Euler-Lagrange equations: change of coordinates

Euler-Lagrange equations for a set of generalized coordinates $q$:

\begin{equation}
\frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}}=0
\end{equation}

Change of coordinates to $s$:

\begin{equation}
q_i=q_i(s_1,…,s_N,t)
\end{equation}

Lagrangian as a function of the $s$ coordinates:

\begin{equation}
L=L\left(q(s, t), \frac{d}{dt}q(s, t), t\right)
\end{equation}

Show that the Euler-Lagrange equations are also obeyed in coordinates $s$:

\begin{equation}
\frac{\partial L}{\partial s_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{s_i}}=0
\end{equation}

Taking the total time derivative of $(2)$:

\begin{equation}
\dot{q_i}=\sum_j\left(\frac{\partial q_i}{\partial s_j}\dot{s_j}\right)+\frac{\partial q_i}{\partial t}
\end{equation}

Applying the chain rule:

\begin{equation}
\frac{\partial L}{\partial s_i}=\sum_j\left(\frac{\partial L}{\partial q_j}\frac{\partial q_j}{\partial s_i}+\frac{\partial L}{\partial \dot{q_j}}\frac{\partial \dot{q_j}}{\partial s_i}\right)=\sum_j\left[\frac{\partial L}{\partial q_j}\frac{\partial q_j}{\partial s_i}+\frac{\partial L}{\partial \dot{q_j}}\frac{d}{dt}\left(\frac{\partial q_j}{\partial s_i}\right)\right]
\end{equation}

\begin{equation}
\frac{\partial L}{\partial \dot{s_i}}=\sum_j\left(\frac{\partial L}{\partial q_j}\frac{\partial q_j}{\partial \dot{s_i}}+\frac{\partial L}{\partial \dot{q_j}}\frac{\partial \dot{q_j}}{\partial \dot{s_i}}\right)=\sum_j\left(\frac{\partial L}{\partial \dot{q_j}}\frac{\partial q_j}{\partial s_i}\right)
\end{equation}

where the the order of differentiation was switched in $(6)$. We can simplify $(7)$ because $\frac{\partial q_j}{\partial \dot{s_i}}=0$ from $(2)$ and $\frac{\partial \dot{q_j}}{\partial \dot{s_i}}=\frac{\partial q_j}{\partial s_i}$ from $(5)$.

Taking the total time derivative of $(7)$:

\begin{equation}
\frac{d}{dt}\frac{\partial L}{\partial \dot{s_i}}=\sum_j\left[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_j}}\right)\frac{\partial q_j}{\partial s_i}+\frac{\partial L}{\partial \dot{q_j}}\frac{d}{dt}\left(\frac{\partial q_j}{\partial s_i}\right)\right]
\end{equation}

Evaluating and canceling out terms:

\begin{equation}
\frac{\partial L}{\partial s_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{s_i}}=\sum_j\left[\left(\frac{\partial L}{\partial q_j}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_j}}\right)\right)\frac{\partial q_j}{\partial s_i}\right]=0
\end{equation}

from equation $(1)$. We see that the form of the equations are the same in all coordinates.