Noether's Theorem is perhaps the most beautiful mathematical theorem in physics •It states that: Every continuous symmetry (T) of the Lagrangian has a corresponding conservation lawpic.twitter.com/RrsS1QKm4A
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Isn't it continuous symmetries of the action (which can change the Lagrangian up to a total derivative of time -for discrete degrees of freedom- or spacetime -for Lagrangian densities-)?
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I find it interesting to contrast how clean and simple the theorem is when stated generally against how nuanced and complicated it becomes when applied to gravitational theories. I was wrestling with this recently in https://arxiv.org/abs/1912.01030
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There are two theorems which go by the name of "Noether's theorem." The one in this post applies to rigid symmetries. When finding conservation laws in GR, one needs to apply Noether's "second" theorem, which applies to gauge transformations (those which depend on spacetime).
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