Noether's theorem -- Time inversion

In summary, Noether's theorem states that the law of conservation of energy exists due to the homogeneity of time. This also implies the validity of time inversion symmetries such as ##t \rightarrow -t##. This symmetry is only applicable in cases where there is conservation of energy, which is a result of homogeneity in time. However, Noether's theorem only applies to differential symmetries, not discrete ones like time reversal. An example of this is the function ##\cos(t)##, which is symmetric under time reversal but not time translations.
  • #1
LagrangeEuler
717
20
Noether's theorem said that because of homogeneity in time the law of conservation of energy exists. I am bit of confused and I am not sure is also time inversion some consequence of this. For example in the case of free fall we have symmetry ## t \rightarrow -t##. I am sometimes confused of that. Symmetry ## t \rightarrow -t##. is valid when we have conservation of energy and conservation of energy we have because of homogeneity in time. Could please give me some more insight on this problem. Thanks.
 
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  • #2
Noether's theorem applies to differential symmetries like time translation, and not to discrete symmetries like time reversal.
 
  • #3
Yes but we will not have time reversal symmetry if we did not have symmetry under the time translations. Right?
 
  • #4
Also when we have ## t \rightarrow -t## didn't we did translation of time ##t## of some kind?
 
  • #5
No. Consider the function ##\cos (t)##. It is symmetric under time reversal ##\cos(t)=\cos(-t)## but not time translations ##\cos(t)\ne \cos(t+\delta t)##
 
  • #6
DaleSpam said:
No. Consider the function ##\cos (t)##. It is symmetric under time reversal but not time translations.
Thanks Sir. Very nice example.
 

1. What is Noether's theorem?

Noether's theorem is a fundamental principle in physics that relates the symmetries of a system to its conserved quantities. It was first proposed by mathematician Emmy Noether in 1915 and has since been applied in various fields of physics, including classical mechanics, quantum mechanics, and general relativity.

2. How does Noether's theorem relate to time inversion?

Noether's theorem states that for every continuous symmetry of a physical system, there exists a conserved quantity. Time inversion is one such symmetry, which means that the laws of physics remain the same when time is reversed. This symmetry leads to the conservation of energy, one of the fundamental principles of physics.

3. Can you explain the mathematical concept behind Noether's theorem?

Mathematically, Noether's theorem states that for a Lagrangian system (a system described by a set of generalized coordinates and their corresponding velocities), if the Lagrangian remains unchanged under a continuous transformation, then there exists a conserved quantity associated with that transformation.

4. What are the practical applications of Noether's theorem in physics?

Noether's theorem has many practical applications in physics. For example, it has been used to derive the conservation of energy, momentum, and angular momentum in classical mechanics. In quantum mechanics, it has been used to explain the conservation of charge and other quantum numbers. In general relativity, it has been used to derive the conservation of energy-momentum tensor.

5. Are there any limitations or exceptions to Noether's theorem?

While Noether's theorem is a powerful tool in physics, it does have some limitations. It only applies to systems that exhibit continuous symmetries, and it cannot be used to derive all conservation laws. Additionally, Noether's theorem does not apply to systems with non-conservative forces, such as friction. There are also some exceptions to the theorem, such as in quantum field theory, where certain symmetries may not always lead to conserved quantities.

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