Syzygies in the two center problem. (9th March 2016)

• Record Type: Journal Article
• Authors: Dullin, Holger R; Montgomery, Richard
• Is Part Of: Nonlinearity. Volume 29:Number 4(2016:Apr.)
• Page Start: 1212
• Page End: 1237
• Publication Date: 2016-03-09
• Subjects: symbolic dynamics -- integrable systems -- periodic orbits; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/4/1212
• Languages: English
• ISSNs: 0951-7715

Abstract

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.