Consider a collection of particles scattered with approximate uniform density in a sphere. Each particle travels in a circular orbit around a central point mass. The particles do not influence each other gravitationally. This is easy to set up in simulation. For each particle, pick an orbital inclination uniformly randomly.

Next, consider a collection of particles all with the same speed but traveling in uniformly random directions. Easiest first is probably to consider a spherical shell of such particles. Place a point mass at the center and let gravity do its work. Again, none of the particles interact with each other so this is a simple 2-body problem repeated for each particle. Some particles may escape. For a given initial velocity and shell radius, what is the spatial distribution of (non-escaped) particles averaged over time, as a function of distance from the central point? Add some more shells to get a somewhat uniform density collection of particles (within a certain radius) buzzing around the central mass in elliptical orbits.

The point of all of this is then to add another massive body to make it a 3-body problem. Dust will get ejected from certain regions, collect in other regions (Lagrange points).

Useful are synplectic integrators, e.g., Verlet.

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