SW VandeCarr said:
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
