Divergence: 0
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Initial Angles
120°
120°
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Chaos Demonstration
Trail
Physics
9.81 m/s²
Chaos from Newtonian mechanics
The double pendulum follows Newton's laws exactly—no randomness involved. Yet two pendulums with imperceptibly different starting positions (differing by 0.001°) will soon move in completely different ways. The physics is deterministic, but the outcome is effectively unpredictable.
The Lyapunov exponent λ measures how fast nearby trajectories diverge. For the double pendulum, λ ≈ 7-8 s⁻¹. This means initial differences are amplified by a factor of e ≈ 2.7 every ~0.13 seconds. After just a few seconds, tiny errors become enormous.
This is one of the simplest mechanical systems imaginable—just two rigid rods and gravity. Yet it exhibits the same "butterfly effect" as weather! No matter how precisely we measure the initial state, our predictions become worthless after a short time. Simplicity breeds complexity.
Despite the chaos, the pendulum's motion is bounded—it stays on a "strange attractor" in phase space. Total energy is conserved (in the ideal case). The chaos lies not in unbounded growth, but in the exponential sensitivity to initial conditions within these bounds.