The only syzygy-free solution is Lagrange’s.

Absract. A syzygy in the three-body problem is a collinear instant. We prove that with the exception of Lagrange's solution every solution to the zero angular momentum Newtonian three-body problem suffers syzygies. The proof works for all mass ratios. We consider the Newtonian three-body problem with zero angular momentum and negative energy. Masses are positive, but arbitrary. A 'syzygy' means an eclipse: an instant at which the three masses are collinear. Theorem 0.1.. Every solution admits a syzygy except one: the Lagrange homo-thety solution. Solutions are defined over their maximal interval of existence and analytically continued through binary collisions a la Levi-Civita [2]. Binary collisions counts as syzygies. A solution cannot be extended past a finite time t = b if and only if as t → b the three positions of the three bodies tend to the same point. In other words, a solution fails to exist past a certain time if and only if it ends in triple collision at that time. (See [11], [5] or [10]) The Lagrange homothety solution [1] begins and ends in triple collision. At every other instant of its existence the masses form an equilateral triangle. This triangle evolves by homothety (scaling). Halfway through its evolution the three bodies are instantaneously at rest, forming an equilateral triangle whose size is determined by the value of the negative energy. In [6] I proved theorem 0.1 upon imposing two additional hypotheses on solutions: that they are bounded, and that they do not end in triple collision. The contribution of the present paper is to dispense with these hypotheses. I first dispense with the hypothesis on collision, keeping the boundedness hypothesis. Again, in [6] I proved that bounded solutions which do not end in collision have syzygies. The same proof, plus invariance of the equations and zero angular momentum condition under time reversal proves existence of syzygies for solutions which are bounded and do not begin in triple collision. All that remains of the bounded soltuions are those, excluding Lagrange, which begin and end in triple collision. The proof for these solutions will follow the same qualitative lines as [6]. According to Moeckel [1989] (see the corollary at the top of p. 53) there are, for generic mass ratios, an infinite number of these finite-interval solutions bi-asymptotic to triple collision. Moeckel, Chenciner and others have pointed out that dispensing with the bound-edness hypothesis on solutions ought …