### Linear Deficit

I’m teaching advanced mechanics, for juniors and seniors, this semester. And, once again, I’m driven bonkers by one of those educational peculiarities of UT (at least, I *think* it’s peculiar to UT).

I’m lecturing about damped, coupled harmonic oscillators (surely, the most basic problem in all of advanced mechanics), and I say, “OK, by introducing the velocities as independent dynamical variables, we can write Newton’s 2^{nd} Law as a system of coupled 1^{st}-order linear differential equations.” We can cast those equations in matrix form

where

and $B$ is a certain $2n\times 2n$ matrix. The solution is

and I then go on to explain that we can compute the exponential of a matrix if we know how to diagonalize it. *Blank stares.* And we can diagonalize $B$, if we know its eigenvectors and eigenvalues. *Uncomfortable rustling.* “OK,” I ask this collection of junior and senior physics majors, “who knows how to find the eigenvectors and eigenvalues of a matrix?”

Slightly less than half the class raises their hands.

*Whoa!*

The problem, you see, is that, unlike most places, where linear algebra is bundled into 2^{nd}-year calculus, the UT Math Department has unbundled it into a separate course. And, nowhere in the curriculum of the UT Physics Department is that course listed as a prerequisite. Thus students can hit my course or, G-d forbid, the Quantum Mechanics course, without so much as a passing acquaintance with linear algebra.

Having taught both courses before, I was *prepared* for this debacle. A brief review of finding the eigenvectors and eigenvalues of a matrix ensued. And then we exponentiated $B t$ and found the general solution to the coupled harmonic oscillator problem.

Finally, we got to case where some of the eigenvalues of $B$ coincide. In this case (called “critical damping” in the case of the 1D harmonic oscillator), I explained that $B$ *cannot* be diagonalized. Instead, the best we can do (but still good enough for exponentiating it) is to put $B$ in Jordan Canonical Form.

“So, who knows what Jordan Canonical Form is?”

No one raised their hand …

Posted by distler at January 29, 2005 1:49 AM
## Re: Linear Deficit

Jacques,

I know the feeling. I am teaching basic mechanics to engineering students at the University of Ottawa and some of them never saw some concepts of linear algebra and calculus. In Ontario, the provincial Government chopped one year of the high school curriculum two years ago to save money… What a bad mistake! Students can get into science and engineering without a single high school physics course and without a full calculus course…

Cheers!

Charles