A function $g(r)$ is chosen to make the integral

$$\int_{a}^{b} f\left(r, g, g^{\prime}\right) d r$$

stationary, subject to given values of $g(a)$ and $g(b)$. Find the Euler-Lagrange equation for $g(r) .$

In a certain three-dimensional electrostatics problem the potential $\phi$ depends only on the radial coordinate $r$, and the energy functional of $\phi$ is

$$\mathcal{E}[\phi]=2 \pi \int_{R_{1}}^{R_{2}}\left[\frac{1}{2}\left(\frac{d \phi}{d r}\right)^{2}+\frac{1}{2 \lambda^{2}} \phi^{2}\right] r^{2} d r$$

where $\lambda$ is a parameter. Show that the Euler-Lagrange equation associated with minimizing the energy $\mathcal{E}$ is equivalent to

$$\frac{1}{r} \frac{d^{2}(r \phi)}{d r^{2}}-\frac{1}{\lambda^{2}} \phi=0$$

Find the general solution of this equation, and the solution for the region $R_{1} \leqslant r \leqslant R_{2}$ which satisfies $\phi\left(R_{1}\right)=\phi_{1}$ and $\phi\left(R_{2}\right)=0$.

Consider an annular region in two dimensions, where the potential is a function of the radial coordinate $r$ only. Write down the equivalent expression for the energy functional $\mathcal{E}$ above, in cylindrical polar coordinates, and derive the equivalent of (1).