Noether's theorems/Explainers/Nudging a path and seeing what changes

Mathematical counterpart: The Lagrange expressions and the first variation.

Hang a chain between two hooks. It settles into one shape. Why that shape and not another?

The nudge

Give the chain a small push somewhere in the middle and let it go. It comes back. Push it anywhere at all, by any small amount, and it comes back to the same curve.

That is the test. The real shape is the one where a small nudge anywhere makes no difference to the score. Not a small difference — no difference at all, to first order. Every nearby shape scores the same, which is what it means to sit at the bottom of a valley rather than on a slope.

Now do the accounting. When you nudge the chain at some point, the score changes by an amount that depends on two things: how big your nudge was, and how sensitive the score is at that point.

Add up those contributions along the whole chain and you have the total change in the score. Each point contributes its own sensitivity multiplied by its own nudge.

The sensitivity has a name

Those sensitivities — one number for each point, and one for each thing you could nudge there — are the objects Noether calls the Lagrange expressions.

Write down what they are and you have written down the equation of the chain, because the real shape is exactly the shape where all of them are zero. No sensitivity anywhere means no nudge changes anything.

Here is the step that matters, and it is easy to walk past.

Noether does not set them to zero.

She writes down the sensitivities and leaves them standing. They are quantities you can compute for any shape at all — the real one, a silly one, one you made up. For the real shape they happen to be zero. For every other shape they are not.

Why leaving them standing is the whole trick

A statement about the sensitivities is a statement about every conceivable shape. A statement about the real shape is a statement about one.

The first kind is much stronger in reach and much weaker in content. It is also the kind both theorems produce. Noether's results are true of chains hanging wrongly, of paths a ball would never take, of arrangements that violate every law of physics. That is not a loophole. It is where the strength comes from.

The moment you set the sensitivities to zero, you have thrown away every shape but one, and you cannot get them back. So she does that step last, separately, and visibly. See True everywhere against true for real motions.

The leftovers at the ends

One more piece of bookkeeping, and it becomes important later.

The accounting above quietly assumed the nudge dies away at the two hooks. If you allow the chain to be nudged at the ends as well, extra terms appear, and they depend only on what happens at the ends.

Those end terms are usually treated as a nuisance to be assumed away. In this subject they are not a nuisance. They are where the conserved quantities come from. See What happens at the edges.

Not the smallest score, the unchanging one

A word of warning about how this is usually said.

You will often read that nature picks the path with the smallest score. That is wrong often enough to matter.

What nature picks is the path where a nudge makes no difference. Usually that is the smallest, in the way the bottom of a valley is the lowest point nearby. But a nudge also makes no difference at the top of a hill, or at a saddle where the ground falls away in one direction and rises in the other.

All three happen in physics. Light reflecting off the inside of a curved mirror can take the path of greatest time among nearby paths, and it is still the path light takes.

So the test is not "smallest". The test is "a nudge changes nothing", and that is a weaker and more useful condition. The chain hanging between two hooks happens to sit at a minimum. Do not assume the next case will.

One nudge at a time

One more point of care. The nudges are independent: you may push the chain here without pushing it there, and the accounting has to hold for every such push separately.

That is what makes the sensitivities well defined point by point, rather than a single overall number. It is also what forces there to be as many equations as there are things you could push.

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