Mechanical & Aerospace
Double Pendulum Lab
Add a second rod to a pendulum and something remarkable happens: the motion becomes CHAOTIC. The equations are exact and deterministic — no randomness — yet the system is so sensitive to its starting point that two releases differing by a millionth of a degree soon swing completely differently. This lab integrates the real nonlinear equations of motion with RK4, so the trail of the lower bob is physically faithful, and the total-energy readout stays essentially constant — proof that the wild behaviour comes from the geometry, not from any noise or friction. Tune the two masses and rod lengths, set a release angle, and hit release twice: the divergence is the whole point. It's the textbook demonstration of deterministic chaos, and a vivid contrast to the tidy, repeating single pendulum.
0.00 J
Total energy
120°
Release angle
Chaotic
Behaviour
The method, with your numbers
Coupled nonlinear dynamics
- 1
Set the release state
θ₁ = θ₂ = θ₀ ω₁ = ω₂ = 0
θ₀ = 120° · π/180 = 2.09 rad ω₁ = ω₂ = 0 rad/s
Angles are measured from the hanging-down vertical. Both rods start aligned and at rest — the sim works in radians.
- 2
Energy budget at release (all potential)
E = −(m₁+m₂)·g·L₁·cos θ₁ − m₂·g·L₂·cos θ₂
E = −(1+1)·9.81·1·cos 120° − 1·9.81·1·cos 120° = 14.71 J
Released from rest, KE = 0 — this number is the system's entire budget. Heights are measured from the pivot, so E is negative while the bobs hang below it.
- 3
Angular accelerations at release (the coupled EOM)
θ̈₁ = [−g(2m₁+m₂)sin θ₁ − m₂g·sin(θ₁−2θ₂) − 2m₂·sin(θ₁−θ₂)·(ω₂²L₂ + ω₁²L₁cos(θ₁−θ₂))] / (L₁·den), den = 2m₁+m₂−m₂cos(2θ₁−2θ₂)
θ̈₁ = -8.5 rad/s² θ̈₂ = 0 rad/s²
At the instant of release the rods are aligned and at rest, so every coupling term vanishes: θ̈₂ = 0 and θ̈₁ = −g·sin θ₀ / L₁ — a simple pendulum for exactly one instant.
- 4
Integrate with RK4 (what the canvas is doing)
step: s ← RK4(s, dt) — no closed-form solution exists
E(now) = 0 J vs E(release) = 14.71 J
From this release angle the motion is fully chaotic: two starts a millionth of a degree apart soon trace completely different paths. Energy is the one invariant left — if E(now) drifts from E(release), that's integrator error, not physics.
Drag any control above — every number here recalculates. Want this method for any problem? Step Sheets →
Two rods, one chaotic system. Unlike the single pendulum, the double pendulum is exquisitely sensitive — two releases a millionth of a degree apart soon trace completely different paths. Total energy stays (nearly) constant, proving it's the geometry, not friction, doing this. Release it twice from the same angle and watch.
How to use this simulation
Add a second rod to a pendulum and something remarkable happens: the motion becomes CHAOTIC. The equations are exact and deterministic — no randomness — yet the system is so sensitive to its starting point that two releases differing by a millionth of a degree soon swing completely differently. This lab integrates the real nonlinear equations of motion with RK4, so the trail of the lower bob is physically faithful, and the total-energy readout stays essentially constant — proof that the wild behaviour comes from the geometry, not from any noise or friction. Tune the two masses and rod lengths, set a release angle, and hit release twice: the divergence is the whole point. It's the textbook demonstration of deterministic chaos, and a vivid contrast to the tidy, repeating single pendulum.
Everything runs in your browser — no sign-up, no download. Change a value and the result updates instantly, so you can build a feel for how each input shapes the outcome. It pairs with Crameleon's practice exams and step sheets when you want to go from intuition to working the problems.