Re: RE: Models, etc.

[PPARKER@TWSUVM.UC.TWSU.EDU]

> Date: Thu, 30 Oct 97 13:01:09 MST
> From: "Alex. F. Burr" <PHYS010@NMSUVM1.NMSU.EDU>
>
> Can anybody think of a process in nature (excepting quantum mechanical
> phenomina) which can be modeled by an equation which is not continuous
> and which does not have continuous derivitives of all orders?
> -
> What is the highest order deriv[a]tive which has physical significance?
> (ex ac[c]elleration, a second order deriv[a]tive undoubtedly has significance.
>    jerk, a third order deriv[a]tive has debatable or only minor
> significance

It has great significance to railroads and their insurers.  Control of
the third derivative (specifically, getting low bounds on it) is the
reason that the transition from straight track (called tangent in the
railroad surveying literature) to curved track is *NEVER* circular for
routes traversed at any useful speed.  Instead, the transition is
done with cubic splines, and has been since the middle 1800s.  I have
a copy of an 1886 railroad survey book with tables for doing the cubic
splines (called "easements") and extensive guides to determining how
much of the curve should be splined and how much, if any, can be
circular.  (This is in addition to computations for appropriate "super-
elevation" as they call it; banking is the common term for roads and
racetracks.)  All this was developed originally by "seat of the pants"
techniques to reduce the damages-in-shipping to goods transported, then
refined until it got to the state in the book I have.
   The "jerk" is easily felt in a car at a modest 15-20 mph: it's the
difference between turning the wheel, say, 1/4 turn ccw as fast as
possible to make a left-hand turn, versus turning it more *smoothly*
[math technical for more continuous derivatives] as almost every
beginning driver very soon learns to do in order to get a license.
   I leave additional examples to the reader as an exercise.

> I am sure that the fouth order deriv[a]tive has been
> named but I can think of no place where it is used.)

I leave it as an exercise for the reader to estimate which extant
railways are interested in fourth derivatives, when they developed
this interest, and how many more they might become interested in
and when.

---------------------------------------------
Phil Parker        pparker@twsuvm.uc.twsu.edu
Random quote for this second:
  Books aren't written; they're rewritten.---Michael Crichton