I have been reading a really good book:

• Structure and Interpretation of Classical Mechanics (SICM)
  

by Gerald Jay Sussman, Jack Wisdom (and Meinhard E. Mayer). The authors teach an “unusual” graduate course at MIT using this book:

• Classical Mechanics: A Computational Approach

A few years ago I had an undergraduate course on Lagrangian Mechanics. Though I could derive the equations of motion of various mechanical systems, I confess that I did not understand the Principle of Stationary Action (aka: “Principle of Least Action”). After years using Newtonian Mechanics, the variational approach of Lagrangian Mechanics seemed both incredibly sexy and utterly mysterious: instead of forces and accelerations, one would think in terms of continuous paths that make the action functional integral stationary.

Truth be told, I never understood the principle. I could apply the principle and get good results, but I did not understand it. I used to think that my limited intellect was the only one to blame, but I found solace on the preface of SICM, where the authors quote Prussian mathematician Karl Gustav Jacobi (1804-1851):

In almost all textbooks, even the best, this principle is presented so that it is impossible to understand.

I am in really good company then!

Much to my enjoyment, SICM is a different kind of book: better than most books on Classical Mechanics. In fact, I would say that the book is a masterpiece, a joy to read, and that each single sentence is meaningful and profound.

-/-

Jonathan Yedidia’s posts on the SICM:

• Using Unambiguous Notation

• SICM on Mac OS X

• Algorithms for Physics

I suppose it would be no exaggeration to say that almost all physical laws are written in the form of partial differential equations (PDEs). I am very far from being an expert on numerical PDEs, but I have played a bit with the finite difference method (FDM) in order to simulate diffusion processes. The FDM framework is (conceptually speaking) rather straightforward: discretize space into a regular lattice, discretize time. However, implementation is not exactly straightforward, as all practitioners know.

However, there’s something I don’t like much about the FDM: it easily introduces artificial (i.e., non-physical) anisotropy (the same happens often with cellular automata). Let’s remember the most basic principle of symmetry:

Physics cannot depend on our choice of reference frame.

But when one uses the FDM, one must choose an arbitrary orientation for the regular lattice, which likely “destroys” some of the original symmetries. At some point, I became interested in the finite element method (FEM) framework, but unfortunately I never had the time to study it.

In Summer 2006, I had the somewhat crackpottish idea of modeling diffusion processes over irregular lattices. If such an idea were actually good, it was extremely likely that someone had already thought of it. I then talked with a friend who is a graduate student in Computational Mechanics, and his reaction to my idea was kind of “so what?”. I decided to put that “idea” aside and move on.

[ image courtesy of Erik Rauch ]

My interest in modeling physical phenomena using irregular discrete structures was recently revived when I accidentally found this paper by Erik Rauch:

• Discrete, Amorphous Physical Models, International Journal of Theoretical Physics, Volume 42, Number 2, February 2003 (available online).

Erik addressed the same topic on his M.Sc. thesis (MIT 1999). Since I am not an expert on Computational Physics, I can’t judge the novelty (or lack thereof) of Erik’s ideas. What I do know is that his paper’s approach is quite atypical: it seems to be more Computer Science than Numerical Analysis or Computational Physics.

The paper’s abstract:

Discrete models of physical phenomena are an attractive alternative to continuous models such as partial differential equations. In discrete models, such as cellular automata, space is treated as having finitely many locations per unit volume and time is discrete, whereas continuous models (e.g. Schroedinger’s equation, and most field theories) specify detail down to infinitesimal spatial and time scales. But all existing discrete models depend critically on a regular (crystalline) lattice, as well as the global synchronization of all sites.

We should ask, on the grounds of minimalism, whether the global synchronization and regular lattice are inherent in any discrete formulation. Is it possible to do without these conditions and still have a useful physical model? Or are they somehow fundamental?

What Erik proposes is rather bold. Basically, he speculates on the existence of discrete physical models which:

• do not require regular spatial lattices.
• do not require time synchronization.

In other words, Erik’s idea is to look for minimal discrete physical models: models that allow one to simulate physical phenomena, but with a minimum number of “structural constraints”. This raises some rather exciting possibilities! Nevertheless, a sexy possibility is not necessarily feasible. Note that in Physics:

Beauty does not imply Truth, though Truth usually implies Beauty.

OK, just for kicks, let’s dare to consider the possibility that some physical process can be modeled according to Erik’s paradigm. Erik included in his web page a few animations depicting waves propagating in 2-dimensional domains. All videos are very cool, but I particularly like this one depicting a plane wavefront colliding with a denser circular object (an example of refraction):

Please do note that the video displayed above was downloaded from Erik’s website. I took the liberty to upload it and (obviously) claim no authorship.

Some quick comments:

• Erik was a graduate student in Prof. Gerald Sussman’s group, whose main focus is Computer Science, not Physics. I would dare to say that Erik’s discrete, amorphous paradigm resembles a distributed computing approach to Physics.
• Prof. Gerald Sussman was one of the creators of the amorphous computing paradigm.
• The “discrete” part is clear. What does the word “amorphous” mean?! In my opinion “amorphous” means non-crystalline: an amorphous lattice is non-regular. I would guess that the term “amorphous” comes from the field of Condensed Matter Physics.

Erik’s approach has some interesting properties:

• locality: each “node” in the lattice interact only with its neighboring nodes. The lattice is, in general, represented by a non-regular graph. The irregular lattice should introduce less anisotropy than the regular one. How is that possible? Note that locally there’s anisotropy, but globally that anisotropy might average out. Isotropy may thus emerge from an irregular discrete structure.
• no synchronization: each node updates its internal state in an asynchronous manner. To be precise, according to Erik’s approach, nodes update their internal states synchronously within a cluster of nodes: all nodes in the same cluster update their state in a synchronous fashion.

However, Erik’s paper has some flaws as well. From a mathematician’s point of view, the paper is somewhat heretic and sinful: it presents no rigorous proofs! Indeed, the paper is quite empirical and “experimental” in the sense that its approach is kind of “hey, we simulated this with an irregular lattice and it seemed to agree with analytical results”. In my humble opinion, the paper does not answer some fundamental questions:

• which physical phenomena can one model using Erik’s approach? Would it work for phenomena other than diffusion and wave propagation?
• would it work for 3-dimensional lattices? Would it work for -dimensional lattices?
• what kind of structure should the irregular lattice have? How can we build such an amorphous lattice? What algorithms should we use?

These seem to be unanswered questions. I would love to talk with Erik Rauch about this topic, but unfortunately that won’t be possible. Sadly, Erik died on July 13, 2005 in a hiking accident in Sequoia National Park, in California. He was 31 years old. Here’s a rather brief biography.

-/-

Remark: if you are interested in distributed numerical methods, you might want to take a look at Prof. Dimitri Bertsekas’ book: Parallel and Distributed Computation (free download!). Still on physical models, a more well-known discrete paradigm is cellular automata; you might want to read Prof. Tommaso Toffoli’s papers on the topic. Last but not least, if you know of papers related to this topic, please let me know. Thanks!

-/-

Disclaimer: Please note that I am NOT a professional physicist. I am just a passionate (yet harmless) amateur. I am fully aware that what I have written in this post might be inaccurate. Moreover, please note that this post is admittedly speculative. If you are a professional physicist, please bear with me.