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On some exceptional cases in the integrability of the three-body problem

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Abstract

We consider the Newtonian planar three-body problem with positive masses m 1, m 2, m 3. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases ∑m i m j /(∑m k )2 = 1/3, 23/33, 2/32 where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in papers [Tsygvintsev, A.: Journal für die reine und angewandte Mathematik N 537, 127–149 (2001a); Celest. Mech. Dyn. Astron. 86(3), 237–247 (2003)] and based on the Morales–Ramis–Ziglin approach.

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  1. Unité de mathématiques pures et appliquées, Ecole Normale Supérieure de Lyon, 46, allée d’Italie, Lyon, Lyon Cedex 07, 69364, France

    Alexei V. Tsygvintsev

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  1. Alexei V. Tsygvintsev
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Correspondence to Alexei V. Tsygvintsev.

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Tsygvintsev, A.V. On some exceptional cases in the integrability of the three-body problem. Celestial Mech Dyn Astr 99, 23–29 (2007). https://doi.org/10.1007/s10569-007-9086-5

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  • DOI: https://doi.org/10.1007/s10569-007-9086-5

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