## Archive for the ‘Math/Physics’ Category

### Mechanics In One Dimension

Friday, August 7th, 2009

Finally I post something about physics. Here are notes from lectures on mechanics to freshmen at the IISER-TVM. I didn’t have time to edit it. Any corrections ( spelling mistakes, algebraic errors etc.) are welcome. It usually takes me a few iterations to get everything right.

### India Gets Spy Satellite From Israel

Saturday, March 21st, 2009

So that is the payoff. A few months ago India launched a spy satellite for Israel, using its PSLV rocket. So why take the risk of doing a favor for Israel, when the political situation in South Asia is so inflammatory?

Now we know. Indian satellites lack the Synthetic Aperture Radar that can see through clouds and at night. The Mumbai terrorist attack highlights the importance of being able to track small vessels in the Indian ocean and to watch terrorist training camps within Pakistan. The two countries can fill the gaps in each other’s capabilities. And then there is the whole enemy of my enemy thing happening also.

It is not something either side wants to talk about much: (more…)

### A Random Pursuit

Saturday, December 1st, 2007

A missile is launched when an airplane is directly overhead, at a height $h$. The missile moves at a constant speed $v$ always heading directly towards the aircraft, which is moving along a straight line at constant velocity $u$. What is the shape of the missile’s trajectory? How far will the aircraft fly before it is hit by the missile? What if the plane takes evasive action by randomly changing its direction, but heading in the original direction on the average? What is the best strategy for the missile to maximize the probability of a hit?

### Complex Time in Quantum Tunneling

Sunday, October 21st, 2007

Perhaps the most spectacular early prediction of quantum mechanics was tunneling: that particles can do things that are forbidden in Newton’s mechanics, although with a small probability. (more…)

### The Geometry of Thermodynamics

Monday, October 8th, 2007

Thermodynamics is the study of heat. Originally developed to understand steam engines and such, it led to a revolution in physics. It showed that time has a preferred direction. Also, that physics is not fully deterministic: the best we can do for large systems is to predict averages of physical quantities and probabilities of events. But with the even greater revolutions of quantum mechanics and relativity that happened soon after , thermodynamics lost some of its original wonder. Nowadays it is thought of a staid old field, barely taught in physics departments anymore ( except as a preparation for a Stat Mech course). This is a pity, because thermodynamics is perhaps the most remarkable of all physical theories. We have none other than Albert Einstein vouching for this1: (more…)

### Reduction or Emergence

Wednesday, September 5th, 2007

Earnest Rutherford used to say that all science is either physics or stamp collecting. This could have been a dig at the biologists of his time, who were still collecting samples and classifying species. He probably would have thought more highly of modern molecular biology, which is a lot like his physics in outlook: everything is determined by the DNA. It is said that Rutherford’s worst insult for a student who had done something stupid was–Chemist. The chemists had the last laugh though: Rutherford was awarded the Nobel Prize not in Physics but in Chemistry for having achieved the transmutation of elements.

Should we understand the world bottom up or top down? Which is the proper scientific view? (more…)

### Matrix Diagonalization by Sampling

Saturday, September 1st, 2007

It is hard to think of a problem more ubiquitous than the diagonalization of a matrix.I will discuss today a statistical approximation method for finding the eigenvalues of a symmetric matrix. (more…)

### Macroscopic Fluid Mechanics 3: References

Friday, June 1st, 2007

Part 1 Part 2 SIAM Talk
A recent review,closest to our point of view :

### Macroscopic Fluid Mechanics 2

Tuesday, May 29th, 2007

Part 1
To understand the origin of this non-commutativity, let us again consider the example of a hurricane. It is an extended object, whose radius is of the order of 100 km. It wouldn’t make sense to have two such objects within a 100 km of each other: the two hurricanes will interact strongly with each other and combine into a single one. (This phenomenon of a `reverse cascade’ can been seen clearly in some simulations.) Thus there is a limit to the resolution of the co-ordinates of a hurricane, given by the area of a hurricane. This is reminiscent of the uncertainty principle of quantum mechanics, except that there it is the co-ordinates of phase space that is fuzzy: the analogue of the area is Plank’s constant.
(more…)

### Macroscopic Fluid Mechanics 1

Tuesday, May 29th, 2007

Introduction to Talk at SIAM DS07 Conference Snowbird Utah May 28th-June 1

The equations of motion of a fluid are obtained by averaging over the equations of motion of the large number of molecules that occupy even a small volume: we are not for the most part interested in the details of the motion of individual molecules. The equations of a fluid so obtained ( Euler or Navier-Stokes) are quite different from those of particle mechanics, being partial differential equations. Nevertheless the fundamental symmetries (translation and rotation invariance) of particle mechanics are preserved in this reformulation.
The conservation laws (energy, momentum,angular momentum) are preserved for ideal flow (Euler). In the next approximation, the effect of the transport of these conserved quantities to molecular scales are incorporated (viscosity). Even higher order corrections from molecular scales can be incorporated ( Chapman and Enskog) but are rarely needed.
(more…)