T = 4√(L/g)·K(sin(θ₀/2)) — elliptic integrals, separatrix, tautochrone
Exact pendulum period: T = 4√(L/g)·K(k) where K is the complete elliptic integral of the first kind, k=sin(θ₀/2).
Small-angle approx T₀ = 2π√(L/g) fails above ~20°. Near separatrix (θ₀→180°): T → ∞ logarithmically.
Phase portrait shows libration orbits (θ₀ < 180°) and rotation (θ₀ > 180°) separated by the separatrix (homoclinic orbit).
Tautochrone: The cycloid curve ensures period independent of amplitude (Huygens 1659) — unlike the pendulum, where T(θ₀) grows with amplitude.