I keep reading that cusps are one of the most "robust" predictions of N-body simulations of dark matter halos. But papers I've found on the topic make it sound like these simulations are complicated and slow and that the cusps are just an observed result of the simulations rather than that the cusps have some understandable physical origin. But my immediate reaction was that the cusps' origin is trivial (just the result of phase-space considerations) and that interaction-less dark matter simulations should be one of the easiest things in the world.
I remember writing a program to simulate non-interacting particles orbiting each-other with an inverse square law on an old PC over a decade ago, and I recall being able to get a pretty good picture of the density profile by tracing the paths of just a few particles. This only took a minute or two on a very slow PC. Indeed I remember something of a cusp being obvious, simply because the orbits of each particle tended to intersect near the CM, an effect that overrode the countervailing fact that individually each spent less time near the CM.
In fact, even without doing a simulation, I would assume this would be the case because Keplerian motion clearly indicates that the time spent near the CM drops close to linear in R while volume goes as R^3. Anyway, I'm just wondering if these observations are basically correct, or if the origin of the cusps is somehow more obscure/complicated.