Conservation of angular momentum in a linearly accelerated particle

Question

The angular momentum of a linearly accelerated particle in an inertial systems in which the line of motion passes through the origin is $\vec{L}=m\vec{r}\times\vec{v}=0$. But if I move my system of coordinates a distance $d$ in a direction perpendicular to the line of motion then I have $L=dv$, or $\dot{L}=da$. So angular momentum is conserved in one inertial system but not in another. Am I wrong on something?

Answer 1

It's an interesting observation but it should not worry you too much.

The general rule is that if there is acceleration then a force is acting so we do not in general expect angular momentum to be constant. But, as in your example, one can find an accelerated motion where the angular momentum about one particular axis is constant.

My answer consists in saying that this is ok, and it is also ok that in this example the angular momentum about some other axis is not constant. After all, for circular motion at constant speed one also has constant angular momentum about the centre of the circle, but not about some other place.

A somewhat related fact is that a motion which conserves kinetic energy in one inertial reference frame does not necessarily conserve kinetic energy in all reference frames (e.g. circular motion in one frame).