Noether's theorems/Explainers/Groups of continuous motions

Mathematical counterpart: Lie groups and Lie algebras.

A Lie group and its algebra as the tangent at the identity

Noether's theorems rest on a body of work by Sophus Lie, and she cites him three times. This page covers only the part her proofs actually use.

Order matters, and the difference is the point

Take a globe. Turn it a quarter turn about the vertical axis, then a quarter turn about an axis pointing at you.

Now start again and do the same two turns in the other order.

The globe ends up somewhere else. Rotations do not commute, and you can check this with any object to hand.

The difference between the two outcomes is itself a rotation. That is not obvious and it is the fact everything is built on: doing two motions in each order and comparing gives you back a motion of the same kind, not something new.

Why that is surprising

Each of the two sequences involves doing something twice. You might expect the comparison to be a second-order thing — twice as complicated as a single turn.

It is not. When you subtract, the complicated parts cancel exactly, and what survives is a plain single motion.

The cancellation happens for a reason you already know in another form: it does not matter which way round you take two derivatives. Slope-of-the-slope is the same either way. That symmetry is what kills the complicated part, and it is worth knowing because it is the entire technical content of the result.

A closed family

A collection of continuous motions is called a group when doing two of them in a row lands you inside the collection.

Combine that with the fact above and you get the structure Lie discovered. Take your smallest motions — one for each knob, in the language of the earlier page. Compare any two in both orders. The result must be one of your smallest motions again, or some combination of them.

That single requirement is enormously restrictive. It fixes how the motions can fit together, and the pattern of how they combine is a complete fingerprint of the symmetry. Two symmetries with the same fingerprint behave identically no matter what they are symmetries of.

How Noether uses it

At one place, and she uses it as a test rather than as a fact.

Proving the reverse direction of her first theorem, she has recovered a family of transformations and needs to know whether they form a proper closed family. Her argument runs by contradiction: if they did not, then comparing two of them in both orders would produce a motion outside the family. But her scoring rule survives that motion too — so it would have to contribute another conserved quantity, and she would end up with more than the count she started with. That contradicts her assumption, so the family is closed.

She also relies on a result of Lie's guaranteeing that a family of smallest motions of a certain kind always belongs to a genuine family of full-sized ones. She cites it by theorem number and page, which tells you she was using it carefully rather than in passing.

What this page leaves out

Almost all of Lie theory. The classification of the possible fingerprints is a major piece of mathematics and neither theorem needs any of it. If you want that, go to a book on the subject; this page exists only so that Noether's three citations mean something when you meet them.

How much symmetry can a thing have

The pattern of how motions combine is a complete fingerprint. Two symmetries whose motions combine the same way are, for every purpose in this subject, the same symmetry — even when the objects they act on have nothing in common.

That is why a single piece of mathematics covers rotating a globe, rotating a compass needle, and rotating the internal properties of a particle, none of which resemble each other. They share a fingerprint.

It also puts a ceiling on things. The fingerprint has to be consistent with itself: compare three motions in every possible pairing, and the results have to agree. That requirement is severe enough to rule out most patterns you might invent, and the surviving ones have been catalogued.

For this volume the practical consequence is the count. Ask how many independent continuous symmetries a situation has and you get a definite number — three for rotations in space, one for a rotation of a pair, four for the coordinate freedom of spacetime. That number is what both of Noether's theorems return results against, so getting it right is the first step in applying either of them, and the fingerprint is what makes it a matter of fact rather than of counting by hand and hoping.

No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.