Abstract
We present a proof of the meromorphic non--integrability of theplanar $N$-Body Problem for some special cases. A simpler proof isadded to those already existing for the Three-Body Problem witharbitrary masses. The $N$-Body Problem with equal masses is alsoproven non-integrable. Furthermore, a new general result onadditional integrals is obtained which, applied to these specificcases, proves the non-existence of an additional integral for thegeneral Three-Body Problem, and provides for an upper bound on theamount of additional integrals for the equal-mass Problem for$N=4,5,6$. These results appear to qualify differential Galoistheory, and especially a new incipient theory stemming from it, asan amenable setting for the detection of obstructions to Hamiltonianintegrability.
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