Non-integrability of restricted double pendula

TL;DR

This paper investigates the chaotic behavior of two restricted double pendula models, providing numerical and analytical evidence that these systems are non-integrable due to their chaotic dynamics.

Contribution

The paper introduces an analytical proof of non-integrability for specific restricted double pendula models using differential Galois theory.

Findings

Numerical simulations show chaotic behavior in both models.

Bifurcation diagrams and Poincaré sections illustrate complex dynamics.

Analytic proof confirms non-integrability through differential Galois group analysis.

Abstract

We consider two special types of double pendula, with the motion of masses restricted to various surfaces. In order to get quick insight into the dynamics of the considered systems the Poincar\'e cross sections as well as bifurcation diagrams have been used. The numerical computations show that both models are chaotic which suggest that they are not integrable. We give an analytic proof of this fact checking the properties of the differential Galois group of the system's variational equations along a particular non-equilibrium solution.