Abstract
Relations between the rectilinear, collinear and syzygy solutions of the N-body problem are first pointed out. It is shown that, along a solution, the set of the non-collinear syzygy configuration instants is formed by isolated points. Then we restrict the study to the planar 3-body problem and prove that for Dirichlet-stable solutions, a non-syzygy solution cannot be as close as possible to a syzygy one. It is also true that, in the case of a syzygy solution, the orbit of one particle crosses the line of the other two and can not be tangent to this line in the transition point. Finally we prove that the set of initial conditions leading to non-collinear syzygy solutions is non-empty and open.
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Diacu, F.N. On the planar syzygy solutions of the 3-body problem. Celestial Mech Dyn Astr 46, 119–128 (1989). https://doi.org/10.1007/BF00053042
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DOI: https://doi.org/10.1007/BF00053042