Question

Show that Kepler's law obeys the principle of conservation momentum.

Answer 1

The first Kepler's law states that all planets orbit on the ellipses. This mean that orbit is plane curve. It corresponds to conservation of the projection of momentum on the axis that is perpendicular to the plane of orbit.

The second Kepler's law says that sector velocity is constant, or

$$\frac{dA}{dt}=\rm const$$

Here

$$A=\frac{1}{2}|{\bf r}(t)\times{\bf r}(t+dt)|=\frac{1}{2}rv\sin\theta dt=\frac{L_z}{2m}dt$$

Thus, we get

$$\frac{dA}{dt}=\frac{L_z}{2m}=\rm const$$

or

$$L_z=\rm const$$