So I looked at the invariance of the Lagrangian under the Gallilei Transformations.
So for the free fall we have the Lagrangian
$$L = \frac{m}{2}\dot{z}^2 -mgz$$
Then I applied the transformation
$$x\mapsto x' = x+\delta x$$
which gives me the following transformed Lagrangian
$$L' = \frac{m}{2}\dot{z}^2-gmz-gm\delta z = L+(-gm\delta z)$$
as the Lagrangian can have an extra addition of a Constant that doesnt change the Euler Lagrange Equations which I also extra verified you get the usual $z = \frac{1}{2}gt^2$
So there is now the Noether theorem which states that when we have an symmetry aka. invariance under a transformation we have a conservation law according to my research if you have translation invariance you get the conservation of momentum.
Now my question is how do we have conservation of Momentum in free falling conditions am I missing something there.