T-18 · Oscillations

Everything swings.

A pendulum is only simple for small swings; at any amplitude its period is an elliptic integral, and Gauss found it by walking down a sequence of right triangles. A chain of springs moves as a sum of perpendicular modes, exactly computable when the angles are multiples of 15°. Drive a damped spring and its response is F over a hypotenuse.

Amplitude θ₀ = 120°Time t = 2.2

The period, four ways

At amplitude θ₀ the period is 4K(k), a complete elliptic integral with k = sin(θ₀/2). Gauss's arithmetic–geometric mean gets it in a handful of steps. Quadrature must agree, and so must a stopwatch on the simulation; the classical series agrees too, eventually.

small swings	2π = 6.283185
2π / AGM(1, cos θ₀/2)	…
quadrature, 24 × 24 points	residual …
timing Runge–Kutta 4	residual …
the series 2π(1 + k²/4 + …)	…
longer than 2π by	37.288%

Gauss's right triangles

The AGM starts from a right triangle: hypotenuse 1, legs cos(θ₀/2) and sin(θ₀/2). Each step makes another, with a the mean of the last two sides, b their geometric mean, and the third side c = (a − b)/2 shrinking quadratically: the digits double every step.

…

aₙ² = bₙ² + cₙ², every step	residual …

Energy

E = ω²/2 − cos θ starts at −cos θ₀ and should never change. Runge–Kutta 4 is not built to keep it, but at this step it holds to about ten digits.

E	0.5
largest drift	residual …