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arXiv · 1705.00723

Free time minimizers for the planar three-body problem

Rick Moeckel·Richard Montgomery·Hector Sanchez Morgado
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Abstract

Free time minimizers of the action (called"semi-static" solutions by Ma\~ne) play a central role in the theory of weak KAM solutions to the Hamilton-Jacobi equation (see Fathi). We prove that any solution to Newton's three-body problem which is asymptotic to Lagrange's parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange's solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz-Maderna who had shown that every free time minimizer for the N-body problem is parabolic, and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler's central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.

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Rick Moeckel, Richard Montgomery, Hector Sanchez Morgado. 2017-05-01. Free time minimizers for the planar three-body problem. https://doi.org/10.1007/s10569-018-9823-y

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