Explain how geodesics of a Riemannian metric

$$g=g_{a b}\left(x^{c}\right) d x^{a} d x^{b}$$

arise from the kinetic Lagrangian

$$\mathcal{L}=\frac{1}{2} g_{a b}\left(x^{c}\right) \dot{x}^{a} \dot{x}^{b}$$

where $a, b=1, \ldots, n$.

Find geodesics of the metric on the upper half plane

$$\Sigma=\left\{(x, y) \in \mathbb{R}^{2}, y>0\right\}$$

with the metric

$$g=\frac{d x^{2}+d y^{2}}{y^{2}}$$

and sketch the geodesic containing the points $(2,3)$ and $(10,3)$.

[Hint: Consider $d y / d x .]$