June 14, 2018 Thursday
Bedtime Story
Framing Alternative Parsimonious Question
Last night I had left you with the
Principle of Least Action.
I am sure many of my generation may never
have been taught or if at all taught, in all likelihood only mentioned
fleetingly so as to be lost and forgotten for good as an adult.
So what exactly is this principle?
It is a technique devised by mathematical
savvy apes to find solutions to functions by using extreme quantities of values
that depend on those functions.
This may not make much sense to you.
So let me try to make it clear by considering
a very simple problem: You wish to know the shape of a chain that is suspended
at both ends.
One way is to tackle the question directly
the Newtonian way by considering the various forces acting upon the desired
object.
The alternative way to arrive at the
solution of this problem is to approach it indirectly by posing another
question and use extreme value of a factor on which the shape of the chain
would depend.
Using the variational principle, you can
frame the question by bringing in the function of potential energy and using
its extreme value: What would be the shape of such a suspended chain that would
minimize its potential energy?
Thus we can determine the state of the
chain or even its dynamics (with a different question and functional parameter)
by identifying the extremum (minimum, maximum, or saddle point) of a function
or functional that has a direct bearing to it, which in our case is the
potential energy.
But how did this idea originate?
Well, almost the since the birth of
thinking and pondering humans who were keen observers and who applied their
minds on something more than mating and survival, there was a vague realization
that nature works on the principle of parsimony, or it takes the “simplest”
path.
One common instance of this principle
enacting out is the way light travels in different mediums; whenever it comes
across a denser medium, its slows down.
And almost as if to make up for the slowed
speed, it takes a shorter path in the denser medium.
So in a manner of speaking, we see light
enacting out the parsimony principle by taking the shortest route in the rarer
medium.
Pierre de Fermat in the seventeenth century
almost made it a law by saying that “light that travels between two given
points along the path of shortest time” and this statement now popularly goes
by the name of principle of least time or Fermat’s principle.
This is not just applicable to refraction
but to reflection as well which was noted by scientific minds of older
civilizations.
They also used similar principles of
parsimony in the field of land surveying where modern surveyors use elements of
geometry, trigonometry, regression analysis, physics, engineering, metrology,
programming languages and of course civil law.
In ancient Egypt, there were special
professionals called rope-stretchers who like today’s surveyors measured real
property demarcations and foundations using knotted cords.
We shall continue with these Egyptian
rope-stretchers and see what connection they have with the Principle of Least
Action.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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