Double pendulum

A double pendulum is a famous example of a chaotic system.

Links per pendulum(1–5)

Pendulums(1–12)

3 chains of 2 rods. Each copy is released 0.0001 rad further along than the last. Past one rod the motion is chaotic, and that initial difference separates exponentially. Changing any setting restarts the run. Drag on the canvas to pose the chain — the joint nearest your pointer follows it, and the rods below swing along rigidly — then let go to drop it from there.

The controls above set how many links each pendulum has and how many are released together. With a single link the system is not chaotic: the copies stay locked on top of one another indefinitely. Add a second link and the same experiment quickly descends into chaos.

You can also set the starting position by hand: drag anywhere on the canvas to swing the chain into a pose, and let go to release it from rest. Try dropping it from almost the same place twice.

The setup

A chain of $n$ massless rods, each of length $\ell_i = 1$ and carrying a point mass $m_i = 1$ at its far end, hanging from a fixed pivot with $g = 9.81$ . Every copy is released from rest with all rods horizontal.

Taking the angles $\theta_i$ from the downward vertical, the Lagrangian gives one equation per rod,

\sum_j c_{ij}\,\ell_j\cos(\theta_i-\theta_j)\,\ddot\theta_j = -\sum_j c_{ij}\,\ell_j\sin(\theta_i-\theta_j)\,\dot\theta_j^{\,2} - g\,c_{ii}\sin\theta_i

where $c_{ij} = \sum_{k \ge \max(i,j)} m_k$ is the mass hanging at or below both rods. The equations are solved using a Runge-Kutta (RK4) integrator."