oscillations and the metric

In class, I was solving the normal-mode problem for a solid object near equilibrium, using generalized coordinates, in the usual manner. This starts by orthogonalizing the coordinates to make (what I call) the "mass tensor" (the tensor that comes in to the quadratic kinetic-energy term) proportional to (or identical to) the identity. This operation was annoying me: Why do we have to get explicit about the coordinates? The whole point is that the coordinates are general and we don't have to get specific about their form!

In my anger, I solved the problem without this orthogonalization. It turns out that this solution is easier! Of course it is: I can do everything with pure matrix operations.

I had two other in-class epiphanies about the problem. The first is that the solution you get when you don't do the orthogonalization is more analogous to the simple one-dimensional problem in every way. The second is that, in a D-dimensional problem with D generalized coordinates, the tensor that goes in to the kinetic energy term is some kind of spatial metric for a D-dimensional dynamical problem. (Or proportional to it, anyway.) That is simultaneously obvious and deep.

many-body systems; composite objects

Every time I teach mechanics (and this is something like the 21st year I have taught it at the undergraduate level) I learn something new. This week we are talking about many-body systems; I had two epiphanies (both trivial, but still): The first is that the description of the object in terms of a center-of-mass vector and then many difference vectors away from the center of mass (one per "atom") is purely a coordinate transform. Indeed, it is just generalized coordinate system that is related to the Newtonian coordinates by a holonomic transformation. Awesome! So when the Lagrangian separates into external and internal terms, this is just a result of the appropriateness of that transformation.

The second is that the definition of the many-body system is completely arbitrary. It should be chosen not on the grounds of being bound or solid or connected but rather on the grounds of whether choosing it that way simplifies the problem solution. Both of these realizations are simple and obvious, but it took a lot of teaching for me to get them fully. I am reminded as I realize these things that the physics concepts we expect first-year undergraduates to manipulate and be comfortable with are in fact pretty damned hard.

air resistance, again

I should stop complaining about air resistance, but I can't help myself! I am teaching this semester from Kibble & Berkshire, and in Chapter 3 there are problems about air resistance that use speeds of around 100 meters per second and an atmospheric drag law that is proportional to velocity to the first power. I don't think there is any physical system that could have these properties: If you are small enough to have viscosity matter, you can never go 100 meters per second. Well, I guess molecules can go that fast, but (a) that isn't what the authors have in mind, and (b) molecules aren't really well described by continuum mechanics!