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.