= Solution
The <principle of stationary action> says that the physical path makes the action stationary under smooth variations $q+\eta h$ with fixed endpoints $h(t_1)=h(t_2)=0$. It need not minimize the action. Differentiating and integrating by parts gives
$$
0=\delta S=\int_{t_1}^{t_2}\left(L_qh+L_{\dot q}\dot h\right)dt
=\int_{t_1}^{t_2}\left(L_q-\frac d{dt}L_{\dot q}\right)h\,dt.
$$
The <fundamental lemma of the calculus of variations> then gives the <Euler-Lagrange equation>
$$
\boxed{\frac d{dt}\frac{\partial L}{\partial\dot q}-\frac{\partial L}{\partial q}=0}.
$$
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