June 18, 2018 Monday
Bedtime Story
Maupertuis's Principle
As we saw last night, Maupertuis was the
person to be given credit for enunciating the principle of least action.
It is interesting to read once again in his
very own words how Maupertuis had framed this principle:
“The law of movement and of rest deduced
from this principle being precisely the same as those observed in nature, we
can admire the application of it to all phenomena.
The movement of animals, the vegetative
growth of plants…are only its consequences; and the spectacle of the universe
becomes so much grander, so much more beautiful, the worthier of its Author,
when one knows that a small number of laws, most wisely established, suffice
for all movements.”
Clearly Maupertuis was no atheist.
The principle of least action has been more
generalized as Maupertuis’s principle which has a mathematical formulation in
terms of an integral equation.
I am stating the mathematical formulation
now, but its explanation will come later so please bear the lack of intelligibility
with me for now.
For a system of N generalized coordinates
(even this will be explained later) Maupertuis’s principle states that the true
path of a system between two specified states q1 and q2
is an extremum (which means a stationary point, minimum, maximum or a saddle
point) of the abbreviated action functional:
Here p is conjugate momenta of the generalized
conjugates where p = (p1, p2,…,pN)
The p or the conjugate momenta can also be
defined in terms of Lagrangian function L which again we shall hopefully
discuss in the nights to come.
pk = δL/(δ.qdot.k)
Here the letter L is the Lagrangian
function of the system and it is often accompanied with letters q, qdot and t
like this:
L(q,qdot,t)
where we know that q represents the state of the system in generalized
coordinates and q is expressed mathematically as
q = (q1, q2,…qN)
The q with a dot on top is the time
derivative and is equivalent to the other mathematical form dq/dt which is
equal to (dq1/dt, dq2/dt,… dqN/dt)
Once again I apologize for not laying down
the ground work for these scary equations (scary because of the fact that they
have not been suitably introduced) and I request you to bear with me some terms
which will not be familiar to you and on which I shall do stories later on.
As later development in physics showed,
that the concept of ‘action’ as the product of mass, velocity and distance
never materialized though the concept of action as a functional is very much
valid till this day in analytical mechanics.
The abbreviated action functional is
denoted by the script
0 and is defined as
the integral of the generalized momenta along a path in the generalized
coordinates.
So what this principle is saying is that
the true path of a system is the path for which the abbreviated action
0 is stationary.
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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