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In order to simulate rigidly rotating polytropes we have simulated systems of$N$ point particles, with $N$ up to 1800. Two particles at a distance $r$interact by an attractive potential $-1/r$ and a repulsive potential $1/r^2$.The repulsion simulates the pressure in a polytropic gas of polytropic index$3/2$. We take the total angular momentum $L$ to be conserved, but not thetotal energy $E$. The particles are stationary in the rotating coordinatesystem. The rotational energy is $L^2/(2I)$ where $I$ is the moment of inertia.Configurations where the energy $E$ has a local minimum are stable. In thecontinuum limit $N\to\infty$ the particles become more and more tightly packedin a finite volume, with the interparticle distances decreasing as $N^{-1/3}$.We argue that $N^{-1/3}$ is a good parameter for describing the continuumlimit. We argue further that the continuum limit is the polytropic gas of index$3/2$. For example, the density profile of the nonrotating gas approaches thatcomputed from the Lane--Emden equation describing the nonrotating polytropicgas. In the case of maximum rotation the instability occurs by the loss ofparticles from the equator, which becomes a sharp edge, as predicted by Jeansin his study of rotating polytropes. We describe the minimum energy nonrotatingconfigurations for a number of small values of $N$.

Rigidly rotating gravitationally bound systems of point particles, compared to polytropes