  A model for synchronization of globally coupled phase oscillators including
``inertial'' effects is analyzed. In such a model, both oscillator frequencies
and phases evolve in time. Stationary solutions include incoherent
(unsynchronized) and synchronized states of the oscillator population. Assuming
a Lorentzian distribution of oscillator natural frequencies, $g(\Omega)$, both
larger inertia or larger frequency spread stabilize the incoherent solution,
thereby making harder to synchronize the population. In the limiting case
$g(\Omega)=\delta(\Omega)$, the critical coupling becomes independent of
inertia. A richer phenomenology is found for bimodal distributions. For
instance, inertial effects may destabilize incoherence, giving rise to
bifurcating synchronized standing wave states. Inertia tends to harden the
bifurcation from incoherence to synchronized states: at zero inertia, this
bifurcation is supercritical (soft), but it tends to become subcritical (hard)
as inertia increases. Nonlinear stability is investigated in the limit of high
natural frequencies.
