Question about Noether's Theorem.

[lugita15]

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.

 

[Rainbow Child]

Noether's Theorem talks about continuous groups of transformations.

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


[tex]\hline [/tex]

[tex]^1[/tex] Along with the identity transformation of course.

 

[charng]

Hello, thank you for your question about Noether's Theorem. I am happy to provide some insight and clarification on this topic.

Firstly, let me explain that Noether's Theorem is a fundamental principle in physics that connects symmetries of a system to conservation laws. It is named after mathematician Emmy Noether, who derived the theorem in the early 20th century.

In response to your question about the corresponding conserved quantity for T-symmetry, it is important to note that Noether's Theorem only applies to continuous symmetries. Time reversal is considered a discrete symmetry, meaning it is not continuously changing. Therefore, Noether's Theorem does not directly apply to this symmetry.

However, in classical mechanics, the laws of physics are T-symmetric, meaning they are symmetric with respect to time reversal. This means that if we reverse the direction of time, the laws of physics would still apply. In this case, the corresponding conserved quantity is the total energy of the system.

The conservation of energy can be derived from the Lagrangian by considering the invariance of the Lagrangian under time reversal. This means that if we reverse the direction of time, the Lagrangian remains the same. From this, we can use the Euler-Lagrange equations to show that the derivative of the Lagrangian with respect to time is equal to zero, resulting in the conservation of energy.

I hope this helps to clarify the relationship between Noether's Theorem, symmetries, and conservation laws. If you have any further questions, please do not hesitate to ask. Thank you.