Equipartition of Energy for Nonlinear Systems

Abstract

A system of harmonic oscillators weakly coupled by nonlinear forces will not achieve equipartition of energy as long as the uncoupled frequencies ωk are linearly independent on the integers, i.e., as long as there is no collection of integers {nk} for which Σnkωk=0 other than all nk=0. This result is shown to follow from the general form of the Kryloff and Bogoliuboff series solution to the equations of motion. Physically, the linear independence of the uncoupled frequencies means that none of the interacting oscillators drives another at its resonant frequency, and this lack of internal resonance precludes appreciable energy sharing in the limit as the coupling tends to zero. It is shown that the lack of equipartition of energy observed by Ulam, Fermi, and Pasta for certain nonlinear systems may be explained in terms of the preceding remarks. Moreover, a Kryloff and Bogoliuboff series solution to the appropriate equations of motion is shown to yield qualitative agreement with the Ulam, Fermi, and Pasta computer solution. Finally, a particular system of linear differential equations is solved which illustrates a mechanism whereby oscillator systems may achieve equipartition of energy.