Oscillating Pendulum

phase portrait · nonlinear period · separatrix

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Physics: Nonlinear pendulum: θ̈ + (b/mL²)θ̇ + (g/L)sinθ = 0. Period T = 2π√(L/g)·[1 + θ₀²/16 + 11θ₀⁴/3072 + ...] (exact: elliptic integral K). For θ₀ → π the period T → ∞ (infinite time to reach unstable equilibrium — the separatrix). Galileo's isochronism (1602) holds only for small angles. The phase portrait (θ, θ̇) shows closed orbits (libration), open orbits (rotation), and the separatrix (homoclinic orbit at energy E = 2mgL).