Appendix

Table of equations

Quantity	Equation
Ellipse	\(r=\dfrac{p}{1+e\cos\theta}\)
Area law	\(\dot A=\frac12 r^{2}\dot\theta =\frac{h}{2}= \text{const}\)
Harmonic law	\(T^{2}\propto a^{3}\)
Gravity	\(\vec F_{21} = -\dfrac{Gm_{1}m_{2}}{r^{2}}\hat{u_r}\)
Reduced EoM	\(\ddot{\vec r}=-\dfrac{\mu}{r^{3}}\vec r\)
Specific angular momentum	\(\vec h=\vec r\times\dot{\vec r}\)
Laplace–Runge–Lenz vector	\(\vec e=\dfrac{\dot{\vec r}\times\vec h}{\mu}-\dfrac{\vec r}{r}\)

Identities derivation

1.

We know that: \(\vec{r}\cdot\vec{r}=r^2\). Then : \[ \frac{d}{dt}(\vec{r} \cdot \vec{r})=2r \frac{dr}{dt}=2r\dot{r} \]

But,

\[ \frac{d}{dt}(\vec r\cdot \vec r)= \vec r\cdot \frac{d\vec r}{dt}+\frac{d\vec r}{dt}\cdot \vec r= 2\vec r\cdot \frac{d\vec r}{dt}=2\vec r\cdot \dot{\vec r} \]

Hence:

\[ \boxed{\vec r \cdot\dot{\vec r}=r\dot r} \]