Following your discovery of the Divine Proportion
in spiral phyllotaxy you become very famous, a real
guru in the New Age Mysticism crowd. You write a
book called "Seven Roads to the Divine Proportion"
published by DoublyDilbert Books. Your fame lands
you on the lucrative lecture circuit taking you
all over the world.
One of your stops is in a very large city where you
have many lectures, book signing events etc. Since
you are a visitor you get back and forth by taxi
cab. In this city, all the taxi drivers charge
according to a two-fold formula. The fare increases
as you move closer to the center of the city and
increases with speed. Speed is particularly important
since increased speeds result in traffic tickets,
accidents etc. For this reason the fare increases
as the square of the speed. The taxi drivers are also
very considerate of your time. You specify to the driver
not only your destination, but when you want to arrive.
The driver never fails to arrive at exactly the time you
specify.
As you are driving along, you happen to have a map of
the city and notice that the driver seems to be
following a rather curious path. Now your suspicions
are aroused. Hey, there must be some huge conspiracy
among taxi drivers to take advantage of poor, unsuspecting
tourists. This must be exposed, so you spend many weeks
collecting data. You go from various locations in the
city to various destinations. Sometimes you repeat the
same trip, specifying a different time interval and
note that in each of these cases you are taken on a
different path.
After collecting mountains of data you isolate yourself
in your motel room for several more weeks trying to
make sense of it all. You plot each of your trips on
the city map. In each case you consider alternate
routes that could have been taken, computing the fare
in each case. What you notice, again to your great
horror :), is that in every single case the taxi
driver followed a path that minimized the fare! You
had these guys figured all wrong. Not only are they
considerate of your time, they are also considerate
of your pocket book.
Ok, the physical application for which this is an
analogy is probably not readily apparent, though there
are enough clues above that I'm sure some have gotten
it. Here's another couple of clues. You find that every
path that you traveled is a segment of an ellipse
with focus at the center of the city. You also notice
that if you draw a line from the center to your location
that equal areas are swept out in equal times. IOW,
the cost due to speed is the kinetic energy T and the
positional cost is the work done by gravity, the negative
of the potential energy V. The total cost is the integral
over the path of the quantity (T-V), more commonly
called the action. So, the taxi problem is an analogy
for the principle of least action. The basic ingredients
of the analogy I have borrowed from some book whose
author and title escapes me for the moment. The main
purpose of the analogy is to illustrate the teleological
nature of the principle of least action.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"It is not certain that all is uncertain,
to the glory of skepticism." -- Pascal