Introducing Noether’s Theorem in Conceptual Physics

With little advanced notice, I was assigned a section of conceptual physics to teach this semester (because second semester astronomy was cancelled due to low enrollment and replaced with a section of conceptual physics with the same low enrollment..go figure). After teaching Matter & Interactions for over a decade and after much enlightenment I have completely changed my views on what the focus of introductory courses should be, I decided to modernize the content by beginning with special relativity and particle physics as presented in the corresponding chapters in Hewitt. I also decided not to present content with the traditional chapter-by-chapter approach and instead to tell a story, with the chapters from Hewitt to be used as reference reading. For special relativity, I gave the class the same handout from Arons that I use in my calculus-based course, but only for the historical information and for the superbly articulated operational definitions of simultaneity and measurement.

So after a non-traditional introduction, we then turned to more conventional material. I knew I wanted to take a systems approach a la Matter & Interactions, but I also felt weirdly obligated to do something with motion, and we did. However, this time, we used Vernier motion sensors and handheld LabQuests (dammit Vernier…you’re changing my attitude toward experimental physics!) and I dare say I’ve never seen a class have such fun with matching position and velocity curves! That by itself was a breakthrough for me.

But then, with only a few weeks left in the semester (this was a late start course), I had to decide how to bring chapters two through ten in Hewitt together coherently, and I decided to do something I’d thought about for years: introducing Noether’s theorem as the foundation for introductory physics. If you don’t know what Noether’s theorem is all about, start with this paper by Hill and Lederman (for an overview) and this paper by Hanc, Tuleja, and Hancova (for undergraduate intro level approach).

So I began by defining symmetry as it’s used in physics: something that remains unchanged when Nature allows some process to happen (that definition is of course subject to improvement). We discussed reflection symmetry, of which most students are already aware and then I demonstrated rotational symmetry with a nondescript blue beach ball. The ball looked the same regardless of how I rotated it. But then I said some really strange things (as I’m known to frequently do). I noted that the motion experiments we did previously could be done today, a different day, and should we expect the results to change just because we did the experiments on a Thursday compared to a Tuesday? Everyone agreed that we should expect the same results. I then noted that when we did the motion experiments, there were three groups, each working in a different part of the room, and each group doing the same experiment with the same instructions and with the same apparatus. Should we expect the results to vary just because the groups are working in different locations? Everyone agreed that this is nonsense. Finally, I noted that our class meets in a room with walls that face each of the four cardinal directions but we could just as well hold class in a building or in a room for which the corners, not the walls, face the four cardinal directions. Should we expect different results from our motion experiments in a room that’s rotated compared to our current room? Everyone once again agreed that that would be a silly expectation. Then I hit them with this: each of these three scenarios displays a symmetry of nature. Experiments are independent of the time, location, and classroom orientation. It’s that simple. Read that again because it really is just that simple. Read it once again to make sure you see how simple this is.

Okay, so what? Well, Noether’s theorem formalizes this into a simple statement:

For each symmetry, there exists a corresponding conserved quantity. 

And there it is. Getting the same experimental results at different times means energy is conserved. Getting the same experimental results in different locations means momentum is conserved. Getting the same experimental results in differently oriented rooms means angular momentum is conserved. Ack! What do these words mean?

Then at this point, rather than focus on energy, momentum, and angular momentum, I focused on invariant, constant, and conserved. Beginning a course with special relativity inevitably introduces the concept of invariance (something being the same in different reference frames), but be careful! Invariant, constant, and conserved have different meanings. Constant simply means “having a value that doesn’t change.” Conserved means “having the same value before and after something happens, like an interaction.” I mentioned this in a previous post. There is still one piece missing, and that is the concept of “system.” I defined system to be “the part of the Universe that you conceptually isolate for study.”

Okay, so then we sat down in the middle of the room with four cue balls (purchased in a clearance sale for one dollar each at a local sporting goods store). I defined the system to be “two balls” and I rolled them toward each other. We’d previously discussed vectors (the language of the arrows as I call it) and everyone got the idea of a zero initial momentum (we’d previously defined momentum too). With very little effort on my part, everyone saw that for this system (both balls, oppositely directed initial momenta with equal magnitudes) had the same total momentum throughout the interaction (collision). Of course we ignored friction from the floor. Then I redefined the system to be just one of the cue balls, and everyone agreed that the new system’s momentum wasn’t conserved. BOOM! Newton’s second law appears! Then we rolled the cue balls into a bowling ball and looked at what momentum conservation looks for unequal masses. Great! Then I pulled out my rotating platform, rolled the bowling ball toward it, and we saw that angular momentum (“spinniness” for rotational angular momentum and “aroundness” for translational angular momentum) were conserved for certain choices of system just like ordinary (translational) momentum. I was asking various check questions all along, and I couldn’t stump the class. Next week, we’ll address energy conservation and with that, we will have stitched together all the ideas in Hewitt’s chapters two through ten with one single, but exceedingly powerful, foundational concept (this is a CONCEPTUAL course for crying out loud, right?) that exists in all aspects of contemporary physics. No inclined planes were necessary.

Not surprisingly, momentum, energy, and angular momentum are concepts that all predate Noether’s theorem. These quantities were well defined decades (centuries even) ago and can certainly be operationally defined independently of each other, and that’s basically how classical physics treats them. But here we see quite explicitly that these quantities are the “Big Three” for a reason, a deep underlying reason that forms the bedrock of all contemporary physics but one that students rarely ever hear of. There’s a reason Matter & Interactions focuses on the Momentum Principle, the Energy Principle, and the Angular Momentum Principle (okay, and the Fundamental Assumption of Statistical Mechanics, but whatever…). That reason is Noether’s theorem.

I intend to use this same approach next year if I’m assigned a section of conceptual physics (as I’ve requested), but here’s the thing. This could also be done in the introductory calculus-based course too! Noether’s theorem makes a superb preface to Matter & Interactions and to Six Ideas That Shaped Physics (Moore’s innovative text). And yes, it even gives us a reason to bring Lagrangians in to the intro course! Just do it!

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