dc.contributor.author Acosta Humánez, Primitivo Belén dc.contributor.author Álvarez Ramírez, Martha dc.contributor.author Delgado Fernández, Joaquín dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I dc.date.accessioned 2008-11-17T19:31:39Z dc.date.available 2008-11-17T19:31:39Z dc.date.issued 2008-11-16 dc.identifier.uri http://hdl.handle.net/2117/2370 dc.description.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. dc.format.extent 33 p. dc.language.iso eng dc.rights Attribution-NonCommercial-NoDerivs 3.0 Spain dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/ dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística dc.subject.lcsh Hamiltonian systems dc.subject.lcsh Differential algebra dc.subject.lcsh Differential equations dc.subject.lcsh Lagrangian functions dc.subject.lcsh Dynamics dc.subject.lcsh Nonlinear operators dc.subject.other n-body problem dc.subject.other Morales-Ramis theory dc.subject.other Kovacic's algorithm dc.subject.other Kimura's theorem dc.subject.other Non-integrability dc.title Non-integrability of some few body problems in two degrees of freedom dc.type Article dc.subject.lemac Hamilton, Sistemes de dc.subject.lemac Àlgebra diferencial dc.subject.lemac Equacions en diferències dc.subject.lemac Lagrange, Funcions de dc.subject.lemac Partícules (Física nuclear) dc.subject.lemac operadors no lineals dc.contributor.group Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions dc.subject.ams Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems dc.subject.ams Classificació AMS::12 Field theory and polynomials::12H Differential and difference algebra dc.subject.ams Classificació AMS::34 Ordinary differential equations::34M Differential equations in the complex domain dc.subject.ams Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics dc.subject.ams Classificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanics dc.subject.ams Classificació AMS::47 Operator theory::47J Equations and inequalities involving nonlinear operators dc.rights.access Open Access local.personalitzacitacio true
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