Rajeev’s Almanack

A missile is launched when an airplane is directly overhead, at a height . The missile moves at a constant speed always heading directly towards the aircraft, which is moving along a straight line at constant velocity . 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?

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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…)

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…)

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…)

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…)

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

B. Khesin Topological fluid dynamics. Notices of the AMS, 52:1
(2005), 9-19
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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.
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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.
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Terrence Tao has made some deep observations on why the regularity of three dimensional Navier-Stokes is such a hard problem. Although he has gone on to many other equally interesting topics, I remain fascinated by his main point there: that Navier-Stokes is supercritical. The nonlinearities become stronger at small distance scales, making it impossible to know (using present techniques) whether solutions remain smooth for all time. Thus, it is crucial to understand the scale dependence of non-linearities in fluid mechanics.
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It appears that the instability that we described in the last article is indeed new; at least not known to experts such as Persi Diaconis and Richard Mongomery. At their encouragement I describe in more detail the effect of small pertrubations on a coin. We can look at this as a problem in Riemannian geometry or one in Classical Mechanics. Each brings its own insights and informs the other. Let us begin with the geometric point of view.

The Riemannian Submersion

In the orthogonal basis used above,

where we choose units such that

. Since is a Killing vector whose orbits are circle, we can quotient by it to get a manifold . We work in the covering space of the orthogonal group for convenience.

There is an induced metric on ; it sectional curvature is given by O’Neill’s formula for Riemannian submersions ( Petersen p. 58)

Thus the induced metric on the sphere is the standard one. In fact in this point of view is the circle bundle of . But it is not the circle of unit tangent vectors; their length is .

Geodesics on do not go project to geodesics in the base. The bundle carries a natural connection; the horizontal vectors are orthogonal to the vertical vectors. Note that this connection is independent of : the horizontal component is the same no matter what the radius of the fiber is. Thus the induced curves on the sphere are not geodesics, but are the same as what one would get for an isotropic rigid body.

The fact that may not be one only affects the lift of these curves to the total space; that is where the negative sectional curvature can come in. When we are `stretching’ the vectors in the vertical direction. If this stretching is too large, it leads to an instability.

More coming…

Classical Mechanics

Landau Lifshitz sections 33 and 35 is a standard discussion of the rigid body.

The Lagrangian is, in Euler co-ordinates,

This leads to the hamiltonian

and are conserved. So is the total angular momentum:

The equation for is not affected by the fact that . This is what we saw geometrically also.

More coming …

Tags: coin+tossing,+mathematics,+physics,+probability

The standard example of a random event we give in probability courses is the tossing of a coin. On the other hand, in mechanics courses we teach that rigid body mechanics is one of the few examples of an integrable system. Not only are its equations deterministic, there is an explicit solution in terms of elliptic functions. In other words, a rigid body is one of the most predictable of all mechanical systems.
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