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.

Kepler's second law

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

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

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