The physics of the familiar

Dear Mahadevan,

Had a look at your Web site.... Nice work, but a thought crossed my mind...

It's been about two decades that I have been reading attractive reports about the following kinds of recent extensions to mathematics through a somewhat new kind of mechanics and physics: (i) catastrophe theory (ii) nonlinear systems (iii) dynamic instability (iv) chaos (v) fractals.

However, if an engineer wants to understand these topics, and so, if he takes a look at the shelves of a decent library or a bookstore, all that he can actually find are only those few popular-science titles---not textbooks as such. Further, most such treatments are written in a modern (or is it post-modern?) style: they serve to confound the reader more than to enlighten him. In fact, I find that invariably, they are addressed to those souls who carry an extremely poor method of mental functioning---chaotic. The Web page that I once mentioned here at iMechanica is an exception, but this article is just an introduction; it is not comprehensive enough to tackle even chaos as a full topic in its own right. (Then there are some other, more advanced kind of books, but mostly on the mathematics of nonlinear differential equations.)

Against this backdrop, I have a few questions:

1. Why are there no good text-books on this range of topics. (Compare, for instance, the first two decades of FEM *after* it had been reinvented by engineers in the late 1950s. You already had a dozen texts by late 1970s.)

2. If you had to design a good course that was actually understandable to a fresh graduate of a typical Indian university what would it be like? I would appreciate it if you can give detailed points: more detailed than a typical syllabus is. Thus, apart from the syllabus some explanatory notes about the depth and the range of the course should also be included.

3. What online course materials and tutorials would you recommend to the same student?

4. Could you design a course for the final year undergraduate student of engineering? If yes, what would it look like?

Before you answer, please note a few points:

(a) The student is assumed to be an engineering graduate, say, civil or mechanical engineering, not someone with science as his major.

(b) I deliberately said Indian universities. Please do not design your answer for an American university student (or the copycats of that system, namely, the IITs). Unlike the US schools, Indian students have to cover far greater breadth and therefore may lack depth, esp. in certain modern topics. An American professor can easily say: "My student did so and so in these A B and C areas," but the same student may not possess even a surface knowledge of any of the areas from D through Z. For instance, ask the typical biomechanics majoring mechanical engineering undergraduate of a US univeristy to just name the societies that design/maintain standards for use in industries like foundries, bearings, and pumps. All that you will get is a quizzical look at least half the time---of incomprehension, almost---on his face. 12 time-zones apart, a mechanical engineering graduate of an Indian university would be expected know about such things, but, now, this Indian student may not be aware about the qualitative theory of differential equations to a great extent (i.e. to an extent significantly beyond Kreyszig or similar books), unless you are in an IIT. The issue is not necessarily competence (or lack thereof), but the areas in which it is systematically developed by the educational system that the student finds himself in. Accordingly, please understand the difference before you answer.

(c) Your course work should prepare the student enough that he can develop an informed kind of understanding about your research (though, there could be some work still left to be done before he becomes productive in a group like yours).

Thanks in advance for answers from Mahadevan and/or managers.

BTW, teachers from other countries, e.g. UK or France or China or Brazil, may please feel free to respond and share their experiences too.

I am honestly interested, and frankly speaking, honestly sick of finding that chaos still is considered a "new" topic by so many people, esp., engineers, and worse, those half-baked science reporters. (I don't mean any adverse implication for the reporter of this issue of this Harvard newsmagazine, though; I was just talking in general terms.)

-- Ajit