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.

parallax distance

The parallax distance to an object is the distance you get by moving your (single) eye's position by a transverse distance x, measuring the angular displacement θ of the object given that move, and dividing the transverse distance by the angle.

Problem-set problem: You are outside after the rain late in the day and you see a rainbow. What is the parallax distance to the rainbow?

(Thanks to Andrei Gruzinov at NYU.)

vector subleties

It is vectors all week this week, in my class, and in two classes I have taught for others. It is understandable that they confuse students, even physics majors with good backgrounds. Here are some subleties that I like to point out:

• Vectors have a magnitude and a direction, but that is not sufficient. They also have a coordinate-free existence or description, and they form a linear space (with the usual linear operators). In this sense, despite what every textbook says, the unit vectors that define the coordinate system are not vectors!
• Although vectors carry around all this geometric baggage, they have a magnitude and a direction and nothing else. I can still confuse the physics majors by sliding around vectors on the board. There is no position associated with a velocity vector, and we confuse the students by always drawing the velocity as coming from the object that is moving.
• Multiplication of a vector by a scalar is usually conceived as changing the magnitude of the vector, which it does, but it also changes the units, in many cases of interest (for example when a displacement is multiplied by an inverse time to make a velocity). So it often produces a new vector that is not longer than the original vector, nor shorter, but really incomparable.
• There is a perfect symmetry between the relationship between velocity and position and the relationship between acceleration and velocity. However, it is far harder for students to understand that the acceleration vector can point perpendicular to the velocity vector than it is to understand that the velocity vector can point perpendicular to the position vector. No amount of class time spent on this point is wasted, in my experience.