Is it possible for a point-like system to behave like $x(t) = \frac{t}{2}\log(t^2)$ near $t=0$? (infinite speed) [closed]

Question

Is it possible for a point-like system to behave like $x(t) = \frac{t}{2}\log(t^2)$ near $t=0$? (infinite speed)

I know beforehand that relativity theory forbids anything with mass from travel faster than speedlight, so please stay with reading the full question first before answer - is more a conceptual/mathematical issue.

For generality issues, I will work with dimensionless units (is explained later).

This question is a modification of a previous one already closed and deleted (at least I tried to, but the system didn't allow it), because of a misused of the term "particle" to explain a point-like system described on time as a scalar variable, and I still don´t know the precise technical terminology, so I hope you get the idea. Apologizes in advance to everyone for any inconvenience caused.

Imagine a point-like system which position in time is described by the equation: $$x(t)=\frac{t}{2}\log(t^2)$$ and the point starts moving at time $t_0=-0.2$ with initial speed and the experiments ends measuring its position at time $t_f=0.2$ following the equation $x(t)$.

From the plot it look like something quite feasible to be done, but if I focus on their derivative I will note that there is a point in space where it maximum speed will rise to infinity without any discontinuity on the position of the point-like element: $$\frac{d}{dt}x(t) = 1+\frac{\log(t^2)}{2}$$

Since the maximum speed diverges $\lim\limits_{t\to 0}\|\dot{x}(t)\|_\infty \to \infty$, near zero is a neighborhood where the variable is "softly/continuously" (but not smoothly) achieving speeds that could be higher than the speed of light, but it never "teleport" since the function is continuous and is only singular in a zero-measure point of time (even, $\int\limits_{-0.2}^{0.2} |\dot{x}|^2\,dt < \infty$ and $\int\limits_{-0.2}^{0.2} |\dot{x}|\,dt < \infty$).

So, I am looking for examples of alike systems, if there are any, that could behave likewise $x(t)$ (on the Motivation I extend on which kind of systems I believe could be candidates).

1. Is this kind of system attainable under current interpretation of physics laws?
2. Where I am messing up with the concept of speed and its divergence?
3. If this is an achievable system, Do you know a mechanic system that behaves like this?
4. If no mechanical system is possible, Any other kind of physical model where some variable behaves as reaching infinite speed continuously in finite time?

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Motivation

I would like to made an explanation of why this question is interesting in my opinion:

In this video I found by accident on youtube, Terence Tao, one of the most important nowadays mathematician, is reviewing finite-time blow up solutions to differential equations. There, he talk about the possibility of having this kind of solutions which reaches infinity in finite time for the Navier-Stokes equation, which is indeed a widely known physical system.

Thinking in the speed limits rise by Relativity Theory due to Einstein, I was wondering if this talk is just a mathematical curiosity which cannot be happening in real life, but then I saw this video about the Euler´s disk toy, where, for the wobbling rate the solutions indeed shows to have a finite time blow up for a inertial system (it just a frequency ratio so inertia restrictions indeed are not applying).

On this question, also I figure out that sometimes a finite time blow up system could be thought as the reciprocal of a solution of finite duration to a differential equation (meaning here, it achieves zero by it own dynamics and stays there forever after - some examples of finite duration solutions here, here, and and here).

So if some people are researching this kind of things, I was thinking that maybe something that achieving the blow up behavior in their speed profile could also be possible (at least is possible in the math).... this is why I keep unitless/dimensionless the differential equations, since maybe a model where inertia is not involved could fit the presented situation.

Another examples, the point-like system could be the displacement of a shadow, or the geometric point where scissors close as the example shown on this video. With this, inertia is not needed to be involved in the point-like system, but as the example of the Euler's Disk toy, the system could indeed be a realizable experiment with a variable showing a finite time blow up behavior (I don´t know if, maybe, by integration of this wobbling rate I could found a variable which do the trick).

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Added later

I don't know why I am receiving so much upvotes, since the question is honest, have explained why is been done, and I put some examples of what kind of things I am thinking about could be candidates. For people that should like science, try to thinking "out of the box" should be something exciting, and not something insulting - by doing always the same things you will never push the borders of knowledge.

Recently I found on these papers here and here that there exists finite duration solutions to differential equations, and also that no Lipschitz ODE will support them, neither an analytic solution since at best they will vanish at infinity. So, if you do believe that there exist experiments which reaches and ending time, you must think of non-linearities of these kinds since no linear model neither non-piecewise power series will work.

Now, since everyday phenomena have ending times in my experience, let say a scalar system described by $r(t)$, since after the ending time $T$ it becomes zero forever, my intuition says that taking the variable $z(t)=1/r(t)$ it should behave as having a finite time blow up, and on the already linked question I found that it could be the case, but it also there is examples when it doesn't work (giving in the answers).

As a known example, the Norton's Dome which is kind of a rarity of system, its dimensionless equation $\ddot{r}=\sqrt{r}$ for real valued initial condition $r(0)=T^4/144>0$ and $r'(0)=-T^3/36<0$ it will stand finite duration solutions $r(t)=\frac{1}{144}(T-t)^4\theta(T-t)$ with $\theta(t)$ the standard Heaviside unitary step function (I think of this as pushing the ball from below the dome such it just stops at the higher point - but I am not fully sure if the right representation). Now, taking the system $z(t)=1/r(t)$ will lead to the differential equation $\ddot{z} +z\sqrt{z}-2\frac{(\dot{z})^2}{z}=0$ which indeed can stand a finite time blow up solution $z(t) = \frac{144}{(T-t)^4}$.

So after seeing a video where a worldwide expert is researching of blow ups in a wide known physical system as the Navier Stokes equation, and thinking that everyday phenomena indeed stops moving, I start wondering if there other kind of physics systems that behaves like having blow up that could be converted on finite duration solutions, or other finite duration models (in which I am interesting to find).

But as the example giving on this answer about the behavior of the collapsing bubble on somnoluminescence phenomena which radius behaves as $R(t) \sim (A-\alpha t)^{2/5}$ and $\dot{R}(t) \sim \frac{2}{5}\alpha (A-\alpha t)^{-3/5}$, and the Norton's Dome example also, they ends sharply as the square root function, I am looking if there are other examples where the singularity point is achieved "softly" (instead of "sharply"), as is done in the function of the question.

Since everyday phenomena haves ending times at first I thought it will be many examples, but looks like everything has been "linearized" in classic physics. Actually is disappointing to see all the negativity, but I still believe is not a loaded question but instead an interesting one.

And also I would like to thank to whom have already answer this question with really interesting and developed insights, I really appreciated.

Answer 1

Usually, an infinity in a solution to an equation in physics is either telling us that the equation breaks down, and must be replaced with something else, or that you made an unphysical assumption in deriving the solution. In the examples you give about fluid mechanics and Euler disks, we certainly don't expect real fluids and real disks to blow up in finite time. If such behavior is observed mathematically, it is a sign that the equations used to derive those solutions are not good models of the physical situation in that regime.

In your situation, you have drawn a curve which could be a trajectory that a particle follows. Let's forget about special relativity for a moment, which just complicates things here (I'll touch on it at the end of the answer). In order for it to be a possible solution to the equations of classical physics, that curve $x(t)$ should satisfy Newton's second law \begin{equation} F = m \frac{d^2x}{dt^2} \end{equation} Let's evaluate the right hand side for your proposed motion (by the way, as a small correction, the argument of the log should always be dimensionless, so we should add a constant $t_0$ to make this work, and to make the dimensions work out, I will also multiply your expression by $x(t)$ by a constant $x_0$ with units of length) \begin{equation} m\frac{d^2}{dt^2} \left[x_0 \frac{t}{2 t_0} \log \left(\frac{t^2}{t_0^2}\right)\right] = \frac{m x_0}{t_0} \frac{1}{t} \end{equation} Now as $t\rightarrow 0$, we see that your particle has an infinite acceleration. This requires that the force applied to this particle is also infinite. (As an aside, this also makes this case less mathematically interesting than fluid mechanics -- of course if we put in an infinite force, than the solution of the equation will blow up. The case of fluid mechanics is interesting because from apparently innocuous initial conditions, the solution's natural evolution may cause it to blow up, without us having to putting that blow up in by hand.)

Realistically, we expect that any physical force acting on your particle will be bounded by some finite amount. So this is not a plausible motion. However, you can have motions that are arbitrarily close to the one you proposed, with a finite force. For example, suppose your motion was actually \begin{equation} x(t) = \begin{cases} x_0 \frac{t}{2 t_0} \log\left(\frac{t^2}{t_0^2}\right), & |t| \geq \epsilon \\ p_\epsilon(t), & |t| < \epsilon \\ \end{cases} \end{equation} for some $\epsilon>0$, and where $p_\epsilon(t)$ is some polynomial which is chosen so that $x(t)$ is continuous and at least twice differentiable at $t = \pm \epsilon$. Then, your motion will have a finite acceleration throughout the trajectory. By taking $\epsilon$ arbitrarily close to $0$, we can make your curve arbitrarily close to the one you wanted, with a finite force. This explains why when you look at a plot of your $x(t)$, the motion is apparently fine. It is hard to see from the curve the one point where the solution is blowing up, and curves that are arbitrarily close to the one you drew are ok (in classical physics).

If you want to generalize this situation to special relativity, the equations of motion of classical physics get modified, in such a way that is not possible to find a solution where the velocity accelerates from a value below $c$, to $c$ or a value above $c$, without expending an infinite amount of energy accelerating the particle. Again, this infinity is a sign that something is wrong, physically -- in the case of special relativity, the problem is that massive particles must travel slower than $c$.