June 25, 2018 Monday
Bedtime Story
Exploiting the Degrees of Freedom
Last night we were talking about the
degrees of freedom a point particle would have in a three-dimensional Euclidean
space.
At the minimum we will need to specify
three numbers that would attest to its translational position.
Besides this, the particle can also undergo
rotation and vibration operation or transformation.
A vibration of a point particle at room
temperature is miniscule and hence negligible and hence its degree of freedom
with respect to vibration is zero.
It gains significance in thermodynamics as
the temperature is one of principle quantities in the study of thermodynamics.
In statistical mechanics temperature is a measure
of the mean kinetic energy of the classical fundamental particles of the
system.
Hence while at higher temperatures the
degree of freedom from vibration becomes critical, at room temperatures it can
be ignored and can be assigned the value of zero.
Similarly, a particle as understood in the classical
mechanics, will remain unchanged if it undergoes rotation and hence the degree
of freedom due to rotation can also be assigned zero to a single point particle
in space.
So the total degree of freedom of a point
particle in a three-dimensional space adds up to 3 + 0 + 0 = 3.
Now for a moving particle in a
three-dimensional space at a certain velocity the calculation of degrees of freedom
gets trickier.
We know that a point particle would require
three position co-ordinates to define its position.
Now it also has a certain velocity with speed
and direction which has three components too.
This is a system that in respect to
evolution of time is deterministic; meaning that if you know its location and
velocity at one point then from it you can either go backward and know its past
position or go forward and determine its future position.
Such a system will have six degrees of
freedom; three from our previous calculation and three from the vector
component at each point.
The analytical mechanics makes use of this
degree of freedom concept as each system has constrains that limits its degree
of freedom and this helps to reduce the number of coordinates to solve the
equation of the system.
Analytical mechanics has been broadly
divided into Lagrangian mechanics that is the subject of out bedtime story and
Hamiltonian mechanics.
Lagrangian mechanics is that mechanics that
rests on generalized coordinates and corresponding generalized velocities in
configuration space.
Hamiltonian mechanics, on the other hand,
rests on coordinates and corresponding momenta in phase space.
Both formulations are equivalent to each
other and with Newtonian mechanics and hence contain the same information and
yield the same solutions while working out the dynamics of a classical 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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