Consider a particle on a trajectory $x^{a}(\lambda)$. Show that the geodesic equations, with affine parameter $\lambda$, coincide with the variational equations obtained by varying the integral

$$I=\int_{\lambda_{0}}^{\lambda_{1}} g_{a b}(x) \frac{\mathrm{d} x^{a}}{\mathrm{~d} \lambda} \frac{\mathrm{d} x^{b}}{\mathrm{~d} \lambda} \mathrm{d} \lambda$$

the end-points being fixed.

In the case that $f(r)=1-2 G M u$, show that the space-time metric is given in the form

$$\mathrm{d} s^{2}=-f(r) \mathrm{d} t^{2}+\frac{1}{f(r)} \mathrm{d} r^{2}+r^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right)$$

for a certain function $f(r)$. Assuming the particle motion takes place in the plane $\theta=\frac{\pi}{2}$ show that

$$\frac{\mathrm{d} \phi}{\mathrm{d} \lambda}=\frac{h}{r^{2}}, \quad \frac{\mathrm{d} t}{\mathrm{~d} \lambda}=\frac{E}{f(r)},$$

for $h, E$ constants. Writing $u=1 / r$, obtain the equation

$$\left(\frac{\mathrm{d} u}{\mathrm{~d} \phi}\right)^{2}+f(r) u^{2}=-\frac{k}{h^{2}} f(r)+\frac{E^{2}}{h^{2}}$$

where $k$ can be chosen to be 1 or 0 , according to whether the particle is massive or massless. In the case that $f(r)=1-G M u$, show that

$$\frac{\mathrm{d}^{2} u}{\mathrm{~d} \phi^{2}}+u=k \frac{G M}{h^{2}}+3 G M u^{2}$$

In the massive case, show that there is an approximate solution of the form

$$u=\frac{1}{\ell}(1+e \cos (\alpha \phi)),$$

where

$$1-\alpha=\frac{3 G M}{\ell} .$$

What is the interpretation of this solution?