Consider the functional

$$F[y]=\int_{\alpha}^{\beta} f\left(y, y^{\prime}, x\right) d x$$

of a function $y(x)$ defined for $x \in[\alpha, \beta]$, with $y$ having fixed values at $x=\alpha$ and $x=\beta$.

By considering $F[y+\epsilon \xi]$, where $\xi(x)$ is an arbitrary function with $\xi(\alpha)=\xi(\beta)=0$ and $\epsilon \ll 1$, determine that the second variation of $F$ is

$$\delta^{2} F[y, \xi]=\int_{\alpha}^{\beta}\left\{\xi^{2}\left[\frac{\partial^{2} f}{\partial y^{2}}-\frac{d}{d x}\left(\frac{\partial^{2} f}{\partial y \partial y^{\prime}}\right)\right]+\xi^{\prime 2} \frac{\partial^{2} f}{\partial y^{\prime 2}}\right\} d x$$

The surface area of an axisymmetric soap film joining two parallel, co-axial, circular rings of radius a distance $2 L$ apart can be expressed by the functional

$$F[y]=\int_{-L}^{L} 2 \pi y \sqrt{1+y^{\prime 2}} d x$$

where $x$ is distance in the axial direction and $y$ is radial distance from the axis. Show that the surface area is stationary when

$$y=E \cosh \frac{x}{E},$$

where $E$ is a constant that need not be determined, and that the stationary area is a local minimum if

$$\int_{-L / E}^{L / E}\left(\xi^{\prime 2}-\xi^{2}\right) \operatorname{sech}^{2} z d z>0$$

for all functions $\xi(z)$ that vanish at $z=\pm L / E$, where $z=x / E$.