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## Homework Statement

Two-body problem given as

$$\ddot{\textbf{r}}+\frac{GM}{r^2}\frac{\textbf{r}}{r}=0$$

$$\textbf{h}=\textbf{r}\times\dot{\textbf{r}}$$

where the moment of the momentum vector mh

## Homework Equations

The vector solution to the above equation may be obtained by first taking the cross-product with the constant h and integrating once with respect to time. This yields

$$\dot{\textbf{r}}\times\textbf{h}=GM(\frac{\textbf{r}}{r}+\textbf{e})$$

The final solution to the equation is then obtained by taking the dot product of above equation with r, is

$$r=\frac{h^2/(GM)}{1+ecos\theta}$$

## The Attempt at a Solution

I have no idea what the author is doing. How does cross product with the momentum and then dot product with r solve the equation?

I try following his step but I get a different integration result

$$\dot{\textbf{r}}\times\textbf{h}=GM(\frac{\textbf{r}}{r^3}t\times\textbf{h}+\textbf{e})$$

I have no idea how his integration w.r.t. time for the GM/r

^{2}reduces it to GM/r and how to cross product GM/r

^{2}with h since they are unknown variables?

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