Integrability and Non Integrability of Some n Body Problems

  • Thierry CombotEmail author
Part of the Mathematics for Industry book series (MFI, volume 23)


We prove the non integrability of the colinear 3 and 4 body problem, for any 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.


Morales-Ramis theory Homogeneous potential Central configurations Differential Galois theory Integrable systems 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.IMBUniversié de BourgogneDijon CedexFrance

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