## Kepler's Second Law

Kepler's second law of planetary motion states that the planets, as they orbit the Sun following an ellipse, sweep out equal areas in equal times, and is a consequence of the conservation of angular momentum.

\begin{aligned}\vec{A} &= \frac{1}{2} r \vec{r} \times \vec{v} dt \\ &= \frac{\vec{r}}{2m} \times m \vec{v} dt \\ &= \frac{\vec{L}}{2m}\end{aligned}

Hence
$\frac{dA}{dt} = \frac{L}{2m}dt$
and since angular momentum
$L$
is conserved, Kepler's second law is proved.