July 09, 2018 Monday
Bedtime Story
Lagrangian Function as the basis of Lagrangian Mechanics
Last night we got a taste of how Lagrangian
mechanics with generalized coordinates works without altering any of the
fundamentals of Newtonian physics.
In fact not only does it contradict any of
Newton’s laws but on the other hand the fundamental laws of Newton are very
much applicable to it.
We got the velocity vector of each
individual particle of the system.
Using the second law of Newton on those
particles we can get the net force as:
And this general formula is applicable to
each particle.
Since ours was presumed to be a three
dimensional system with N number of particles, there would be 3N second order
ordinary differential equations in the positions of the particles to solve for.
Differential equations that are most
commonly encountered in mathematical physics are either ordinary or partial.
An ordinary differential equation has only
one independent variable and its derivatives.
This is in contrast to the partial
differential equations that are functions containing more than one independent
variable.
This is how the Newtonian mechanics works
for such a physical system comprising of N particles in three dimensions.
But we I had told you earlier, the whole idea
of Lagrange was to develop a mathematical method that does away with vectors
and forces.
So let us consider how the mathematics of
Lagrange would work for this same system.
One thing is, as was told earlier, that it
completely does away with forces.
The next thing that it does is to bring in
the idea of energies, may it be potential or kinetic.
The most important or the key quantity that
this novel system depends on is the Lagrangian, which is a function.
You will often come across the term
‘Lagrangian function’ in Lagrangian mechanics and it is such a function that
summarizes the dynamics of the entire system.
It is a pretty cool stuff if you care to
think about it.
Since the Lagrangian function uses energy
as its working tools, its unit will be same as that of energy.
You need to understand that there is no one
specific function that can be termed as Lagrangian; any function that
summarizes the dynamics of an entire mechanical system is by definition a
Lagrangian function.
Having said that, one can still give a
general definition of Lagrangian function as L = T – V
Here T is the total kinetic energy of the
system and V is the potential energy of the system.
We shall continue our story of this amazing
Lagrangian function 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:
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recommend this large collection of Halloween Songs for Kids:
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