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I am confused, in the proof of Noether's theorem, by the change of boundary in the action integral during the transformation of coordinates. I have seen on[ Wikipedia ][1] that along with the change of Field, they also change $\Omega$ to $\Omega'$, where $\Omega$ is the space-time boundary of the action integral.

If we change the fields and boundaries both due to coordinate transformations then wouldn't that constitute a zero change? (I am keeping the intrinsic changes in the field apart)

Don't we consider a fixed region (arbitrary but unchanging during the flow) of space-time and then see the changes on Lagrangian due to only the flow of fields and some intrinsic change of fields, before and after the flow? (as shown below) The coordinates should be treated as dummy variables. $$\int_\Omega \delta L\ d^4x$$ I don't think we should move our boundary with the flow, am I right? Moreover, in the proof shown by [joshphysics][2] he didn't consider action at all. He worked only with the variation of Lagrangian and so there was no integral and hence, no boundary.

So, why do some proofs change the boundary and some do not? I mean how are these equivalent?

Another question: If we prove Noether's Theorem as [joshphysics][2] did, using only Lagrangian and not action, do we miss some conservations compared to the proof done in [ Wikipedia ][1] using the action integral? [1]: https://en.wikipedia.org/wiki/Noether%27s_theorem#Field-theoretic_derivation [2]: https://physics.stackexchange.com/a/56905/143440

I am confused, in the proof of Noether's theorem, by the change of boundary in the action integral during the transformation of coordinates. I have seen on Wikipedia that along with the change of Field, they also change $\Omega$ to $\Omega'$, where $\Omega$ is the space-time boundary of the action integral.

If we change the fields and boundaries both due to coordinate transformations then wouldn't that constitute a zero change? (I am keeping the intrinsic changes in the field apart)

Don't we consider a fixed region (arbitrary but unchanging during the flow) of space-time and then see the changes on Lagrangian due to only the flow of fields and some intrinsic change of fields, before and after the flow? (as shown below) The coordinates should be treated as dummy variables. $$\int_\Omega \delta L\ d^4x$$ I don't think we should move our boundary with the flow, am I right? Moreover, in the proof shown by joshphysics he didn't consider action at all. He worked only with the variation of Lagrangian and so there was no integral and hence, no boundary.

So, why do some proofs change the boundary and some do not? I mean how are these equivalent?

Another question: If we prove Noether's Theorem as joshphysics did, using only Lagrangian and not action, do we miss some conservations compared to the proof done in Wikipedia using the action integral?

I am confused, in the proof of Noether's theorem, by the change of boundary in the action integral during the transformation of coordinates. I have seen on[ Wikipedia ][1] that along with the change of Field, they also change $\Omega$ to $\Omega'$, where $\Omega$ is the space-time boundary of the action integral.

Don't we consider a fixed region (arbitrary but unchanging during the flow) of space-time and then see the changes on Lagrangian due to only the fields, before and after the flow? (as shown below) The coordinates should be treated as dummy variables. $$\int_\Omega \delta L\ d^4x$$ I don't think we should move our boundary with the flow, am I right? Moreover, in the proof shown by [joshphysics][2] he didn't consider action at all. He worked only with the variation of Lagrangian and so there was no integral and hence, no boundary.

So, why do some proofs change the boundary and some do not? I mean how are these equivalent?

Another question: If we prove Noether's Theorem as [joshphysics][2] did, using only Lagrangian and not action, do we miss some conservations compared to the proof done in [ Wikipedia ][1] using the action integral? [1]: https://en.wikipedia.org/wiki/Noether%27s_theorem#Field-theoretic_derivation [2]: https://physics.stackexchange.com/a/56905/143440

I am confused, in the proof of Noether's theorem, by the change of boundary in the action integral during the transformation of coordinates. I have seen on[ Wikipedia ][1] that along with the change of Field, they also change $\Omega$ to $\Omega'$, where $\Omega$ is the space-time boundary of the action integral.

If we change the fields and boundaries both due to coordinate transformations then wouldn't that constitute a zero change? (I am keeping the intrinsic changes in the field apart)

Don't we consider a fixed region (arbitrary but unchanging during the flow) of space-time and then see the changes on Lagrangian due to only the flow of fields and some intrinsic change of fields, before and after the flow? (as shown below) The coordinates should be treated as dummy variables. $$\int_\Omega \delta L\ d^4x$$ I don't think we should move our boundary with the flow, am I right? Moreover, in the proof shown by [joshphysics][2] he didn't consider action at all. He worked only with the variation of Lagrangian and so there was no integral and hence, no boundary.

So, why do some proofs change the boundary and some do not? I mean how are these equivalent?

Another question: If we prove Noether's Theorem as [joshphysics][2] did, using only Lagrangian and not action, do we miss some conservations compared to the proof done in [ Wikipedia ][1] using the action integral? [1]: https://en.wikipedia.org/wiki/Noether%27s_theorem#Field-theoretic_derivation [2]: https://physics.stackexchange.com/a/56905/143440