PARAMETERS
T exact: — s
T₀ (linear): — s
T/T₀ ratio: —
k = sin(θ₀/2): —
Exact period of the nonlinear pendulum:
T = 4√(L/g) K(sin²(θ₀/2))
where K(m) = ∫₀^{π/2} dφ/√(1−m·sin²φ) is the complete elliptic integral of the first kind.
As θ₀ → π (separatrix): K(1) = ∞, so T → ∞ logarithmically. An unphysical infinite period for a perfectly poised inverted pendulum.
The phase portrait shows librations (closed curves) below the separatrix, and rotations (open curves) above.