mjhjmtu

I have a question about angular momentum:
Is it possible to have a system where angular momentum is conserved relative to 1 point,but not conserved relative to another?

Consider central-force motion, such as a planet moving around a (very massive) star. The angular momentum of such a planet is constant if we take the origin as the center of the star. It is not constant if we take the origin to be any other point.

Is it possible to have a system where angular momentum is conserved relative to 1 point,but not conserved relative to another?

Total angular momentum will be conserved but the angular momentum of any part of the system will have a value dependent on where you take your base point.

Angular momentum relative to an origin $mathcal O_1$

$$ mathbfL_mathcal O_1 = mathbfr_mathcal O_1 times p_mathcal O_1$$

where $mathbf r_mathcal O_1$ is the position vector to the particle relative to some origin $mathcal O_1$.
Now suppose that angular momentum is conserved in $mathcal O_1$. Then

$$ fracd mathbf L_1dt = mathbfdotr_1 times p_1 + mathbfr_1 times dotp_1 = frac1m mathbfp_1 times p_1 + mathbfr_1 times dotp_1 =0 $$

but since the direction of momentum is frame-independent, the first term vanishes (that is, $mathbfp_1 = mathbfp$). It then follows that

$$ mathbfr_1 times F_1 =0 . $$

Now, let's look at some other origin $mathcalO_2$, given that $L$ is conserved in $mathcal O_1$. Well the first term much vanish again, that's fine but what about the second term? Does

$$mathbfr_2 times F_2 stackrel?=0. $$

Well, no not necessarily. Namely, just choose an origin in which the force is perpendicular to your position vector.