Suppose in a system of $n$ dynamic degrees of freedom $q_i$, $i=1,\dots,n$, there is only kinetic energy: $$L=\frac{1}{2}\sum_{a,b}g_{ab}(q_i)\dot{q_a}\dot{q_b}$$
where $g_{ab}(q_i)=g_{ba}(q_i)$ are the components of a symmetric $n\times n$ matrix. Show that the Euler-Lagrange equations for this system are $$\ddot{q_a}+\sum_{b,c}\Gamma^{a}_{bc}\dot{q_b}\dot{q_c}=0$$ Where $$\Gamma^{a}_{bc}=\sum_{d}\frac{1}{2}g_{ad}^{-1}\Big(\frac{\partial g_{bd}}{\partial q_c}+\frac{\partial g_{cd}}{\partial q_b}-\frac{\partial g_{bc}}{\partial q_d}\Big)$$ I know there is a similar question in mse but that lacks some explanations so I cannot fully understand. May anyone help?