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Following the discussion Here, in particular this answer by Qiaochu Yuan.

If we have a set $X$ acted by an arbitrary "time monoid" $M$ and $G$ is a group of symmetries of $X$ as in the answer's setup, could we reasonably write this generalization of Noether's Theorem like this: $$X^M/G = X/G$$

I would read this as 'the set of $G$ orbits of $X$ time shifted by $M$ is the same as the set of $G$ orbits of $X$'.

Does that seem right?

I really like how simple it is compared to other statements of Noether's Theorem, but is it missing something important?

Matthew Miller
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  • $X$ is the whole state space and $X/G$ is the set of all $G$-orbits, so this formulation seems to lose the main point that $M$ preserves each individual orbit. – Karl Nov 26 '24 at 16:41
  • How about $(X/G)^M=X/G$ ? The $G$-orbits of $X$ time shifted by $M$ are the same $G$-orbits of $X$ – Matthew Miller Nov 26 '24 at 18:28
  • You need to express that each orbit is mapped to itself, not just that the set of orbits is the same before and after. – Karl Nov 26 '24 at 19:56

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