A Parameter-Free Closed-Form Approximation for the Nonlinear Pendulum Period

A closed-form approximation for the period of a nonlinear pendulum is derived from first principles, anchored at both limiting regimes of the exact period integral. Using x = cos(θ₀/2) as the natural variable, the normalized function q(x) = K(k)/ln(4/x) is smooth and bounded on [0,1], with all four boundary values and slopes expressible in π and ln 4 alone. A two-term ansatz satisfying all four anchor conditions exactly yields maximum absolute error 0.01304%, occurring as a negative deviation near θ₀ = 96.51°, with a maximum positive excursion of +0.000018% near θ₀ = 168.99° — a 13.03-fold improvement over Chachiyo (2025), the best comparable parameter-free formula. All constants are composed exclusively of π and ln 4, with no free parameters.

Part of a series: AGM · Ellipse perimeter · Quartic anharmonic oscillator · Asymmetric cubic oscillator · Relativistic harmonic oscillator