The motion of planets in the solar system is an example of the conservation of

From Kepler's second law a line joining any planet to the sun sweeps out equal areas in equal times, that is, the areal speed of the planet remains constant.

dA =  area of curved triangle SAB
      = $\frac{1}{2}(\mathrm{AB}\times \mathrm{SA})$

      = $\frac{1}{2}(\mathrm{rd\theta } \times \mathrm{r}) = \frac{1}{2}{\mathrm{r}}^2\mathrm{d\theta }$

The instantaneous areal speed of the planet is
$\frac{\mathrm{dA}}{\mathrm{dt}} = \frac{1}{2}{\mathrm{r}}^2\frac{\mathrm{d\theta }}{\mathrm{dt}} = \frac{1}{2}{\mathrm{r}}^2\mathrm{\omega }$

where $\mathrm{\omega }$ is an angular speed. Let J be angular momentum of planet about sun
$\mathrm{J} = \mathrm{I\omega } = {\mathrm{mr}}^2\mathrm{\omega }$

$\frac{\mathrm{dA}}{\mathrm{dt}} = \frac{\mathrm{J}}{2\mathrm{m}}$

From Kepler's law areal speed is constant, therefore angular momentum J is constant.
Hence, Kepler's second law is equivalent to conservation of angular momentum.
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