For the differential equation with two real-valued positive parameters $\{a,\,b\}>0$: $$\ddot{y}+a\cdot\text{sgn}(\dot{y})\sqrt{|\dot{y}|}\left(1+|\dot{y}|^{\frac{3}{2}}\right)+b\cdot\sin(y)=0,\,y(0)>0,\,\dot{y}(0)=0 \tag{Eq. 1}\label{Eq. 1}$$
I would like to know if is possible to find its extinction time $0 < T < \infty$ such their solutions become exactly $y(t)=0,\,t>T$, as a function of their parameters and initial conditions $\{a,\,b,\,y(0),\,\dot{y}(0)\}$, being in principle, $y(t)$ a real-valued scalar function.
Motivation - not needed for finding the solution
Recently I learned that a ordinary differential equation (ODE) require to have a point in time where is locally non-Lipschitz (LNL) in order to been able of having solution with a finite extinction time (details here), which have interesting consequences in using ODEs as a tool for model reality:
- An ODE that stands solutions of finite duration cannot hold uniqueness of solutions at the same time (or at least, violates Picard-Lindelöf Theorem's conditions due the LNL singular point).
- If a solution is of finite duration, it don't going to hold time-reversal symmetry (property is commonly only attributed to entropy).
- Since traditional physics models holds uniqueness of solutions, their solutions, accurately speaking, are always never-ending in time: this means that everything have ever happen before in time could be, in principle, affecting any experiment in the present time, which is in conflict with locality and causality.
With this in mind, I am exploring how to modify the classic nonlinear pendulum with friction in order of having a finite extinction time, and I am using a LNL drag force: $$F_d = a\cdot\text{sgn}(\dot{y})\sqrt{|\dot{y}|}\left(1+|\dot{y}|^{\frac{3}{2}}\right) = a\cdot\text{sgn}(\dot{y})\sqrt{|\dot{y}|} + a\cdot\dot{y}|\dot{y}| \tag{Eq. 2}\label{Eq. 2}$$ which behaves similar to the classic quadratic drag force for high speed $F_d \approx c\cdot (\dot{y})^2$, also behaves similar the classic Stokes' Law for non-zero-low-speeds $F_d \approx 2c\cdot \dot{y},\,0.2 <\dot{y}<1.2$, and makes a singular LNL point for $\dot{y}\to 0$ so finite duration solutions could arise (see details here).
Issues with numerical examples:
If the classic nonlinear pendulum with friction equation is reviewed as example, in Wolfram-Alpha it can be seen it have decaying solutions as expected: $$\ddot{x}+2\cdot0.021\,\dot{x}+0.2\sin(x)=0, x(0)=\frac{\pi}{2}, \dot{x}(0)=0 \tag{Eq. 3}\label{Eq. 3}$$
But following the analysis for T-Symmetry, under the transformation $\hat{t} \to - t$ the position and acceleration remains the same, but the velocity profile change in sign: this applied to \eqref{Eq. 3} shows it change under this transformation, so it not fulfill T-Symmetry.
If instead the standard drag force $F_{\text{drag}}\propto (\dot{x})^2$ is used as is shown here for the equation: $$\ddot{x}+0.021(\dot{x})^2+0.2\sin(x)=0, x(0)=\frac{\pi}{2}, \dot{x}(0)=0 \tag{Eq. 4}\label{Eq. 4}$$ their solution aren't showing the expected decay one can see on the experimental pendulums. This is commonly solve using an ansatz for the drag force $F_{\text{drag}}\propto \dot{x}|\dot{x}|$, which as can be seen here for the equation: $$\ddot{x}+0.021\dot{x}|\dot{x}|+0.2\sin(x)=0, x(0)=\frac{\pi}{2}, \dot{x}(0)=0 \tag{Eq. 5}\label{Eq. 5}$$ their solution are indeed having the expected decaying behavior for a pendulum with friction.
Since \eqref{Eq. 4} fulfills T-Symmetry but fails to proper model the pendulum with friction, but both \eqref{Eq. 3} and \eqref{Eq. 5} don't fulfill T-Symmetry but modeled properly the amplitudes, its look like for having a decaying behavior it was required to choose a differential equation that is not fulfilling the time-reversal-symmetry.
But, since the function $f(x)=x|x|$ is locally Lipschitz (see here), it cannot be used for having a singular point on a finite time where uniqueness could be broken, so this ansatz wouldn't have solutions of finite duration, this is why I chose to work instead with $F_d$ which also shows to have the decaying solutions (but also without T-Symmetry): $$\ddot{x}+0.021\,\text{sgn}(\dot{x})\sqrt{|\dot{x}|}(1+|\dot{x}|^{\frac{3}{2}})+0.2\sin(x)=0, x(0)=\frac{\pi}{2}, \dot{x}(0)=0 \tag{Eq. 6}\label{Eq. 6}$$
Analysis of the decay
Since for some parameters $\{a,\,b\}$ the equation \eqref{Eq. 1} is having decaying solutions as can be seen for the example of \eqref{Eq. 6}, when the time is reaching the extinction time, the behavior of \eqref{Eq. 1} will be driving by:
- For the position the small-angle approximation could be used so $\sin(y)\approx y$
- For the drag force $F_d$, the quadratic term $\dot{y}|\dot{y}| \to 0$ for small speeds, so the remaining term $\text{sgn}(\dot{y})\sqrt{|\dot{y}|}$ will be only present.
With this, in the small-angles low-speeds regimen the equation will be behaving as:
$$\ddot{y}+a\cdot\text{sgn}(\dot{y})\sqrt{|\dot{y}|}+b\cdot y=0 \tag{Eq. 7}\label{Eq. 7}$$
Which, as for the previous examples without T-Symmetry, the equation show its having decaying solutions: $$\ddot{x}+0.021\,\text{sgn}(\dot{x})\sqrt{|\dot{x}|}+0.2\,x=0, x(0)=\frac{\pi}{14}, \dot{x}(0)=0 \tag{Eq. 8}\label{Eq. 8}$$
I have found the following papers under the term sublinear damping that shows example in physics of equations really similar to \eqref{Eq. 7} and \eqref{Eq. 8}:
- "A note on the dynamics of an oscillator in the presence of strong friction" - H.Amann & J.I.Diaz
- "Behavior of Solutions of Second-Order Differential Equations with Sublinear Damping" - J. Karsai & J. R. Graef
So I believe they will be achieving a finite extinction time, but I have not being able to formally prove it.
Added later
I found this paper:
where the autors, in equation $(19)$ use a similar drag force of the form $F_d = b\ |\dot{y}| + c\ (\dot{y})^2$, so at least a description through two components doesn't look like a complete insane approach. Also on equations $(59)$ and $(60)$ they introduce a Drag Force $F_d = b\ \dot{y} + c\ \dot{y}|\dot{y}|$ for a more accurate description of the air effects on the pendulum.
