Integrability of the double pendulum -- the Ramis' question

TL;DR

This paper investigates whether a double pendulum in a constant gravity field is integrable, concluding non-integrability through algebraic and numerical methods involving monodromy group analysis.

Contribution

It demonstrates the non-integrability of the double pendulum using a novel combination of algebraic and numerical techniques, specifically analyzing the monodromy group.

Findings

Double pendulum is non-integrable in constant gravity.

Monodromy group analysis confirms non-integrability.

Algebraic and numerical methods complement each other in this proof.

Abstract

In this short note we address the problem of integrability of a double pendulum in the constant gravity field. We show its non-integrability using the combination of algebraic and numerical approaches, namely we compute the non-commuting generators of the monodromy group along a particular solution obtained numerically.