On the Meromorphic Non-Integrability of Some Problems in Celestial Mechanics

Abstract

In this thesis, we present a proof of the meromorphic non-integrability for some problems arising from Celestial Mechanics, as well as a new necessary condition for partial integrability in a wider Hamiltonian setting. First of all, 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. Second of all, a further strengthening of a prior existing result allows us to detect obstructions to the existence of a single additional first integral for classical Hamiltonians with a homogeneous potential. Third of all, using the aforementioned new result, we have proven the non-existence of an additional integral both for the general Three-Body Problem (hence generalizing, in a certain sense, Bruns'Theorem) and for the equal-mass Problem for N=4,5,6. Fourth of all, finally, we have proven the non-integrability of Hill's problem using the most general instance of the Morales-Ramis Theorem. A varying degree of theoretical complexity was involved in these results. Indeed, the proofs involving the given instances of the N-Body Problem required nothing but the exploration of the eigenvalues of a given matrix, with the advantage of knowing four of them explicitly. Thus, not all variational equations were needed but those not corresponding to these four eigenvalues -- this is exactly what transpires from the system reduction and subsequent introduction of normal variational equations, as done by S. Ziglin, J. J. Morales-Ruiz, J.-P. Ramis, and others. Hill's Problem, however, required the whole variational system since only thanks to the special functions introduced in the process of variation of constants was it possible to assure the presence of obstructions to integrability. 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 total and partial Hamiltonian integrability.