On some exceptional cases in the integrability of the three-body problem |
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| Authors: | Alexei V Tsygvintsev |
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| Institution: | (1) Unité de mathématiques pures et appliquées, Ecole Normale Supérieure de Lyon, 46, allée d’Italie, Lyon, Lyon Cedex 07, 69364, France |
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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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| Keywords: | Meromorphic first integrals Non-integrability Ziglin’ s lemma Three-body problem |
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