of two fixed centers. By analogy with the 3-body problem we use the collinearities (or syzygies) of the three bodies as symbols. We show that motion without collision on regular tori of the regularised integrable system are given by so called Sturmian sequences. Sturmian sequences were introduced by Morse and Hedlund in 1940. Our main theorem is that the periodic Sturmian sequences are in one to one correspondence with the periodic orbits of the two center problem. Similarly, finite Sturmian sequences correspond to collision-collision orbits.