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.


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......