Let's say I have found the path for which the action:
$$ S = \int_{x_1}^{x_2}\mathcal{L}(y(x),y'(x),x) dx $$
Is minimized. Does this mean that this path minimizes the integral of the square of the Lagrangian? I have started sketching a proof by considering the variation of the integral of $\mathcal{L}^2$, but have hit a wall trying to integrate by parts. If this does not hold in general, are there conditions on $\mathcal{L}$ for which this is true?