  We analyze the equilibrium properties of a chain of ferromagnetically coupled
rotators which interact through a force that decays as $r^{-\alpha}$ where r is
the interparticle distance and $\alpha\ge 0$. Our model contains as particular
cases the mean field limit ($\alpha=0$) and the first-neighbor model ($\alpha
\to \infty$). By integrating the equations of motion we obtain the
microcanonical time averages of both the magnetization and the kinetic energy.
Concerning the long-range order, we detect three different regimes at low
energies, depending on whether $\alpha$ belongs to the intervals $[0,1)$,
$(1,2)$ or $(2,\infty)$. Moreover, for $0 \le \alpha < 1$, the microcanonical
averages agree, after a simple scaling, with those obtained in the canonical
ensemble for the mean-field XY model. This correspondence offers a
mathematically tractable and computationally economic way of dealing with
systems governed by slowly decaying long-range interactions.
