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Double Pendulum — Chaotic Motion

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Kinetic Energy

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Potential Energy

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Total Energy

Phase Space (θ₁ vs θ₂)

Explain the Physics

The double pendulum is one of the simplest physical systems that exhibits deterministic chaos — extreme sensitivity to initial conditions. Even a 0.001° difference in starting angle leads to completely different trajectories after a few seconds.

We derive the equations of motion using the Lagrangian formulation:

$$\mathcal{L} = T - V = \frac{1}{2}(m_1+m_2)l_1^2\dot{\theta}_1^2 + \frac{1}{2}m_2l_2^2\dot{\theta}_2^2 + m_2 l_1 l_2 \dot{\theta}_1\dot{\theta}_2\cos(\theta_2-\theta_1) + (m_1+m_2)gl_1\cos\theta_1 + m_2gl_2\cos\theta_2$$

This yields two coupled second-order ODEs which we integrate using the 4th-order Runge–Kutta method (RK4).

Try this: set both initial angles to 120° and 100°, then change θ₂ to 100.1° — after 30 seconds, the trajectories are completely different. This is chaos.