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

Abstract

To follow up recent work of Xiao-Song Yang on the Nosé-Hoover oscillator we consider Dettmann's harmonic oscillator, which relates Yang's ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator 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. 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 {\it 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.