Many physics problems involve finding the configuration of a system that minimizes a quantity. It could be as easy as “How much does a mass \(m\) stretch a spring \(k\)?”, in which case the system is specified by a single variable height \(y\) and the quantity to minimize is energy \(E\), which is just a function of \(y\)
$$E(y)=\frac{1}{2}ky^2-mgy$$
and we can find the equilibrium position simply by setting the derivative to 0
$$\frac{\partial E}{\partial y}|_{y_o}=0 \Rightarrow y_o=\frac{mg}{k}$$
Unfortunately, life is not always so simple and we could be faced with a problem like “What’s the shape of a soap film stretched between two rings”. In this case, the configuration of the system is described by a function \(y(x)\) (revolved around the x axis) and the quantity to minimize is surface area, which is a functional of \(y(x)\).
$$ A[y]=\int_{x_1}^{x_2}a(y,y’)dx=\int_{x_1}^{x_2}2\pi y\sqrt{1+{y’}^2}dx $$
equilibrium is found by setting the functional derivative to 0
$$ \frac{\delta A}{\delta y}|_{y_o}=\frac{\partial a}{\partial y}-\frac{d}{dx}\frac{\partial a}{\partial y’}=0 \Rightarrow $$
$$\frac{d}{dx}\left( a-y’\frac{\partial a}{\partial y’} \right)=0 \Rightarrow$$
$$y_o(x)=\kappa\cosh(\frac{x+a}{\kappa})$$
This procedure doesn’t look too bad on the surface, but that’s because I hid all the subtleties. For starters, the definition of a functional derivative as shown above only works in very special cases, namely \(a(y,y’)\) is implicit and the variation has fixed-ends. Even in cases where the simple form of functional derivative does apply, it is not at all obvious how to solve the resultant differential equation in general. I was only able to solve the above problem fairly easily due to Noether’s theorem. That’s why a lot of what we did in the rest of the semester had do with solving various differential equations.
We shall next study the beast to tame – differential operators, and some tools to tame it – Fourier expansion, Green’s function.