  A harmonic oscillator linearly coupled with a linear chain of Ising spins is
investigated. The $N$ spins in the chain interact with their nearest neighbours
with a coupling constant proportional to the oscillator position and to
$N^{-1/2}$, are in contact with a thermal bath at temperature $T$, and evolve
under Glauber dynamics. The oscillator position is a stochastic process due to
the oscillator-spin interaction which produces drastic changes in the
equilibrium behaviour and the dynamics of the oscillator. Firstly, there is a
second order phase transition at a critical temperature $T_c$ whose order
parameter is the oscillator stable rest position: this position is zero above
$T_c$ and different from zero below $T_c$. This transition appears because the
oscillator moves in an effective potential equal to the harmonic term plus the
free energy of the spin system at fixed oscillator position. Secondly, assuming
fast spin relaxation (compared to the oscillator natural period), the
oscillator dynamical behaviour is described by an effective equation containing
a nonlinear friction term that drives the oscillator towards the stable
equilibrium state of the effective potential. The analytical results are
compared with numerical simulation throughout the paper.
