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Non-integrability of some hamiltonians with rational potentials
Acosta Humánez, Primitivo Belén; Blázquez Sanz, David
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
In this paper we give a mechanism to compute the families of classical hamiltonians of two degrees of freedom with an invariant plane and normal variational equations of Hill-Schr\"odinger 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-Schr\"odinger 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.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística
Hamiltonian systems
Differential algebra
Difference algebra
Hamiltonian dynamical systems
Lagrangian functions
Algebrization algorithm
Autonomous Hamiltonian systems
Hill-Schrödinger equation
Kovacic algorithm
Morales-Ramis theory
Non-integrability of Hamiltonian systems
Virtually abelian groups
Hamilton, Sistemes de
Àlgebra diferencial
Hamilton, Sistemes de
Lagrange, Funcions de
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::12 Field theory and polynomials::12H Differential and difference algebra
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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