The lagrangian is always phrased as $L(t,q,\dot{q})$. If you magically knew the equations $q(t)$ and $\dot{q}(t)$, could the Lagrangian ever be written only as a function of time? Take freefall for example. $$y(t)=y_0 + v_0t -(1/2)gt^2$$ $$\dot{y}(t)=v_0 -gt$$ Can the Lagrangian now be written a...