June 30, 2018 Saturday
Bedtime Story
Foundations of Lagrangian Mechanics
Tonight we shall continue with the last
four reformulations of Mach’s principles.
Statement 8 – If you take away all matter,
there is no space.
Statement 9 – Ω = 4ϖρGT2 is a
definite number, of order unity, where ρ is the mean density of matter in the
universe, and t is the Hubble time (which is inverse of Hubble’s constant and
rounds up to 14.4 billion years)
Statement 10 – The theory contains no
absolute elements
Statement 11 – Overall rigid rotations and
translations of a system are unobservable
With this I shall end my digression on
Ernst Mach and return back to Lagrangian mechanics.
Now we have enough background material to
discuss the foundations of Lagrangian mechanics.
Any moving mechanical object, for instance,
a swinging pendulum that has always been the subject of study for the classical
physicists (and a favorite of quite a many examiners in many competitive
entrance exams world over), can be tracked very accurately using Newtonian
mechanics.
Newtonian solution of moving objects
involves time-varying constraint forces that keep the body in the constrained
motion.
Such forces include reaction force exerted
by the wire on the pendulum or it could be the tension on the swinging rod.
Lagrangian mechanics does away with
tackling forces on such moving objects.
It considers the path the particle (bodies
are generally assumed as particles for simplifications) can take and then picks
out some independent generalized coordinates that would suitably characterize
its motion.
Generalized coordinates are not exactly the
type of coordinates that you study in Cartesian geometry; they allude to
certain parameters that describe the configuration of the system with respect
to some reference configuration.
These parameters satisfy mathematical
constraints such that they represent a specific manifold/space.
This may be confusing to you very
understandably since to most of us Euclidian space is how we perceive the
world.
Even in our daily lives knowing that the
Earth is a sphere, for all practical purposes on our small local scale, it is
flat.
This makes our planet Earth a topological
manifold which can be generalized to be flat locally in spite of being a sphere
factually.
Manifold is a mathematical equivalent of
the physical space but it is a more general term implying topological space.
The concept of manifold allows our mind to
look into space in a way that is not intuitive to our evolved minds since for
its survival and procreation it never had to consider any more dimensions than
three.
To explain this idea in a more visual way,
we will need to consider a big thick book of the beautiful Bartholomew Atlas
and a globe of the earth often seen in many plush offices often accurately
depicted with the axial tilt.
I shall proceed to give you a fairly-intuitive
satisfactory understanding of the concept of the manifold 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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