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Is it correct that for a motion in a central force field, e.g. a gravitational field, the absolute value of the total angular momentum of the particle and the component of the perpendicular to the plane of motion are conserved, but the other components of the angular momentum in general not?

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I believe all three components are constant.

First, be sure you are careful about the point from which you are measuring angular momentum. The definition of angular momentum is $\vec{L}=\vec{r}\times\vec{p}$, where $\vec{r}$ is the position of the object measured from your point of origin. This point can be whatever you want, but it's most convenient to measure it from the "source" of your force.

To answer your question, the rotational analog of Newton's second law can help: $$ \vec{\tau}_\text{net} = \frac{d\vec{L}}{dt}. $$ As long as the net torque is zero, then $\vec{L}$ is constant – all of the three components! In the case where the position vector $\vec{r}$ is measured from the source of the central force, then the corresponding torque is indeed zero (since force and position would be (anti)parallel, causing a zero cross product).

Note that using the convention for the position vector described above, the direction of angular momentum is perpendicular to the plane of motion. Use the good ol' right hand rule to convince yourself of this. So I suppose the "other" two components you mention are constantly zero.

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    $\begingroup$ Note that the if the point of origin is taken to be something other than the center where the force is directed, then angular momentum will not be conserved. $\endgroup$ Nov 4, 2013 at 0:01

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