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

To follow up recent work of Xiao-Song Yang [1] on the Nosé-Hoover oscillator [2-5] we consider Dettmann's harmonic oscillator [6,7], 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 (ζ, ξ) vary the energy.Dettmann's description includes a time-scaling variable s, as does Nosé's original work [2,3].Time scaling controls the rates at which the (q, p, ζ) variables change.The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable s.Yang considered qualitative features of Nosé-Hoover dynamics.He showed that longtime Nosé-Hoover trajectories change energy, repeatedly crossing the ζ = 0 plane.We use moments of the motion equations to give two new, different, and brief proofs of Yang's long-time limiting result.