# Fourth derivative of position?

The first three derivatives of position are velocity, acceleration and (some say) jerk. I've read that the fourth derivative is position again. This is in the context of interpreting the third derivative as control in a so called control field. Can anyone shed some light on this?

Staff Emeritus
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The first three derivatives of position are velocity, acceleration and (some say) jerk. I've read that the fourth derivative is position again.

Not true in general.

Isn't the fourth derivative of position "acceleration of accelerate" ?

like :
g is the third derivative of position in falling

g = GM / R^2

dg / dt = dg/dR * dR/dt = -2GM/R^3 * V

(vector that points top is + for Velocity)

so g is chanching with dg/dt rate ?

robphy
Homework Helper
Gold Member
The first three derivatives of position are velocity, acceleration and (some say) jerk. I've read that the fourth derivative is position again. This is in the context of interpreting the third derivative as control in a so called control field. Can anyone shed some light on this?

No. The fourth derivative of position isn't position. Look at robphy's second link. It says that the fourth derivative of position with respect to time can be called a "jounce" or a "snap". Either one will work I guess. There really isn't an official name for the fourth and higher derivatives of position because they really aren't used like the the first three derivatives. I don't even think the third derivative (a jerk) is used that often.

Not true in general.

I gave a specific context. Apparently it's related to Control Theory. I'm familiar with the usual interpretation given by the other posters.

No. The fourth derivative of position isn't position. Look at robphy's second link. It says that the fourth derivative of position with respect to time can be called a "jounce" or a "snap". Either one will work I guess. There really isn't an official name for the fourth and higher derivatives of position because they really aren't used like the the first three derivatives. I don't even think the third derivative (a jerk) is used that often.

I had already read that link. It assumes that position has infinite qualitatively unique derivatives. As I indicated in post 6, my context is a control space as it's apparently defined in Control Theory. I'm not very familiar with CT other than what I read on the Wiki.

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robphy
Homework Helper
Gold Member
The first three derivatives of position are velocity, acceleration and (some say) jerk. I've read that the fourth derivative is position again. This is in the context of interpreting the third derivative as control in a so called control field. Can anyone shed some light on this?

If the derivatives are with respect to time,
then the fourth time-derivative of position doesn't carry the units of a position.

There must be more to the situation.

If the derivatives are with respect to time,
then the fourth time-derivative of position doesn't carry the units of a position.

There must be more to the situation.

I'm thinking that the control variable (third derivative in this interpretation) has something to do with inducing a change in the acceleration of a 'particle' (anything being controlled). The value of the third derivative in turn is sensitive to the position of the particle either at any point in time or to the ultimate destination of the particle. I don't know which. It's not clear to me after reading the Wiki article and I'm not about to master Control Theory (CT). I thought there might be some knowledgeable people here who could help me. CT has a lot do to with dynamical systems. (This post was edited at 3:12 pm 5/28/09)

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berkeman
Mentor
I'm thinking that the control variable (third derivative in this interpretation) has something to do with inducing a change in the acceleration of a 'particle' (anything being controlled). The value of the third derivative in turn is sensitive to the position of the particle either at any point in time or to the ultimate destination of the particle. I don't know which. It's not clear to me after reading the Wiki article and I'm not about to master Control Theory (CT). I thought there might be some knowledgeable people here who could help me. CT has a lot do to with dynamical systems. (This post was edited at 3:12 pm 5/28/09)

BTW, if the link is to wikipedia articles, you might want to find more reliable sources of info, or at least some indepentl corroboration of what the wikipedia article is saying. Do you have access to a university library that may have CT textbooks?

I do controls, we dont use 4th derivatives of anything...

sylas
I do controls, we dont use 4th derivatives of anything...

Speculation...

Control theory sometimes does things in a frequency domain, and there's a lot of fun and games with periodic functions.

... and the fourth derivative of a sin curve is the original sin curve again -- but scaled with the fourth power of frequency.

Doesn't work in general, of course; as soon as you have two or more frequencies in your signal, the fourth power looks totally different, since each frequency component scales differently. The scale also accounts for the units difference.

But the fourth derivative of sin(x) is sin(x) again, and I wonder if that somehow lies behind the question of the thread.

Cheers -- sylas

berkeman
Mentor
Speculation...

Control theory sometimes does things in a frequency domain, and there's a lot of fun and games with periodic functions.

... and the fourth derivative of a sin curve is the original sin curve again -- but scaled with the fourth power of frequency.

Doesn't work in general, of course; as soon as you have two or more frequencies in your signal, the fourth power looks totally different, since each frequency component scales differently. The scale also accounts for the units difference.

But the fourth derivative of sin(x) is sin(x) again, and I wonder if that somehow lies behind the question of the thread.

Cheers -- sylas

Um, so what the heck is your point? And how does this help the OP?

sylas
Um, so what the heck is your point? And how does this help the OP?

The OP says "I've read that the fourth derivative is position again." He asks if anyone can shed light on this.

Now I don't know what he actually read, but I wonder if perhaps he actually read that the fourth derivative of a simple frequency is the same as what you started with. (Assuming unit frequency.)

If so, this may help sort out the whole basis for the question in the OP, particularly as the author says there's some connection to control theory. The author asks an open question to see if anyone can shed light on this, and so I propose one possibility, given in good faith to see if it helps. If so, great! If not, then no problem. Is that any better?

Cheers -- sylas

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The OP says "I've read that the fourth derivative is position again." He asks if anyone can shed light on this.

Now I don't know what he actually read, but I wonder if perhaps he actually read that the fourth derivative of a simple frequency is the same as what you started with. (Assuming unit frequency.)

If so, this may help sort out the whole basis for the question in the OP, particularly as the author says there's some connection to control theory. The author asks an open question to see if anyone can shed light on this, and so I propose one possibility, given in good faith to see if it helps. If so, great! If not, then no problem. Is that any better?

Cheers -- sylas

Thanks for your reply. I've read something on this years ago which stated the fourth derivative of position is position again and I never could understand it. It was also discussed in "The Reflexive Universe" (1978) by Arthur M Young (1905-1995), a mathematician and aeronautical engineer of specializing in helicopters. I also saw on unattributed article on the net of dubious worth: http://community-2webtv.net/SkyVessel/FreeEnergy/ referring to work done at General Motors. This all seems to be something outside the mainstream

sylas
Thanks for your reply. I've read something on this years ago which stated the fourth derivative of position is position again and I never could understand it. It was also discussed in "The Reflexive Universe" (1978) by Arthur M Young (1905-1995), a mathematician and aeronautical engineer of specializing in helicopters. I also saw on unattributed article on the net of dubious worth: http://community-2webtv.net/SkyVessel/FreeEnergy/ referring to work done at General Motors. This all seems to be something outside the mainstream

I don't know about the https://www.amazon.com/dp/1892160110/?tag=pfamazon01-20; but it looks a bit like one of the attempts to make deep connections between conventional science and philosophical speculations about consciousness and some kind of grand purpose or destiny or innate tendency of the universe. Sometimes such writing can propose deep significance or meaning in certain patterns or cycles, like harmonic functions.

Mathematically, the only functions that satisfy
$$\frac{d^4y}{dx^4} = y$$​
are linear combinations of
$$e^{kx} \;\mbox{where}\; k^4 = 1 \;\mbox{that is}\; k \in \{ \pm 1, \pm j\}$$​
In real numbers, that means there are constants A, B, C, D such that
$$y = Ae^x + B e^{-x} + C \sin(x) + B \cos(x)$$​
or (if you like to make it look even more evocative for pop philosophy) use the hyperbolic functions in place of exponential functions:
$$y = A\sinh(x) + B \cosh(x) + C \sin(x) + B \cos(x)$$​
There's no sensible general notion in which the fourth derivative of position is the same as position; unless for some reason position happens to be defined by such a function, and even then the units don't match. But still... simple harmonic motion, or circular motion, can be defined so that for that case, you do have the fourth derivative being position again, as long as the angular frequency is set to 1 radian per unit time. You can always pick your time unit to help frequency have the value where this works out. I don't think there's any deep meaning to this, and certainly no general principle about fourth derivatives of position.

As for your http://community-2.webtv.net/SkyVessel/FreeEnergy/) which has some weirdness on fourth derivatives. Again, this seems to be in the context of AC currents, or circular motions, which are just the case in which you happen to have the fourth derivative of a function equal to itself. There's no general principle here or useful association with real control theory. Just a lot of barely coherent strangeness on perpetual motion.

This was not done at General Motors. There a mention of work on "jerk" at General Motors -- which is indeed a sensible use of third derivatives of position relating to passenger comfort. Just as you suggest, I agree that perpetual-motion-man is not a reliable source on the implications of this work. He coins the term "control" for what the rest of us would call "jerk", but it is not in fact anything usefully associated with what engineers or control theory call control. It looks a lot like he really is looking at harmonic motions and circular motions, and goes from there into chains of reasoning that are beyond all limits of rational thought.

Thanks for the links... most delicious!

Cheers -- sylas

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Thanks for the links... most delicious!

Cheers -- sylas

Your welcome and thanks for the comprehensive reply. At least now I know why I couldn't make sense out of the fourth derivative of position being position again.

FredGarvin
I really can't comment on the 4th derivative stuff because I abhor controls stuff. However it doesn't seem to make much sense to me either.

On the other hand, you read some good books. Arthur Young is a helicopter legend. Among other things, he came up with the stabilizing bar concept for rotor heads. I never knew he got into the whole metaphysics stuff later in life.

I really can't comment on the 4th derivative stuff because I abhor controls stuff. However it doesn't seem to make much sense to me either.

On the other hand, you read some good books. Arthur Young is a helicopter legend. Among other things, he came up with the stabilizing bar concept for rotor heads. I never knew he got into the whole metaphysics stuff later in life.

There's nothing wrong with "good" metaphysics as long as we draw a clear boundary between metaphysics (not verifiable or falsifiable) and science. In this case, I was drawn in by the "mathematical" Arthur Young and generally took his metaphysics for what it was worth. As far as the other reference, I only read the first part dealing with harmonic and circular motions and never got into the perpetual motion stuff. As sylas indicated, you can make a very narrow special case for the fourth derivative as iterated position, but there's no general principle involved and apparently no connection with formal Control Theory.

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