A Riemannian manifold $\mathcal{X}$ contains, for certain coordinate chart $(U, \varphi)$, the (local) coordinate map $\varphi$ that maps a neighborhood of coordinate $p$ to an Euclidean space $\tilde{U}$, which represents its tangential space $T_p \mathcal{X}$. My question regards a geodesic differential equation in local coordinates.
In case of a submerged surface $\mathcal{X}$ in $\mathbb{R}^3$, the geodesic equation over $\mathcal{X}$ is given by the flow solution $\phi(t)$ of differential equation below:
$\ddot{c} + \Gamma(c, \dot{c}) = 0$
Is there a way to represent the equality above in coordinates $U$ such that the flow $\omega(t)$ gives the flow $\phi(t)$ by equality $\phi = \varphi(\omega)$?
$\ddot{u} + I(u, \dot{u}) = 0$