Write down the Euler-Lagrange (EL) equations for a functional

$$\int_{a}^{b} f\left(u, w, u^{\prime}, w^{\prime}, x\right) d x$$

where $u(x)$ and $w(x)$ each take specified values at $x=a$ and $x=b$. Show that the EL equations imply that

$$\kappa=f-u^{\prime} \frac{\partial f}{\partial u^{\prime}}-w^{\prime} \frac{\partial f}{\partial w^{\prime}}$$

is independent of $x$ provided $f$ satisfies a certain condition, to be specified. State conditions under which there exist additional first integrals of the $\mathrm{EL}$ equations.

Consider

$$f=\left(1-\frac{m}{u}\right) w^{\prime 2}-\left(1-\frac{m}{u}\right)^{-1} u^{\prime 2}$$

where $m$ is a positive constant. Show that solutions of the EL equations satisfy

$$u^{\prime 2}=\lambda^{2}+\kappa\left(1-\frac{m}{u}\right)$$

for some constant $\lambda$. Assuming that $\kappa=-\lambda^{2}$, find $d w / d u$ and hence determine the most general solution for $w$ as a function of $u$ subject to the conditions $u>m$ and $w \rightarrow-\infty$ as $u \rightarrow \infty$. Show that, for any such solution, $w \rightarrow \infty$ as $u \rightarrow m$.

[Hint:

$$\left.\frac{d}{d z}\left\{\log \left(\frac{z^{1 / 2}-1}{z^{1 / 2}+1}\right)\right\}=\frac{1}{z^{1 / 2}(z-1)} . \quad\right]$$