Deterministic chaos: A pedagogical review of the double pendulum case

Abstract

This review aims to provide a comprehensive evaluation of the dynamics of the double pendulum, with a particular emphasis on its chaotic behavior. It examines the complicated and unpredictable behavior of the double pendulum. It highlights the pedagogical value of the double pendulum as it bridges concepts across physics, mathematics, and computer science, providing a tangible demonstration of chaotic dynamics. Numerical methods, such as the 4th-order Runge-Kutta method, estimate solutions to the autonomous Hamiltonian equations that govern a system. We simulate the motion of a double pendulum, enabling the visualization of intricate trajectories and the analysis of emerging patterns with the Python and Fortran programming languages. We discuss the importance of Poincaré’s maps and the Lyapunov exponents in characterizing and quantifying the rate at which trajectories diverge in phase space to elucidate the chaotic nature of the system. The educational significance of the double pendulum is emphasized in teaching key concepts in Classical Mechanics, Differential Calculus, Linear Algebra, and Numerical Methods for solving ordinary differential equations (ODEs). We also explore the interdisciplinary teaching opportunities presented by the double pendulum.