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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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    Do you have any knowledge of differential equations? If not, I think the first step is to take a course in that subject.2011-10-11
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    I've taken Calculus 1 to 3 during my undergrad, and was suffering pretty badly by Cal3. I'm doing my best to engage the math here, but I am by no means proficient.2011-10-11
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    Note that "two-body problem" refers to a general problem of two bodies under arbitrary forces. The reason you didn't find a concrete solution in that article is that the article is only about that very general problem. Since you say "orbiting", I'm wondering whether you're in fact interested in the [Kepler problem](http://en.wikipedia.org/wiki/Kepler_problem)? (If so, see also [Kepler orbit](http://en.wikipedia.org/wiki/Kepler_orbit).)2011-10-11
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    Those links were helpful, and also this question: http://math.stackexchange.com/questions/21864/solving-short-trigo-equation-with-sine-need-some-help However, the mathematical machinery for kepler orbits seems to assume that you're looking for time since perihelion, and you already know the semi-major axis, etc. In other words, given an already-defined orbit, where on it will the planet be at time t? Ultimately, I'm hoping to obtain a function that takes a general m1, m2, x1(t), v1(t), x2(t), v2(t), and dt as input and spits out x1(t+dt), v1(t+dt), x2(t+dt) and v2(t+dt) as a result.2011-10-11
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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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