Double pendulum

How it works

A double pendulum follows exact, deterministic equations: the same start always gives the same motion. At small angles it moves regularly, but at large angles it is chaotic. A tiny difference in the starting angle then grows roughly exponentially until the two motions are completely different. This is the butterfly effect. The simulation uses the fourth-order Runge–Kutta method with a 1/480 s step and continuously shows how well the total energy is conserved.

E = Ek + Ep = const.divergence in chaos: |Δ(t)| ≈ |Δ₀|·e^(λt)

Try it yourself

1. Choose “Experiment: gentle motion” (20° and 25°) and turn on “Butterfly effect” with a 0.1° difference. The motion is regular and the two pendulums stay almost together.
2. Switch to “Experiment: butterfly effect” (140° and 140°). The “Watch the paths diverge” graph shows the moment the motions break apart.
3. Reduce the “Angle difference” to 0.01°. Predict first: the split comes only a little later. With 0.1° the pendulums separate after about 3.5 s; with a difference ten times smaller, after about 6 s.
4. Watch “Numerical energy error A”. It stays very small, so the divergence is not a calculation error but a property of chaotic motion.

Model limitations

The model uses point masses and rigid massless rods with no friction or air resistance, so the pendulum never stops. A real pendulum would gradually lose energy to friction.

Related experiments

• Levers and torque
• Gravity and orbits
• Fractals