  We investigate the stationary and dynamic properties of the celebrated
Nos\'e-Hoover dynamics of many-body interacting Hamiltonian systems, with an
emphasis on the effect of inter-particle interactions. To this end, we consider
a model system with both short- and long-range interactions. The Nos\'e-Hoover
dynamics aims to generate the canonical equilibrium distribution of a system at
a desired temperature by employing a set of time-reversible, deterministic
equations of motion. A signature of canonical equilibrium is a single-particle
momentum distribution that is Gaussian. We find that the equilibrium properties
of the system within the Nos\'e-Hoover dynamics coincides with that within the
canonical ensemble. Moreover, starting from out-of-equilibrium initial
conditions, the average kinetic energy of the system relaxes to its target
value over a {\it size-independent} timescale. However, quite surprisingly, our
results indicate that under the same conditions and with only long-range
interactions present in the system, the momentum distribution relaxes to its
Gaussian form in equilibrium over a scale that {\it diverges with the system
size}. On adding short-range interactions, the relaxation is found to occur
over a timescale that has a much weaker dependence on system size. This
system-size dependence of the timescale vanishes when only short-range
interactions are present in the system. An implication of such an ultra-slow
relaxation when only long-range interactions are present in the system is that
macroscopic observables other than the average kinetic energy when estimated in
the Nos\'e-Hoover dynamics may take an unusually long time to relax to its
canonical equilibrium value. Our work underlines the crucial role that
interactions play in deciding the equivalence between Nos\'e-Hoover and
canonical equilibrium.
