  In response to W. G. Hoover's comment [arXiv:1204.0312v2] on our work
[arXiv:1203.5968], we show explicitly that the divergence of the velocity field
associated with the Nos\'e-Hoover equations is nonzero, implying that those
equations are not volume preserving, and hence, as often stated in the
literature, are not Hamiltonian. We further elucidate that the trajectories
{q(t)} generated by the Nos\'e-Hoover equations are generally not identical to
those generated by Dettmann's Hamiltonian. Dettmann's Hamiltonian produces the
same trajectories as the Nos\'e-Hoover equations only on a specific energy
shell, but not on the neighboring ones. This fact explains why the
Nos\'e-Hoover equations are not volume preserving. The Hamiltonian that we put
forward with [arXiv:1203.5968] instead produces thermostated dynamics
irrespective of the energy value. The main advantage of our Hamiltonian
thermostat over previous ones is that it contains kinetic energy terms that are
of standard form with coordinate-independent masses and consequently is readily
matched in laboratory experiments.
