Imagine there is a mechanical system described in unitary units by the equation: $$\dot{x} = -\text{sgn}(x)\sqrt{|x|},\quad x(0)=1 \tag{Eq. 1}$$ such it has a finite duration solution: $$x(t) = \frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2 \tag{Eq. 2}$$
Is this enough information to reconstruct its Kinetic and Potential Energies to obtain its Lagrangian of this System and its Least Action Principle's integral? What are these values in terms of $x(t)$?
Motivation
Recently I have learned about the existence of finite duration solutions of differential equations on these papers: Finite Time Differential Equations and Finite Time Controllers by Vardia T. Haimo, and since everyday phenomena are of finite duration, I want to know how will behave the Energy and the Least Action Principle on this kind of system with finite duration solutions, and this is the only example I have so far of an autonomous system that stands finite duration solutions (maybe $\dot{y} = -\sqrt{y},\,y(0)=1$ also works if the solutions is restrained to the reals, since after $(y,\,\dot{y})=(0,\,0)$ the derivative is never going to rise up again since the square root is positive).
I am trying to make a mechanical system with $x(t)$ the solution to their equation of motions, not in the other way.