Non-integrability of the generalised spring-pendulum problem

TL;DR

This paper examines a three-dimensional spring-pendulum system with two parameters, demonstrating its non-integrability except in a specific case, using advanced mathematical tools like Morales-Ramis theory and higher order variational equations.

Contribution

It establishes the non-integrability of the generalized spring-pendulum system for most parameter values and applies higher order variational equations to analyze the special case where integrability conditions are met.

Findings

System is non-integrable when k ≠ -a

Special case k = -a satisfies necessary integrability conditions

Higher order variational equations used to prove non-integrability in the special case

Abstract

We investigate a generalisation of the three dimensional spring-pendulum system. The problem depends on two real parameters $(k,a)$, where $k$ is the Young modulus of the spring and $a$ describes the nonlinearity of elastic forces. We show that this system is not integrable when $k\neq -a$. We carefully investigated the case $k= -a$ when the necessary condition for integrability given by the Morales-Ramis theory is satisfied. We discuss an application of the higher order variational equations for proving the non-integrability in this case.