Symmetry that can differ from place to place
Mathematical counterpart: Groups depending on arbitrary functions.
Noether's second theorem takes a different kind of symmetry from the first. This page says what kind, and separates it from two things it gets confused with.
One clock, or a clock each
Give a class of thirty students one stopwatch between them and a shared signal. Everyone starts at the same moment. If the experiment they are timing does not care what that moment was, you have a symmetry — the kind on the earlier page. One knob: when the signal went.
Now give every student their own stopwatch and let each start whenever they like. Thirty different starting times. Then a hundred students. Then a student at every point of the room, so that "when did you start" is answered not by a list of numbers but by a picture — a value at every location, which you can draw however you please.
If the physics still comes out the same, you have the second kind of symmetry. Physicists call it a local symmetry, because the choice is made afresh at each place, and a symmetry of the first kind global, because one choice covers everything.
A hundred knobs is not the same as a free choice
This is where the distinction usually gets blurred, so it is worth being blunt.
Adding more knobs to a global symmetry never turns it into a local one. Thirty students with thirty stopwatches still gives you thirty numbers. A thousand students gives you a thousand. However many you add, you are still handing over a list.
A local symmetry hands over a shape. You are not picking numbers off a list; you are drawing a curve, and any curve will do.
Noether is explicit about this. She puts the case of infinitely many numbers in between the two, as its own category, and notes that her first theorem still applies to it. Her second theorem does not. Infinitely many numbers is not what the second theorem needs.
Why does the difference matter so much? Because of one move the proof makes. With a free shape, you can choose it to be busy in the middle of a region and to fade to nothing near the edges. A number cannot do that. A number is the same everywhere by definition. That one move is what the second theorem's proof turns on, and it is unavailable no matter how many numbers you have.
How fussy is the rule
There is one more thing to count.
Some rules care only about your choice. Others care about how quickly your choice changes from one place to the next — and some care about how quickly that rate changes, and so on.
Noether keeps track of how far this goes, and the answer feeds straight into the theorem's output. A rule that cares only about the choice gives a simpler result than one that cares about how the choice varies.
So the theorem needs two counts: how many free shapes there are, and how deep the rule looks into how they vary. The first fixes how many results come back. The second fixes how complicated each one is.
Two examples, and one warning
Noether gives two before she proves anything, which is a habit worth copying.
Tracing a curve. Draw a curve on paper. You can describe it by saying where you are at each moment as you trace it — but you were free to trace it fast, slow, or with a pause halfway. The curve is the same. Your freedom to choose the pace is a free shape, and it is a local symmetry.
Choosing coordinates. In general relativity you are free to label the points of spacetime however you like, and you can relabel differently here than there. Same freedom, one shape for each dimension.
The warning. It is easy to come away thinking the second theorem is about general relativity. It is not. Noether names general relativity as an example of a theory with this kind of symmetry, and near the end of her paper she constructs another such symmetry with nothing physical attached to it at all, precisely to show that the physics was never the point.
Where this goes
Every force in modern physics is described by a theory with a local symmetry. That was not known in 1918 — the idea arrived in the 1950s. The second theorem was waiting for it.
What it says about such theories is on the next page, and the answer is not a conservation law.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.