Solving Noether's Theorem: Examining "Reverse" Transformation

In summary, the conversation discusses a transformation from (t, x) to (t', x') using generators ζ and τ, and the question is posed on how to write the reverse transformation in terms of the original generators. It is also questioned if the functional is invariant under the "forward" transformation, is it also invariant under the "reverse" transformation? The answer is that for the inverse transformation, ζ and τ change signs, and the invariance holds for both the "forward" and "reverse" transformations due to the group theory structure of symmetries of the Lagrangian.
  • #1
facenian
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Missing template due to originally being placed in different forum
Hello, I've reading "Emmy Noether's wanderfull therorem" by Neuenschwander and he asks this question as exersice:
We described a transformation that takes us from (t, x) to (t', x') with
generators ζ and τ . How would one write the reverse transformation from (t', x')
to (t, x) in terms of the original ζ and τ? If the functional is invariant under the
‘forward” transformation, is it also invariant under the “reverse” transformation?

I would like to Know whether my answer is correct.
For the first question I found that for the inverse tansformation ζ and τ change signs, as for the second question I found that the invariance stands using either the "forward" or "reverse" transformation.
 
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  • #2
Your statements are correct, they are connected to the group theory structure of symmetries of the Lagrangian.
 

FAQ: Solving Noether's Theorem: Examining "Reverse" Transformation

What is Noether's Theorem?

Noether's Theorem is a fundamental principle in physics that relates symmetries of a physical system to conserved quantities. It was first discovered by mathematician Emmy Noether in 1915 and has since become an essential tool in understanding the laws of nature.

What is the "reverse" transformation in Noether's Theorem?

The "reverse" transformation in Noether's Theorem refers to the process of finding the symmetries of a physical system by examining the equations of motion for that system. This is in contrast to the traditional approach, where symmetries are first identified and then used to derive the equations of motion.

Why is solving Noether's Theorem important?

Solving Noether's Theorem allows us to identify and understand the underlying symmetries of a physical system, which in turn allows us to predict and explain the behavior of that system. This is crucial in the development of new theories and in testing the validity of existing ones.

What are some applications of Noether's Theorem?

Noether's Theorem has been applied in many areas of physics, including classical mechanics, electromagnetism, and quantum field theory. It has also been used in other fields such as economics and biology to understand systems with symmetries.

What are the limitations of Noether's Theorem?

Noether's Theorem is limited in its application to systems with continuous symmetries. It also assumes that the underlying equations of motion are known, which may not always be the case. Additionally, the interpretation of conserved quantities can be challenging in some cases.

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