Reading the theorem backwards

Mathematical counterpart: The converse of the first theorem.

Noether proved the theorem in both directions. Textbooks give one. This page is about the other one.

The forward and the reverse

Forward: there is a symmetry, therefore there is a conserved quantity.

Reverse: there is a conserved quantity, therefore there is a symmetry.

Both are in the 1918 paper. The forward direction gets a few lines; the reverse gets a whole section, because it is harder.

Why the reverse is the useful one

Footprints in snow.

The forward statement is: a person walked here, so there are footprints. True, and not much use — you already knew about the person.

The reverse is: there are footprints, so a person walked here. That is the one you use, because footprints are what you actually find.

Physics is in the same position. Conserved quantities are what turns up. Something holds steady when you expected it to drift, and you want to know why.

The reverse direction says: there is a symmetry, it is not a coincidence, and here is how to find it. Without the reverse, a conserved quantity is consistent with a symmetry and evidence of nothing.

How the proof goes

Backwards, and that is not a joke — it is the actual structure.

Noether takes the forward argument and runs each step in reverse. The conserved flows come first, they are combined back into the single relation the forward proof had produced, and from that relation the transformation is reconstructed explicitly. Not shown to exist. Written down.

Then one more thing has to be checked, and it is not automatic: that the transformations you have recovered form a proper family — that doing two of them in a row lands you inside the family rather than outside it. Noether checks it.

Where she stops

There is a case she cannot settle, and she says so.

When the transformation depends on speeds as well as positions — the situation on the prolongation page — one possibility remains that her argument does not eliminate. Her text says the matter must remain undecided.

This volume reports that rather than smoothing it over. The reverse direction is not unconditional, its author knew it, and a modern account that states it flatly has added a certainty she declined to claim.

One thing the reverse direction settles for free

Early in the paper Noether makes an assumption to keep things manageable: that turning a knob twice as far does twice as much, at least for the smallest steps. It looks like a real restriction.

The reverse direction shows it is not. If the knobs behaved otherwise, you would get relations of the same kind, just more of them — and feeding those back through the reverse direction returns you to a symmetry whose knobs behave simply after all. Nothing was lost by assuming it.

This is a small point and a good example of what the reverse direction is for. It answers a question about the setup that the forward direction cannot reach.

A conserved quantity nobody had a reason for

The best illustration of why the reverse direction earns its keep is a planet.

A planet orbiting the sun conserves energy, because time is uniform. It conserves angular momentum, because space has no preferred direction. Both of those were understood.

It also conserves a third thing — a quantity that points along the long axis of the ellipse and keeps the orbit from slowly rotating. This was noticed in the eighteenth century and had no explanation for a very long time. There was no obvious symmetry to blame. Rotations were already used up on angular momentum.

The reverse direction says there must be one anyway. And there is: a symmetry that does not move the planet around in space at all, but mixes together the different ways of describing where it is and how fast it is going. It is invisible if you only look for symmetries that move things about a room.

That is the reverse direction doing exactly what it is for. Somebody found a conserved quantity, could not account for it, and the theorem said: there is an account, keep looking. There was.

The full story is on the Runge-Lenz page.

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