Re: Acceleration's higher orders

On 03/09/2024 10:38 AM, Ross Finlayson wrote:

On 03/09/2024 08:34 AM, gharnagel wrote:

Volney wrote:

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Ross wrote:

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So I'm wondering about v', v'', v''', that being
acceleration and its higher orders, out to v^prime-infty,
that at an instant, help figure this out.

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For what it's worth, some higher derivatives have (somewhat whimsical)
names. The derivative of acceleration with respect to time is called
jerk, the derivative of jerk is called snap or jounce, the derivative
of snap is crackle, the derivative of crackle is pop. Someone was a
breakfast cereal fan. The highest derivative I know of that's actually
used is snap, when designing the transition of roads or railroads from
straight to a curve they try to minimize the 'snap' of a vehicle
following the transition segment.

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I'd heard of jerk.  Many years ago, Norman Dean "invented" the Dean
drive,
a system of rotating masses with the center of rotation of the masses
being moved at particular times in the rotation cycle.  He showed that
the
weight of the assembly was decreased when running - on a bathroom scales.
>
William O. Davis analyzed the system which was referred to by John W.
Campbell, Jr. as "the fourth law of motion" - i.e., jerk.  Davis and G.
Harry Stine got together and tested the invention.  They hung it from a
wire and oriented it so the supposed thrust would be horizontal.  There
was no net thrust.  The "weight loss" was due to nonlinearities in the
bathroom scales because of the thumping around of the weights.

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These days that includes "pseudomomentum not dead again",
"McIntyre's arguments to be re-read and re-though, re-visited",
"rest-exchange momentum", "balleton", these kinds of things.
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SUSY not dead again, ....
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>

Of course anybody who sat physics class and remembers it
and internalized the concepts, knows that "scales" and "balances"
are two different things, and we measure "mass" with "balances".
(These days scales vary quite a bit, ..., day to day.)
Here the idea that non-linearity starts as an infinitesimal
impulse and results linearity, and vice-versa, is very much
represented in "Einstein's Bridge", the converse, while
the forward case very much is that an infinitesimal
change cascades down to dv/dt, and, about dt/dv.
With "meters per second, or, seconds per meter",
is that it's "meters per second, and, seconds per meter".
So, how it's usually figured is that f = ma and
then that it's always integrated, here with the
idea that the fuller integration, is to result
what are currently neglected terms, to analyze
the contributions of the non-linear or neglected
terms, so it results all the real analytical character,
includes the infinite series.
The infinite series most always starts at the big end.
Yet, change starts at the little end. The usual idea
of infinite, is that there is no end. So, it becomes
usual to work up finitely many higher orders of acceleration,
and result that there's a big end and a little end.
(In Gulliver's Travels there's an account of that in
two land, the inhabitants ate hard-boiled eggs. Eggs
have two ends, a big end and a little end. In
one land, the inhabitants started from the big end,
in the other, the inhabitants started from the little
end, that what is trivial to one who'd eat from either end,
resulted a cultural divide to the point of conflict.
These days this is reflected for example in the computer
architecture of most-significant byte B or bit b, to
least-significant, in terms of a bit-sequence representing
an integer, with the 1's place being least-significant
and higher places more-significant, reflecting writing
the numbers in order according to the bits of the digit
and the digits as moduli, MSB-to-LSB, LSB-to-MSB, and
msb-to-lsb, and lsb-to-msb, the "Big-Endian" and "Little-
Endian", with regards to read-out is easier Big-Endian,
while addressal is easier or aligned, Little-Endian.)
So here, the usual higher orders of acceleration are
usually under-defined after the first order, acceleration
itself, "instantaneous" or "constant", f = ma, what
results that f is a linear vector, and over time is
what results force applied and work done, and all
usually with a notion that conservation is energy.
Yet, we have the great classical exposition of Zeno,
in which we can being to frame all things with respect
to the dialectic, of rest and motion, and relative motion,
and uniform motion.
Here it's sort of the idea, that a cylinder is
standing upright, only most-minimally locally stable,
then a feather lands on it, and it tips, converting
all of its potential energy in its oriented stable
configuration, to kinetic energy, what with regards
to reaching another, more, yet still locally, stable
configuration, lieing down.
So, the feather, is an infinitesimal, and it's
the little end, of the cylinder's minimization of
potential energy, just as an example of the sort
of thing, that the cylinder is arbitrarily stable
and the feather while arbitrarily small is arbitrarily
large, with respect to the arbitrarily stable configuration
of the cylinder, or obelisk, which is arbitrarily high,
thin, narrow, or wide, keeping things simple in the
configuration space, while general as these things are.
Then, for Zeno, is this notion of, "meeting in the
middle", "middle of nowhere", this is the sort of
accompaniment to "21'st century Zeno", which not
only models ancient Zeno, but every edition and
each variation between.
So, in physics, there's singularity theory. One of
the usual most usual notions of applied physics,
is that "singularity's don't exist", then, though,
what results is "they do", then, furthermore,
"singularities are multiplicities", vis-a-vis,
"singularities are either origins or attractors".
Then, for the infinite and being down at both ends,
is much about, being around. (And through.)
So, "acceleration's infinitely-many higher-orders",
is a fundamental concept that reflects the very
notion of state, configuration, and change, itself,
and of course is what must follow from a very thorough
and didactic deconstructive account of "Zeno: then and now".