Noether’s Theorem is something I wish was introduced a lot earlier than it ever seems to be: I think a simple, restricted, but still enlightening form of it can be introduced as soon as students have a basic mastery of derivatives and the idea that a derivative of zero means the function is constant with respect to that variable. You don't need to introduce the idea of generalized coordinates to give a flavor of how powerful this notion is.
IIRC when I studied physics I think they tried to tell us that there was something profound about the fact that time-invariant systems conserve kinetic energy and space-invariant systems conserve momentum, … during the first year, but I don't think anyone was mindblown. Then during the second year we started learning about generalized coordinates and Lagrangians and I think we did learn the Noether theorem then (probably just about one semester later).
However I think I really fell in love with it when I understood gauge invariance.
In general I think you can always try to tell your students things a little early to impress them, but it rarely works until they can work out the math themselves. I remember our professor in introductory quantum mechanics saying something along the lines of: "and now you see why quantum mechanics is just Markov chains in imaginary time", but until another professor showed us the Wick rotation in the context of path integrals nobody really appreciated that even if it could have been in our grasp earlier.
I guess this has come out way denser than I meant it to be, but my point is that you can always try to introduce something a little bit earlier, but you'll often find that your students don't want to learn some diluted crap, they want the real deal!
I didn't learn to appreciate Noether's theorem until my second semester in graduate school. I learned the math in undergrad and knew if you did fancy derivatives, you could obtain a conserved current. I thought it was just a way to get the conservation laws for cheap (kind of like how you got the equations of motion from Lagrangians for cheap, as was my understanding at that point).
Then, I learned about trying local U(1) symmetry and trying to make the Langrangian invariant under that and wham, you get EM for free, without even postulating it a priori. I was amazed--all of a sudden, I realized the reason why all these particle physicists were so bent on postulating this or that symmetry, hitherto not seen, might exist in nature; I realized how fundamental symmetries were and how deeply intertwined with conservation laws they were.
Though I didn't find Scott Aaronson to be as clear as the professor who finally showed us the Wick rotation. Unfortunately he passed away a few years back right after having missed being awarded the Nobel prize by an inch.
Eugene Hecht introduces it conceptually at the beginning of his algebra-based College Physics book. It's nice to capture the attention of the women in the class with this person (woman) of enormous import that "nobody ever heard of." Bravo professor Hecht!
Here's my favorite way of thinking about Noether's theorem:
In physics, we have to assume the laws of nature are the same everywhere and for all time. Otherwise, we're wasting our time. Because of Noether's theorem, this means there are some conserved quantities. Whatever these happen to be is what we _call_ momentum and energy. If the laws of physics change (they are still the same everywhere, just modified based on an improved theory), then we have to change our definitions of what is conserved.
If you define kinetic energy as mv^2/2, then when you take into account relativity, you realize that's not conserved any more. However, because the physical law that relatively predicts still does not change over time, then _something_ must be conserved. So we call that energy instead. Until the next iteration, of course.
I actually think it may be a little more prosaic than that.
In order for us to have a scientifically verifiable theory, or at least a consequence of a theory that is scientifically verifiable, that theory/consequence must be invariant under translations in space, time and rotation. Otherwise there would be no repeatability of any related experiment.
As such, by Noether's theorem, that theory/consequence must conserve momentum, energy and angular momentum. At the very least.
So any theory that didn't conserve these quantities couldn't be scientifically verifiable. So it's sort of a tautology that scientific theories conserve these quantities.
It is a matter of perfectly ordinary empirical fact that we live in a universe that, so far as we can tell, is translationally, temporally and rotationally invariant. There is absolutely nothing in Noether's theorem that forbids us from having theories that violate momentum, energy or angular momentum conservation if they happen to describe a universe that has aspects that violate those invariance conditions.
There have been theories that do so, like Dirac's weird large number thing. These are completely legitimate theories, and are entirely subject to absolutely ordinary observational, experimental and inferential verifiability (which is why we know Dirac's large number thing is likely false.)
Yes indeed. However, for us to be able to scientifically verify those theories, something about them must be invariant in space, time and rotation. That's pretty much the definition of a scientifically verifiable theory.
My point isn't that there can't be theories without those properties nor that the universe necessarily has them (or not). Just that, if it doesn't, we couldn't verify it scientifically.
According to Feynman there once were people saying that since experiments need to be repeatable, to be scientific the laws of physics must be deterministic, by pure logic. Oh, quantum mechanics? Oops.
Note that Noether's theorem does not apply to the universe as a whole. Expanding universe is not invariant under shifts in time. Energy is not conserved in expanding universe.
... and therefore, anyone peddling a perpetual motion machine must be scientifically illiterate or believe that their audience is, because what they claim to have goes against all of the physical theories which get verified every time we launch a rocket, build a bridge, or throw a damned baseball.
I think we should teach a lot more things using the basic framework of "baloney detection": Use what you know to prove that this claim is either bogus or potentially true. Show your work.
Or maybe what the theorem really tries to say is that the universe is self-consistent in time and space - it does not consist of multiple, fractured domains.
Conversely, a violation of NT might indicate just such a fracture.
She was also very instrumental in the discovery of the functorality of homology in algebraic topology, which led to the explosion of homological algebra. She was definitely a great polymath
Now, who can explain the Noether's theorem binding conservation law with an invariant better than "Structure and Interpretation of Classical Mechanics"? :)
When these periodic "let's put a woman on a bill" come up Noether is my choice for the subject. Her work had a huge influence on the shape of the world today and few people outside physics and mathematics have heard of her.
And she's dead and has a name, which is good enough for the Department of the Treasury. That's the only legal restriction I could find on who may appear on US currency and securities [1]:
United States currency has the inscription “In God We Trust” in a place the Secretary decides is appropriate. Only the portrait of a deceased individual may appear on United States currency and securities. The name of the individual shall be inscribed below the portrait.
See the FAQ [2] at the Bureau of Engraving and Printing site for some interesting information on how the current choices came about.
Noether made important contributions worth talking about, but I'd reserve the "changed the course of physics" qualification for Einstein and those oh his level.
Precisely. She didn't "change the course of physics" at all. Nothing in her work was a massive paradigm shift. She didn't shape the field for decades to come. Einstein did that, with special and general relativity. Him, Planck, de Broglie and all that crowd did that with quantum theory. Noether's theorem is beautiful, powerful and amazingly profound, and she definitely deserves way, way more recognition than she gets. But it's wrong to say that she "changed the course of physics".
> Nothing in her work was a massive paradigm shift. She didn't shape the field for decades to come.
I disagree. Quantum field theories are often expressed as a collection of symmetries. The ever-successful Standard Model is:
SU(3) × SU(2) × U(1)
Everything else follows because of Noether's theorem. And by "everything", I mean, "every phenomena in the universe that we are aware of except for gravity".
Well, to be fair, you actually also have to put in some numbers, too: masses, mixing angles, coupling constants, and the Higgs vacuum expectation value. (See https://en.wikipedia.org/wiki/Standard_Model#Construction_of... .) But, yeah, that symmetry description is incredibly powerful.
I have no idea about physics but Noether(with Artin) certainly changed the mathematical landscape. She was probably the first person to really do modern algebra(building on Dedekind) and is the first person whose works really seem modern.
Here's a few simple examples: http://www.sjsu.edu/faculty/watkins/noetherth.htm
Nothing about this requires math beyond undergrad calculus.