On some exceptional cases in the integrability of the three-body problem
Alexei Tsygvintsev
Abstract
We consider the Newtonian planar three--body problem with positive masses m1, m2, m3. 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 Σ mi mj/(Σ mk)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 our previous papers and based of the Morales-Ramis-Ziglin approach.
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