About the proof of Noether Theorem

[princeton118]

In the general Noether Theorem's proof, it is required that the action before transformation is equal to the action after transformation: I = I'.
Who can tell me why this condition has to be used. In my opinion, we can obtain the same form of the motion equation after the transformation only if the Lagrangian density has the same functional form. We don't have to use this scale-invariance condition. If we don't use this condition, can we still obtain the same form of the equation?PS:Which book gives a clear proof of the Noether Theorem?

 

[siddharth]

In the general Noether Theorem's proof, it is required that the action before transformation is equal to the action after transformation: I = I'.
Who can tell me why this condition has to be used.

Because, the numerical value of the Lagrangian and the action integral is invariant under translation of cyclic coordinates.

In my opinion, we can obtain the same form of the motion equation after the transformation only if the Lagrangian density has the same functional form. We don't have to use this scale-invariance condition.

How would you obtain the same form of the motion equation?

PS:Which book gives a clear proof of the Noether Theorem?

Have you read the discussion of the proof in Goldstein?

 

[princeton118]

I have read the book written by Goldstein, but I still have some questions.
In Goldstein's book, he gave two restricts about the symmetry. What puzzles me is the relationship between the form invariance of equations and these two restricts.

Thanks for your explanation!

 

[blechman]

The Lagrangian is only defined up to a total time derivative - such terms do not affect the equations of motion. Therefore, the Lagrangian need not be invariant for Noether's theorem to work, but be invariant up to total derivatives. By making the action invariant, you automatically take this into account. Furthermore, the final "total derivative" that the Lagrangian changes by is nothing more than the Noether current - that quantity which is conserved by the symmetry. So the proof is constructive - it gives you the conservation law for free.

 

[samalkhaiat]

In the general Noether Theorem's proof, it is required that the action before transformation is equal to the action after transformation: I = I'.
Who can tell me why this condition has to be used. In my opinion, we can obtain the same form of the motion equation after the transformation only if the Lagrangian density has the same functional form.

If you are dealing with INTERNAL symmetries, then yes, form-invariant Lagrangian is sufficient for proving Noether theorem. Indeed, in this case, the invariance condition is given by

\delta \mathcal{L} = 0 \ \ (1)

However, for space-time symmetries, Eq(1) does not lead to the correct Noether current; The missing term comes partly from the change in the spacetime region overwhich the action integral is taken. So, for spacetime symmetries, the invariance condition is that of the action integral.

We don't have to use this scale-invariance condition.

What do you mean by scale-invariant?

PS:Which book gives a clear proof of the Noether Theorem?

See post #12 in

https://www.physicsforums.com/showthread.php?t=172461

regards

sam

 

[princeton118]

I see, thank you all.

 

[micomaco86572]

I have the same problem. There are 3 conditions in the Goldstein's proof. If we break the second condition, we cannot obtain the same equation form after the variables' transformation. What if we break the third condition? Why we have to introduce this condition? Thx.