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About this tool
Real-time chaos theory visualization with interactive double pendulum physics and glowing motion trails.
The Double Pendulum Chaos Simulator integrates the standard double-pendulum equations of motion with a fourth-order Runge-Kutta solver and draws the resulting swing in real time with fading motion trails. You set both masses, both rod lengths, gravity, damping and the two starting angles, then watch θ₁, θ₂, ω₁ and ω₂ update live. A comparison mode runs a second pendulum whose lower angle is offset by just 0.001 radians so you can see sensitive dependence on initial conditions with your own eyes.
Open Double Pendulum Chaos Simulator on AltFTool — it loads instantly in your browser.
Set the Parameters sliders — Mass 1 and Mass 2 (0.5–5 kg), Length 1 and Length 2 (0.5–3 m), Gravity (0.5–30 m/s²), Damping and Speed — plus the Initial Angles θ₁ and θ₂ in radians.
Press Start, use Apply & Restart after moving the angle sliders or Randomize for a random pair, and toggle Compare to run a second pendulum offset by 0.001 rad.
Watch the trails draw while the Live Stats panel updates FPS, θ₁, θ₂, ω₁ and ω₂; in comparison mode separate Pendulum A and Pendulum B stats show the divergence.
Each frame advances the state with a four-stage Runge-Kutta step, so the motion stays coherent far longer than a naive forward-Euler simulation.
Comparison mode draws two pendulums whose only difference is 0.001 rad in the second angle, making the exponential separation visible rather than theoretical.
Masses from 0.5 to 5 kg, lengths from 0.5 to 3 m, gravity from 0.5 to 30 m/s² and damping from 0 to 0.5 all take effect without restarting.
Because its equations of motion are nonlinear and coupled, so the difference between two nearby starting states grows exponentially rather than staying small. In comparison mode a 0.001 rad offset — under a twentieth of a degree — produces completely unrelated motion within seconds.
Both masses are 1 kg, both rods are 1 m, gravity is 9.81 m/s², damping is 0.01 and both arms start horizontal at 90°. Reset all returns to exactly these values.
It solves the standard Lagrangian equations for a double pendulum with point masses and massless rigid rods, using RK4 stepped by the real-time frame delta capped at 0.05 s (that cap is applied before the Speed slider scales it, so the integrator's actual step is min(frameDelta, 0.05 s) x Speed - up to 0.15 s at the 3x Speed maximum). It is a faithful model of that idealisation, but it ignores rod mass, air resistance and joint friction beyond the linear damping term you set.
It subtracts a term proportional to each angular velocity from that arm's angular acceleration, which is a simple linear drag model. Set it to 0 for an energy-conserving pendulum that swings indefinitely; the default 0.01 makes the motion settle slowly so the trails stay readable.