If you can take an infinite number of derivatives of position, then why is it so hard to visualize/think of real world examples beyond the jerk? Does the function essentially become meaningless to the physical world?
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You can take the derivative of jerk, you end up with something called “jounce”
You can indeed take theoretically infinite derivatives of position but last a certain point, knowing what they are doesn’t give you a whole lot more useful information
Basically yeah, it stops corresponding to anything with a physical interpretation.
Position is a directly-observable quantity. First derivative is proportional to momentum and second derivative is proportional to kinetic energy. Even if the derivative is kind of an abstract operation, momentum and energy are conserved quantities so they are “real” to the laws of physics.
The Jerk is a number you can calculate, but even that isn’t meaningful to physics the way that conserved quantities are. Human perception happens to map to it, but it doesn’t describe reality the way that acceleration (kinetic energy) does. It’s an emergent description of the behavior of a system.
Further derivatives don’t represent physical quantities, and they also don’t map to any definite human perception. They’re just increasingly abstract numbers.
Each derivative is a correlate of something real. The zeroth derivative is position itself. The first derivative is velocity, the second acceleration.
Position is the distance between things, constant motion is what happens when you’re interacting with no one, acceleration happens when there’s energy being converted at a constant rate, jerk happens when you’ve just come into contact with something else that’s accelerating, or accelerating at an accelerating pace (like say a muscle is contracting but its contraction is getting stronger as it approaches the midpoint of the contraction).
To find physical analogies of higher order derivatives, we need more and more elaborate physical setups.
And the physical mechanisms we have around us just max out at a certain complexity, because there’s not much to be gained from having evolved higher complexity.
Animals for instance have bones with muscles attached and the system only needs so much complexity to get around the environment which is all mostly static a pervaded with a gravitational field that provides first order acceleration. We have our muscles which can accelerate our bones at a variable rate (more effort, more adrenaline, etc allows us to contract our muscles quicker). That gives us a one-up on gravity.
I guess a mechanism like a whip might introduce higher order derivatives, ie accelerating at an accelerating rate, because you’ve got a wave traveling to the end of a line and that’s sort of like a singularity right there. As the wavelength approaches zero the frequency approaches infinity causing the tip of the whip to move faster than any muscle could move it.
There must be animals that use that kind of action in their bodies. I would bet money that wherever they do it’s for the purposes of fighting, because fighting introduces a new evolutionary bar that continuously raises as the animals you’re fighting against evolve.
So in a way, evolution itself is the source of higher-order derivatives, because evolution is filtered by systems containing the lower-order derivatives, and evolution rewards the edge.
Thank you everyone for all the answers!
It’s just rate of change.
Rate of change of position is speed.
Rate of change of speed is acceleration.
Rate of change of acceleration is jerk.
Rate of change of jerk … you can see that, but it’s on the order of microseconds even for contrived examples. You can give it a name, and then look for rate of change of that.
Basically, at some point, it just becomes a rate of change you don’t care about, and becomes harder to see, harder to measure and harder to assign some kind of analogy too.
Same way that you can do multiple dimensions but at some point they’re still there but their physical analogy is just irrelevant to you because you don’t really “sense” or deal with them in the physical world you live in.
If your position as a function of time is a polynomial at some point the derivative becomes zero.
If you integrate backwards from jerk position becomes a cubic polynomial O( x^3 ), this is sufficient for almost all uses. Sometimes [fourth, fifth, and sixth derivatives](https://en.wikipedia.org/wiki/Fourth,_fifth,_and_sixth_derivatives_of_position) are useful for considering vibration and shock.
Mostly because we don’t have any way of reliably identifying experiences that correspond to those higher derivatives.
Humans primarily feel *acceleration* (i.e., we feel force, which is related to acceleration by Newton’s second law of motion). We can’t directly feel position or velocity.
We can feel the change in acceleration (jerk) fairly clearly. But derivatives beyond that? I have no clue what I’m trying to feel for.
Do you know what a change in the change in acceleration feels like? I don’t. The derivative of jerk has no physical meaning to me, because I can’t feel it happen. I only feel its effects. The instantaneous value found by taking the derivative of jerk is a purely conceptual tool to me that I would only bother with it if helped me solve a problem.
Higher order derivatives basically give us better and better approximations to things, but you usually only need a few to get a pretty decent approximation and so that is what happens.
However, there is a sense in which higher order derivatives yield less and less information. You can tell by trying to make sense of an “infinite order derivative”. That is, what would it mean to take infinitely many derivatives and what information do you get from them? If you know the answer to this question, then you can, in a sense, recognize higher and higher order derivatives as getting closer and closer to that infinite-order information.
Well we can, in a way, make sense of the notion of an “infinite order derivative” (see [here](https://math.stackexchange.com/questions/362873/is-there-any-meaning-to-an-infinite-derivative)). The information you get from it, however, is basically nothing because, for any function f(x) (such that the following values are defined) the “infinite order derivative” will always look like f^([∞])(x) = A*e^(x) for some number A. This number A is the limit of the derivatives f^([k])(0) as k goes to infinity. This means that the infinite order derivative smooths out the entire function, basically forgetting all information except that one value. This means that, if you take too many derivatives then eventually you are not getting any new information and are merely zeroing in on this single remaining value. It can be hard to know when this “eventually” is, but it generally does mean that the first few derivatives contain a reasonable chunk of information about a function and so you don’t need to go beyond that.
So, in a way, we *don’t* need more than a few derivatives in physics to understand mostly everything about what is going on – even beyond just the simple approximation results.
In an attempt to actually answer the question in a proper ELI5 manner:
*So much* of the motion we see around us does actually have 4th/5th/6th/whatever derivatives of position, but ignoring it rarely matters – the difference is usually small or takes a long time to add up to something large.
Our ability to see the world is *heavily* influenced by what is useful for passing on our genes – instinctive 5th differentials don’t really help you run away from a tiger or find berries, so evolution hasn’t provided that ability.
Similarly, even though it’s something a suitably smart person probably could learn, that would require experiences where it matters and effort to understand its effects, and *you* don’t find it important enough to have worked on that ability, either.
Because of all this, we don’t have an intuitive (or trained) grasp of how the Nth derivative *looks* or *feels* so we don’t see it, even when it’s under our noses (or under the seat of our pants as our car trundles over rough surfaces – see the [J-damper](https://www.formula1-dictionary.net/j-damper.html) I mentioned in another post).
I’d bet you a shiny penny that the 4th/5th/Nth derivatives of the motion of a leaf fluttering in a stiff breeze aren’t zero until you get up to a pretty high number, but who cares? Certainly not your genes, and probably not you, either, really…
The first reason is theoretical. For some unknown reasons, our physical laws run on the relationship between position and velocity. This means that any higher derivatives don’t provide additional information. You can use one extra level of derivative to make convenient computation, but this usefulness rapidly decrease the more derivatives you use. That’s why you basically don’t even see much beyond acceleration.
The second reason is practical. We don’t get an infinite amount of data, or perfect data without error. Small errors get amplified the more errors you have, and the less data you have, and the more derivative you have. So too many derivatives produces data that are basically indistinguishable from noises.
The third reason is that a position function is an *idealized* version of the real world’s position. Objects don’t actually have specific pinpoint positions for many reasons, and what we assigned to be its position function is just an approximation. This makes infinite many derivatives not that useful.
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