Integrability and non integrability of some n body problems

Thierry Combot

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arXiv · 1509.08233

Integrability and non integrability of some n body problems

Thierry Combot

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Abstract

We prove the non integrability of the colinear $3$ and $4$ body problem, for any masses positive masses. To deal with resistant cases, we present strong integrability criterions for $3$ dimensional homogeneous potentials of degree $-1$, and prove that such cases cannot appear in the $4$ body problem. Following the same strategy, we present a simple proof of non integrability for the planar $n$ body problem. Eventually, we present some integrable cases of the $n$ body problem restricted to some invariant vector spaces.

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Thierry Combot. 2015-09-28. Integrability and non integrability of some n body problems. https://arxiv.org/abs/1509.08233

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