  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 the
total energy $E$. The particles are stationary in the rotating coordinate
system. 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 the
continuum limit $N\to\infty$ the particles become more and more tightly packed
in 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 continuum
limit. 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 that
computed from the Lane--Emden equation describing the nonrotating polytropic
gas. In the case of maximum rotation the instability occurs by the loss of
particles from the equator, which becomes a sharp edge, as predicted by Jeans
in his study of rotating polytropes. We describe the minimum energy nonrotating
configurations for a number of small values of $N$.
