The Pendulum: Small-Angle Lie vs the Real Nonlinear Swing

The real equation, and the approximation everyone remembers

A simple pendulum — a point mass on a massless rod of length L — obeys one equation from torque and Newton's second law for rotation:

θ'' = -(g/L) · sin(θ)      ← the real, nonlinear equation

for small θ, sin(θ) ≈ θ, so:

θ'' ≈ -(g/L) · θ           ← simple harmonic motion, solvable exactly
T   = 2π√(L/g)             ← the famous formula, valid only in this limit

That substitution, sinθ ≈ θ, is what turns an unsolvable nonlinear equation into the clean linear one everyone learns first. It's an excellent approximation for small swings — and a genuinely wrong one for large ones.

Why there's no closed-form solution

sinθ is a nonlinear function of θ, and that nonlinearity is what breaks the usual algebra. The exact period of the real pendulum can be written down, but only in terms of the complete elliptic integral of the first kind, K — a special function that itself has no elementary closed form:

T(θ₀) = 4 √(L/g) · K( sin(θ₀/2) )

θ₀ → 0     ⇒  T → 2π√(L/g)      (the small-angle formula, recovered exactly)
θ₀ → 180°  ⇒  T → ∞             (a pendulum balanced upright never quite falls)

The small-angle formula is exactly the θ₀ → 0 limit of the true answer — not a different, competing formula, but the leading term of a series expansion of K. Every additional term in that series is a correction that grows with amplitude.

How much the period actually grows

Expanding K in powers of the amplitude gives a series correction to the naive period:

T(θ₀) ≈ T₀ · [ 1 + (1/16)θ₀² + (11/3072)θ₀⁴ + ... ]     (θ₀ in radians)

θ₀ = 10°   →  period ≈ 0.19% longer than T₀
θ₀ = 30°   →  period ≈ 1.7%  longer than T₀
θ₀ = 90°   →  period ≈ 18%   longer than T₀

Real pendulums swing slower than the textbook formula predicts, and the gap grows quickly once you go past about 20-30 degrees — a pendulum clock built for small swings will visibly lose time if someone gives it a big push.

Solving it numerically: RK4

Rather than evaluate elliptic integrals, the practical route — and the one this site's simulation uses — is to integrate the real equation directly with a fourth-order Runge-Kutta step. Splitting the second-order ODE into two coupled first-order ones (angle θ and angular velocity ω) and stepping forward:

state = [θ, ω]
f(state) = [ω, -(g/L) * sin(θ)]

k1 = f(state)
k2 = f(state + h/2 * k1)
k3 = f(state + h/2 * k2)
k4 = f(state + h   * k3)
state += h/6 * (k1 + 2*k2 + 2*k3 + k4)

RK4's fourth-order accuracy makes the numerical period converge to the true elliptic-integral answer to well within floating-point precision at any reasonable step size, which is why the simulation can show the small-angle prediction visibly drifting apart from the real swing as the amplitude grows — two pendulums released together, one obeying the linear approximation and one obeying physics.

Frequently asked questions

Try it live

Everything above runs in your browser — open Simple Pendulum and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.