On some exceptional cases in the integrability of the threebody problem
Abstract
We consider the Newtonian planar threebody 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, 2^{3}/3^{3}, 2/3^{2} where the linearized equations are shown to be partially integrable. This result completes the nonintegrability analysis of the threebody 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.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 September 2007
 DOI:
 10.1007/s1056900790865
 arXiv:
 arXiv:math/0610951
 Bibcode:
 2007CeMDA..99...23T
 Keywords:

 Meromorphic first integrals;
 Nonintegrability;
 Ziglin's lemma;
 Threebody problem;
 Mathematics  Dynamical Systems;
 Mathematical Physics;
 37J30
 EPrint:
 7 pages