[NOTES/CM-05005] Runge Lenz Vector

For the Kepler problem there is an additional constant of motion, given by
\begin{equation}
\vec{N} = \vec{v} \times \vec{L} - \frac{k\vec{r}}{r}
\end{equation}
Also the orbit equation can be derived using conservation of Runge Lenz vector.

\paragraph*{Proof:} For attractive Coloumb potential, \(V(r)=-k/r, k>0\), and
the equations of motion are
\begin{equation}
\frac{d}{dt} m \vec{v} = - \nabla \left( - \frac{k}{r} \right) \\
= - \frac{k \vec{r}}{r^{3}}
\end{equation}

\begin{eqnarray}
\frac{d \vec{N}}{dt}
&=& \frac{d \vec{v}}{dt} \times \vec{L} + \vec{v} \times
\left( \frac{d \vec{L}}{dt} \right) - \frac{k}{r} \frac{d \vec{r}}{dt} + k
\vec{r} \left( \frac{\dot{r}}{r^{2}} \right)\\
&=& -\frac{1}{\not m} \left( \frac{k \vec{r}}{r^{3}} \right) \times (\vec{r}
\times \not m \vec{v}) - \frac{k \dot{\vec{r}}}{r} + k \vec{r} \left(
\frac{\vec{r}\cdot \vec{v}}{r^{3}} \right)\\
&=& -\frac{k}{r^3} \vec{r}\times(\vec{r}\times\vec{v}) - \frac{k
{\vec{v}}}{r} + k \vec{r} \left(
\frac{\vec{r}\cdot \vec{v}}{r^{3}}\right)\\
&=& \frac{k \,\vec{r}^{\,2}\vec{v} }{r^{3}} -
\frac{k(\vec{r}. \vec{v}) \vec{r}}{r^{3}} - \frac{k \vec{v}}{r} +
\frac{ k \vec{r}(\vec{r}\cdot \vec{v} ) }{r^{3}} \\
&=& 0
\end{eqnarray}