Two rods, one hinge on another, and a law you can write on a napkin — yet the motion is unpredictable forever. The equations are perfectly deterministic; a difference of one part in a hundred million in the start grows to something of order one within seconds. Down the center, data flows: the angles go in, the Lagrangian engine integrates, the energy and the divergence come out. The blue team builds and defends it; the red team tries to break it.
source J.-L. Lagrange, Méchanique analitique (Paris, veuve Desaint, 1788) — the method, not the pendulum: a figure-less algebraic recasting of mechanics from the Lagrangian L = T − V. archive.org/details/meychaniqueanal00lagr. AMBER — 1788 first edition, French, no figures; the double pendulum is a standard modern application of Lagrange's equations, not a worked example in the original. Rendered, not quoted.
Two point masses m on massless rods of length L, angles θ₁, θ₂ from vertical, Δ = θ₁−θ₂. The Lagrangian is kinetic minus potential:
T = ½mL²ω₁² + ½m(L²ω₁² + L²ω₂² + 2L²ω₁ω₂cosΔ) · V = −2mgL·cosθ₁ − mgL·cosθ₂
Euler–Lagrange, d/dt(∂L/∂ω) − ∂L/∂θ = 0, gives a coupled linear system for the accelerations, M(θ)·α = b(θ,ω):
| α₁ | α₂ | = b | |
|---|---|---|---|
| row θ₁ | (m₁+m₂)L₁² | m₂L₁L₂cosΔ | −m₂L₁L₂sinΔ·ω₂² − (m₁+m₂)gL₁sinθ₁ |
| row θ₂ | m₂L₁L₂cosΔ | m₂L₂² | +m₂L₁L₂sinΔ·ω₁² − m₂gL₂sinθ₂ |
The engine solves this 2×2 every step. The off-diagonal m₂L₁L₂cosΔ is the coupling — the whole story lives there.
Add one joint to a pendulum and determinism stops meaning predictability. The double pendulum is the simplest mechanical system with a positive Lyapunov exponent — chaos small enough to build on a desk.
At small angles the coupling barely matters and it is just the spring–mass's two normal modes, ω² = (g/L)(2±√2). Swing it hard and the same equations turn violent. Each sphere is the next one's premise: the linear cousin, taken nonlinear.
The blue team's live check: re-integrate two trajectories a hair apart, confirm energy stays bounded and that they diverge by many orders of magnitude. If red drops the coupling, this badge is where it shows.
The state is four numbers: two angles and two angular velocities, (θ₁, θ₂, ω₁, ω₂). Release from rest (ω=0) and only the angles matter. That is the entire input — no forcing, no noise, no friction.
Constants: m₁=m₂=1, L₁=L₂=1, g=9.81. Everything downstream is computed live from these four numbers and the Lagrangian above.
The pink pendulum and its cyan ghost start a thousandth of a radian apart. Watch them track — then split.
Every frame steps the coupled equations with an implicit-midpoint integrator (symmetric, time-reversible) — energy holds, nothing is looked up.
What the machine proves, computed at boot over a fixed chaotic run: energy is conserved to the integrator tolerance despite the wild motion; the closed-form accelerations satisfy M·α=b exactly; two states 10⁻⁸ apart diverge to order one; and at small angles the modes are (g/L)(2±√2).
The blue team's witness (left) re-runs the divergence live; the red team (right) tries to make chaos disappear.
And it is worse: the "energy conserved" claim is only as good as the integrator. Any explicit scheme (forward Euler, plain RK) leaks energy secularly and eventually invents motion the physics forbids. The choice of integrator is a modelling assumption, marked and defended below — not a free lunch.
"Chaotic means random." Cut. The motion is fully deterministic and repeatable to the bit for identical input; the engine proves it by re-running. Chaos is sensitivity, not randomness.
"Sensitive dependence breaks energy conservation." Cut. Orthogonal facts. Trajectories diverge on a fixed energy surface — the witness checks both at once.
"Two pendulums, so two independent swings." Kept, corrected. Drop the coupling and you do get two independent pendulums — regular, non-chaotic, wrong energy. That deletion is exactly the tamper.
The red team's move: drop the coupling term — zero the off-diagonal m₂L₁L₂cosΔ and the cross-forcing — so the two arms swing as independent pendulums. The witness (window 7) is watching.
Decoupled, the energy of the true system is no longer conserved and the two near-identical starts stop diverging — chaos vanishes. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.