10.5 Newton’s Planetary Motion

$$\begin{align*} \vec{r} &= \left[ x, y \right] \\ \vec{v} &= \left[ v_x, v_y \right] \\ \vec{s} &= \left[ \vec{r}, \vec{v} \right] = \left[ x, y, v_x, v_y \right] \\ d \vec{r} &= \vec{v} dt \\ d \vec{v} &= \vec{a} dt \\ \vec{v} &= \frac{d \vec{r}}{dt} \\ \vec{a} &= \frac{d \vec{v}}{dt} = \frac{ \vec{F} }{m} = - \frac{GM}{ \left| r \right| {}^3 } \vec{r} \\ \because \vec{F} &= - \frac{GmM}{ \left| r \right| {}^3 } \vec{r} \end{align*}$$