Angular momentum conserved for central forces not at origin?

[OmegaKV]

My textbook says that for a central force at the origin, the angular momentum is constant, because the derivative rxF is zero since F points radially outwards so it is in the same direction as r. Ok, but what about the angular momentum about a point other than the origin, or the angular momentum with respect to the origin of a central force that is not located at the origin? Would angular momentum still conserved in these cases?

 

[Ibix]

What would it mean if angular momentum were not conserved in those cases?

Can you do the maths to see what would happen?

 

[vanhees71]

Of course then you have to relate the angular momentum to the new point of rotational symmetry. So you have to define
$$\vec{J}=(\vec{r}-\vec{r}_0) \times \vec{p},$$
if $\vec{r}_0$ is the position vector of the point around which the situation is rotatinal symmetric (isotropic). It's very obvious in view of Noether's theorem.

 

[rcgldr]

Angular momentum with respect to the origin of a central force that is not located at the origin?

Momentum is conserved in a closed or isolated system. If there is a central force not located at the origin, then from the frame of the origin, that force must be exerting a torque on something with respect to the origin, and would need to be taken into account.