Noether's theorems/Explainers/Why the moment map resists plain language

Noether's theorems: map
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Related Hamilton's principle On Physical Lines of Force

Mathematical counterpart: The symplectic form of the theorem and the moment map.

This page is partly an explanation and partly an admission. The two cases you can see, you can see plainly. The general construction you cannot, and this page says what stands in the way rather than pretending otherwise.

What can be said plainly

The previous page set out the double life of every measurable quantity: it is a number, and it is also an instruction for moving things.

Read that backwards. Start with a motion — sliding sideways, turning — and ask which quantity issues that instruction.

For sliding, the answer is momentum. For turning, angular momentum.

The rule that goes from a motion to the quantity issuing it is called the moment map. In these two cases it hands back exactly the conserved quantities the first theorem gives, arrived at by a different road.

Both cases can be checked in a line, and the counterpart page checks them.

What cannot

Beyond those examples the subject becomes geometry, and the geometry does not survive translation.

The general question is whether the rule exists at all. Given a symmetry acting on a space of states, is there always a quantity issuing each instruction? The answer is: not always, and the obstruction to its existence is a genuinely geometric object with no everyday counterpart. It is not that it is hard to say simply. It is that the thing itself is a statement about the shape of a space, and there is no shape in ordinary experience it resembles.

There is a further question about whether the rule can be chosen to respect the symmetry's own internal structure, and the answer there involves the way a symmetry acts on its own description — a construction with, again, no everyday analogue.

This page could produce sentences about both. They would not transmit anything. A reader who came away from them believing they understood the moment map would be worse off than one who came away knowing they did not.

What the ceiling costs, which is nothing

Worth saying plainly: nothing else in this volume depends on the general construction.

Both of Noether's theorems are proved without it. Every conserved quantity in the standard cases comes out of her argument directly. The moment map is a second, more elegant route through territory this volume has already crossed on foot.

One thing that is not true

You will read that the moment map is Noether's theorem in another language. It is not, and the difference is worth carrying even from this page.

The moment map lives in a framework built out of states: where the system is now, and how fast. Noether's theorems are about histories: whole stories with a beginning and an end. And her results hold for configurations that are not solutions at all, which a framework built from actual states has no way to talk about.

Where both apply, they agree. That is a statement about an overlap, and it is less than an identification.

What the general construction is for

It is fair to ask why anyone builds the general machinery if the two cases you can see already give the familiar answers.

Two reasons, both practical.

Coordinates. Writing momentum and angular momentum down the ordinary way requires choosing coordinates first — which direction is sideways, which axis you are turning about. The general construction does not. It produces the same quantities as objects that do not depend on the choice, which matters as soon as the space of states is not a flat sheet of paper.

Reduction. If a system has a symmetry, some of its description is redundant, and you would like to work with a smaller description that has the redundancy removed. The moment map is the tool that does the removing: it identifies precisely which part of the description the symmetry is moving around, so it can be quotiented away. This is how the two-body problem gets reduced to a one-body problem, and the general version of the same move handles far harder cases.

Neither of these is a result about Noether's theorems. They are results about working with symmetric systems, and the moment map is the standard tool for the job. This volume names it, checks it in two cases, and stops there.

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