Non-integrability of some few body problems in two degrees of freedom
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hdl:2117/2370
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Data publicació2008-11-16
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Reconeixement-NoComercial-SenseObraDerivada 3.0 Espanya
Abstract
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 me\-thod 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.
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