The Nose-Hoover, Dettmann, and Hoover-Holian Oscillators

To follow up recent work of Xio-Song Yang [1] on the Nose-Hoover oscillator [2-5] we consider Dettmann's harmonic oscillator [6l,7], which relates Yang's ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator [8] to relate our mechanical studies to Gibbs' statistical mechanics. All three oscillators are described by a coordinate q and a momentum p. Additional control variables (\zeta, \xi) vary the energy. Dettman's description includes a time-scaling variable s, as does Nose's original work [2,3]. Time scaling controls the rates at which the (q, p, \zeta) variables change. The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable s. Yang considered qualitative features of Nose-Hoover dynamics. He showed that longtime Nose-Hoover trajectories change energy, repeatedly crossing the \zeta = 0 plane. We use moments of the motion equations to give two new, different, and brief proofs of Yang's long-time limiting result.