Modeling friction with Non-Lipschitz ODEs - Pendulum example

Question

The main objective of this question is to figure out if the following differential equation have [finite-duration] solutions: $$ \ddot{\theta}+0.021\,\operatorname{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|} + 0.02\sin(\theta)=0,\quad \theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \tag{Eq. 4}$$

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Introduction

I am trying to understand the dynamics of the non-linear ODE of the classic pendulum with friction, which is modeled by: $$ \ddot{\theta}+a\,\dot{\theta} + b\sin(\theta)=0,\quad \theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \tag{Eq. 1}$$ for some real constants $a$ and $b$ both different from zero.

From the experiments, I believe is natural to expect that the "exact solution" for Eq. 1 should have an "ending time" from where the solution becomes exactly zero forever, this given by the fact that the experimental pendulum does indeed stop moving in reality (so far I know there is no known exact solution for Eq. 1).(For a discussion about the asumption of the pendulum movement stopping in a finite time due pendulum dynamics you can see this question).

But from what I am understanding now from this other question I have made, I am realizing that since Eq. 1 is a Lipschitz ODE, it stands to only have a unique solution. So finite-duration solutions can't be sustained by these kinds of ODEs- even if an analytical solution is obtainable, at best the solutions will lead to functions that "vanish at infinity" instead of having a "true ending time". This is because a function that reaches zero and stays there can't be analytical for the whole real line. As an example, this is what happens with bump functions $\in C_c^\infty$ (I am probably wrong, but it looks like from the Picard-Lindelöf theorem, that its solutions are going to be analytic since the Picard's iterations build a Taylor Series - this is not a fact, but is what I am interpreting of the theorem, so if I am wrong please comment it).

I have found recently a paper Finite Time Differential Equations (V. T. Haimo - 1985) where scalar ODEs with finite-duration solutions are studied. There, the author set all the equations as having an "ending time" at $t=0$, and says: "One notices immediately that finite time differential equations cannot be Lipschitz at the origin. As all solutions reach zero in finite time, there is non-uniqueness of solutions through zero in backwards time. This, of course, violates the uniqueness condition for solutions of Lipschitz differential equations." Thus I believe now that to have models that achieve finite-duration solutions, one has to leave the "comfort zone" of Lipschitz ODEs and Uniqueness of solutions (which could be also quite problematic, as the only example I know is the Norton's Dome, which is kind of a fictitious system built over a singularity and indeed there is a lot of discussion related to its solution's definition).

As an example of what I mean with a finite-duration solution, consider this ODE: $$ \frac{y'}{y} = \frac{-2\,t\,(1+|1-t^2|)}{(1-t^2)^2},\quad y(0)=1\tag{Eq. 2} $$ which will have, as a solution $$ y(t) = \frac{(1-t^2+|1-t^2|)}{2} \exp\left[-\frac{t^2}{1-t^2}\right] $$ see plot, which behaves as a smooth-non-analytic bump function $\in C_c^\infty$ within $t=(0,\,1)$ and stays at $0$ after time $t=1$ (there are some definition issues with the differential equation from where the solution becomes zero, and also issues with the solution at $t=1$ but solvable by extending the function through limits keeping its continuity - see details here).

With this idea in mind, if I plot the following equation (this is Eq. 1 with some "arbitrary" values): $$ \ddot{\theta}+0.1\,\dot{\theta} + 0.02\sin(\theta)=0,\quad \theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \tag{Eq. 3}$$ which can be seen plotted here, its phase diagram $(\theta,\,\dot{\theta})$ looks like it converges to the point $(0,\,0)$, but because it is a Lipschitz solution, the oscillations will decrease but continue forever in time without arriving at the point $(0,\,0)$ (only achieved at infinity).

Now, using "what is done" in the paper, there are only examples of equations of a specific form: if I introduced a similar Non-Lipschitz component for the friction, "arbitrarily" chosen here just to use it as an example (I don't have any physical explanation for it - if you can give it one it would be awesome), Eq. 3 will look like: $$ \ddot{\theta}+0.021\,\operatorname{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|} + 0.02\sin(\theta)=0,\quad \theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \tag{Eq. 4}$$ which can be seen plotted here, where it "looks like" it is indeed behaving similar to Eq. 3, having damped oscillations, but it is also achieving the phase space point $(\theta,\,\dot{\theta}) = (0,\,0)$ in finite time, as my intuition says it should be for a classic mechanic system.

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Main Questions

But since these things, being interesting, so far are on-purpose-made and quite speculative, I have the following questions:

1. Is Eq. 4 actually achieving its ending time in finite time? (a math proof of it)... Here to be explicit for what I am asking for, I have added an example: lets think in the equation $y'=-\sqrt{y},\,y(0)=1$ which has as solution $y(t)=\frac{1}{4}\left(t-2\right)^2$, here the solution is not of finite duration, since after reaching zero at $t=2$, it will start to rising again. But the equation $x' = -\text{sgn}(x)\sqrt{|x|},\,x(0)=1$ will have as solution $x(t) = \frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$ which indeed is of finite duration reaching zero at $t=2$ and staying there forever (see plot here). I want to know if the solution $\theta(t)$ behaves at the end like the solution $x(t)$, reaching zero at some ending time and staying there forever.
2. Are there examples of physical models that already use non-Lipschitz representations of friction? For any other kind of model (apart from the Norton's Dome), please move to this more general question.
3. Is there a more accurate representation of the pendulum with friction, that achieves a finite-duration solution? So far, even though Eq. 1 is widespread as the traditional equation for the "realistic" pendulum, the linear dependence of friction on its derivative is still an approximation, at least from what it is said on Wikipedia page for Drag... so, surely there are other attempts.
4. Does it make sense for you to work with these finite-duration solutions when modeling classical mechanics? (these question only apply for physicists). Do you ever hear of finite-duration solutions before? Or has Physics forgotten about them?

To be honest, I don´t fully understand what it is said on the paper, but I think that the examples on Wolfram-Alpha will speak for themselves.

PS: About the "arbitrarity" of the selection of the non-lipschitz component, the traditional pendulum model friction as proportional to the rate of change as is explained on Wikipedia as the Newtonian Law of Viscosity, but in the same section is explained that are other models, like the Power Law Model which has components actually similar with which I am using here. I hope that knowing how to model finite duration solution will lead to adapts these classic models to show solutions that indeed behaves as having an ending time $T < \infty$, which I believe is the case on everyday phenomena, as a example, the Euler's Disk toy that show its ending time when the sound stops - I have a related question here).

PS2: An interesting extension of this discussion was made in this other question, hope you can visit to see the details and comment.

PS3: I have improved the model of the drag force into this other question, if you are interested in this question, surely you are going to be checking also this another.

Answer 1

for this differential equation

$$\ddot \theta+a\sqrt{\dot\theta}+b\,\sin(\theta))=0\tag 1$$

with $~\dot\theta \gt 0$

to obtain the time t for $~\theta(t)\mapsto 0~$ put $~\ddot \theta(t)=0$ and solve for $~\theta(t) ~$

$$-a^2\,\frac{d\theta}{dt}=b^2\sin^2(\theta)\quad\Rightarrow\\ t=-\int \frac{a^2}{b^2\,\sin^2(\theta)}\,d\theta+c =-{\frac {{a}^{2}\cos \left( \theta \right) }{{b}^{2}\sin \left( \theta \right) }}+c $$ $$\theta(t)=-\arctan\left(\frac{a^2}{b^2(c-t)}\right)\tag 2$$ where c is integration constant . $$\theta(0)=\frac{\pi}{2}\quad\Rightarrow\,c=0$$

solving equation (2) for $~t_f~$ with $~\theta(t=t_f)=\epsilon~$ and $~\epsilon~$ is a small value. you obtain

$$t_f={\frac {{a}^{2}}{\tan \left( \epsilon \right) {b}^{2}}}$$

with your data and $~\epsilon=0.01~$ you obtain that $t_f=90~[s]~$

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numerical result of

$$\ddot \theta+a\,\text{sgn}(\dot\theta)\,\sqrt{|\dot\theta|}+b\,\sin(\theta))=0\tag A$$ $~a=0.02~,b=0.021~$