What is the Precise Heuristic Argument that Leads to Noether's Theorem

[liorde]

Hi,
I'm confused about the exact interpretation of Noether's theorem for fields. I find that the statement of the theorem and its proof are not presented in a precise manner in books.
My main question is: what is the precise heuristic argument that leads to Noether's theorem?

The question is presented in the attached pdf document.

Thanks

 

[andrien]

what is the precise heuristic argument that leads to Noether's theorem?

precise argument is that any continuous symmetry of lagrangian i.e. which can be build up from infinitesimal ones implies a conserved quantity.Also you have forgotten the action in your two dimensional case i.e. where is time?

 

[liorde]

I excluded time because I used a 2D space analogy, which was easy to illustrate in a figure. I'm interested in the idea, so it doesn't matter if I use time or not.

 

[andrien]

if you are looking for some elementary treatment then this might help.I am not sure what is really the problem.
http://www.mathpages.com/home/kmath564/kmath564.htm

 

[PJC01]

for your question. Noether's theorem is a fundamental result in mathematical physics that has been used to explain the symmetries underlying the laws of physics. It was first formulated by German mathematician Emmy Noether in 1915 and has since been generalized to various mathematical frameworks, including fields. The theorem states that for every continuous symmetry of a physical system, there exists a corresponding conserved quantity. This means that if a physical system remains unchanged under a certain transformation, then there is a corresponding physical quantity that remains constant throughout the system's evolution.

The heuristic argument that leads to Noether's theorem is based on the concept of a Lagrangian, which is a function that describes the dynamics of a system in terms of its coordinates and their derivatives. The Lagrangian is used to derive the equations of motion for a system, and it is also used to determine whether a system has any symmetries.

The key idea behind Noether's theorem is that symmetries of a system correspond to invariance properties of the Lagrangian. In other words, if the Lagrangian remains unchanged under a certain transformation, then the system has a corresponding symmetry. This can be seen in the fact that the equations of motion derived from the Lagrangian will also remain unchanged under the same transformation.

The precise heuristic argument that leads to Noether's theorem can be summarized as follows:

1. Start with a physical system described by a Lagrangian.
2. Consider a transformation that leaves the Lagrangian invariant.
3. Use the invariance of the Lagrangian to derive the equations of motion for the system.
4. If the transformation is a continuous symmetry, then the equations of motion will also remain unchanged.
5. From the equations of motion, identify a physical quantity that remains constant throughout the system's evolution.
6. This quantity is the conserved quantity corresponding to the symmetry.

In summary, Noether's theorem is based on the idea that symmetries of a system correspond to invariance properties of the Lagrangian, and it provides a powerful tool for understanding the fundamental laws of physics. I hope this helps to clarify the precise heuristic argument that leads to Noether's theorem.