Computer Studies of Energy Sharing and Ergodicity for Nonlinear Oscillator Systems

Weakly coupled systems of N oscillators are investigated using Hamiltonians of the form $$\mathrm{H=}\frac{1}{2}{\displaystyle \sum _{\text{k=1}}^N}{\mathrm{(p}}_k^2{\mathrm{+\omega }}_k^2{q}_k^2\mathrm{)+\alpha }{\displaystyle \sum _{\text{j,k,l=1}}^N}{A}_{\mathrm{jkl}}{q}_j{q}_k{q}_l,$$ where the A_jkl are constants and where α is chosen to be sufficiently small that the coupling energy never exceeds some small fraction of the total energy. Starting from selected initial conditions, a computer is used to provide exact solutions to the equations of motion for systems of 2, 3, 5, and 15 oscillators. Various perturbation schemes are used to predict, interpret, and extend these computer results. In particular, it is demonstrated that these systems can share energy only if the uncoupled frequencies ω_k satisfy resonance conditions of the form $${\displaystyle \sum }{n}_k{\omega }_k\mathrm{\hspace{0.17em}\lesssim \alpha }$$ for certain integers n_k determined by the particular coupling. It is shown that these systems have N normal modes, where a normal mode is defined as motion for which each oscillator moves with essentially constant amplitude and at a given frequency or some harmonic of this frequency. These systems are shown to have, at least, one constant of the motion, analytic in q, p, and α, other than the total energy. Finally, it is demonstrated that the single‐oscillator energy distribution density for a 5‐oscillator linear and nonlinear system has the Boltzmann form predicted by statistical mechanics. Thus, these nonlinear systems are shown to have many features in common with linear systems. In particular, it is unlikely that they are ergodic. From the standpoint of statistical mechanics, it is argued that this lack of ergodicity may be a welcome feature.