On the dynamics and integrability of the Ziegler pendulum
Dynamical Systems
2024-01-04 v4 Chaotic Dynamics
Exactly Solvable and Integrable Systems
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.
Keywords
Cite
@article{arxiv.2209.03724,
title = {On the dynamics and integrability of the Ziegler pendulum},
author = {Ivan Polekhin},
journal= {arXiv preprint arXiv:2209.03724},
year = {2024}
}