June 19, 2018 Tuesday
Bedtime Story
Euler's Principle
Last night the storytelling chimpanzee had
introduced to you the Maupertuis’s principle in terms of abbreviated action
functional that is represented by
0.
The principle is very elementary – it states
that the true path of any system is the path for which the abbreviated action
functional is zero.
This mathematical formulation can also be
expressed with Lagrangian function L and this thus brings us back to Lagrange
who we had left to talk or rather write about the principle of least action.
Before we return back to Lagrange, who
along with his foundational work is a very important character on the story of
Conservation Laws particularly the one pertaining to the Law of conservation of
the angular momentum, I wish to point out that after Maupertuis, Euler,
Hamilton and Lagrange gave their own formulation of this principle.
Euler’s mathematical formulation was much
simpler as he only used the three familiar variables of mass of the object M,
its speed or velocity v and its movement over the infinitesimal distance ds.
Let us how Euler wrote it down in his
treatise of 1744:
“Let the mass of the projectile be M, and
let its speed be v while being moved over infinitesimal distance ds.
The body will have a momentum Mv, that when
multiplied by the distance ds, will give Mv ds, the momentum of the body
integrated over the distance ds.
Now I assert that the curve thus described
by the body to be the curve (from among all other curves connecting the same
endpoints) that minimizes
Or, provided that M is constant along the
path
M ∫
So to complete the whole idea
mathematically, we can write this idea in terms of generalized coordinates p
and state of the system q as integral function of the product of p and dq
(infinitesimal change of state from q1 to q2)
δ ∫
This is known as Euler’s principle, very much
equivalent to Maupertuis’s principle.
I shall not go it into the mathematical
formulation of the principle of least action worked out by Hamilton.
This Hamilton is William Rowan Hamilton of
the quaternions fame who on 16th of October, 1843 while taking a
stroll with his wife along the Royal Canal in Dublin in a flash of sudden
inspiration carved out the solution with his pen knife on the stony walls of
the Broom Bridge:
i2 = j2 = k2
= ijk = -1
Lagrange while in Turin, as I had stated
earlier, was heavily corresponding with Euler who was in Berlin then.
Alongside with Euler, Lagrange was also in
communication with Maupertuis, and both these men were very impressed with his
mathematical talent.
We shall continue with the story of
Lagrange and his intellectual interactions with both Euler and Maupertuis in
the nights to come.
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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