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I'm trying to understand the equations of two-body motion. Namely, given the position, velocity and mass of two orbiting bodies at time $t$, how can I explicitly find their position and velocity for any arbitrary time?

First place I looked was the Wikipedia article, which I followed until it got to "Solving the equation for $\mathbf r(t)$ is the key to the two-body problem; general solution methods are described below." Below, it talked about the motion being planar and/or a "central force", but I couldn't figure out how to get an $\mathbf r(t)$ function out of anything there.

The question two-body problem circular orbits seems relevant, but only answers a specific sort of case.

Finally, I found this article. I feel like what I'm looking for might be hidden in here, possibly equations (17) and (18). But I can't manage to get an $\mathbf r(t)$ out of them (is there a relationship between $\mathbf r(t)$ and $\dfrac{\mathrm d \mathbf r}{\mathrm dt}$?)

Any help would be appreciated. Please forgive me if this is blindingly obvious. Many thanks.

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    For the time being, I'll keep working at this. I guess it involves constructing an orbital definition from the general position, velocity and mass of the bodies.2011-10-11

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