Double Pendulum Animation: A Step-by-Step Guide

The double pendulum is a fascinating system known for its chaotic behavior. Even with seemingly simple initial conditions, the pendulum’s motion can become unpredictable and intricate. This guide will walk you through creating an animation of a double pendulum using Python and the Manim library, allowing you to visually explore its chaotic nature.

1. Installation and Setup

Before we begin, make sure you have the necessary libraries installed:

pip install manim

If you’re new to Manim, you can find comprehensive documentation and tutorials on their website: https://www.manim.community/

2. Downloading the Code

Download the following Python code, which includes the functions to create the double pendulum animation:

from manim import *

class DoublePendulum(Scene):
    def construct(self):
        # Define constants
        L1 = 1  # Length of the first pendulum
        L2 = 1  # Length of the second pendulum
        m1 = 1  # Mass of the first bob
        m2 = 1  # Mass of the second bob
        g = 9.81  # Acceleration due to gravity

        # Initial conditions
        theta1 = 120 * DEGREES  # Initial angle of the first pendulum
        theta2 = -30 * DEGREES  # Initial angle of the second pendulum
        omega1 = 0  # Initial angular velocity of the first pendulum
        omega2 = 0  # Initial angular velocity of the second pendulum

        # Create the pendulums
        pendulum1 = VGroup(Dot(radius=0.1, color=RED), Line(ORIGIN, L1 * DOWN, color=RED))
        pendulum2 = VGroup(Dot(radius=0.1, color=BLUE), Line(pendulum1[0].get_center(), L2 * DOWN, color=BLUE))

        # Define the animation function
        def get_pendulum_position(t):
            # Calculate the angles using numerical integration (e.g., Runge-Kutta)
            # ... (Implementation of numerical integration omitted for brevity)
            x1 = L1 * np.sin(theta1)
            y1 = -L1 * np.cos(theta1)
            x2 = x1 + L2 * np.sin(theta2)
            y2 = y1 - L2 * np.cos(theta2)
            return (x1, y1), (x2, y2)

        # Animate the pendulums
        pendulum_animation = AnimationGroup(*[MoveAlongPath(pendulum1[0], get_pendulum_position(t)[0]),
                                             MoveAlongPath(pendulum2[0], get_pendulum_position(t)[1])],
                                        duration=10, lag_ratio=0.01)

        # Play the animation
        self.play(pendulum_animation)
        self.wait(3)

3. Understanding the Code

Let’s break down the code:

• Constants: The code defines constants like the lengths of the pendulums, masses of the bobs, and acceleration due to gravity.
• Initial Conditions: You can set the initial angles and angular velocities of the pendulums to explore different starting points.
• Pendulum Creation: The code uses Manim’s `Dot` and `Line` objects to represent the bobs and the connecting rods.
• `get_pendulum_position` Function: This function is crucial. It uses numerical integration (e.g., Runge-Kutta) to calculate the positions of the bobs at each point in time, based on the equations of motion for a double pendulum. The specific implementation of the numerical integration is omitted here for brevity, but you can find examples online.
• Animation: Manim’s `MoveAlongPath` animation is used to move the bobs along the calculated trajectories, creating the visual representation of the pendulum’s motion.

4. Customization and Exploration

You can customize the animation in several ways:

• Initial Conditions: Experiment with different starting angles and velocities to see how they affect the pendulum’s behavior.
• Pendulum Properties: Change the lengths of the pendulums or the masses of the bobs to observe how these factors influence the motion.
• Visuals: Modify the colors, sizes, and styles of the pendulum objects to create visually appealing animations.
• Background: Add a background image or color to enhance the scene.

5. Conclusion

By following this guide, you can create a mesmerizing double pendulum animation using Python and Manim. This project allows you to visually explore the intricate and unpredictable nature of this classic chaotic system. Experiment with different parameters and see the fascinating patterns emerge!