🌀 Triple Pendulum — Extreme Chaos
Three pendulums connected in series form a 3-degree-of-freedom chaotic system. Even a difference of 0.0001 rad in the initial angle leads to completely different trajectories within seconds — the defining signature of chaos. Uses RK4 integration of the full Lagrangian equations of motion.
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Physics
A triple pendulum has 3 degrees of freedom (angles θ₁, θ₂, θ₃). The equations of motion are derived from the Lagrangian L = T − V, where T is kinetic energy and V is potential energy. The result is a coupled system of 3 nonlinear second-order ODEs. This simulation integrates them with a 4th-order Runge-Kutta (RK4) method with fixed timestep Δt = 1 ms for accuracy.
Energy should be conserved (E = const). Any visible drift is integration error — smaller Δt gives better conservation. The Lyapunov exponent λ > 0 quantifies how fast nearby trajectories diverge: |δ(t)| ≈ |δ₀|·e^(λt). For a triple pendulum λ ≈ 3–7 s⁻¹, much larger than a double pendulum.
vs. Double Pendulum
A double pendulum (2 DOF) is the classic chaos demonstration. The triple pendulum (3 DOF) is even more chaotic: longer Lyapunov exponent, richer attractor structure, and the bob-3 trail almost immediately appears fully random. This makes it ideal for visualising the butterfly effect.
Key facts
- Degrees of freedom: 3
- Lyapunov exponent λ ≈ 3–8 s⁻¹ (double pendulum ≈ 1–3)
- Tiny Δθ = 1×10⁻⁴ rad → variants visibly diverge in < 5 s
- Phase space dimension: 6 (3 angles + 3 angular velocities)