# Finding $\mathbf r(t)$ for the parameterized two-body equations of motion

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. – Raskolnikov Oct 11 '11 at 9:49
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. – Cephron Oct 11 '11 at 9:52
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? (If so, see also Kepler orbit.) – joriki Oct 11 '11 at 10:36
Those links were helpful, and also this question: math.stackexchange.com/questions/21864/… 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. – Cephron Oct 11 '11 at 20:57
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. – Cephron Oct 11 '11 at 20:59