Exploring the chaos: a journey with the double pendulum
Following my post here on the dynamics of the double pendulum, I created a simulation that vividly illustrates the complex and chaotic movements that define this fascinating physical system. The double pendulum, a seemingly simple setup with two pendulums attached end to end, manifests chaotic behavior under specific conditions, making it a captivating subject for study in physics and mathematics.
The foundation
The simulation starts with defining the physical parameters and initial conditions of the double pendulum system. By employing Python’s capabilities, particularly NumPy for mathematical computations and the Runge-Kutta method for solving differential equations, I established a foundation for the simulation:
from scipy.integrate import solve_ivp
import numpy as np
from types import SimpleNamespace
# Physical data and initial conditions setup
m1, m2 = 1, 1 # Masses of the pendulums
p = SimpleNamespace(g=9.81, m1=m1, m2=m2, l1=4.0, l2=4.0, mu=1.0 + m1 / m2)
m = SimpleNamespace(
th1=90 * np.pi / 180.0, th2=180 * np.pi / 180.0,
thdot1=0.0, thdot2=0.0, timedelta=0.03)
The simulation core
The core of the simulation lies in the double_pendulum function, which calculates the derivatives of the system’s state variables at any given time. This function, alongside the rk4singlestep method for numerical integration, forms the backbone of the simulation process, allowing for the accurate portrayal of the pendulum’s dynamics over time.
Visualization and animation
Creating a visualization of the double pendulum’s motion was paramount. I developed a class, MyPlotter, to handle the plotting and animation of the system’s behavior over time. Utilizing Matplotlib, I crafted detailed animations that reveal the intricate paths taken by the pendulums under the influence of gravity and their initial conditions.
Conclusion
The double pendulum simulation highlights the unpredictable and sensitive nature of chaotic systems. By tweaking initial conditions, one can observe a wide array of behaviors, from simple oscillations to complex, seemingly random motions.
Look at the results here. This simulation not only serves as a tool for education and exploration but also as a reminder of the beauty inherent in the laws of physics and mathematics that govern our universe.