Correction to period of pendulum: range of amplitude

Question

Question Statement (from Kleppner and Kolenkow "Introduction to Mechanics"):

Let us change variables as follows:

$\sin u = \sin(θ/2)/ \sin(θ_0/2)$

The motivation for this is that although $ θ$ is periodic, as the pendulum swings through a cycle, θ varies between $−θ_0$ and $ θ_0$. On the other hand, $u$ varies between $−π$ and $+π$.

How come $u$ varies between $−π$ and $+π$ ? If we replace $θ$ with $−θ_0$ and $ θ_0$ in the given equation we get the value ranging from $-1$ to $+1$ i.e. angle ranging from $−π/2$ and $+π/2$.

I was also confused why we could suppose "$\sin u = \sin(θ/2)/ \sin(θ_0/2)$"? Elliptical integral is totally new to me.

Answer 1

From energy conservation of the pendulum system, the period can be deduced from evaluation of the following integral, where $\ell$ is the length of the pendulum, $$T \propto \sqrt{\frac{\ell}{g}} \int_0^{\theta_0} \frac{\text{d}\theta}{\sqrt{\cos \theta - \cos \theta_0}}.$$

The motivation for the clever substitution you presented is such to recast this integral into the form of an elliptic integral of the first kind, $F(1,\sin \theta_0/2) = K(\sin \theta_0/2)$ making apparent the underlying mathematical structure of the problem in terms of this non elementary function.

See e.g https://en.wikipedia.org/wiki/Elliptic_integral for an introduction.

Yes, you are correct, the bounds on the transformed variable are such that $u \in [-\pi/2, \pi/2]$.