Write down the Euler-Lagrange equation for the variational problem for $r(z)$

$$\delta \int_{-h}^{h} F\left(z, r, r^{\prime}\right) d z=0$$

with boundary conditions $r(-h)=r(h)=R$, where $R$ is a given positive constant. Show that if $F$ does not depend explicitly on $z$, i.e. $F=F\left(r, r^{\prime}\right)$, then the equation has a first integral

$$F-r^{\prime} \frac{\partial F}{\partial r^{\prime}}=\frac{1}{k},$$

where $k$ is a constant.

An axisymmetric soap film $r(z)$ is formed between two circular rings $r=R$ at $z=\pm H$. Find the equation governing the shape which minimizes the surface area. Show that the shape takes the form

$$r(z)=k^{-1} \cosh k z .$$

Show that there exist no solution if $R / H<\sinh A$, where $A$ is the unique positive solution of $A=\operatorname{coth} A$.