Let's say that $y = f(x)$ is not a function of position - but rather some function of an arbitrary data feature $x$.
$$ \frac{\mathrm{d} \mathcal{L}}{\mathrm{d} y} -\frac{\mathrm{d}}{\mathrm{d} x} \left(\frac{\mathrm{d} \mathcal{L}}{\mathrm{d} y'}\right) = 0 $$
If we solve for $\mathcal{L}$ what properties does the functional guarantee, and how to interpret it in plain english (not in terms of the action integral)?