Noether's theorems/Explainers/True everywhere against true for real motions

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
Sources Sources Bibliography Chronology Further reading
Related Hamilton's principle On Physical Lines of Force

Mathematical counterpart: On-shell and off-shell.

On-shell and off-shell as nested regions

Two statements have been in play in this volume and they are not the same statement. This page separates them. It is the least glamorous page here and probably the most useful.

Two kinds of true

A company's books satisfy a rule: what the company owns equals what it owes plus what the owners hold. That is true of every set of books ever written, including fraudulent ones, badly kept ones, and ones for a company that does not exist. It is true because of what the words mean.

The same company made a profit last year. That is also true. But it is true only of this company, in this year. Change the facts and it stops being true.

The first is an identity. The second is a claim about the actual case.

Noether's theorems produce the first kind. Conservation laws are the second kind.

What the theorem actually delivers

The first theorem's output is an identity. It says a certain combination of sensitivities — the leftovers from the nudging page — flows without accumulating.

This is true of any configuration you care to name. A chain hanging in a shape no chain would ever hang in. A planet on an orbit that violates gravity. The statement holds because of how the scoring rule is built, not because of what the system is doing.

Then you add one condition: the system is obeying its equations of motion. The sensitivities go to zero, the left-hand side of the identity vanishes, and what is left says the flow balances.

That is the conservation law, and it arrived at the second step, not the first.

The trade

The identity is true of everything and says less. The conservation law is true of real motions only and says something you can measure.

Neither contains the other. You get the second from the first by adding physics. You cannot get the first back from the second, because the second says nothing about the configurations you threw away.

Physicists have names for the two situations. On the actual solutions is on-shell. True regardless is off-shell. Noether has no such words; she just says what she is doing, which is passing to the variational problem.

Three reasons to keep them apart

The reverse direction needs the identity. Running the theorem backwards starts from the identity and reconstructs the symmetry. Start from the conservation law alone and there is no route back.

The non-isolated case needs the identity. When something outside is feeding the system, the sensitivities are not zero — they equal whatever is being fed in. That case is available only because the underlying statement never assumed otherwise.

The second theorem never makes the crossing at all. Its output is an identity too, and adding physics does not turn it into a conservation law. It turns it into a statement about the equations themselves.

That third reason is the important one. A reader who has only ever met the first theorem in its on-shell form has nowhere to put the second theorem, and will reach for the nearest available shape — a conservation law — which is the wrong one. Almost every mis-statement of the second theorem starts here.

The habit worth forming

When you meet a claim in this subject, ask which kind it is. True of every configuration, or true of the real one?

Noether was careful about this on every page. Most accounts of her are not, and the carelessness is not visible in the formulas. It is visible only in what people then believe the theorems say.

How to tell which one you are being told

A practical test, for reading anyone's account of this subject.

Find the statement. Then ask: is this true of a configuration that breaks the laws of physics?

If yes, it is an identity. If no, it is a claim about actual solutions.

Now check whether the author told you which. Most do not, because in the first theorem's case the two look almost identical on the page — one has some terms on the left, the other has zero there, and everything else is the same.

If an account never distinguishes them, it has silently chosen the second, and its reader has no way to reach the second theorem from there.

A note on the words

Physicists say "on-shell" for true of actual solutions and "off-shell" for true regardless. The words come from an unrelated corner of particle physics and are not intuitive.

Noether has no such words. She simply says what she is doing: she passes to the variational problem, meaning she now assumes the system is doing what the scoring rule says it should. Her way of putting it is clearer, and it is a small loss that the modern terms replaced it.

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