Double Pendulum

Lagrangian Mechanics: The double pendulum Lagrangian L=T−V gives two coupled nonlinear ODEs via the Euler-Lagrange equations. The system has two degrees of freedom (θ₁,θ₂) with conserved energy but no analytic solution — it exhibits deterministic chaos for large angles. The phase portrait shows (θ₁,θ̇₁). Sensitivity to initial conditions: a twin pendulum (yellow) started δ=0.001 rad away diverges exponentially — the Lyapunov exponent λ≈1/time for separation to grow by e.