$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$
I don't understand partial derivative by "function" (e.g. $q_j$). $q$ can be displacement. Then $\dot{q}$ is velocity. Both can be represented in terms of $t$. So by eliminating $t$, $q$ can be represented in terms of $\dot{q}$ vice versa. Hence, all $q$ terms in $L$ can be replaced with $\dot{q}$ terms making RHS of the above equation $0$. What's wrong with this?
