Hamilton's principle/Explainers/Symmetry and conservation: Difference between revisions
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This page explains why conserved quantities exist. It uses no mathematics. The technical page it goes with is named above.
Things that do not change
Some quantities in physics stay the same no matter what happens.
Energy is one. Momentum is another. Angular momentum, the quantity that keeps a skater spinning, is a third.
For a long time these were facts. Careful experiments showed the numbers did not change, and physics recorded that they did not. Nobody could say why these particular quantities and not others.
Emmy Noether answered that in 1918, and the answer is one of the most useful in physics.
The answer
Every continuous symmetry gives a conserved quantity.
A symmetry means something you can change about your description without changing the physics.
Do your experiment a metre to the left. If it behaves identically, the laws do not care about position. Do it tomorrow. If it behaves identically, the laws do not care about time. Turn the whole apparatus to face north. If it behaves identically, the laws do not care about direction.
Each of those indifferences produces a conserved quantity.
| the laws do not care about | what is conserved |
|---|---|
| where you are | momentum |
| what time it is | energy |
| which way you face | angular momentum |
The three great conservation laws are three statements about what the universe is indifferent to.
Why it works
The argument, in outline and without any symbols, runs like this.
The whole of this volume rests on a total cost for a path. A symmetry means moving the path — shifting it sideways, or later in time — leaves that total unchanged.
Now put those two facts together. Nudging a true path leaves the total unchanged, because the path is stationary. Shifting the path leaves the total unchanged, because of the symmetry.
Two different reasons for the same conclusion. Combining them produces a constraint, and that constraint says a particular quantity cannot change as the motion proceeds.
The full working is on the technical page. The shape of it is what matters here: two independent reasons for one fact, combined.
Why this changed things
Before 1918, conservation laws were experimental results. You checked them.
After, they became consequences. If you believe the laws of physics are the same everywhere, you must believe momentum is conserved. They are not separate claims.
The reverse direction is at least as useful. Find a new conserved quantity in an experiment and you have found a symmetry you did not know about — and symmetries tell you about the structure of the theory.
That reversal is how much of modern particle physics is done. Conservation of electric charge, for instance, comes from a symmetry that has nothing to do with space or time at all, but with a quantity that can be shifted without physical effect. Charge is conserved because of that indifference.
Where the volume already used it
This theorem explains things this volume derived earlier without knowing why.
A skater on flat ice keeps their momentum, because the ice is the same everywhere. A pendulum keeps its energy, because the laws do not change from moment to moment. A planet sweeps out equal areas in equal times, because space has no preferred direction — that is Kepler's second law, and it is angular momentum conservation wearing seventeenth-century clothes.
Each of those was found separately, by its own calculation. They are one theorem.
Two cautions
The symmetry has to be a symmetry of the cost rule.
That is a stronger requirement than it sounds. The equations of motion can have symmetries that the cost rule does not, and those produce nothing. Only indifferences of the underlying rule count.
Independent symmetries need not give independent quantities.
Sometimes two symmetries yield conserved quantities that turn out to be related, so the second tells you nothing new. Noether pointed this out herself. Counting symmetries does not straightforwardly count useful conservation laws.
The limits
The theorem needs the symmetry to be continuous — something you can do a little bit of.
You can shift by a millimetre, or a micrometre, or any amount you like. That is continuous, and the theorem applies.
Some symmetries are not like that. A snowflake looks the same after turning by sixty degrees, but not after turning by one degree. That is a discrete symmetry, and it gives no conserved quantity by this route. Discrete symmetries matter in physics for other reasons.
The theorem also assumes the system has a cost rule at all. Systems with friction usually do not, in the simple form, and their energy is genuinely not conserved — it leaks into heat, which is a description of parts the model left out.
A second theorem
Noether's paper contains two theorems, not one.
The second treats a different kind of symmetry: one where the change can be chosen independently at each point in space, rather than being the same everywhere.
It gives a completely different conclusion, and it is the theorem she wrote the paper to prove. It is far less known than the first, which misrepresents the paper.
It is at Hamilton's principle/Explainers/When symmetry is local.
Where to go next
The local case, and what it produces instead of conservation laws, is at Hamilton's principle/Explainers/When symmetry is local.