As known, if $S\subset \mathbb{R}^3$ is a surface, $\sigma: U\subset \mathbb{R}^2 \rightarrow V\cap S$ a parametrization, and \begin{align}\alpha:I& \rightarrow S\\ t& \mapsto \sigma(u(t),v(t)) \end{align} a curve on $S$, then $\alpha$ is a geodesic if, and only if, resolve que defferential equation
\begin{align}u'' + \Gamma_{11}^1 (u')^2 + 2 \Gamma_{12}^{1}u' v' + \Gamma_{22}^{1} (v')&=0, \qquad(1) \\ v'' + \Gamma_{11}^2 (u')^2 + 2 \Gamma_{12}^{2}u' v' + \Gamma_{22}^{2} (v')&=0, \qquad(2) \end{align}
where $\Gamma_{ij}^{k}$ are the Christoffel symbols.
I need to show that:
QUESTION: When the differential equation of the geodesics are referred to the arc length then the second equation (2) is, except for the coordinate curves, a consequence of the first equation (1).
The book suggests write each $\Gamma_{ij}^{k}$ in function of $E$, $F$, $G$ (where $E(u,v)= <\sigma_u(u,v),\sigma_u(u,v)>$, $F(u,v)= <\sigma_u(u,v),\sigma_v(u,v)>$), $ G(u,v)= <\sigma_v(u,v),\sigma_v (u,v)>$) i. e. \begin{align} \Gamma_{11}^{1} &= \frac{G E_u - 2 F F_u + F E_v}{2(E G - F^2)}, \quad \Gamma_{11}^{2} =\frac{2EF_u - E E_v - F E_u}{2(EG - F^2)},\\ \Gamma_{12}^{1} &= \frac{G E_v - F G_u}{2(E G - F^2)}, \hspace{2cm} \Gamma_{12}^{2} =\frac{EG_u - F E_v}{2(EG - F^2)},\\ \Gamma_{22}^{1} &= \frac{2GF_v - G G_u - F G_u}{2(EG - F^2)}, \quad \Gamma_{22}^{2} =\frac{E G_v - 2F F_v + F G_ u}{2(EG - F^2)}, \end{align}
and use that $\sigma(u(t),v(t))$ is parametrized by arc length, i.e.
$$1= E (u')^2 + 2F (u'v') + G (v')^2 $$ $$0= E_u (u')^3 + E_v (v')(u')^2 + 2 E u'' u' + 2F_u (u')^2 v' + 2F_v u' (v')^2 + 2F u'' v' + 2F u' v'' + G_u u' (v')^2 + G_v (u')(v')^2 + 2 G v'' v', $$
to conclude the result, but I wasn't able to manipulate this equations in order to prove the required. Any help?