June 12, 2018 Tuesday
Bedtime Story
Euler-Lagrange Equation
Just to give you a glimpse into the world
of mathematical physics that is two hundred and thirty years old, I am going to
show you how the Euler-Lagrange equation.
It will not be of much use to you but it
should dazzle you with the ingenuity that the human mind is capable of; that of
describing the movements of mechanical bodies in terms of elegant letters
scribbled on a piece of paper.
For a function q of a real argument t,
which is a stationary point of the functional
S(q) = a∫b
Here q is the function to be found
The q with dot
is the derivative of q
L is a real-valued function with continuous
first partial derivatives
For this, then, the Euler-Lagrange equation
would be
Lx (t, (q(t), qdot
Here Lx and Lv denote the partial
derivatives of L with respect to the second and third arguments respectively.
I know this looks like a foreign language
to most of us because the Lagrange equation is indeed written in the
international foreign language of mathematics which sadly most of us untrained
in.
Yet this is just to show you that this was
the kind of pioneering work that Lagrange was doing.
I say work because such mathematical
equations generally do not arise spontaneously in the minds of mathematicians
(though sometimes they do), but are often painstakingly derived using the
precise rules of mathematics along with high level of mathematical intuition.
In modern mathematics, the derivation of
the one-dimensional Euler-Lagrange equation is one of the classical proofs; as
classical as say the Basel problem once again solved by Leonhard Euler and
named after the hometown of both Euler and the Bernoulli family (who tried but
failed at arriving at the solution to this problem).
The derivation of the one-dimensional
Euler-Lagrange equation relies on the fundamental lemma of the calculus of
variations which states that a variation
f
of a function f can be concentrated on an arbitrarily small interval, but not a
single point.
The lemma’s most basic version is stated as
follows:
If a continuous function f of an open
interval (a, b) satisfies the equality
for all compactly supported smooth
functions h on (a, b), then f is identically zero.
The word ‘smooth’ is obviously not in any
way related to what is meant in the natural English language, but it refers to
a function that is ‘infinitely differentiable” or even “continuously
differentiable” .
The term “compactly supported” refers to
‘vanishing outside (c, d) for some c, d such that a < c < d < b’.
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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