Instantaneous cause

Question

The position of an object has a rate of change called velocity. The velocity of an object has a rate of change called acceleration.

Force applied is proportional to the acceleration of the body (according to Newton). However it is known that a force can't be applied instantly. So acceleration has a rate of change too (called?).

Will the cause of the "force acceleration" also have a rate of change? Is this infinite, meaning that there will be always something accelerating something else, or is there any cause in nature that is truly instant?

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Edit. To make myself more clear. Is there acceleration of the jerk? Is there acceleration of the acceleration of the jerk, and so on...

The time derivative of acceleration is called jerk. While I'm not sure an exact duplicate of your question exists, searching this site for jerk finds many closely related questions. — John Rennie, Sep 12 '15 at 11:00
Of course you can take further time derivatives. What's the question about that? — ACuriousMind, Sep 12 '15 at 14:30
I think you may profit from reading up on functional analysis: https://en.wikipedia.org/wiki/Functional_analysis. It's the mathematician's theory of why physics can live with ugly monsters like Dirac-Delta-functions and live to tell tall tales. — CuriousOne, Sep 12 '15 at 15:17

score 1 · Answer 1 · answered Sep 12 '15 at 12:29

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The rate of acceleration is called jerk. The reason you don't see a name for derivatives of acceleration in classical mechanics text is that they are not particularly important concepts for the study of Newtonian mechanics. It is not important if we can or can't apply acceleration instantaneously. However allowed jerk rates are analyzed and studied in mechanism design and ergonomics.

answered Sep 12 '15 at 12:29

QuantallicA

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Yes but this is not my question. The question is, is this finite? Or there are infinte derivatives? – user3634713 Sep 12 '15 at 13:09
@user3634713: Physics doesn't really care much about differentiability. Indeed, all we need is for our functions to live in $L^2$, which makes life easy for us and rather hard for the mathematicians who have to clean up our Dirac-Delta-function mess all the time. – CuriousOne Sep 12 '15 at 15:15

score 0 · Answer 2 · answered Sep 12 '15 at 13:32

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However it is known that a force can't be applied instantly.

There is no fundamental factor limiting the rate of change of a force. There are only practical limitations. How short EM pulse can you create? That fast can current-carriers in surrounding bodies experience pulse in force.

answered Sep 12 '15 at 13:32

Ján Lalinský

37,229

These are not just practical limitations. They are "real" limitations, and I am asking whether they are finite. The fact that the most used models do not take into account more derivatives, does not meen that they do not exist. – user3634713 Sep 12 '15 at 14:18
Also I am not talking about the duration of the pulse, but about its slew rate, and what causes that slew rate. – user3634713 Sep 12 '15 at 14:22

score -1 · Answer 3 · edited Apr 13 '17 at 12:19

You can differentiate certain types of functions an infinite number of times and they never will become zero. See this related question: https://math.stackexchange.com/questions/1210819/infinite-number-of-derivatives.

After reading your edit of the question, I'm thinking this may be more a philosophy than physics question. What comes to mind is Zeno's paradox.

Zeno's teacher Parmenides taught that existence is a motionless continuum; that all movement is illusion. He illustrated this with the paradox of Achilles racing a turtle. As the Turtle moved, Achilles had to move also. As Achilles caught up with the Turtle, he covered half the remaining distance to the turtle. And then half of that distance, and half of that, and so on. But because half the distance always remained, Achilles never would catch the turtle. The distance between Achilles and the turtle is asymptotic, if the distance function is smooth and continuous.

You seem to me to be asking for a resolution of something like Zeno's paradox with regard to time. Is there a quantized unit of time that would make the rate of acceleration discontinuous?

The Planck time is the time duration for a photon to travel one Planck length in a vacuum. At the Planck length quantum mechanical effects become important. Does this mean the Planck time is the closest to instantaneous one can get? That may be what your question is about. According to the Wikipedia article linked above, a theory of quantum gravity is needed to determine a time scale shorter than the Planck time. So, your question is not resolvable currently.

$f\colon \mathbb{C} \to \mathbb{C}, f(z) = \Re z$ is not even (complex-)differentiable once! — Sebastian Riese, Sep 12 '15 at 16:16

Instantaneous cause

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