Integrability and Non Integrability of Some n Body Problems

Abstract

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.

Appendix A. Integrable Series Expansions at Order 4

The results are written in the following way. We give a series expansion of the form (9) of V, such that the k-th order variational equation of V near c has a virtually Abelian Galois group if and only if $(u_{3,0},u_{3,1},u_{3,2},u_{3,3})\in \mathcal {I}_k^{-1}(0)$. The sequence of ideals $\mathcal {I}_k$ is growing, and we compute these conditions up to order 4. For the eigenvalues in (8), they are given below. Remark that the Hilbert dimension of the ideals $\mathcal {I}_4$ greatly depend on eigenvalues, and that sometimes exceptional possible solutions appear in the 4-th order variational equation. In particular, the restriction of these series expansions to the planes in $(q_1,q_2)$ and $(q_1,q_3)$ does not always lead to integrable series expansion at order 4 on these planes.