Lagrangian dynamics: The equations of motion come from d/dt(∂L/∂q̇) - ∂L/∂q = 0 with L = T - V. The double pendulum has two degrees of freedom and is non-integrable for large angles. The Lyapunov exponent λ ≈ 3–7 s⁻¹ means nearby trajectories diverge as e^(λt) — after a few seconds, all predictability is lost. Multiple pendula started with tiny angle differences quickly explore completely different regions of phase space.