Syzygies in the two center problem (Q2805233)

Consider the planar three-body problem, i.e., the dynamics of three massive point particles \(p_1,p_2,p_3\) evolving under the influence of mutual gravitational interactions. There can be times \(t_k\) where the configuration of the system sees the three particles aligned along a straight line, with particle \(p_{m(k)}\) in the middle (note this avoids to choose a direction on the alignment line). Such an alignment is known as a sygyzy.NEWLINENEWLINEOne could wonder, for a given initial condition, what is the sequence of sygyzies which are realized by the dynamics, i.e., what is the sequence of the middle particles \(M = \{ m(1) , m(2),\dots\}\). A less ambitious question to ask is what are the possible sequences \(M\) which could be realized for suitable initial conditions.NEWLINENEWLINEIt turns out that this question, despite long efforts involving eminent mathematicians, is still too hard. The paper considers a related question, namely what are the possible sequences \(M\) of sygyzies in the planar two-center problems (first investigated by Euler in 1760). This is an integrable system, so surely the required information could in principle be extracted from the dynamics, albeit it turns out that such extraction is far from trivial.NEWLINENEWLINEThe work provides a full classification of the possible sequences \(M\) for regular periodic orbits (Theorem 1) and for regular non-collision orbits (Theorem 2).NEWLINENEWLINEIn particular, for regular periodic orbits the possibilities are (up to an exchange of 1 and 2) the following: NEWLINENEWLINE\noindent \textit{L type}: \(1 3^{n_1} 2 3^{n_2} 1 3^{n_3} 2\dots3^{n_s}\) (thus 1 and 2 alternate); NEWLINENEWLINE\noindent \textit{S type}: \(1 3^{n_1} 1 3^{n_2} 1 3^{n_3} 1 \dots 3^{n_s}\) (or the same with 1 instead of 2); NEWLINENEWLINE\noindent \textit{P type}: \((12)^q\).NEWLINENEWLINEHere \(n_k\) are sequences of positive integers, known as Sturmian sequences (see Definition 1 in the paper) associated to the rotation number \(W=p/q\) for the torus identified by initial conditions; the length \(2(p+q)\) of the fundamental sequence as well as \(q\) are determined in terms of \(W\).NEWLINENEWLINEFor regular non-collision orbits, one has: NEWLINENEWLINE\noindent \textit{L case}: \(1 3^{n_1} 2 3^{n_2} 1 3^{n_3} 2\dots\); NEWLINENEWLINE\noindent \textit{S type}: \(1 3^{n_1} 1 3^{n_2} 1 3^{n_3} 1 \dots \); NEWLINENEWLINE\noindent \textit{P type}: \(121212\dots\).NEWLINENEWLINEThe proofs of these results are delicate and demanding, giving an idea of how difficult the extension to the full three-body problem could be. The paper also includes some corollaries to these main results, conjectures and open problems.NEWLINENEWLINEThe authors state that the main motivation for this work is to ``shed light on the honest three-body problem'', but surely it has an interest in itself and is a beautiful piece of symbolic dynamics.