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

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We present a proof of the meromorphic non--integrability of the planar -Body Problem for some special cases. A simpler proof is added to those already existing for the Three-Body Problem with arbitrary masses. The -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 . 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.
Original languageEnglish
Pages (from-to)1225-1273
Number of pages49
JournalDiscrete and Continuous Dynamical Systems
Volume24
Issue number4
DOIs
Publication statusPublished - 2009

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