I know angular momentum $L=q \times p$ is conserved, where $p=L_{\dot q}$ is linear momentum. How to apply this to a planet orbiting the star, described by the position vector $q$ relative to the star. Here $L=\frac{1}{2}M|\dot q ^2|+\frac{GM}{|q|}$.

To deduce the Kepler's law

A planet always stays in one fixed plane and the radius vector $q$ sweeps equal areas in equal time.

My attempt:

I compute $L_{\dot q}=M\dot q$, so the given condition means $q \times \dot q$ is conserved. I think the cross product can denote the area the planet sweeps in $\triangle t$.

But I don't know how to show the planet always stays in the same plane.

Can anyone help? Thanks very much!

I know angular momentum $L=q \times p$ is conserved, where $p=L_{\dot q}$ is linear momentum. How to apply this to a planet orbiting the star, described by the position vector $q$ relative to the star. Here $L=\frac{1}{2}M|\dot q ^2|+\frac{GM}{|q|}$.

To deduce the Kepler's law

A planet always stays in one fixed plane and the radius vector $q$ sweeps equal areas in equal time.

My attempt:

I compute $L_{\dot q}=M\dot q$, so the given condition means $q \times \dot q$ is conserved. I think the cross product can denote the area the planet sweeps in $\triangle t$.

But I don't know how to show the planet always stays in the same plane.