Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory : Part II

As in Part I of this paper, we consider the problem of the energy exchanges between two subsystems, of which one is a system of ν harmonic oscillators, while the other one is any dynamical system of n degrees of freedom. Such a problem is of interest both for the realization of holonomic constraints of classical mechanics, and for the freezing of the internal degrees of freedom in molecular collisions. The results of Part I, which referred to the particular case ν=1, are here extended to the more difficult case ν>1. For the rate of energy transfer we find exponential estimates of Nekhoroshev's type, namely of the form exp (λ*/λ)1/a, where λ is a positive real number giving the size of the involved frequencies, and λ* and a are constants. For the particularly relevant constant a we find in general a=1/ν however, in the particular case when the ν frequencies are equal (collision of identical molecules), we find a=1 independently of ν, as conjectured by Jeans in the year 1903.