Conflict of domain and endpoints in Noether's theorem

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• Van Ladmon
In summary, the conversation discusses the concept of energy conservation and the transformation rule ##q(t)\rightarrow q'(t)=q(t+\epsilon)##, which has fuzzy endpoints. The speakers also debate the correctness of differentiating between variations used in Hamilton's principle and Noether's theorem, as well as the need to express everything in terms of "old coordinates." Ultimately, the topic revolves around the application of appropriate boundary conditions for Hamilton's principle and the symmetry condition of ##\partial_t L=0## for time translations.
Van Ladmon
TL;DR Summary
Conflicts arise on boundary when proving energy conservation using Noether's theorem. Different statement appear in Physics from Symmetry and Kleinert's Particles and Quantum Fields.
In the derivation of energy conservation, there is the transformation ##q(t)\rightarrow q'(t)=q(t+\epsilon)##, whose end points are kind of fuzzy. The original path ##q(t)## is only defined from ##t_1## to ##t_2##. If this transformation rule is imposed, ##q'(t_2-\epsilon)=q(t_2)## to ##q'(t_2)=q(t_2+\epsilon)## is not defined in the original path. Then how could the Lagrangian be integrated?

On P.98 of Jakob Schwichtenberg's book Physics from Symmetry, he stated that ##\delta q(t_1)=\delta q(t_2)=0## whereas Kleinert stated in his Particles and Quantum Fields ##\delta q_s(t_a)## and ##\delta q_s(t_b)## are not necessarily ##0##. Who's correct?

This question is different from the endpoint questions since it is already clear that ##q(t_2+\epsilon)\neq q(t_2)##.

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You have to express everything in terms of the "old coordinates" including time. Then there's no problem with imposing the appropriate boundary conditions ##\delta q(t_1)=\delta q(t_2)=0## for Hamilton's principle. For time translations the symmetry condition is ##\partial_t L=0##, i.e., ##L=L(q,\dot{q})## and the conserved quantity is ##H=p \cdot q-L## with ##p=\partial L/\partial \dot{q}##.

vanhees71 said:
You have to express everything in terms of the "old coordinates" including time. Then there's no problem with imposing the appropriate boundary conditions ##\delta q(t_1)=\delta q(t_2)=0## for Hamilton's principle. For time translations the symmetry condition is ##\partial_t L=0##, i.e., ##L=L(q,\dot{q})## and the conserved quantity is ##H=p \cdot q-L## with ##p=\partial L/\partial \dot{q}##.
But why Kleinert differentiates the two kinds of variations: ##\delta q## used in Hamilton's principle and ##\delta_s q## in Noether's theorem? He says that ##\delta_s q## need not be ##0## on the boundaries. Also, what do you mean by expressing everything in terms of "old coordinates"? Could you please give an example? Thanks.

I don't know, why Kleinert does it this way.

I have my treatment of Noether's theorem unfortunately only in a German manuscript. I hope the formula density is high enough, so that you can understand the argument:

https://itp.uni-frankfurt.de/~hees/publ/theo1-l3.pdf

1. What is Noether's theorem?

Noether's theorem is a fundamental principle in physics that relates symmetries in a physical system to conserved quantities, such as energy or momentum.

2. What is the conflict of domain and endpoints in Noether's theorem?

The conflict of domain and endpoints in Noether's theorem refers to the fact that the symmetries of a physical system may not always be well-defined or applicable at the boundaries or endpoints of the system.

3. How does the conflict of domain and endpoints affect the application of Noether's theorem?

The conflict of domain and endpoints can limit the applicability of Noether's theorem, as it may not be possible to define or calculate the conserved quantities associated with certain symmetries at the boundaries or endpoints of a physical system.

4. Can the conflict of domain and endpoints be resolved?

In some cases, the conflict of domain and endpoints can be resolved by considering the symmetries and conserved quantities in a larger or more general system. However, in other cases, it may be necessary to modify or extend Noether's theorem to account for the boundaries or endpoints of the system.

5. How does understanding the conflict of domain and endpoints in Noether's theorem contribute to our understanding of physics?

By understanding the limitations and challenges posed by the conflict of domain and endpoints in Noether's theorem, we can gain a deeper understanding of the symmetries and conserved quantities that govern physical systems. This can also lead to further developments and advancements in the application of Noether's theorem and its implications for physics.

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