Angular momentum and Potential field

Question

In addition to total energy, which components of the angular momentum $(mathbf{L}_x,mathbf{L}_y,mathbf{L}_z)$ are conserved in the presence of an external potential field $V(x,y,z)$? I've learnt that it is usually the $mathbf{L}z$, but how should I prove it? There seems no way to find a relation between $V$ and $mathbf{L}$. Please help !!

Michael Seifert · Answer

The general statement is that if $V$ is rotationally symmetric about a given axis, then the component of $vec{L}$ parallel to that axis is conserved.  This is easiest to demonstrate by rewriting the potential in terms of polar coordinates and showing that the torque is always perpendicular to the axis of symmetry.
Alternately, this can be viewed as a consequence of Noether's theorem, which I am honor-bound to mention in this context because it's a fascinating theorem.

Angular momentum and Potential field

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