Non-integrability of some Hamiltonians with rational potentials

TL;DR

This paper investigates the non-integrability of certain two-degree-of-freedom Hamiltonians with rational potentials by analyzing the Galois groups of associated Schrödinger-type equations using Kovacic's algorithm and introduces an algebrization method to facilitate this analysis.

Contribution

It provides a systematic approach to determine non-integrability of Hamiltonians with rational potentials through Galoisian obstructions and introduces an algebrization technique for equations with transcendental coefficients.

Findings

Identifies Galoisian obstructions to rational first integrals.

Computes Galois groups for Schrödinger-type equations with polynomial potentials.

Develops an algebrization method to analyze equations with transcendental coefficients.

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.