# Surface Optimization

Ok, so here's the problem:

Two circular wire hoops of radius $R$ are spaced a distance $2\ell$ apart. Consider a soap film stretching beatween the two hoops. Due to surface tension the film's equlibrium form is a surface of minimal area. Use the calculus of variations to find this minimal surface.

What happens when $\ell/R$ is small? What happens when $\ell/R$ becomes large? Is there a critical value of $\ell/R$? If so, explain why; find in numerically; and say what happens when one exceeds it.

Calculus of Variations is simple: To "optimize" a quantity, $y(x)$, (i.e. find its minima/maxima), you find the value of $x$ which satisfies $y'(x)=0$.

If the independent is a function which optimizes a quantity (as in the case of this problem),
$$A=\int F(y(x),\,y'(x);\,x)\,dx$$​
the optimized $y(x)$ is given by the solution to the differential equation below (Euler's Equation):
$$\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right)=0$$​

Here's what I did. It makes most sense to set this problem in cylindrical coordinates. So, the surface area of the soap film is given by

$$A=\int_0^{2\ell}2\pi r(z)\sqrt{1+r'(z)^2}dz.$$​

Here, we are assuming that there is no angular dependence of $r$. So in anticipation of applying the calculus of variations (Euler's Equation), I evaluated the following quantities for $F(r(z),\,r'(z);\,z)=r(z)\sqrt{1+r'(z)^2}$:

$$\frac{\partial F}{\partial r}=\sqrt{1+r'^2}\,,\,\,\,\,\,\,\,\,\frac{\partial F}{\partial r'}=\frac{r(z)r'(z)}{\sqrt{1-r'^2}}\,,\,\,\,\,\text{and}\,\,\,\,\,\,\frac{d}{dz}\left(\frac{\partial F}{\partial r'}\right)=\frac{1-r'^4+rr''}{(1+r'^2)^{3/2}}$$​

Therefore Euler's equation becomes,
$$\frac{1+r'^2-rr''}{(1+r'^2)^{3/2}}=0.$$​
If we disallow the denominator from vanishing, we can simplify this:
$$1+\left(\frac{dr}{dz}\right)^2-r(z)\frac{d^2r}{dz^2}=0.$$​

I have no clue how to approach this. Mathematica gave me some really wierd ugly answer, which when I asked it to FullSimplify, gave an unmanagable expression with hyperbolic sines and cosines. What's the solution to this differential equation, and how do I analytically apply the boundary conditions?