Overview
The double pendulum is one of the simplest systems that produces chaos. I derived its equations of motion from the Euler–Lagrange equation and solved the resulting non-linear ODEs numerically in Python to explore how chaotic it really is.
Time-to-flip fractal
This plot maps every starting position of both pendulums against how long it takes the second pendulum to flip over. The result is a fractal — a signature of chaos. The black region marks starting positions without enough energy for the second pendulum to ever flip.
Sensitivity to initial conditions
I ran 200 simulations with initial angles varying by one billionth of a radian. The trajectories stay synchronized for roughly 40–45 seconds — then diverge completely. By 60 seconds the system is fully chaotic.
Method
- Derived the equations of motion with Lagrangian mechanics (full derivation typeset in LaTeX)
- Solved the coupled non-linear ODEs numerically in Python
- Swept initial conditions to map flip times and divergence behavior
The live trace on this site’s homepage is this same system — an RK4 integration running in your browser.