# Non-integrability of some few body problems in two degrees of freedom - Mathematical Physics

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Abstract: The basic theory of Differential Galois and in particular Morales-Ramistheory is reviewed with focus in analyzing the non-integrability of variousproblems of few bodies in Celestial Mechanics. The main theoretical tools are:Morales-Ramis theorem, the algebrization method of Acosta-Bl\-azquez andKovacic-s algorithm. Morales-Ramis states that if Hamiltonian system has anadditional meromorphic integral in involution in a neighborhood of a specificsolution, then the differential Galois group of the normal variationalequations is abelian. The algebrization method permits under general conditionsto recast the variational equation in a form suitable for its analysis by meansof Kovacic-s algorithm. We apply these tools to various examples of few bodyproblems in Celestial Mechanics: a the elliptic restricted three body in theplane with collision of the primaries; b a general Hamiltonian system of twodegrees of freedom with homogeneous potential of degree -1; here we performMcGehee-s blow up and obtain the normal variational equation in the form of anhypergeometric equation. We recover Yoshida-s criterion for non-integrability.Then we contrast two methods to compute the Galois group: the well known, basedin the Schwartz-Kimura table, and the lesser based in Kovacic-s algorithm. Weapply these methodology to three problems: the rectangular four body problem,the anisotropic Kepler problem and two uncoupled Kepler problems in the line;the last two depend on a mass parameter, but while in the anisotropic problemit is integrable for only two values of the parameter, the two uncoupled Keplerproblems is completely integrable for all values of the masses.

Author: ** Primitivo Acosta-Humanez, Martha Alvarez-Ramirez, Joaquin Delgado**

Source: https://arxiv.org/