The basic theory of Differential Galois and in particular Morales--Ramis
theory is reviewed with focus in analyzing the non--integrability of various
problems of few bodies in Celestial Mechanics. The main theoretical tools are:
Morales--Ramis theorem, the algebrization method of Acosta--Bl\'azquez and
Kovacic's algorithm. Morales--Ramis states that if Hamiltonian system has an
additional meromorphic integral in involution in a neighborhood of a specific
solution, then the differential Galois group of the normal variational
equations is abelian. The algebrization method permits under general conditions
to recast the variational equation in a form suitable for its analysis by means
of Kovacic's algorithm. We apply these tools to various examples of few body
problems in Celestial Mechanics: (a) the elliptic restricted three body in the
plane with collision of the primaries; (b) a general Hamiltonian system of two
degrees of freedom with homogeneous potential of degree -1; here we perform
McGehee's blow up and obtain the normal variational equation in the form of an
hypergeometric equation. We recover Yoshida's criterion for non--integrability.
Then we contrast two methods to compute the Galois group: the well known, based
in the Schwartz--Kimura table, and the lesser based in Kovacic's algorithm. We
apply 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 problem
it is integrable for only two values of the parameter, the two uncoupled Kepler
problems is completely integrable for all values of the masses.Comment: 33 page