// sketch

Double Pendulum

Deterministic chaos in Lagrangian mechanics.

Controls

› No interaction needed — physics are perfectly deterministic

Chaos Theory

A single pendulum swinging follows a perfectly predictable, symmetrical arch in simple harmonic motion. But the moment you attach a second pendulum to the bottom of the first, the system breaks into highly complex, non-repeating, and seemingly random movements. This is called deterministic chaos.

The Butterfly Effect

Because the system is chaotic, it exhibits extreme sensitive dependence on initial conditions (often popularized as the "Butterfly Effect"). If you were to restart this simulation and change the starting angle of the top pendulum by just 0.0001 degrees, the pendulums would initially look identical, but within seconds their paths would diverge completely.

Lagrangian Mechanics

To accurately calculate the acceleration of both swinging arms over time without breaking down complex tension forces, we use the Lagrangian formulation of mechanics (Energy instead of Forces): L = T - V (where T is kinetic energy and V is potential energy).

// Angular Acceleration of Top Arm (θ1):

num1 = -g * (2 * m1 + m2) * sin(θ1)
num2 = -m2 * g * sin(θ1 - 2 * θ2)
num3 = -2 * sin(θ1 - θ2) * m2 * (v2² * L2 + v1² * L1 * cos(θ1 - θ2))
den = L1 * (2 * m1 + m2 - m2 * cos(2 * θ1 - 2 * θ2))
a1 = (num1 + num2 + num3) / den

These differential equations are recalculated on every single frame, adjusting the angular velocities, which adjust the angles, which draw the path on the screen.