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

by Giancarlo Benettin, Luigi Galgani and Antonio Giorgilli

**Abstract:**
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
$\nu$ 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 $\nu=1$, are
here extended to the more difficult case $\nu\gt 1$. For the rate of
energy transfer we find exponential estimates of Nekhoroshev's type,
namely of the form $\exp(\lambda_*/\lambda)^{1/a}$, where $\lambda$ is a
positive real number giving the size of the involved frequencies, and
$\lambda_*$ and $a$ are constants. For the particularly relevant
constant $a$ we find in general $a=1/\nu$; however, in the particular
case when the $\nu$ frequencies are equal (collision of identical
molecules), we find $a=1$ independently of $\nu$, as conjectured by
Jeans in the year 1903.

Comm. Math. Phys, **121**, 557-601 (1989)

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