In the very first post of this thread I used two
URLs from TinyURL.com pointing to a couple of
diagrams on my Beta-atmosphere Yahoo web-site.
Though I tested them and they worked originally,
it would appear that Yahoo have a dynamic system
for referring to their group files and consequently
the TINY URL's work no longer.
I apologise to any Vortexians who might have been
frustrated by this dock-up and I have now put the
diagrams on my own web site.
I have copied the original post below together with
the updated URLs.
Cheers
Frank Grimer
===========================================================
At 07:20 pm 22-02-05 +0000, I wrote:
In previous posts the idea of a series of increasing
orders of derivatives of length with respect to time
has been chewed over.
It will be helpful to recapitulate the discussion by
using excepts from the Vortex archives.
On the subject of imagining what these derivative mean
physically, we have.
========================================================
...Funny you should say that, Richard, because I've been
pondering how one could physically visualize high order
derivatives of distance with respect to time.
dL/dT ......VELOCITY .......moving scenery
- no problem
d2L/dT2 ....ACCELERATION ...being pushed back in ones
seat as the plane takes off
- no problem
d3L/dt3 ....JERK............Mmm..more difficult - being
hit over the head with a
bottle perhaps?
d4L/dT4 ....JOUNCE..........I have no feeling whatsoever
for this or high derivatives.
But the failure to visualise these higher order derivative
is because I am thinking in terms of straight line motion.
If I think instead in terms of circular motion, or better
still, helical motions, then things become very much easier.
If I allow myself to be pinned to the wall of a fairground
centrifuge then I can experience being "pushed back in my
seat on a continuous basis. By imposing a circular motion
on this circular motion to form an open vortex helix I can
visualize the next derivative, though I am well past the
age where I would want to experience it - and so on - and
so forth.
========================================================
On the topic of visualisation Keith Nagel wrote,
======================================================
Also, you mentioned Jerk and Jounce ( sounds like
a b-list rap group ). I've also puzzled over the
physical meaning of these terms. It's rather like
trying to imagine higher dimensional shapes. One
dimension up is about all I can muster, which in
this case is Jerk. Standing on a carousel, with
the speed increasing and decreasing sinusoidally,
ought to do it. Perhaps a better term would be
"projectile vomiting" rather than jerk, huh??? (grin).
======================================================
The discussion then wandered off into considering the
outcome of experiments with coil-coils and coil-coil....
coils - and a host of other topics - as discussions on Vortex
frequently do <g> - which is, of course, one of the delights
of the Group. One is not artificially constrained to narrowly
keep to the point. 8-)
I can now see another way of visualizing the higher order
differentials but before doing that I would like to state
something I have realised in relation to the conservation
laws.
We have:-
=======================================================
SYMBOL DERIVATIVE PROPERTY CONSERVED
dL/dt velocity momentum yes
d2L/dt3 acceleration energy yes
d3L/dt3 jerk angular
acceleration yes
d4L/dt4 jounce rate of change Err...?
of ang.acc.
d5L/dt5 Err...? RoC of RoC Err...?
of ang.acc.
....... ..... ........... .......
dnL/dtn ...
=======================================================
It has become increasingly clear to me that each derivative
is associated with a conservation law of its own. In short
there are an indefinitely large numbers of conservation
laws of motion. The reason we fail to see this is that we
are hag ridden by Cartesian Geometry and its unbounded x,
y and z dimensions. The coil-coil...coil visualization
discussed previously on Vortex avoided this trap because
it implicitly involved upper and lower bounds to the size
of the coils.
It is interesting to note that the calculus also avoids
the Cartesian trap by not associating length with the
Cartesian space. The calculus allows us to have any
number of independent dimensions since no finite amount
of the (n+1)th derivative will give us a nth derivative.
In the Cartesian case we can move from x (length) to
y (area) to z (volume)- but beyond that we are stuffed.
In Keith's post he got as far as the third derivative
(angular acceleration) by "Standing on a carousel, with
the speed increasing and decreasing sinusoidally,..."
but could take it no further.
I have now discovered how to generate physical
equivalents of d4L/dt4 and all higher derivative
motions with devices that anyone who is familiar
with the action of gyroscopes can readily visualise
and understand.
Consider the Mechanism set out in
http://www.grimer2.freeserve.co.uk/pge22.htm
This is a gyro consisting of an outer ring of 8 gyros.
The eight outer ring gyros (2nd order gyros) are free
to rotate around their extended axles. These axles are
fixed rigidly to the central hub so that the rotating
2nd order gyros are forcibly constrained to follow a
circular path when the central hub is rotated.
Now I have chosen 8 gyros for pedagogic purposes since
it is easy to visualise the solid outer ring of a
classic gyro being replaced by multiple small gyros.
However, to take the physical example to d4L/dt4 and
higher order derivative motions it is easier to think
in terms of only a single pair of opposed gyros.
A diagrammatic representation of a 7 order device
is shown in:
http://www.grimer2.freeserve.co.uk/pge23.htm
[scan down the page]
As you can see, it makes a rather nice
Iterative Hierarchical (IH) fractal pattern.
The different orders are indicated by different
colours.
At this point it is worth repeating something from
a previous post. Cartesian geometry as taught at
school is a mathematical abstraction. It can't be
applied willy-nilly to the real world. In that
world real things have upper and lower bounds so one
must use an x, y, and z geometry which also has upper
and lower bounds. This means that we can nest as
many spaces as we like in the manner of Russian dolls
or Chinese boxes.
In this case the upper and lower bounds of IH orders
are quite obvious if you think about it. A gyro of
the nth order is bounded in size by the gyros of the
(n+1)th and the (n-1)th orders. Likewise with every
other order. It's precisely the same situation as
with the "coiled-coiled-coiled-........coil" mentioned
in a previous post. Like the different order of gyros,
each different order of coils implicitly has an upper
and lower bound since a coil cannot be smaller than
twice the diameter of the wire, nor larger than the
next coil up the pecking order.
Interestingly enough the IH fractal can be seen as
an attenuated version of the turbulent structure of
water. In the words of the poet,
-------------------------------------
Big whorls have little whorls
That feed on their velocity,
And little whorls have lesser whorls
And so on to viscosity.
- Lewis F. Richardson -
-------------------------------------
And the power laws discovered for water vapour
are no doubt a reflection of their fractal
dynamic structure.
=========================================================
