July 10, 2018 Tuesday
Bedtime Story
Working out Lagrangian Function
Tonight we shall try to work out the Lagrangian
function for our system that consists of N point particles by calculating its
total kinetic energy and potential energy.
For N number of particles each of mass m1,
m2,…, mk and velocities v1, v2,…,vk,
the total kinetic energy can be formulated as (where k = 1 to N)
T = ½ ∑ (k = 1 to n)
Kinetic energy as you may very well
understand is the energy contained in it by the virtue of the movement of its
individual l particles.
The concept of the kinetic energy is easy
to grasp for such a system of n particles.
But how do we work out the potential energy
of such a system that is so loose?
Where does that come from in such a type of
N-body system?
The potential energy in such a system is
generated out of the interaction between its constituent particles and will
reflect the amount of energy each particle will have due to other particles
interacting with it along with the external forces.
Such forces could be gravitational force or
electromagnetic potential.
Working out the potential energy of such a
system is a complicated affair as you can very well imagine as that amounts to
calculating innumerable interactions.
Interatomic potential of such a system that
is made up of particles that are electrically neutral was worked out by John
Lennard-Jones, an English mathematician in 1924 and hence is known as the
Lennard-Jones potential.
We shall leave out the details as it is
physics that is beyond my technical or educational expertise.
The definition of Lagrangian function that
is stated above is valid for non relativistic physics and is not applicable to
relativistic Lagrangian mechanics where it needs to be replaced with new
function that would be consistent either with general or relativistic
mechanics.
I shall not go into relativistic physics as
that is not currently our aim.
From this Lagrangian mechanics, the
principle of least action can be derived.
The principle of least action in Lagrangian
mechanics uses a term called ‘action’ that is a functional of the Lagrangian.
Now what is this ‘action’ in Lagrangian
mechanics?
Such terms are often very confusing that
has some other general meanings in English language but used for some specific
meaning in specialized subjects such as mathematical physics and mathematics.
With this action one can convert a
trajectory of a body or its path or history to a real number with each path
having a different number and the unit being [energy].[time] or joule-second.
We shall look further into this concept of ‘action’
on Lagrangian mechanics 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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