Double Pendulum: Chaos & Sensitivity to Initial Conditions (2D)

Initial Conditions

Angle 1 (from vertical) 120°

Angle 2 (from vertical) 90°

Twin offset (chaos seed) 0.010°

Arms & Masses

Length 1 (L₁) 2.00 m

Length 2 (L₂) 1.70 m

Mass 1 (m₁) 1.20 kg

Mass 2 (m₂) 1.00 kg

Live Readout

Elapsed time

0.0 s

Total energy

0.0 J

Energy drift

0.00%

Twin separation

0.0°

How it works

Two rigid arms hinged end to end swing under gravity. There is no closed-form solution — the equations of motion are two coupled, nonlinear second-order ODEs derived from the Lagrangian, integrated here with 4th-order Runge–Kutta (RK4) at a fixed 4 ms sub-step so the physics stays accurate even at high on-screen frame rates:

δ = θ1 − θ2
D  = 2m1 + m2 − m2·cos(2δ)
θ1'' = [ −g(2m1+m2)sinθ1 − m2·g·sin(θ1−2θ2)
         − 2sinδ·m2·(θ2'²L2 + θ1'²L1·cosδ) ] / (L1·D)
θ2'' = [ 2sinδ·(θ1'²L1(m1+m2) + g(m1+m2)cosθ1
         + θ2'²L2·m2·cosδ) ] / (L2·D)

The "twin" is a second pendulum with the same masses and lengths, released with angle 2 offset by the tiny amount set by the chaos-seed slider. Because the system is chaotic, that offset amplifies roughly exponentially (Lyapunov instability): the twin trace and the divergence readout below the phase plot show the two trajectories peeling apart even though they started almost identical.

Trail — the last few seconds of bob 2's path (main pendulum solid, twin translucent).
Phase plot — θ2 vs. angular velocity ω2; a chaotic system never retraces the same loop.
Energy drift — total mechanical energy (kinetic + potential) should stay flat; RK4 keeps drift under roughly 0.01% over a full minute at these step sizes, confirming the integration is sound.

Main pendulum Twin (perturbed) Phase space (θ2, ω2)