TITLE: On the Meromorphic Non-integrability of some N-Body Problems

AUTHORS: 
  Juan J. Morales-Ruiz
  Dpt. Matematica Aplicada II, Universitat Politecnica de Catalunya
  Jordi Girona 1-3, 08034 Barcelona

  Juan.Morales-Ruiz@upc.edu
  
  Sergi Simon
  Dpt. Maths Informatique, Universite de Limoges
  XLIM-UMR CNRS n.6172, 123, av. Albert Thomas - 87060 Limoges

  sergi.simon-estrada@unilim.fr, sergi.simon@gmail.com

ABSTRACT:

We present a proof of the meromorphic non-integrability of the planar N-Body
Problem for some special cases.  A simpler proof is added to those already
existing for the Three-Body Problem with arbitrary masses. The N-Body Problem
with equal masses is also proven non-integrable. Furthermore, a new general
result on additional integrals is obtained which, applied to these specific
cases, proves the non-existence of an additional integral for the general
Three-Body Problem, and provides for an upper bound on the amount of additional
integrals for the equal-mass Problem for N = 4, 5, 6. These results appear to
qualify differential Galois theory, and especially a new incipient theory
stemming from it, as an amenable setting for the detection of obstructions to
Hamiltonian integrability.