Consider a spacetime $\mathcal{M}$ with a metric $g_{a b}\left(x^{c}\right)$ and a corresponding connection $\Gamma_{b c}^{a}$. Write down the differential equation satisfied by a geodesic $x^{a}(\lambda)$, where $\lambda$ is an affine parameter.

Show how the requirement that

$$\delta \int g_{a b}\left(x^{c}\right) \frac{d}{d \lambda} x^{a}(\lambda) \frac{d}{d \lambda} x^{b}(\lambda) d \lambda=0$$

where $\delta$ denotes variation of the path, gives the geodesic equation and determines $\Gamma_{b c}^{a}$.

Show that the timelike geodesics for the 2 -manifold with line element

$$d s^{2}=t^{-2}\left(d x^{2}-d t^{2}\right)$$

are given by

$$t^{2}=x^{2}+\alpha x+\beta$$

where $\alpha$ and $\beta$ are constants.