I have a proof for the Coordinate Invariance of E-L equation, that is a little bit messy. the very last step is:
$\frac{d}{dt}(\frac{\partial L}{\partial \dot q_m}) = \sum_{i=1}^N \frac{\partial L}{\partial \dot x_i} \frac{\partial \dot x_i}{\partial q_m} + \sum_{i=1}^N \frac{\partial L}{\partial x_i} \frac{\partial x_i}{\partial q_m}$
$\frac{d}{dt}(\frac{\partial L}{\partial \dot q_m}) = \frac{\partial L}{\partial q_m}$ which the E-L
Now here is my stupid question: what happened to the summation, I guess it became irrelevant, for $x_i$ no longer "counts".. shouldn't we see something like:
$\frac{d}{dt}(\frac{\partial L}{\partial \dot q_m}) = \sum_{i=1}^N \frac{\partial L}{\partial \dot x_i} \frac{\partial \dot x_i}{\partial q_m} + \sum_{i=1}^N \frac{\partial L}{\partial x_i} \frac{\partial x_i}{\partial q_m}$
$\frac{d}{dt}(\frac{\partial L}{\partial \dot q_m}) = c\frac{\partial L}{\partial q_m}$
Although the c would be arbitrary here...I am curious if it is still mathematically correct!
Ok this is what I missed: L is a function of $x_i , \dot x_i ,t$ --> $L(x_i,\dot x_i,t)$.... $x_i$ depends on $q_m$, $\dot x_i$ depends on $q_m$ as well
So by Chain rule (the derivative of L with respect to $q_m$ is: $\frac{\partial L}{\partial q_m} = \sum_{i=1}^N \frac{\partial L}{\partial \dot x_i} \frac{\partial \dot x_i}{\partial q_m} + \sum_{i=1}^N \frac{\partial L}{\partial x_i} \frac{\partial x_i}{\partial q_m}$
Sorry If I confused anybody