Non-integrability Criterium for Normal Variational Equations around an integrable Subsystem and an example: The Wilbeforce spring-pendulum

TL;DR

This paper develops a criterion for non-integrability of normal variational equations around an integrable subsystem and applies it to demonstrate the non-integrability of the Wilbeforce spring-pendulum using Morales-Ramis theory.

Contribution

It introduces a new non-integrability criterion based on Galois group analysis and applies Kovacic's algorithm to a Heun equation in the context of the Wilbeforce pendulum.

Findings

The Galois group of the variational equation is not virtually abelian.

The variational equation reduces to a Heun differential equation with four singularities.

The Wilbeforce spring-pendulum is proven non-integrable.

Abstract

In this paper we analyze the non-integrability of the Wilbeforce pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.