A Century of Noether's Theorem | Hacker News Reader

A Century of Noether's Theorem

arxiv.org

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Noether is one of my heroes. Rising through the ranks to one of the greatest minds we've known, recognized in spite of being a woman in a time where that was unthinkable in science, all odds against her. And yet here she is, the name of one of the most basic, and most beautiful, concepts in physics. The inventor of abstract algebra too (which I hear is as significant, it's just not my domain).

So many great minds have had to fight an uphill battle, but few had it as steep and even fewer were as successful as her doing so.

It really is a shame that she's not as recognized as the Bohrs and Feynmans and Paulis and so on, but at least everyone with a passing interest in theoretical physics ought to know about her.

it was several years between the time I learnt noether's theorem and the time I learnt she was a woman. sad failure of the educational system, it really did deserve to be a more prominent fact since as you say it meant that she had to overcome heavy odds to do what she did.

She is in many ways the George Washington Carver of math/physics women.

I have the greatest respect for both Noether and Carver.

I wish invariant theory was taught outside of physics and maths departments. Us engineers missed out.

A much restricted case is/(used to be) taught to computer vision students. It's needed to generate features that are invariant to shift, rotation and scaling.

https://mitpress.mit.edu/9780262132855/geometric-invariance-...

As an engineer, this and the principle of least action occupy my wall of “things I think are super deep and maybe mysterious* and interesting and I wish I understood deeply”

* interpret generously

A big part of what’s impressive about Noether’s theorem is that it’s not at all mysterious. At its core, it’s a mathematical proof that’s possible to fully understand. It doesn’t depend on any magic constants in our universe, or indeed anything in the universe at all. It should apply to all possible universes in any situation that satisfies its conditions. The PLA is similar.

Some people see a mystery at the point where these mathematical constructs are applied to our physical universe. Eugene Wigner wrote about “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

There are explanations for many of the points he raised in that paper. Perhaps the one that remains most unresolved is the question of why universal law or behavior isn’t messier, more chaotic - why it should so often correspond so neatly to physical phenomena. Intuitively, this doesn’t seem surprising to me, but Wigner correctly points out that we don’t really know why this is the case.

Answering that gets deeper into philosophy: structural realism, the anthropic principle, and so on. But one possible explanation is an extension of ideas like Noether’s: that the various mathematical constraints collapse the space of possibilities enough to make it likely, if not inevitable, that the universe ends up embodying relatively simple mathematical structures.

Indeed. I'd suggest Susskind's Theoretical Minimum: Classical Mechanics if someone wants an introduction. He doesn't explicitly prove Noether but he demonstrates the connection between symmetry and conservation laws building the intuition to properly appreciate Noether.

I've also written a series on Abstract Algebra for computer programmers if you're serious about learning it:

https://xorvoid.com/galois_fields_for_great_good_00.html

As to understanding Hamilton's stationary action deeply: that is accessible.

I have created a resource with interactive diagrams. Move sliders to sweep out variation of a trial trajectory. The diagram shows the response.

https://cleonis.nl/physics/phys256/energy_position_equation....

About the form of the resource:

In physics textbooks the usual presentation is to posit Hamilton's stationary action, followed by demonstration that F=ma can be recovered from it.

Now: we have that in physics you can often run derivations in both directions.

Example: the connection between the Lagrangian formulation of mechanics and the Hamiltonian formulation. The interconversion is by way of Legendre transformation. Legendre transformation is it's own inverse; applying Legendre transformation twice recovers the original function.

Well, the relation between F=ma and Hamilton's stationary action is a bi-directional relation too: it is possible to go _from_ F=ma _to_ Hamilton's stationary action.

The process has two stages:

- Derivation of the work-energy theorem from F=ma

- Demonstration that in circumstances such that the work-energy theorem holds good Hamilton's stationary action holds good also.

Knowing how to go from F=ma to Hamilton's stationary action goes a long way towards lifting the sense of mystery.

General remark: Of course, in physics there are many occurrences of hierarchical relation. Classical mechanics has been superseded by Quantum mechanics, with classical mechanics as limiting case; the validity of classical mechanics must be attributed to classical mechanics emerging from quantum mechanics in the macroscopic limit.

But in the case of the relations between F=ma, the work-energy theorem, and Hamilton's stationary action: the bi-directionality informs us that the relations are not hierarchical; those concepts are on equal par.

I'm a physicist, and I love invariants. In my work, being able to use the optical invariant is a superpower. It governs the statistics of light beams.

(2019)

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