Suggestions for Projects (with and
without references)

Students who wish to receive a grade
in this course are expected to do one of the following:

Hand in a certain amount of homework

Present in the Morse Theory Seminar

Read some additional material related to the Solitons Course and give a presentation on it (either in class or at some other time). Remember if you are interested in this last option, you are urged to get together with other students and work in groups of two or three. These days, most mathematical research is done in groups....If you have an idea for a project, or are already working in some area, you certainly may choose your own project. Everybody's lives will be easier if this is done before the last week of the semester. You should also realize that the minimal requirement is a half-hour presentation. It is hard to keep a presentation on any material that short, so you are not expected to master a lot of material, unless you want to! The references with a * can be borrowed from my office.

Possibilities:

1. Shock waves, traffic
flow and rarefaction waves. The
notes by Palais just mention this topic. There are more details in
the book "An Introduction to the Mathematical Theory of Waves " *
by Roger
Knobel (AMS/IAS subseries, Student Mathematical Library,
Vol 3).

2. Split-stepping.
In the notes, Palais mentions a numerical procedure for solving KdV.
A suggestion for those interested in numerical methods is to look up the
two references (Tappert and Strang) and understand how this procedure is
actually carried out. The ambitious might want to construct a KdV
solver....

3. The n-body (or 3-body) problem: A lot can be learned at any level from going through arguments about the conservations laws and the Hamiltonian system for the n-body problem. Some suggested references (the trick is to choose the one which suits your background and intuition):

Singer, Stephanie: Symmetry in Mechanics, A Gentle Introduction
(Birkhauser, Boston)

* Siegel and Moser: Lectures on Celestial
Mechanics (Springer Verlag) Chapters 5-8 (out of 40!)

*?Guillemin and
Sternberg, Variations on a Theme by Kepler , AMS Colloquium Publications
42 (This is formidable).

I am told that Feynman's lectures
are a very good reference for this in physics....and maybe you have your
own favorite?

4. The Kovaleskaya Top ;This is discussed in Arnold, Mathematical methods of Classical Mechanics, Springer Verlag. You can't have my copy, I might need it. See also chapter I.1 and the beginning of Chapter III of Michelle Audin, Spinning Tops, * A course on integrable systems, Cambirdge Studies in Advanced Mathematics.

5. The Maxwell-Vlasov
system; section 1.5-1.8 Marsden
and Ratiu, Introcudtion to Mechanics and Symmetry, *Springer
Verlag.

This discusses the Poisson Bracket
and flows for an important and iinteresting fluid system at the level which
we talked about field theory February 18-20.

6. A particle in a rotating Hoop; This is a cute unusual problem in mechanics I have only seen in Marsden and Ratiu.*

7. The equivalence of Lagrangian and Hamiltonian Mechanics. We aren't doing this, and it is important...or one could just expllain what Lagrangian mechanics is....most books on mechanics have this....

8. Elliptic integrals and solitons for the periodic KdV...

There is lots more coming, of course......