  We present a pair-wise force law in a system of N particles that produces
analytic solutions for arbitrary number of particles, masses, and initial
conditions. Each pair of particles interacts via a force that is proportional
to the product of their masses and their separation distance, with the force
directed radially. We show that, despite the N-body interaction, each particle
behaves as if it interacts only with the center of mass of the system. This
effective two-body interaction behaves as Hooke's Law with a common frequency
for all particles, with the familiar analytic solutions for the trajectories.
With these analytic solutions, it is possible to efficiently simulate a
collection of these particles and incorporate other external forces. As an
example, we simulate the particles within an adiabatically expanding container
and calculate pressure and temperature in both the attractive and repulsive
cases.
