A pendulum swinging from the end of another pendulum — one of the simplest systems that behaves chaotically, and the wild cousin of our tidy simple pendulum. Release it, then launch a "chaos twin" a hair apart and watch the two paths track together for a heartbeat before diverging completely. The twin separation readout and the log divergence graph below make the butterfly effect visible — the same sensitive dependence you can spin in the Lorenz attractor. Same laws, same start, wildly different fate, all on your device. Free on LK Forge — no sign-up.
You are in the Physics lab.
What the two arms, the twin and the log graph are telling you.
The upper rod swings from the fixed pivot; the lower rod hangs from the tip of the first and swings freely. The Upper angle and Lower angle readouts give each rod's angle from straight down, and the glowing trail follows the lower bob. Near 0° the arms hang and barely move; start them near 180° — inverted over the pivot — and the whole system flails, which the Motion badge flags as chaotic and the Total energy stat confirms.
Turn on the Chaos twin and a faint blue pendulum starts a mere perturbation away — a hundredth of a radian or less. For a second or two the twin hides behind the original, then it peels off and swings completely differently. The Twin separation stat, shown in scientific notation, is the distance between the two lower bobs; watch it climb from around 1e-3 toward the size of the pendulum itself.
The graph plots that separation against time with a logarithmic vertical axis. Exponential growth becomes a straight line, so while the twins diverge the curve climbs steadily — its slope is the Lyapunov exponent, the rate chaos amplifies errors. It then flattens once the gap saturates at the pendulum's own size: the two are now on unrelated swings. Raise the Damping or drop the Gravity and the whole character changes.
Two coupled joints, the exact equations of motion, and a tiny nudge that changes everything.
Four numbers describe the state — the two rod angles and how fast each is turning. From the masses, rod lengths and gravity, the coupled equations give each joint's angular acceleration at every instant. There is no closed-form solution, so the motion has to be computed step by step rather than written as a formula.
The simulator advances the state with fourth-order Runge–Kutta at a small fixed time step, taking several substeps per frame. RK4 samples the accelerations four times per step and blends them, which keeps the total energy stable and the swing faithful far longer than a naive update would — you can watch the Total energy stat hold steady when damping is off.
The twin is a second copy of the same equations, started from angles offset by the perturbation you set. Both obey identical rules with no randomness, yet the gap between their tips grows exponentially. That runaway separation — plotted straight on the log graph — is sensitive dependence on initial conditions, the reason long-range prediction is hopeless even though nothing is random.
Teaching or blogging about chaos? You can embed this simulator on your own site free — one line of code, no sign-up.
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
Describe how quickly two nearly identical starts diverge from each other.
The double pendulum is deterministic yet chaotic: nearby starting states diverge exponentially, so its long-term motion is unpredictable even though the equations are exact. It is a classic, visual demonstration of the butterfly effect.
Educators: link or embed this simulation freely in your LMS or course guide.
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