Noether’s theorem reveals symmetry and conservation. There are spatial translation symmetry, time translation symmetry, rotation symmetry, leading to momentum, energy and angular momentum conservation.

Symmetry leads to conservation proved by Lagrangian, and vice versa? conservation leads to symmetry?

As dQ/dt = {Q, H}, hence dQ/dt = 0 leads to {Q, H} = 0, so this is another form or expression of symmetry? compared to previous concept of differential of Lagrangian (dL) is zero is symmetry. Actually dL = 0 is premised on translation symmetry x –> x + ita, L–> L etc. So why dL = 0 indicates symmetry?