Rajeev’s Almanack

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 Jaataka tales are a collection of parables about the 500 lives of the Buddha until he achieved Nirvana, salvation. After that there are no more re-incarnations. The stories proceed from simple morality tales in which the Bodhisatva ( the soul of the Budha) was alive in the body of a lower life-form: a rabbit, an elephant and so on. Until he attains human form and the stories get more sophisticated. Various versions of these stories have been told and retold over many generations all over the Eastern World.
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Most of what we know of the Spartans is from their arch-rivals, the Athenians. So we have to be a little skeptical of what we hear. Still, we know that they were a city state that was dominated by a small tribe of warriors. They had an underclass of agricultural workers,the helots. These were descendants of the messenians whom they had subjugated in earlier wars. The young men and women of Sparta were separated early on. The men received military training. Women received education as well, unusual in ancient societies. Spartan way of life was austere, based on a system of honor that emphasized valor above all else. They are most famous for the battle of Thermopylae, in which a small band of Spartan braves fought off the invading Persian army.
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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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Living in Rochester, one hears a lot about how science is done at Xerox and Kodak. Xerox was a little better at it, but their experience was still rather painful. It looks like the company has recovered from some of its stumbles and has posted a reasonable record in recent years.
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Mira Nair’s movie `Namesake’ is about a man with an odd name
(Gogol) for an Indian. I have my own situation to deal with.
My name is usually written as Sarada G. Rajeev.
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Vedaanta is the end of all knowledge. End as in goal, or as in the ultimate kind of knowledge. It is a theory of what knowledge itself is. What practical use could it be? Volumes have been written on how to translate the abstract concepts of Vedanta to every day life. The ultimate authority in `modern times’ (only about a few hundred years ago) is Sankara Acharya. His Vivekachoodaamani and Bhajagovindam are attempts to explain this most abstruse of all branches of classical Indian philosophy to the masses; or at least to laymen.
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