I report to you an interesting excerpt from my Physics book. It is an Italian version, so I apologize in advance, as I'm sure I won't give proper justice to its beauty in the translation as the authors would have done.
There is a profound theoretical connection between quantities that are preserved and symmetries of nature. So the principle of conservation of the linear momentum is related to the spatial symmetry of nature, which implies that an experiment done in a place give identical results to an equal experiment performed in any other place. [...] The principle of conservation of energy is related to the temporal symmetry: the result of an experiment today will be equal to the result of the same experiment did yesterday.
Physics I - Resnick, Halliday, Krane
After this words, the authors start talking about the conservations of the linear momentum in the canonical way. However, I was captured by this fly away and I want to find out more. How can I do that?
Spatial symmetry of nature $~\Rightarrow~\ldots?? \ldots~\Rightarrow~$ Conservation of the linear momentum
Temporal symmetry of nature $~\Rightarrow~\ldots?? \ldots~\Rightarrow~$ Conservation of energy
What do you think is in between?