I have this conceptual question: In Landau's book of classical mechanics, about the principle of least action, it's written:
$$\left. \delta S =\frac{\partial L}{\partial v} \delta q \right\rvert_{t_1}^{t_2} + \ \int\limits_{t_1}^{t_2} \ dt \left(\frac{\partial L}{\partial q}\ - \frac d {dt}\frac {\partial L}{\partial v}\right) \delta q \ =0 ,$$
where $q=q(t)$ is the position function, $v=v(t)$ velocity function, $S$ the action, and L the Lagrangian of the system.
There is the condition $\delta q(t_1)=\delta q(t_2)=0$. So the first term is zero, and then it says " there remains an integral which must vanish for all values of $\delta q$. This can be so only if the integrand is zero identically.
Well I can't understand why
$$\left(\frac{\partial L}{\partial q}\ - \frac d {dt}\frac {\partial L}{\partial v}\right) =0 \;.$$