Layman's version of Noether's Theorem (or the intuition behind it)

As part of a science project, I have to give a presentation to my classmates about a topic of my choice (within some constraints) - and I chose symmetry, and it's importance in physics. One important aspect that I think must be talked about is Noether's Theorem.

Now, of course the statement of the theorem itself is very impressive, but I feel like it lacks intuitional backing. I could go up there are start talking about infinitesimal transformations of fields, but even that gets very complicated very quickly (and the mathematics can't be over an Algebra 1 level or I'll completely lose them).

What I'm instead thinking is taking one specific example of a system, and showing that if some symmetry is true, then something else is conserved.

But I can't think of any examples. Is there any specific example of a system where it is easy to explain why translation symmetry implies conservation of momentum in that system, or some other symmetry implies some other conservation? If not, how can I go about demonstrating Noether's Theorem in an intuitive way to a class of 8th graders?

Explaining Noether's theorem to 8th graders may be hard, as even the simplest special case of the theorem requires some calculus to state. However, if we're talking about why one example of an invariance corresponds to one example of a conservation law, there are some calculus-free arguments you can make. I'll mention one example, energy (whose conservation law is a little different from the others); hopefully others will follow.

Imagine if the laws of physics or parameters therein varied over time; imagine, for example, if gravity became stronger tomorrow. Today, lift an object onto a shelf, which requires some work equal to the GPE it's gained. Lower it to its previous position tomorrow, when gravity is stronger. The GPE loss will exceed the GPE gain today. Boom! You've made energy out of nothing.

• Sorry for posting my answer without reading yours carefully. Mine said basically the same as yours so I deleted it. – Diracology Apr 11 '17 at 21:06
• That shows that energy conservation implies time invariance of gravity. It doesn't show that time invariance of gravity (and whatever else is needed) implies energy conservation. – immibis Apr 11 '17 at 23:44
• I'm not sure if that example works. What's to say you can't account for the change in GPE as part of whatever process causes the change in gravity? What if the change in gravity were caused by, say, the planet passing near a black hole? The amount of work needed to lift the object may change in that case, but not because any energy has been created or destroyed. – aroth Apr 12 '17 at 3:34
• @immibis You're right (unfortunately). Though this gravity argument is popular, I'm not sure Noether can be simplified to the desired level without a sleight of hand. I find that is a recurring problem when explaining physical ideas. – J.G. Apr 12 '17 at 7:12
• Time symmetry means the laws don't change with time as long as nothing else is changing with time, i.e., you can't be passing a black hole. We're just trying to prove time symmetry implies energy conserv'n, other things held constant. – murray denofsky Apr 19 '17 at 4:46

The example I like to use for lay people is based on an air hockey table (or pool table). If I hit the puck it moves in a straight line with constant speed. Why? The answer is - because the table is flat. Stated in terms of symmetry, the table is invariant under translations. If the table weren't flat, the puck wouldn't move in a straight line.

Now, getting the analogy with physics in general for a non-flat table requires finding a way to make the force holding it to the table always perpendicular to the table, but that's a detail I haven't fully worked out.

• That's an example of symmetry, but how is it an analogy for Noether's theorem? – immibis Apr 11 '17 at 23:45
• The fundamental content of Noether's theorem is that symmetries lead to conservation laws. In this case, the things conserved include the direction of motion and speed (ie momentum). – Sean E. Lake Apr 11 '17 at 23:48

protected by Qmechanic♦Apr 11 '17 at 18:11

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