Solution to the Two-Body Problem: Cross-Product and Dot-Product Integration

AI Thread Summary
The discussion centers on solving the two-body problem using cross-product and dot-product integration methods. The initial equation involves the acceleration of a body influenced by gravitational force, leading to the momentum vector's definition. The solution process includes taking the cross-product with the momentum vector and integrating with respect to time, but participants express confusion over the integration steps and the treatment of variables. There is a notable disagreement on the integration results, particularly regarding the treatment of the gravitational term and the variables involved. The conversation highlights the complexity of deriving the position and angle functions over time, emphasizing the need for careful variable management during integration.
TimeRip496
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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/r2 reduces it to GM/r and how to cross product GM/r2 with h since they are unknown variables?
 
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What's the objective here? To find ##r(t)## and ##\theta(t)##? If so, the solution is well-known...Landau / Lifshitz "Mechanics" has a good summary if you've never seen it.
 
TimeRip496 said:
I try following his step but I get a different integration result
$$\dot{\textbf{r}}\times\textbf{h}=GM(\frac{\textbf{r}}{r^3}t\times\textbf{h}+\textbf{e})$$
How did you get the first term on the right: ##GM(\frac{\textbf{r}}{r^3}t\times\textbf{h}) \,##?

Keep in mind that ##\mathbf r## and ##r## are functions of time. You cannot treat them as constants when integrating with respect to time.
 
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