Phase space: θ₂ (horizontal) vs ω₂ (vertical). Orderly motion draws loops; chaos smears.
A single pendulum is boring in the best way: it repeats. Bolt a second pendulum to the bottom of the first and the system becomes chaotic — still perfectly deterministic, still obeying a tidy pair of equations, but so sensitive to its starting point that prediction falls apart.
Hit Chaos ghosts. Every ghost starts with the upper angle nudged by about one ten-thousandth of a degree — far smaller than any measurement you could ever make. They stay glued together for a few seconds, then peel apart and never agree again. That divergence, not randomness, is what “chaos” means.
The simulation integrates these equations of motion with a 4th-order Runge–Kutta step at a fixed 2 ms interval:
With friction at zero the total energy readout should hold nearly constant — a good sanity check that the integrator isn't quietly inventing or leaking energy. Crank Friction up and watch the pendulum settle, inevitably, into the one boring state: straight down.
Keyboard: space play/pause · R reset · C clear trail · G ghosts · T trail