Abstract
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) = (H 0, C 0) with (H 0, C 0) ≠ (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.
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Combot, T. Non-integrability of the equal mass n-body problem with non-zero angular momentum. Celest Mech Dyn Astr 114, 319–340 (2012). https://doi.org/10.1007/s10569-012-9417-z
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DOI: https://doi.org/10.1007/s10569-012-9417-z
Keywords
- Non-integrability
- Homogeneous potentials
- Central configurations
- Differential Galois theory
- n-body problem