What is a Double Pendulum?

A double pendulum is exactly what it sounds like: a pendulum hanging from the end of another pendulum. A single pendulum swings back and forth in a predictable arc. Attach a second pendulum to the end of the first, and the system becomes chaotic. Not random—chaotic. The motion is fully deterministic (governed by Newton’s laws), but it is so sensitive to initial conditions that any tiny difference in starting position produces wildly divergent behavior within seconds.

The equations of motion come from the Lagrangian of the system. For two rods of lengths $L_1$, $L_2$ and masses $m_1$, $m_2$, the angular accelerations are coupled nonlinear differential equations with no closed-form solution. The system must be integrated numerically—here using RK4 (Runge-Kutta 4th order).

Try It

Drag the pendulum to set a starting position, then release to let it swing. Or use the angle sliders. Small changes in starting angles produce dramatically different trajectories—this is chaos in action. The trail traces the path of the lower bob.

The Physics

The double pendulum is governed by the Lagrangian $L = T - V$, where T is kinetic energy and V is potential energy. For masses $m_1$, $m_2$ on rods of length $L_1$, $L_2$, the angular accelerations are:

$$\ddot{\theta}_1 = \frac{-g(2m_1+m_2)\sin\theta_1 - m_2 g \sin(\theta_1-2\theta_2) - 2\sin(\theta_1-\theta_2)\,m_2\!\left(\dot{\theta}_2^2 L_2 + \dot{\theta}_1^2 L_1 \cos(\theta_1-\theta_2)\right)}{L_1\!\left(2m_1+m_2-m_2\cos(2\theta_1-2\theta_2)\right)}$$ $$\ddot{\theta}_2 = \frac{2\sin(\theta_1-\theta_2)\!\left(\dot{\theta}_1^2 L_1(m_1+m_2) + g(m_1+m_2)\cos\theta_1 + \dot{\theta}_2^2 L_2 m_2 \cos(\theta_1-\theta_2)\right)}{L_2\!\left(2m_1+m_2-m_2\cos(2\theta_1-2\theta_2)\right)}$$

These equations are derived by applying the Euler–Lagrange equations to the system’s Lagrangian. They cannot be solved analytically. The simulation integrates them numerically using the 4th-order Runge-Kutta method (the same RK4 described in the numerical methods article), which provides excellent accuracy and energy conservation over long runs.

Why It’s Chaotic

A single pendulum is integrable—it has as many conserved quantities (energy) as degrees of freedom (one angle). The double pendulum has two degrees of freedom but only one conserved quantity (total energy). This mismatch is what allows chaos. The system has enough freedom to explore its phase space in complicated ways, but not enough constraints to keep it predictable.

Quantitatively, the double pendulum has a positive Lyapunov exponent, meaning nearby trajectories in phase space diverge exponentially fast. Two pendulums started just 0.001° apart will look completely different within 10–20 seconds. This is not a failure of the simulation—it is a fundamental property of the physics.

What to Try

• Small angles: Start both arms at 10–20°. The motion is nearly periodic—the system is approximately linear and not yet chaotic.
• Large angles: Start at 120° or more. Watch the lower bob trace wild, unpredictable loops.
• Asymmetric masses: Make $m_1$ much larger than $m_2$. The upper arm barely moves while the lower one whips around.
• Short lower arm: Reduce $L_2$. The lower bob moves fast and chaotically while the upper arm swings slowly.
• Near-identical starts: Set angles, note the trajectory, then change $\theta_1$ by just 1° and reset. Watch how quickly the paths diverge.