June 11, 2018 Monday
Bedtime Story
Exploiting the Constraints in Physical Mechanics
So then finally you should be able to understand
what the dot over the letter s in the statement “the kinetic energy will be
proportional to (s with dot)2 because kinetic
energy is proportional to square of the velocity.”
Lagrange figured out that one way the curve
can be an isochrone would be if the curve were to match the curve of a simple
harmonic oscillator wherein the height of the curve is proportional to the
square of arc length.
Applying the calculus of variations to
this, Lagrange ended up with the equation that describes a cycloid, and thus
the solution.
Lagrange while working at this isochrone
problem corresponded heavily with Euler, writing several letters to him between
the years of 1754 to 1756.
This serious mathematical correspondence
eventually led to the development of Euler-Lagrange equation sometimes simply
known as Lagrange equation which is a second order partial differential
equation.
The solutions to this equation gives rise
to functions for which a given functional is stationary.
Although it seems to be a useless
derivation at first, but it turned out to be very useful in the physical world
as a differentiable functional is stationary at its local maxima and minima.
It then becomes very useful in solving
problems of mathematical optimization that are applicable not only to
mechanics, but even fields as wide ranging as electrical engineering, civil
engineering, economics and finance and many more.
The application of mathematical
optimization comes into play in the mechanical dynamics of rigid bodies because
rigid body dynamics can be seen to be making an attempt to solve an ordinary
differential equation in a constraint manifold.
The constraints are nothing dramatic and
even an average ape can work them out to be.
The constraints are of various nonlinear
geometric types such as:
‘This point must always lie somewhere on
this curve’ or
‘This surface must not penetrate any other’
or
‘These two points must always coincide’.
In a way these constraints are something akin
to fundamental property of life or genes that ‘aims to maximize’ replication
and survival.
Why it does cannot be answered but that the
fact that genes display this fundamental behavior is well established.
So it is with mechanics.
The evolution of a physical system (i.e.
physical description of mechanics with flow of time) can be described by the
solutions to the Euler-Lagrange equation for the action of the system.
As stated earlier, it is same or rather
equivalent to Newton’s laws of motion but has certain advantages of using only
the scalar quantities and constraints inherent in the system.
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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