Noether's theorem -- Time inversion

[LagrangeEuler]

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.

 

[Dale]

Noether's theorem applies to differential symmetries like time translation, and not to discrete symmetries like time reversal.

 

[LagrangeEuler]

Yes but we will not have time reversal symmetry if we did not have symmetry under the time translations. Right?

 

[LagrangeEuler]

Also when we have $t \rightarrow -t$ didn't we did translation of time $t$ of some kind?

 

[Dale]

No. Consider the function $\cos (t)$. It is symmetric under time reversal $\cos(t)=\cos(-t)$ but not time translations $\cos(t)\ne \cos(t+\delta t)$

 

[LagrangeEuler]

No. Consider the function $\cos (t)$. It is symmetric under time reversal but not time translations.

Thanks Sir. Very nice example.