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There are already several questions about the intuitive meaning of Noether's theorem, e.g. this & this Phys.SE posts (with not very satisfactory answers to me so far). Therefore, this time let's go in the other direction.

Is there some deep topological/geometrical explanation/reason for Noether's theorems?

To give an example, what I'm looking for is a reason like "the boundary of a boundary is zero" for the Bianchi identities.

Noether's second theorem (the one for infinite-dimensional groups) yields "generalized Bianchi identities" and thus I suspect there could be some similar deep reason?!

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I'd say the deep reason is the product rule. Combining that with the Euler-Lagrange equation, the variation of a Lagrangian can easily be expressed as a total derivative. Somehow, the two results complement each other to that end. Thus each action-preserving transformation, by expressing the Lagrangian's variation as another total derivative, is equivalent to some total derivative being zero.

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    $\begingroup$ Why was this downvoted? $\endgroup$ – J.G. May 18 '17 at 12:49

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