Non-integrability of some hamiltonians with rational potentials

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September 2008, 10(2&3, September): 265-293. doi: 10.3934/dcdsb.2008.10.265

Non-integrability of some hamiltonians with rational potentials

Primitivo Acosta-Humánez ^1, and David Blázquez-Sanz ^1,

1.	Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Giron 1-3 08034, Barcelona, Spain, Spain

Received September 2006 Revised November 2007 Published June 2008

Abstract
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In this paper we give a mechanism to compute the families of clas- sical hamiltonians of two degrees of freedom with an invariant plane and normal variational equations of Hill-Schrödinger type selected in a suitable way. In particular we deeply study the case of these equations with polynomial or trigonometrical potentials, analyzing their integrability in the Picard-Vessiot sense using Kovacic’s algorithm and introducing an algebraic method (algebrization) that transforms equations with transcendental coefficients in equations with rational coefficients without changing the Galoisian structure of the equation. We compute all Galois groups of Hill-Schrödinger type equations with polynomial and trigonometric (Mathieu equation) potentials, obtaining Galoisian obstructions to integrability of hamiltonian systems by means of meromorphic or rational first integrals via Morales-Ramis theory.

Keywords: hamiltonian systems, integrability, Kovacic's algorithm, Picard-Vessiot theory..

Mathematics Subject Classification: Primary: 37J30, 12H05; Secondary: 70H0.

Citation: Primitivo Acosta-Humánez, David Blázquez-Sanz. Non-integrability of some hamiltonians with rational potentials. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 265-293. doi: 10.3934/dcdsb.2008.10.265

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