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Noether's theorem informally states something like "symmetries in the dynamical law imply conserved quantities". However, the theorem is generally stated in terms of physics-specific classes of dynamical systems such as Lagrangian or Hamiltonian dynamics.

I'm not a physicist and I am curious whether the result generalizes to non-physical dynamical systems. E.g. we can ask whether cellular automata satisfy conservation laws. Moreover, I think if I had a more general formulation of Noether's theorem that didn't rely on a lot of physics-specific details, I'd understand it better even in a physics context.

The most general formulation of a dynamical system I know is a tuple $(T, X, \Phi)$ where $T$ is a monoid (representing the time domain, e.g. $\mathbb R$ for continuous time, or $\mathbb Z$ for discrete time), $X$ is a non-empty set (state space) and $\Phi$ is a function $\Phi :(T\times X)\to X$ (the dynamical law).

Is there a generalization of Noether's theorem that doesn't assume anything (or at least makes minimal assumptions) about the dynamical system? Possibly not even that state space and time are continuous, so that the result applies directly to e.g. cellular automata?

user56834
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  • For perspectives on dynamical systems that are even more general than you are suggesting see https://math.stackexchange.com/q/626927/169085 and the second part of my answer at https://math.stackexchange.com/q/4132091/169085. – Alp Uzman Jul 14 '22 at 20:22
  • It might also be good to give more details as to what constitutes an acceptable generalization of Noether's Theorem for you (e.g. in the CA survey you link there are already some references for Noether's Theorems for CA). – Alp Uzman Jul 14 '22 at 20:25
  • Of course it's likely that what a Noether's Theorem really is is part of the question. – Alp Uzman Jul 14 '22 at 20:25
  • A candidate would be of the following form: Let $X,T,L,V$ be objects such that $L:X\to V$, $Aut(L)\leq End(X)$ and let $\alpha^L,\sigma:T\to End(X)$. Then say that a Noether Theorem is satisfied if whenever $\sigma$ takes values in $Aut(L)$, there is an $N(L,\sigma):X/{\alpha^L}\to V$. Here $X,T,L,V,\alpha^L,\sigma,X/{\alpha^L}$ are the space, time, Lagrangian, observable values, the time evolution defined by $L$, $\sigma$ symmetries of L, and the orbit space, respectively. – Alp Uzman Jul 14 '22 at 20:49
  • For a more practical approach, e.g. the classical Krylov-Bogoliubov Theorem says that any homeomorphism of a compact metric space has an invariant probability measure, however for "dynamically significant" invariant measures (e.g. an SRB measure or a measure of maximal entropy) more sophisticated arguments (and hypotheses) are needed. Even though there are certain recipes as to what "dynamical significance" ought to mean (see e.g. Milnor's notes https://www.math.stonybrook.edu/~jack/DYNOTES/dn3.pdf), there doesn't seem to be a recipe for recipes. – Alp Uzman Jul 14 '22 at 20:55
  • "It might also be good to give more details as to what constitutes an acceptable generalization of Noether's Theorem for you (e.g. in the CA survey you link there are already some references for Noether's Theorems for CA)."

    In the most ideal case, there could be a theorem which literally if you apply it to continous-time and continous- (euclidean?) space with a Lagrangian formulation, you get back the standard Noether's theorem, but when you apply it to a cellular automaton gives you some interesting conservation law as well. I don't know if this is possible. Otherwise, ...

     – user56834 Jul 17 '22 at 09:50
  • Otherwise, maybe an "analogous" theorem in cellular automata which basically captures the same idea would be kind of interesting too, as you point to in those references. Those aren't exactly a generalization, just an analogous thing. I honestly couldn't really understand some of those references, but based on superficially reading them they seem not very general. – user56834 Jul 17 '22 at 09:53

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