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November  2009, 24(4): 1225-1273. doi: 10.3934/dcds.2009.24.1225

On the meromorphic non-integrability of some $N$-body problems

Juan J. Morales-Ruiz 1, and  Sergi Simon 2,

1. 

Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain
2. 

Département Maths Informatique, Université de Limoges, XLIM - UMR CNRS n. 6172, 123, avenue Albert Thomas - 87060 Limoges, France

Received  June 2008 Revised  January 2009 Published  May 2009

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.
Keywords: Obstructions to integrability (nonintegrability criteria) in Hamiltonian systems, $N$-body problems., Dynamical systems in classical and celestial mechanics, differential algebra.
Mathematics Subject Classification: Primary: 37J30, 12H05; Secondary: 53C35, 37N05, 70F1.
Citation: Juan J. Morales-Ruiz, Sergi Simon. On the meromorphic non-integrability of some $N$-body problems. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1225-1273. doi: 10.3934/dcds.2009.24.1225
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