# Appendix

By

Matthew Sung

Kepler's second law is in fact conservation of angular momentum L.

Kepler's second law gives

\frac{dA}{dt}=\frac{1}{2}r^{2}\frac{d\theta}{dt}=\text{constant}

Angular momentum L is defined by

\mathbf{L}=\mathbf{r}\times\mathbf{p}=m\mathbf{r}\times\mathbf{v}

The magnitude of angular momentum is

|\mathbf{L}|=L=mrv\sin\alpha

where \alpha is the angle between \mathbf{r} and \mathbf{v}.

From the diagram above, we see that

\begin{align*}
v\sin\alpha\,dt & =rd\theta\\
v\sin\alpha & =r\frac{d\theta}{dt}
\end{align*}

Putting this into the expression for the magnitude of angular momentum, we have

L=mr^{2}\frac{d\theta}{dt}

Since Kepler's second law says that

\frac{dA}{dt}=\frac{1}{2}r^{2}\frac{d\theta}{dt}=\text{constant},

this implies that

L=2m\frac{dA}{dt}=\text{constant}.