Elliptic functions from a pendulum

As readers of this blog might realize, I'm a big fan of elliptic functions and elliptic curves. In addition to having all sorts of practical applications, they actually offer a good motivation to learn all sorts of things that you might otherwise find sort of dry and uninteresting. And they certainly seem to appear almost everywhere.

An example of this is how elliptic functions appear when you look at how pendulums work. The model for this that you learn in high-school physics assumes that the pendulum starts swinging from a relatively small angle. But if we don't assume that this angle is small, elliptic functions appear. Here's why.

Let's assume that we have a pendulum like shown in this picture:


The usual derivation of the equation that describes the motion of this pendulum is to use Newton's Second Law to get

m g sin(θ) L = m L2 d2θ / dt2

so that

d2θ / dt2 = – g / L sin(θ)

If we assume that θ is small so that sin(θ) ≈ θ we get the high-school physics version of pendulum motion.

But what if θ is not small?

Let's write k2 = g / L so that we have

d2θ / dt2 = –k2 sin(θ)

To integrate dθ we look at

(d2θ / dt2) dθ = –k2 sin(θ) dθ

Integrating and assuming that the initial conditions let us drop any constants of integration we find that

(1 / 2) (dθ / dt)2 = k2 cos(θ)


(dθ / dt)2 = 2 k2 cos(θ)

so that

dθ / dt = √2 k √cos(θ)


dθ / √cos(θ) = √2 k dt

Now if we make the substitution

x = tan(θ / 2)

we find that we have

dθ / √cos(θ) = 2 ∫ dx / √(1 – x4)

But wait. That's an integral of one over a cubic or a quartic, and that's an elliptic integral. That means that x is an elliptic function of t, or that tan(θ / 2) is an elliptic function of t. That also means that we can't write θ as a nice, simple function of t like we can if we make the small angle approximation, but it also shows that there's definitely an elliptic function that naturally arises here.

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