Non-integrability of some Hamiltonians with rational potentials

2006/10/06 by Primitivo B. Acosta-Humánez, Acosta-Humanez, Primitivo, David Blázquez-Sanz +1
Mathematics · Physics and Astronomy · #34M15 #37J30 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems

paper · doi:10.48550/arxiv.math-ph/0610010

openalex publication_date 2006/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this work we compute the families of classical Hamiltonians in two degrees of freedom in which the Normal Variational Equation around an invariant plane falls in Schroedinger type with polynomial or trigonometrical potential. We analyze the integrability of Normal Variational Equation in Liouvillian sense using the Kovacic's algorithm. We compute all Galois groups of Schroedinger type equations with polynomial potential. We also introduce a method of algebrization that transforms equations with transcendental coefficients in equations with rational coefficients without changing essentially the Galoisian structure of the equation. We obtain Galoisian obstructions to existence of a rational first integral of the original Hamiltonian via Morales-Ramis theory.

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