Celestial Mechanics and Dynamical Astronomy

, Volume 114, Issue 4, pp 319–340

Non-integrability of the equal mass n-body problem with non-zero angular momentum

Original Article

DOI: 10.1007/s10569-012-9417-z

Cite this article as:
Combot, T. Celest Mech Dyn Astr (2012) 114: 319. doi:10.1007/s10569-012-9417-z


We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree −1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H0, C0) with (H0, C0) ≠ (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided.


Non-integrability Homogeneous potentials Central configurations Differential Galois theory n-body problem 

Mathematics Subject Classification

37J30 70F15 

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.IMCCEParisFrance

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