Functional Relativity, Symbolic Geometry, et al

Bryan Newbold, bnewbold@mit.edu
http://web.mit.edu/bnewbold/Public/sicm-fall08.html

Informal Background

For the fall of 2008 I'm very interested in investigating gravitation and other physical theories using functional programming techniques. I find that formalizing physical systems into a computer model is the best way to solidify my understanding of the system; using functional languages and techniques makes the conceptual wall between mathematical abstraction and programming implementation much lower; the result is a more reusable and general model well suited for experimentation and exploration.

I am planning on getting my undergraduate physics degree in spring 2009, for which I will need a thesis. I am hoping to develop skills and tools this fall with which to accomplish Real Live Science over IAP and in the early spring.

The stimulus for this course of study was the class Classical Mechanics: A Computational Approach taught by G. Sussman and J. Wisdom at MIT. I had trouble with the later sections of the book/course and am hoping that now with an eta of math under my belt I can chip away at it.

Potential Fall Projects

Integration of mit-scheme and scmutils into Sage (yes)
The Sage math system is an open-source alternative to Mathematica, Maple, etc. It provides an easy to learn html notebook interface (as well as command line) and is bundled with a plethora of high performance libraries (like PARI, GMP, MAXIMA, SINGULAR, see this list).
A number of other packages (including common lisp) already have interfaces based around a fake TTY device; this should be easy with mit-scheme. Or a more complete object-style interface could be implemented. There is documentation for writing interfaces here and here
There is a public demo server at sagenb.org, but it's usually slow. Try this server instead (user: ableseaman, password: bottlerum, if you don't want to fill out the form). Sage has been used in math classes at MIT already; Tim Abbot is working on "debianizing" the whole system, after which it should be on Athena.

Exploration of "higher order dynamics" (possible)
I'd like to play with systems involving "higher order dynamics", aka {jerk, yank, snap, crackle, pop}. These dynamics have become interesting to cosmologists?
See arxiv one, two, other chaotic pdf.

General Relativity Simulations: compact bodies, inspirals, precession (possible)
Should talk with Lee Finn @penn, pranesh@mit? Go to mki journal club.

Modified Newtonian Dynamics (possible)
MOND was originally proposed to explain the galactic rotation curve problem; it has been extended as a relativistic field theory as TeVeS (Tensor-vector-scalar gravity, described in 2004).
I think it would be interesting to implement and play with MOND or other alternative gravitational theories in a symbolic computation framework. Assumptions could be checked quickly and easily (eg, behaves like X in the short distance limit, behaves like Y in the high stress-energy limit). The process of formalization could also be a good test; if the theory can't be coded, is it a valid theory? Would also demonstrate that programming tools are general and can be used to explore non-physical theories.
See also Henon-Heiles.

Action Minimization Problems (possible)
Minimization of action over path integrals is a classic hammer in the physics toolbox (everything looks like an oscillating nail). It might be fun to play with some old classics like optics or Ohm-ic resistance.

Basic Quantum Mechanics (unlikely)
Methods with Wilkson-Sommerfeld quantization? I don't know enough QM to go beyond simple, introductory quantum systems, but might be interesting.

Quantum Computation (unlikely)
There is already extensive work done here; see http://tph.tuwien.ac.at/~oemer/qcl.html

Resources

The SICM text book is free online; so is the SICP book.
There is an unofficial SICM mailing list.

Papers to read? (download)