I’ve taught Theoretical Mechanics before using Standards-Based Grading, but I think I want to make some tweaks to the focus of the standards. Here’s the old list. In this post I want to get my current thoughts down about where the big areas should be. Note that the work Danny Caballero has done at the University of Colorado has helped me with these thoughts a lot (though there the course is joined with math methods and spread over two semesters).

Ok, here’s my list of big ideas, that I hope, once refined, will be the main categories of standards for the course (they’re numbered for ease in referral in the comments).

- Differential equations
- Equations of motion are very compact ways of describing physical laws
- I think students should understand the form, solutions, approaches, visual representations, and numeric work surrounding these
- Typical introductions to this include air resistance trajectories and oscillators

- Calculus of variations
- I did this very early last time, and I think it worked out. Having the Lagrangian approach as an early tool comes in handy for oscillations, gravitation, etc.
- I usually only do Lagrangian, not Hamiltonian, as I’m still unconvinced that it provides a tool that can do a lot more than the Lagrangian approach, given the level of abstraction needed to learn it.

- Momentum is king
- Last time I enjoyed the class-wide conversation we had about whether momentum and kinetic energy are both needed to describe motion. This is really a focus on Newton’s laws.

- Oscillations
- damped, driven, etc.

- Central force potentials
- Kepler’s laws
- I don’t like that students don’t know an ellipse when they see the mathematical evidence for it, but it’s a tour-de-force of physics.
- I love Hohmann transfer problems.

- Systems of particles
- usefulness of center of mass
- conservation issues

- Noninertial frames
- Rigid bodies
- Linear algebra is not a pre-requisite for this class, so I’ve got some awkward work arounds for this.

- Normal modes
- Linear algebra is not a pre-requisite for this class, so I’ve got some awkward work arounds for this.

The linear algebra thing is really a problem for some of the cool stuff that happens with rigid bodies and normal modes, but I’m pretty happy with my normal modes solution, and slightly happy with my rigid body approach.

One notable absence is gravitation as a main topic. It’s a chapter in the main books I’ve looked at, but I like putting the focus on general central force stuff.

Any thoughts would be greatly appreciated.

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About Andy Rundquist

Professor of physics at Hamline University in St. Paul, MN

Here’s one big idea that I think encompasses some of these: advanced mechanics is the story of r(t), v(t) and a(t) (or you could replace v(t) and a(t) with p(t)). What I found last quarter is that students were familiar with r, v, and a, but not as functions of time that could be manipulated or that formed the basis of other functions like the Lagrangian and Hamiltonian.

I disagree with your thoughts on H; I would never stop at L. In my experience, the Hamiltonian is much, much more useful than L for working with anything but the simplest problems. Moving away from second order equations to sets of first order equations is a big win both conceptually (e.g. phase space) and practically (numerical solutions). It also ties in nicely with QM, of course.

I really like the r, v, and a approach you’re suggesting here. The idea that they now need to be considered functions is very important, and perhaps can be considered part of my “differential equations” section.

As for L vs H, here are 2 issues: 1) 2nd vs 1st order doesn’t matter if most of what you’re dealing with needs computational approaches, and 2) the connection to QM is not something I have time for in this course. Do you think that connection should be done in a single course, or have the students see the two ends of the connection in two different courses?

For computational purpose, maybe Mma doesn’t care, but most methods I’m familiar with are for sets of first order equations. Is that not true? Maybe it’s just my bias because of my background in dynamical systems, but I really think that getting to H buys you a lot.

I don’t spend much time on the connection to QM, but most of are students take QM and CM together (or in close succession), so I think seeing H in a second context (and seeing q and p on equal footing) is very valuable.

Yeah, I should have been more clear: Mathematica doesn’t care what the order is when numerically integrating differential equations.

I’m still not convinced that the H connection between classical and quantum mechanics is done well in either course. What book do you think takes a stab?

For example, here’s how Griffith’s QM book introduces the Hamiltonian:

“They (solutions to the TI-SWE) are states of difinite total energy. In classical mechanics, the total energy (kinetic plus potential) is call the Hamiltonian.”

Eventually things like commutators are developed, but I don’t think there’s really a strong connection with the Hamiltonian approach in classical mechanics.

“Chaos in Classical and Quantum Mechanics” by Martin Gutzwiller? 🙂 (http://www.amazon.com/Chaos-Classical-Quantum-Mechanics-Gutzwiller/dp/3540971734)

You mean of undergraduate texts? None. But that’s not what I’m trying to say. Seeing H and q and p in two different contexts lends something of a spiral approach that would otherwise be absent.

I would say something similar about finding principal axes or normal modes and the connection to eigenvalues and wavefunctions in QM. I don’t spend a lot of time on it, but I do point out that the same sort of stuff goes on in QM.

I’m also interested that you don’t do the Hamiltonian. While it may not be crucial for the classical mechanics course as you teach it, I think tend to think it is valuable to expose students to the idea of the Hamiltonian before they hit quantum mechanics. I like students to see different pieces of a concept in different courses as it provides an opportunity to make connections, and revisit what they have learned before.

I really don’t like the way that Thornton, Marion deal with the Hamiltonian. It doesn’t really motivate it other than to point out when things are conserved, and it definitely doesn’t give you much purchase on the slope to understanding the quantum application.

Have you looked at how Taylor deals with the Hamiltonian? It’s much better IMO.

I’m embarrassed to admit I haven’t because I don’t have that book. How far down the path to QM does it go?

One point in favor of the Hamiltonian formalism is that it takes you to conjugate variables, and those lead to the link between physical symmetries and conservation laws — Noether’s theorem. And, philosophically, that’s pretty heady stuff. Even if you don’t “do that” in “detail”, throwing it in as a forward-reference lecture bit can help students appreciate that the physics they’re learning is connected to bigger, deeper things that they (hopefully) look forward to learning some day.

My very wise mentor and former teacher, Bill Gerace, likes to say that every course he teaches is 20% review (of stuff they should have learned before but probably don’t really get as well as they should), 60% new material they should get, and 20% forward-reference material that they won’t really get this time around but will get much better in a future course for having seen it now. Honor, don’t hide, the dangling ends at the edges of your course!

I do like the connection with conjugate variables, especially when connecting to the Heisenberg Uncertainty Principle. I also like the 20-60-20 idea. As usual, I’m trying to bring focus and efficiency to my class. I’m so happy I use my blog now instead of blank sheets of paper for these brainstorming sessions.

I also like the idea of introducing the Hamiltonian in mechanics, even if it’s not fully grasped. You can use Poisson brackets to write out the algebra of classical observables and to make the connection to quantum mechanics. The Poisson bracket makes it easy to find how observables change in time: $\frac{\mathrm{d}\mathcal{O}}{\mathrm{d}t} = \{f, \mathcal{H}\} + \frac{\partial\mathcal{O}}{\partial t}$ It’s interesting that quantization is like choosing a the “quadratic” part of the algebra. Hamiltonian mechanics also has so many beautiful connections to the symplectic geometry of phase space. There’s the Hamilton-Jacobi equations.

Utility? Maybe not so much.

Yeah, this is deeper than I would usually choose to go in this course, Brian. I’m really starting to wonder if I should do more of this in quantum, though I haven’t taught that in a while.

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