Let $L:TQ\times \mathbb R\to\mathbb R$ be smooth. A smooth path $q:[a,b]\to Q$ is stationary under all fixed-endpoint variations if and only if, for every subinterval $J\subset [a,b]$ whose image lies in a coordinate chart, the local coordinate expression of $q$ satisfies \begin{align*} \frac{d}{dt}\frac{\partial L}{\partial v_i}(q(t),\dot q(t),t)-\frac{\partial L}{\partial q_i}(q(t),\dot q(t),t)=0, \qquad i=1,\dots,n, \end{align*} for all $t\in J$.