On the dynamics and integrability of the Ziegler pendulum

TL;DR

This paper investigates the conditions under which the Ziegler pendulum, a double pendulum with a follower force, is integrable, revealing that zero spring stiffness and no friction lead to integrability, while non-zero stiffness introduces complex dynamics.

Contribution

It demonstrates that the Ziegler pendulum is integrable only when the spring stiffness at the pivot is zero and no friction is present, and explains the transition to chaotic behavior.

Findings

Integrability occurs with zero spring stiffness and no friction.

Existence of two-parameter families of periodic solutions.

Coexistence of regular and chaotic dynamics when springs have non-zero stiffness.

Abstract

We prove that the Ziegler pendulum -- a double pendulum with a follower force -- can be integrable, provided that the stiffness of the elastic spring located at the pivot point of the pendulum is zero and there is no friction in the system. We show that the integrability of the system follows from the existence of two-parameter families of periodic solutions. We explain a mechanism for the transition from integrable dynamics, for which there exist two first integrals and solutions belong to two-dimensional tori in a four-dimensional phase space, to more complicated dynamics. The case in which the stiffnesses of both springs are non-zero is briefly studied numerically. We show that regular dynamics coexists with chaotic dynamics.