Double Pendulum Chaos
Two double pendulums start with angles differing by 0.001°. Within seconds they diverge completely, tracing entirely different paths — a vivid demonstration of sensitive dependence on initial conditions, the hallmark of chaos.
d²θ₁/dt² = [−g(2m₁+m₂)sinθ₁ − m₂g·sin(θ₁−2θ₂) − 2sin(θ₁−θ₂)m₂(ω₂²L₂+ω₁²L₁cos(θ₁−θ₂))] / [L₁(2m₁+m₂−m₂cos(2θ₁−2θ₂))]
The double pendulum is one of the simplest physical systems that exhibits chaos. Unlike a single pendulum, it has two degrees of freedom coupled through a nonlinear interaction, which allows energy to transfer between the two arms unpredictably.
The Lyapunov exponent measures the rate of separation between nearby trajectories. For chaotic systems it is positive, meaning small differences grow exponentially: Δθ(t) ~ Δθ(0)·e^(λt). Even in exact arithmetic, after a finite time (the Lyapunov time), the two pendulums are completely uncorrelated.
This is not randomness — the equations are deterministic. But in practice, any physical system has finite precision, so the future is genuinely unpredictable beyond the Lyapunov time.