Dynamics of the N-fold Pendulum in the framework of Lie Group Integrators

TL;DR

This paper explores the application of Lie group integrators, specifically Runge--Kutta--Munthe--Kaas methods, to simulate the complex dynamics of an N-fold 3D pendulum, highlighting practical implementation and mathematical foundations.

Contribution

It introduces the use of Lie group integrators for N-fold pendulum systems, including implementation details and mathematical framework, demonstrating their effectiveness on a complex mechanical system.

Findings

Lie group integrators effectively simulate N-fold pendulum dynamics.

Adaptive time stepping enhances simulation accuracy.

Mathematical framework extends to Lagrangian mechanical systems.

Abstract

Since their introduction, Lie group integrators have become a method of choice in many application areas. Various formulations of these integrators exist, and in this work we focus on Runge--Kutta--Munthe--Kaas methods. First, we briefly introduce this class of integrators, considering some of the practical aspects of their implementation, such as adaptive time stepping. We then present some mathematical background that allows us to apply them to some families of Lagrangian mechanical systems. We conclude with an application to a nontrivial mechanical system: the N-fold 3D pendulum.