Noether's theorem and Invariance of Lagrangian under translation Consider a coordinate system in which the point A has coordinate $z_0$ and point B has ccordinate $z$ where there is a particle of mass $m$. Let there be a constant gravitational field of Earth. The Lagrangian of a particle in this system is given by $$L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2)-mg(z-z_0)$$ where the potential energy of point B is measured w.r.t point A. Now consider different coordinate system where the origin is shifted from $x=y=z=0$ to $x=y=0,z=\alpha$. In this coordinate, A has coordinate $z_0+\alpha$ and B has coordinate $z+\alpha$. Therefore, although the coordinate $z$ is not cyclic, the Lagrangian remains invariant under translation in the $z$ direction. Therefore, by Noether's theorem, the momentum $p_z$ should be conserved. But as we know this is not the case due to constant gravitational field along $z$-direction. Is this not a contradiction?
 A: *

*ACuriousMind has already pointed out in comments the crucial point: $z_0$ is an external parameter, not an dynamical variable of the action $S[x,y,z]$.

*OP then ponders in a comment what happens if we artificially promote $z_0$ to a dynamical variable of the action $S[x,y,z,z_0]$? Nevermind this possibly does not make physical sense. What would be the mathematical consequences for Noether's theorem? 

*OK, let's investigate. The translations $$ z ~\longrightarrow ~ z+\alpha, \qquad z_0 ~\longrightarrow ~ z_0+\alpha $$ is indeed an exact off-shell symmetry of the action $S[x,y,z,z_0]$. Noether's theorem then therefore states that the corresponding Noether charge $$p_z~:=~\frac{\partial L}{\partial \dot{z}}$$
is conserved on-shell
$$ \frac{dp_z}{dt}~\approx~0. $$
[Here the $\approx$ symbol means equality modulo eom.]

*Only trouble is that the eom for $z_0$ reads
$$ 0~\approx~\frac{\delta S}{\delta z_0}~=~mg, $$
i.e. if we want to put the system on-shell, we would have to turn off gravity! And then $p_z$ would be conserved on-shell. Therefore Noether's theorem is not violated.
