May 27, 2018 Sunday
Bedtime Story
Breaking Down the Two-Body Problem
In fact, the two-body problem when reduced
two independent one-body problems, the scenario becomes even more familiar in
the form of Newton’s second law.
In case you have forgotten, it would be
worth going through this law once again which now may seem very primitive (after
general relativity) but still serves us wonderfully well in our Middle World.
Middle World quite curiously is a term
coined by the evolutionary biologist Richard Dawkins, introduced first in his
2005 TED talk held at Oxford titled: “Queerer Than we can Suppose: the
Strangeness of Science”.
Middle World is the habitat that we humans
live in, lying between the microscopic world of quarks and subatomic particles
and the cosmic world of giant stars and galaxies.
Newton’s second law states that the rate of
change of momentum of body is directly proportional to the force applied on it.
Moreover, the change in the momentum takes
place in the direction of force applied.
Mathematically, it is represented as:
F = dP/dt = d(mv)/dt
Now the two-body problem seeks to determine
the position of two point particles at various times that only interact with
each other.
In the real world or rather the real
universe, such two-bodies are often encountered in the form of satellite
orbiting a planet, or a planet or orbiting a star or even a star orbiting
around another star when the system is known as binary star.
So let us see how a two-body problem can be
broken down into two independent one-body problems.
So let us say that there are two point
bodies whose vector positions are given by x1 and x2 and
there masses are represented with m1 and m2.
So mathematically, the problem is reduced
to determining the trajectories x1(t) and x2(2) for all
times t given the initial positions x1(0) and x2(0) and
the initial velocities v1(0) and v2(0).
In this case scenario, the force acting on
the first body with mass 1 is only from mass 2 and hence we can label it as F12.
Analogously, the force acting on body 2 of
mass 2 is only from the first body of mass 1 and hence we can label the force
as F21.
This data is enough to apply Newton’s
second law on to this bodies which will give us the following two equations:
F12 (x1, x2)
= m1 
1
1
F21 (x2, x1)
= m2 2
2
I hope you noticed the two dots on top of
the letter x.
They have an interesting significance.
The xs with 2 dots on top represents the
second derivative with respect to time, which in other words means they
represent acceleration vectors.
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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