Noether's theorem -- Time inversion

AI Thread Summary
Noether's theorem establishes a connection between symmetries and conservation laws, specifically linking time homogeneity to the conservation of energy. The discussion highlights confusion regarding time inversion symmetry and its relationship to time translation symmetry. It clarifies that time reversal symmetry (t → -t) is distinct from time translation symmetry, as demonstrated by the example of the cosine function. The cosine function exhibits symmetry under time reversal but does not maintain symmetry under time translations. Understanding these distinctions is crucial for grasping the implications of Noether's theorem in physics.
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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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Noether's theorem applies to differential symmetries like time translation, and not to discrete symmetries like time reversal.
 
Yes but we will not have time reversal symmetry if we did not have symmetry under the time translations. Right?
 
Also when we have ## t \rightarrow -t## didn't we did translation of time ##t## of some kind?
 
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)##
 
DaleSpam said:
No. Consider the function ##\cos (t)##. It is symmetric under time reversal but not time translations.
Thanks Sir. Very nice example.
 
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