Nonlinear Pendulum — Exact Period via Elliptic Integrals

θ̈ = −(g/L) sin θ. The exact period T = 4√(L/g) K(sin²(θ₀/2)) where K is the complete elliptic integral of the first kind. Small-angle gives T₀=2π√(L/g). See how they diverge as θ₀→180°.

Initial angle θ₀ 30° Length L 1.00 m

The complete elliptic integral K(k²) = ∫₀^{π/2} dφ/√(1−k²sin²φ) diverges logarithmically as k→1 (θ₀→π), predicting the infinite period of the separatrix (homoclinic orbit).