The Euler Lagrange equation $\frac{\partial F}{\partial q}-\frac{d \frac{\partial F}{\partial \dot{q}}}{d t}=0$ can also be put in the form $\frac{\partial F}{\partial t}-\frac{d (F- \dot{q}\frac{\partial F}{\partial \dot{q}})}{d t}=0$,
How is the second form of the equation arrived at mathematically, and does equating them and rearranging (as above) lead to any simplifications of the equations? I'm probably missing some elementary rearrangement.