It is possible for a scalar finite-duration continuous system to achieve an infinite speed (in finite-time)? How if it true? Why not if it false? (Please read first the restrictions of the system I am interested in).
Since new information has rise, now I am actually more interested in the "Added Later" part
I have change the term "time-limited" for "finite-duration" since is more accurate and widely used, but it means that the scalar one-variable function $f(t)$ has an starting time $t_0$ for which $f(t) = 0\,\forall t<t_0$, and also an ending time $t_F$ from which $f(t) = 0\,\forall t>t_F$, with $t_0 < t_F$
To avoid some "bad-behaved" functions, I am thinking in functions $f(t)$ as a position of a classical object that change with time, with the function describing its position restricted as:
- $f(t)$ is continuous, including it been so in the following way: $\forall\,t_d,\,f(t_d^-) = f(t_d^+)$ so there is no "jump-like discontinuities" in the function $f(t)$, so no "teleportation" is allowed. With this, I am avoiding the existence of delta functions components $\delta(t)$ in the derivative $f'(t)$ because of jump discontinuities (I don´t know if could have them because of other reasons). Also, I am thinking in well-behaved "classic" continuous functions, not things like a Brownian motion which is continuous and nowhere differentiable (I don´t know how to "formally" described this restriction - I hope you get the idea).
- The function $f(t)$ is of finite-duration, so compact-supported since is a one variable function (scalar). Since is continuous and compact-supported is also bounded, so $\sup\limits_t |f(t)| = \|f(t)\|_\infty < \infty$.
- The function $f(t)$ has a well defined Fourier transform $F(w)$ (analytic, and follows the Riemann-Lebesgue Lemma). Since for a compacted-supported function the Fourier transform could have some "issues" at the boundaries of it domain $\partial t = \{t_0,\,t_F\}$, you could assume that $f(t_0)=f(t_F)=0$ if that makes it more easy to work with (here I found a way to overcome this issues for arbitrary finite $\{f(t_0),\,f(t_F)\}$ so generality is sustained).
- The function $f(t)$ have finite energy $\int_{t_0}^{t_F} |f(t)|^2 dt < \infty$, and is also Lebesgue integrable $\int_{t_0}^{t_F} |f(t)| dt < \infty$ (later I learned that if continuous and compacted supported, then is bounded, and this imply that the Fourier Transform is analytic, and then adding is Lebesgue Integrable implies then is of finite energy since is bounded, and I believe all this implies then the Riemann-Lebesgue follows, but not quite sure about the last affirmation).
I believe that if the function $f(t)$ have sharp-edges/tips/spikes like the absolute-value function at the origin, the first derivative (speed) $f'(t)$ will be discontinuous, but since $f(t)$ is bounded the "jump" in the derivative will be also bounded: as example, let $f(t) = \sin(|t|\pi),\,|t|\leq 1$ (plot here), it have a sharp edge at $t=0$ so its derivative have a jump-discontinuity, but $\sup\limits_t |f'(t)| = \pi$. At first glance, I thought that to have a spike with $\sup\limits_t |f'(t)| \to \infty$ at some time $t_d$ it has to have some $|f(t_d^-)| = |\lim\limits_{t \to t_d^-} f(t)| \to \infty$ and/or $|f(t_d^+)| = |\lim\limits_{t \to t_d^+} f(t)| \to \infty$ making the function $f(t)$ discontinuous, which is not allowed (as counterexample, $f(t) = 1/|t|$ has an "infinite jump" in $f'(t)$), but then I found the function $f(t) = \left|t \cdot \displaystyle{\frac{\log(t^2)}{2}}\right|,\,|t|\leq 1$ which have a sharp edge where an infinite speed is achieved with an infinite-size jump-discontinuity on $f'(t)$ without being $f(t)$ a discontinuous function (it also has bounded variation equal to $4/e$ even given it has a discontinuity on its derivative).
Nevertheless, note that this functions can achieve these infinite speeds only at isolated/disjointed single points (of measure zero), or it will have a sudden position change with zero time, creating a jump-discontinuity in $f(t)$ which is not allowed from the assumptions (continuity part).
With this in mind, the second derivative (acceleration) $f''(t)$, must be allowed to have infinite jump discontinuities, or no abrupt changes of direction could be allowed, like crashing and bouncing from a wall, so the jump-discontinuity in $f'(t)$ will become a delta function $\delta(t) \equiv \infty$ in the acceleration profile (as example for $f(t) = \sin(|t|\pi),\,|t|\leq 1$, its second derivative is $f''(t) = 2\pi\delta(t)-\pi^2 \sin(|t|\pi),\,|t|\leq 1$). So acceleration is necessarily unbounded. Also, speed must be allow to have discontinuities or no sudden changes in direction would be achieved - meaning this that smoothness is a too restrictive for this function, discarding Analytic solutions!. So, If I am right, an unbounded acceleration makes possible to achieve an infinite speed on a finite time, but it doesn´t mean that there are other laws making impossible to it to be unbounded - law which I am looking for.
This behavior can be seen in the following examples:
- $f(t) = \sqrt{1-t^2},\,|t|\leq 1$, which starts/ends with $|f'(t)| \to \infty$ as can be seen on its plot here.
- $f(t) = t \cdot \displaystyle{\frac{\log(t^2)}{2}},\,|t|\leq 1$, which "softly" achieve $|f'(t)| \to \infty$ at $t=0$ as can be seen on its plot here.
- $f(t) = \left|t \cdot \displaystyle{\frac{\log(t^2)}{2}}\right|,\,|t|\leq 1$, which achieve $|f'(t)| \to \infty$ at $t=0$ with an infinite-size jump-discontinuity in $f'(t)$, as can be seen on its plot here.
So the main question of the beginning can be divided as:
- It is possible for a real-life classic mechanical system to behave as the last examples? So achieving and infinite speed in a finite time? Here, I think that a model that can achieve an infinite speed, even if is only in one point in time, will violate every possible physics model with a finite speed for causality, but I haven´t found yet how causality conditions will restrict the derivative of these finite-duration functions.
- Can you think of examples of real life classic mechanic systems that behaves as the last examples? (I can´t made yet by myself an idea of an example).
- There are other ways of achieving an unbounded maximum rate of change for a classic mechanic system that are not included in the scenarios I mention?? (keeping the restrictions over $f(t)$).
- If is not possible, Which physics laws are avoiding it to happen? (here meaning that the maximum rate of change must be bounded because of these laws). Here is where the problem of the bounds of the domain can make struggles in the Fourier Transform, since $\sup\limits_t |f'(t)| \leq \frac{1}{2\pi} \int_{-\infty}^\infty |w F(w)|dw$ which can fictitiously diverge because of the effect of the discontinuities at the "edges" of the time-limited compact-support, but they can be avoided as is explained here by using instead $\sup\limits_t |f'(t)| \leq \frac{1}{2\pi} \int_{-\infty}^\infty |iw F(w)+f(t_F)e^{-iwt_F}-f(t_0)e^{-iwt_0}|dw$.
- If there are mistakes in my argumentation, please let me know what assumptions/lines-of-thought are wrong.
Added Later
I have found recently two papers from the author V. T. Haimo (1985), that analyze finite-duration differential equations [1] and [2].
From them, I realize I could be mixing two things into the question: first, math actually could stand any function cropped at some compact-domain to be treated as a finite-duration function, and second, that it is not really what I want to know, actually what I am looking for are for solutions to differential equations which are of finite-duration.
On the papers the author explain that: "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."
Since, linear differential equations have solutions that are unique, and finite-duration solutions aren´t, finite-duration phenomena models must be non-linear differential eqs. to show the required behavior (non meaning this, that every non-linear dynamic system support finite-duration solutions).
Also, since the system "dies" at the end of the domain, the solutions will have the same issues than compacted-supported functions in this ending point, which will leads that finite-duration solutions cannot be Analytic in the whole time domain (maybe using functions defined piece-wise could work, like common bump-functions $\in C_c^\infty$ are defined, but no restricting the starting point to be also zero - which is a requirement for bump-functions for keeping smoothness).
Note: discarding "whole-domain analytic functions" like Power Series, and also Linear ODEs, actually "discards" almost-all the maths knowledge I acquire on engineering, so this is totally new for me.
The papers also show which conditions must fulfill the non-linear differential equation to support finite-duration solutions, at least for first and second order scalar ODEs.
And in [1] on Theorem 2 point (i), it is said that, without losing generality by considering that the ending time of the finite-duration solution happens at $t_F = 0$, for a second order dynamical system described by $\ddot{x}(t) = g(x(t),\dot{x}(t))$ such $g(0,0)=0$ (the system dynamics "die" at $t_F = 0$), with $g \in C^1(\mathbb{R}\setminus \{0\})$, then for the system to support finite-duration solutions, the following another differential equation must have solutions: $$q(z)\frac{dq(z)}{dz} = g(z,q(z)),\,q(0)=0$$
Honestly the papers are bit advanced to my mathematical skills, but if I didn´t made any mistakes, what the author is doing is splitting the second derivative of the scalar one-variable function $x(t)$ as:
$$\ddot{x} = \frac{d}{dt}\frac{dx(t)}{dt} = \frac{d}{dt}\frac{dx}{dx}\frac{d\,x(t)}{dt} = \frac{dx}{dt}\frac{d}{dx}\frac{dx}{dt} = q(z)\frac{dq(z)}{dz}$$ by using the change of variable $z=x(t)$ and $q(z)=\dot{x}$.
I don´t really understand why this transformation leads to another differential equation that "tells" how the original equation will behave, so the following analysis is probably wrong, but I want to share it with you so you can correct me:
Since I am looking from the maximum speed $\sup_t |\dot{x}|$, and from the papers looks like I can figure out the behavior of $\dot{x}$ from $q(z)$, I believe whatever it achieve a maximum, the values obtained should be the same, so finding $\sup_t |\dot{x}| \equiv \sup_z |q(z)|$. With this, since $q(z)=0$ is not really a value I "care", I could use first order conditions to look for the maximum value of $q(z)$, so I need to find $z$ such $$\frac{dq(z)}{dz}=0 \rightarrow z^* \rightarrow q(z^*)$$
So, since I am interested in $q(z) \neq 0$, looking for the first order conditions is equivalent to looking for $q(z)\frac{dq(z)}{dz}=0$, which is indeed the same that looking for $\ddot{x} = 0$ (if my assumption of interchangeability of equations is right - which I believe is not), so it would be meaning that finite-duration solutions of differential equations only can achieve their maximum speeds at inflection points of the acceleration profile where it is equal to zero $\ddot{x}=0$, which instantly discard situations as the example $f(t) = t \cdot \displaystyle{\frac{\log(t^2)}{2}},\,|t|\leq 1$, which "softly" achieve $|f'(t)| \to \infty$ at $t=0$ but at this points it second derivative is non-zero (actually diverges to infinity).
This is quite an aggressive affirmation (so, probably wrong), since it can be reformulated as: finite-duration solutions to scalar-second-order differential equations, which are of unlimited bandwidth because of having finite-duration so they could achieve infinite speeds in principle, are actually restricted by being solutions of finite-time differential equations (with $g(x, \dot{x}) \in C^1(\mathbb{R}\setminus \{0\})$) so they can achieve their maximum speeds only when it acceleration is zero discarding it of happening at discontinuities, so their maximum speeds are actually bounded.... this is actually too good to be true (maybe it happens because of the restriction on $g(x,\dot{x})$), but it is quite interesting to see which restrictions could rise for mechanical systems described by these finite-time differential equations, and I did not find too much information related to them so I think are quite unknown.
So far I have only found nunerical representatuons of this finite-duration solutions, and maybe the mentioned paper is only reviewing the behavior near the ending point, but anyway, any example of a function that is a finite-duration solution of a differential equation will be preciated (I don't even know if a close-form is possible).
Hope you can comment about.
