elementary orbital mechanics from first principles

Suppose a body at position \( \vec{r} \) is acted on by a large mass \( M \) (at the origin). Then the force on the body is given by

\[ \vec{F} = -\frac{MG}{\verts{\vec{r}}^2}\vec{r} , \]

where \( G \) is the gravitational constant. (The force is negative because it is pulling the body towards the origin). By Newton's second law, we have

\[ \vec{F} = m\vec{a} , \]

where \( m \) is the body's mass and \( \vec{a} \) is the body's acceleration. Since \( \vec{a} = \frac{d^2\vec{r}}{dt^2} \), we have the differential equation

\[ m\frac{d^2\vec{r}}{dt^2} = -\frac{MG}{\verts{\vec{r}}^2}\vec{r} , \]

or

\[ \frac{d^2\vec{r}}{dt^2} = \frac{MG}{m\verts{\vec{r}}^2} \vec{r} . \]