# What's so great about Noether's theorem?

Thomas Rigby
I read about Noether's theorem that says how for every symmetry there is a conserved quantity. Seems kind of obvious. Does anyone understand it well enough that they can explain precisely why that notion is profound?

• • weirdoguy and PeroK

Mentor
I read about Noether's theorem that says how for every symmetry there is a conserved quantity. Seems kind of obvious. Does anyone understand it well enough that they can explain precisely why that notion is profound?
Well, what's "profound" is something of a matter of taste, but...

Noether's theorem contributes a deeper level of understanding. For example we have believed that energy is conserved for centuries, but had no better justification for this belief than that we had never ever observed a violation. Noether's theorem tells us that energy conservation is a consequence of time-translation symmetry, that the universe has to work that way if it has that symmetry.

• vanhees71, PhDeezNutz, russ_watters and 3 others
I read about Noether's theorem that says how for every symmetry there is a conserved quantity. Seems kind of obvious.
And yet, if you phrase Noether's Thm in that too-simplistic way, the statement is false.

Standard phrasing of Noether's Thm involves "leaving the action unchanged". But Kepler's 3rd law arises from a rescaling of space and time (i.e., a nonuniform dilation). This multiplies the Lagrangian by a constant, hence changes the action, but does not affect the equations of motion. Thus it is a symmetry which maps solutions of the equations of motion among themselves. This symmetry does not commute with the Hamiltonian, hence does not correspond to a conserved quantity.

(This is not to denigrate Noether's Thm of course -- it is important in a vast array of physical scenarios.)

Does anyone understand it well enough that they can explain precisely why that notion is profound?
One can deduce various features of a (class of) systems by examining its symmetries, more easily than exhaustively solving its equations of motion. E.g., solving for Kepler orbits leads to a transcendental equation which cannot be solved analytically. But a study of the various symmetries of that problem (energy conservation, momentum conservation, the LRL vectors, and the K3 dilation symmetry) allow many useful features of the orbits to be deduced. E.g., what's the relation between orbital radius and tangential velocity, and so on.

The quantum version of the same problem, i.e., the hydrogen atom, allows various features of the energy and angular momentum spectra to be discovered by study of the Lie algebra of those symmetries, without needing to solve explicitly for the wave function.

• vanhees71, Nugatory and topsquark
Well, what's "profound" is something of a matter of taste, but...
She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics.
https://en.wikipedia.org/wiki/Emmy_Noether

• vanhees71 and Nugatory
Mentor
Who says physical laws need to be profound? That's not the same as important. The first law of thermodynamics seems pretty obvious to me, but it is critically important.

Staff Emeritus
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