Non-integrability of restricted double pendula

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.