Question about Noether's Theorem.

AI Thread Summary
Noether's Theorem establishes a connection between symmetries in the Lagrangian and conservation laws, such as energy conservation linked to time translation and momentum conservation linked to space translation. The discussion highlights that while classical mechanics exhibits T-symmetry, which relates to time reversal, this symmetry does not yield a corresponding conservation law due to its classification as a discrete group of transformations. The theorem primarily addresses continuous transformations, making time reversal an exception. Therefore, no conserved quantity arises from time reversal in the context of Noether's Theorem. Understanding these distinctions is crucial for applying the theorem effectively in physics.
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According to Noether's Theorem, for every symmetry of the Lagrangian there is a corresponding conservation law. For instance, the invariance of the Lagrangian under time translation and space translation correspond to the conservation laws of energy and momentum, respectively. Also, the invariance of the Lagrangian under rotation in space corresponds to the conservation of angular momentum.

In classical mechanics at least, the laws of physics are also T-symmetric, i.e. they are symmetric with respect to time reversal. What is the corresponding conserved quantity, and how is it derived from the Lagrangian?

Any help would be greatly appreciated.
Thank You in Advance.
 
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Noether's Theorem talks about continuous groups of transformations.

The time reversal is a discrete group^1 of transformations, thus it does not corresponds to any conservation law.


\hline

^1 Along with the identity transformation of course.
 
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