= Solution
Consider a smooth variation $q^a(t)+\epsilon\eta^a(t)$ with $\eta^a(t_1)=\eta^a(t_2)=0$. Differentiating the action at zero and integrating the <velocity> term by parts gives
$$
\delta S=\int_{t_1}^{t_2}\left[\frac{\partial L}{\partial q^a}\eta^a+\frac{\partial L}{\partial\dot q^a}\dot\eta^a\right]dt=\int_{t_1}^{t_2}\left[\frac{\partial L}{\partial q^a}-\frac d{dt}\frac{\partial L}{\partial\dot q^a}\right]\eta^a\,dt.
$$
The boundary term vanishes because the endpoints are fixed. Independent compactly supported variations and the <fundamental lemma of the calculus of variations> make every bracket zero. Thus the <Euler-Lagrange equations> are
$$
\boxed{\frac d{dt}\frac{\partial L}{\partial\dot q^a}=\frac{\partial L}{\partial q^a}.}
$$
The required principle is stationarity of the action; an extremal need not be a strict minimum.
Back to article page
