Modeling acceleration due to gravity for large bodies as a function of time

[NJJ289]

Given two bodies where GM/x^2=A and Gm/x^2=a, how might one model the acceleration of either bodies as a function of time?

A simpler version of the problem involves one of the masses being stationary (just for the sake of simplicity), so that GM/x^2=A and Gm/x^2=0.

A more complicated version would involve more than two bodies in multiple planes of space such that x1, x2, x3... etc. are vector quantities in 3 dimensions.

An even more complicated and accurate version would involve using integrated mass (taking into consideration the density distributions) instead of point mass.

I smell some serious calculus here but I can't wrap my head around how to do it. If anyone could explain how to do the simplest version or even what is involved that would be awesome. Also, any web resources on the kind of math involved or similar problems would be appreciated. Thanks!

 

[timthereaper]

You could use the differential form of Gauss' Law of Gravity:

$$\nabla \cdot g = -4 \pi G \rho$$

You would have to specify the mass density at each point on the masses, but then you can obtain the gravitational vector field for each mass. That might help you get started.