I'm entirely clueless with this problem. No formal training in variational methods.
Show that for function $\phi\left ( x \right )$ with
$$\phi\left ( a \right )=\phi\left ( b \right )=0$$
and with the further constraint
$$I = \int_{a}^{b}r\left ( x \right )\left ( \phi'\left ( x \right ) \right )^{2} dx=1 $$
that the functional
$$J\left ( \phi\left ( x \right ) \right )=\int_{a}^{b}\left [ p\left ( x \right )\left ( \phi'\left ( x \right ) \right )^{2}-q\left ( x \right )\left ( \phi^{2}\left ( x \right ) \right ) \right ]dx$$
is rendered stationary by the solution $y\left ( x, \lambda \right )$ of the Sturm-Liouville system
$$\left ( p\left ( x \right )y'\left ( x \right ) \right )'+\left ( q\left ( x \right ) +\lambda r\left ( x \right )y\left ( x \right ) \right )=0$$
in $a< x< b$ with $y\left ( a \right )=y\left ( b \right )=0$.
and that the minimum value of J is given by $\lambda_{1}$, the smallest eigen value of the Sturm-Liouville system
This popped up in a past year paper so I'm expecting something like this to spring up on the exam paper.
Help is appreciated