Brownian motion and G.P.Rajaratnam.

G.P.Rajaratnam iyengar (1908-1979) popularly known as Rajaratnam is a well known figure in Kannada literature. He has written many, many books, but he is best-known for his work "Ratnana Padagalu" (ರತ್ನನ ಪದಗಳು) --- life as seen from the eyes of Yendkuduka Ratna (ಹೆಂಡ ಕುಡುಕ ರತ್ನ) (Drunkard Ratna). Ratnana Padagalu has been popularized by well-known Kannada singers, and it is still appreciated and admired. In fact many kannda movie songs are taken from Rajaratnam's "Ratnana Padagalu".

In one of his writings he says there are many things one can learn from a drunkard except drinking. He says a poor drunkard having just enough money to buy a peg of liquor will offer a poorer drunkard standing outside the shop who cannot even afford that to share his peg but a man with all his senses will not share anything in life.

In another of his plays there is a very comical scene where a completely drunk man will be trying to figure his way home. He tells a passer by that "Brother, i have been walking for a very long time towards my home keeping a street light as my reference but after such a long walk the street light is still where it was when i started my walk, the light seem to be going away. Please help me". Comical indeed particularly if you watch this play. But there is an important mathematical concept hidden in this statement. (Long read from here).

Brownian motion is one of the very difficult mathematical concepts to understand. It all began with the english botanist Robert Brown (1773-1858) who in 1827 noticed that wild flower pollen grains suspended in water jiggled about under the lens of the microscope, following a zigZag path. How could these small tiny little things moved about, perpentually and erratically, by itself?

This is very complicated and only statistical mechanics provides an explanation.
Einstein understood that this happened because the suspended particles are being constantly hit by molecules in the sorrounding fluid. These molecules hit from all possible directions and change the path of the particles instantaneously at the moment of impact.

Mathematically , if T is the average time between two collissions, then in time t, the particle suffers N=t/T random collissions. If R is the distance travelled by the particle after N such random collissions , then R=(a){root(N) } [a x square root of N] where a is the average distance covered between two successive collissions. This is famously called the Einstein-Smoluchowski formula.

Atleast for me this is very difficult to understand. Thankfully G.P.Rajaratnam's statement quoted above is for our rescue.

Brownian motion is analogous to an ideal drunkard who has no memory of the previous step.(He has no idea about his relative position with reference to the street light) This lack of memory is the basis for Einstein-smoluchowski formula. Had the steps been corelated, then R=aN after N steps.

Consider each step of length 1 unit. i.e.a=1. If each step is corelated , meaning walker always follows the same direction, then after 10,000 steps, the walker would be 10,000 units from the starting point.
For a random walker, this does not apply. He has no idea of the previous step. He moves randomly. For an ideal drunkard, with a=1 unit, after 10,000 steps, the drunkard is R=1x{root(10000)}= 100 units away.

The drunkard has no destination nor can he return to the starting point. This is the cornerstone of irreversibility of physical system. Matter, or matter in motion exhibits this at every instance by virtue of thermal effects. This is the message that first came from einstein's paper in 1905.

Similarly in the trans-- state, high on alchohol, athma and paramathma (in a crude sense mind and body) collide to produce bodily motion that is random and irreversible with relation to the starting point.

Certainly as G.P.Rajaratnam said, there are many things to learn from the drunkard and one of them is Brownian motion.
Thanks to Rajaratnam, its a little easy for a layman like me to get an idea of Brownian motion.

I have not tried liquor to really experience brownian motion first hand. May be its worth it to give a try to experience brownian motion.

Bedke Neeche Rehnewali; Truly a masterpiece

Me and my Cousin Pavan did not have any work to do this evening so we decided to watch a play at Ranga shankara. When i went there to get tickets around 6:00 PM, i saw the poster that it was for kids aged 10+. I called Pavan to reconfirm if we were really so jobless to see a Play for kids in Hindi. He said we shall go. Also this was the first Hindi play we were watching at Ranga shankara.

When i asked for the tickets, i was given two tickets and asked to pay 300 rupees. I thought to myself that this play better be good since we are spending 300 rupees to watch a Hindi play for kids.

Pavan, for a change came to my house before time and we went to Ranga Shankara well ahead of schedule. So we decided to go to the theater and take a "Pretty" seat (near pretty people). But as usual Bad luck. Forget pretty people, there were hardly any people. Theater was just about half full.(Normally we go just in time and miss occupying "pretty" seats.)

The review said that anyone who had an invisible friend in their childhood is sure to enjoy this comic play. I never had an invisible friend but thoroughly enjoyed the play. It was truly a masterpiece. Acting by kids was exemplary, Music was outstanding. Story was superb. All in all fantastic.

The story is about Ira aged about 14 and her imaginary friend (in her childhood) Manu who lives beneath Ira's bed (Bedke Neeche Rehnewali) .The story starts from Manu crawling from beneath Ira's bed and reading Ira's personal dairy. Manu reads about Ira's boyfriend Gynanesh who is called roger by Ira. Ira's brother Chintu joins her and enjoys reading his sister's personal dairy. Then enters Ira totally annoyed by what is happening. She does not recognize Manu and complains to her mom that Manu is a stranger. The whole sequence is marvelously enacted by three kids and one must see the play to enjoy it. It is impossible for a bad writer like me to describe the scene.

Ira makes repeated complaints to her mother that Manu is a stranger and she does not know how she came into her room. But Manu repeatedly insists that she knows Ira and has been staying beneath her bed for a very long time. But Ira's mom does not take it seriously and in fact likes Manu. She leaves the scene asking the kids to play. Here Manu tells Ira about the latter's birthday, the night when Ira got scared about the ghost, about Ira's conversation with her mother about her career. Ira gets perplexed about how this stranger knows so much about her. Manu repeats that she is her friend and they used to share everything. Ira gets flummoxed and cries for her mother. Again Mother does not take things seriously.

Ira then gets scared of Manu and asks her mother to drive manu away. So Ira's mother(forgot the name) calls the police. When the policeman makes his entry into the scene, all the other characters will be dancing and the policeman is forced to join them. The policeman tries to convince Manu that she was lost in some stampede and her parents were looking for her. She refuses to believe. When Manu is dragged by the policeman, she escapes and goes under the bed and disappears.

Manu is an imaginary (abstract) character in this play. Ira will be sharing all her secrets with her imaginary friend manu in her childhood. As Ira gets older, like her toys, she puts Manu also beneath her bed and forgets her.

The play ends with Ira writing in her dairy about the next day's date with Gyanesh.As she pens down her excitement about the next day, she feels something missing. She remembers Manu and is filled with nostalgia. Manu crawls from beneath the bed and makes Ira comfortable and puts her to sleep.

As i said before, i am hopelessly incapable of writing about this marvelous masterpiece. Three kids Manu,Chintu and Ira performed out of the world. music was Exemplary.,Lighting outstanding and acting truly awesome.

With all my incompetence to write about an excellent piece of art, i have tried to give a gist of the play. Please do see it whenever you get an opportunity. Believe me its really fantastic. Don't miss it.

Interestingly, there were no kids to see this play. Me and my cousin pavan both at 26 were the youngest in the audience.

Making Logically Simple things mathematically complicated:

There are few physical laws that form the bedrock for engineering mechanics and design. One of which is the famous Pascal Law established by Blaise Pascal, a French mathematician in 1653.


The law states that “Pressure at any point in a fluid at rest has the same magnitude in all directions.”

This is one of the most fundamental laws in Engineering Mechanics. I don’t know how Pascal derived this law. But below I have shown how this law is derived in engineering text books and in engineering classes. Later I will show how simple is the law and how the engineering professors and textbooks have complicated the derivation. First read on how it is derived in engineering classes.

Consider an infinitesimal wedge shaped element of fluid at rest as free body. The element is arbitrarily chosen and has dimension as shown below.


Since the fluid is at rest there can be no shear forces, the only forces acting on the free body are the normal pressure forces exerted by surrounding fluid on the plane surfaces, and the weight of the element.
As the element is in equilibrium, the sum of force components on the element in any direction must be equal to zero. (Px, Py and Ps are normal pressures acting on the three faces of the wedge as shown and W is the specific weight of the fluid acting vertically downwards at the Centre of gravity of the wedge)
Therefore, Equilibrium equation in X direction is
Px(dy)(dz) – Ps (ds) (dy) Sin a = 0..................(1)

From the above figure, (dz)= (ds) Sin a ;

So the above equation reduces to
Px(dy)(dz) – Ps (dz) (dy) =0 ;

Dividing the equation by (dz) (dy)
Px – Ps = 0

Therefore:
Px =Ps………………………………………………(A)

Similarly, Equilibrium equation in X direction is

Pz(dy)(dx)–Ps (ds)( dy Cos a)–(W/2)(dy)(dx)(ds) = 0.................(2)

From the figure ;
(ds)( Cos a)= (dx)

Therefore the above equation (2) reduces to;

Pz(dy)(dx)–Ps(dx)(dy)–(W/2)(dy)(dx)(ds)=0.....................(3)

Since (W/2)(dy)(dx(ds) is the product of three infinitesimally small quantities ; this term can be ignored; So (3) reduces to

Pz(dy)(dx) – Ps(dx)(dy)= 0

Dividing by (dy)(dx)
Pz – Ps= 0

Therefore: Pz =Ps………………………………………………..(B)

Combining equations (A) and (B);

Pz = Ps = Px………………………………………………………..(C)

This is the standard derivation followed in all engineering reference books. Obviously most students follow the same derivation. (This also forms a standard 10 marks question in every fluid mechanics examination)

As such this does not look complicated. But even this much complication and mathematics is not required to derive this equation.

A basic property of liquid learnt in primary school tells that liquids do not have a shape and it takes the shape of the container. In that case, it is obvious that the liquid contained in the container must exert equal pressure in all directions (be it 3 or 16 or 10000 directions).
Put other way, if the walls of the container containing a liquid are removed, same volume of liquid flows out in every direction implying that the force exerted at any point in the fluid is same in all direction.

Most simply put, as the liquid takes the shape of the container containing it implies that force exerted at any point in the fluid is same in all directions. (This is a one mark question in 5th or 6th grade).

The point here is that, sometimes obvious things are made mathematically complicated by textbook writers and professors. Unnecessary sophistication is burdened on students, in the process also compromising their inquisitive thinking and creativity.

Swine flu panic

Swine flu panic achieved what terrorists could never and will never achieve. It has brought the city of Mumbai to a halt. The pandemonium created by the media and to a certain extent the people have brought Mumbai to a halt. No minor or major bomb blast in mumbai has created so much confusion and panic among the people.

Experts say that swine flu is just another strain of influenza. It is perfectly curable. Its just another flu. It has Medicines and the situation can out of control only if the patient's immunity levels are low. This is true with pneumonia or any other infection.

Whatever it may be, its stupid to close down cinema halls and schools because of this virus. What is likely to change if the schools are closed for 1 week or 10 days. The Virus will not take a break or evaluate the cost benefit ratio and move into the next city. Virus cannot and will not be eliminated in one week. Its ridiculous to shut public places. This will only add to the fear of the people and create more confusion.

Its high time that people and particularly media behave more pragmatically. They are simply blowing the situation out of proportion. There is a TB related death every 2 minutes in India. More than 1000 children die in India everyday from Vitamin A deficiency. Number is almost same for the number of diarrhoea deaths every day in India. Annually about 45000 pregnant women die because of under nutrition. About 7 million babies are born every year with iodine deficiency. More than 2 million babies have folic acid deficiency. None of this gets reported in the main stream media.

Media is obsessed with numbers. Headlines in most news channel today said "Swine flu toll up to 15" Another leading channel says "Swine flu meter running fast". This is ridiculous. Are the news channels competing in reporting deaths.

Even if they are, there are other deaths everyday in India. As i said before, they can report a case of malnutrition or under nutrition death every 5 minutes. They seem to be turning a blind eye towards this grave problem.

This is not the first strain to have hit the country not this will be the last. Media must be careful not to exaggerate the issue and cause panic among people.