When can the Lagrangian be squared without changing the stationary path? Assume that the path $y(x)$ makes the functional
$$ S[y] = \int _a ^b L(y, y', x) dx$$
stationary. Under what conditions does 
$$  \int _a ^b L(y, y', x)^2 dx$$
have the same stationary path? And what other functions can be applied to $L$ without changing the stationary path?
The question is inspired by ordinary calculus: The $x$ that makes $f(x)$ extremal also makes $f(x)^2$ extremal, and often it is convenient to minimize the latter. 
Oh, and this very technique is used in this script on page 13 and 14. To give a brief summary, the goal there is to find the geodesic curve that minimizes 
$$ \int (g_{i j} \frac{dx^{i}}{dt}\frac{dx^{j}}{dt})^{1/2} dt $$
where $x^i$ are curvilinear coordinates $g_{i j}$ is the metric. And then the author says that it is easy to realize that we can also simply use 
$$ \int g_{i j} \frac{dx^{i}}{dt}\frac{dx^{j}}{dt} dt$$
 A: *

*OP is asking:

When does the Lagrangian $L(q,\dot{q},t)$ lead to the same solutions for the Euler-Lagrange (EL) eqs. as the squared Lagrangian $L^2$?

Well, that's a good question. Let's calculate:
$$\begin{aligned} E_i [L^2] -2L~ E_i[L]~:=~&
\left(\frac{\partial }{\partial q^i}- \frac{d}{dt} \frac{\partial }{\partial \dot{q}^i}\right)L^2 \cr
~-~& 2L\left(\frac{\partial }{\partial q^i}- \frac{d}{dt} \frac{\partial }{\partial \dot{q}^i}\right)L \cr
~=~&-2\frac{dL}{dt} \frac{\partial L}{\partial \dot{q}^i}.\end{aligned}\tag{1} $$
In eq. (1) $E_i[L]$ denotes the $i$th EL eq. for the Lagrangian $L$.
Sufficient conditions are apparently:


*

*$L$ is a constant of motion (COM), and the constant of motion $\neq 0$ is not zero.

*$L(q,t)$ does not depend on velocities $\dot{q}$, and $L\neq 0$ is not zero.




*But how can we guarantee that $L$ is a COM? Here is a common strategy.

*

*If $L$ does not depend explicitly on $t$, then the energy
$$h~:=~\left(\dot{q}^i\frac{\partial }{\partial\dot{q}^i} -1\right)L \tag{2}$$
is a COM, cf. Noether's theorem.




*

*If moreover $L$ is homogeneous in the velocities $\dot{q}$ of weight $w\neq 1$, then $$L~\stackrel{(2)}{=}~\frac{h}{w-1}\tag{3}$$ is also a COM as well!



*Main example. How does the theory from sections 1 & 2 apply to the square root Lagrangian
$$L~:=~\sqrt{ g_{ij}(q) \dot{q}^i \dot{q}^j}\tag{4}$$
for geodesics? Well, it fails because of a tiny but important detail: The weight $w=1$ turns out to be exactly one! This is related to the fact that the solutions to the EL eqs. for $L$ in eq. (4) are all parametrized geodesics, but the solutions to the EL eqs. for $L^2$ are only all affinely parametrized geodesics.
See e.g. this Phys.SE post for details.

