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Non-integrability of some few body problems in two degrees of freedom. (English) Zbl 1200.37051

The paper under review is devoted to a study of integrability for some Hamiltonian systems with two degrees of freedom, related with few body problems: a) the elliptic restricted three body problem in the plane with collision of the primaries; b) the rectangular 4 body problem; c) the anisotropic Kepler problem. This can be made through the analysis of the linearization of the Hamiltonian system, that is, variational equations and normal variational equations. Two main tools are used: 1) the differential Galois theory through the Moralez-Ruiz approach; 2) the algebrization method through the Kovacic’s algorithm.

MSC:

37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
37N05 Dynamical systems in classical and celestial mechanics
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