  We investigate the off-equilibrium dynamics of a spin system with O($N$)
symmetry in $2 < d < 4$ spatial dimensions arising by the presence of a slowly
varying time-dependent magnetic field $h(t,t_s) \sim t/t_s$, $t_s$ is a time
scale, at the critical temperature $T = T_c$ and below it $T < T_c$. After
showing the general theory, we demonstrate the off-equilibrium scaling and we
formally compute the correlation functions in the limit of large $N$. We derive
the off-equilibrium scaling relations for the hysteresis loop area and for the
magnetic work done by the system when the magnetic field $h(t,t_s)$ is varied
across the phase transitions cyclically in time. We also investigate the first
deviations from the equilibrium behavior in the correlation functions checking
the consistence for an exponential approach.
