Forgive me if the title sounds ignorant, but I’m genuinely trying to understand this concept. Everything I’ve read about Noether’s theorem says basically the following: If a system remains the same (in some way) after a transformation, then there must be type of quantity or property in the system that remains constant.

But isn’t that just common sense, or already implied by the premise? As I understand it, it basically states that “If something is the same, then something about it has not changed”.

In: Physics

Noether’s theorem gives us a direct way to calculate the conserved quantities of a system given its continuous symmetries like rotation, spatial and temporal translation. i.e. if you have the explicit expression for a transformation that leaves the system unchanged, you can plug it into Noether’s theorem and get an expression for a conserved quantity.

For example if you know how rotation changes a system that has rotational symmetry, you can right away get an expression for the angular momentum of the system.

This is very useful because often theories are constructed with the symmetries already known at the outset. All theories in particle physics for example are constructed by proposing a symmetry and looking for systems that respect said symmetry.

One thing you are missing is that the *way* the system is supposed to be symmetric (unchanged) may not have an obvious connection to the numerical quantity that stays constant (is conserved).

Example: that the laws describing a physical system are preserved by a change in time leads to the conservation of energy. I don’t think this link is obvious at all: if you told someone learning physics that conservation of energy is due to the physics of the system being the same at all times, that person is unlikely to say “yeah, that’s obvious”.

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