Consider the integral

$$I=\int f\left(y, y^{\prime}\right) d x$$

Show that if $f$ satisfies the Euler-Lagrange equation, then

$$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant. }$$

An axisymmetric soap film $y(x)$ is formed between two circular wires at $x=\pm l$. The wires both have radius $r$. Show that the shape that minimises the surface area takes the form

$$y(x)=k \cosh \frac{x}{k}$$

Show that there exist two possible $k$ that satisfy the boundary conditions for $r / l$ sufficiently large.

Show that for these solutions the second variation is given by

$$\delta^{2} I=\pi \int_{-l}^{+l}\left(k \eta^{\prime 2}-\frac{1}{k} \eta^{2}\right) \operatorname{sech}^{2}\left(\frac{x}{k}\right) d x$$

where $\eta$ is an axisymmetric perturbation with $\eta(\pm l)=0$.