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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.

Joako
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1 Answers1

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  1. A 1st-order ODE $$\dot{x}~=~f(x,t)\tag{A}$$ cannot be a Euler-Lagrange (EL) equation if we are only allowed to use a single real variable $x(t)$, cf. e.g. this Math.SE post.

  2. With several variables it is easy. We can e.g. use a Lagrange multiplier $$ S[x,\lambda]~=~ \int_{t_i}^{t_f}\!dt~ \lambda(\dot{x}-f(x,t)) \tag{B}$$

  3. Care should be taken wrt. boundary conditions, so that they are compatible with the physical system at hand, i.e. one might need to add boundary terms to the action (B).

Qmechanic
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