FM 92-1, Annales de l' Institut Poincar\`e, B 60, 1--144, 1994,
Title: Drift and diffusion in phase space
Authors: Luigi Chierchia, Giovanni Gallavotti
Abstract: The problem of stability of the action variables (\ie of the
adiabatic invariants) in perturbations of completely integrable (real
analytic) hamiltonian systems with more than two degrees of freedom is
considered. Extending the analysis of [A], we work out a general
quantitative theory, from the point of view of {\sl dimensional
analysis}, for {\sl a priori unstable systems} (\ie systems for which
the unperturbed integrable part possesses separatrices), proving, in
general, the existence of the so--called Arnold's diffusion and
establishing upper bounds on the time needed for the perturbed action
variables to {\sl drift} by an amount of $O(1)$.
The above theory can be extended so as to cover cases of {\sl a priori
stable systems} (\ie systems for which separatrices are generated near
the resonances by the perturbation). As an example we consider the
``D'Alembert precession problem in Celestial Mechanics" (a planet
modelled by a rigid rotational ellipsoid with small ``flatness" $\h$,
revolving on a given Keplerian orbit of eccentricity $e=\h^c$, $c>1$,
around a fixed star and subject only to Newtonian gravitational forces)
proving in such a case the existence of Arnold's drift and diffusion;
this means that there exist initial data for which, for any $\h\neq 0$
small enough, the planet changes, in due ($\h$--dependent) time, the
inclination of the precession cone by an amount of $O(1)$. The
homo/heteroclinic angles (introduced in general and discussed in detail
together with homoclinic splittings and scatterings) in the D'Alembert
problem are not exponentially small with $\h$ (in spite of first order
predictions based upon Melnikov type integrals).
Keywords: perturbed hamiltonian systems, stability theory,
Arnold's diffusion, homoclinic splitting, heteroclinic trajectories, KAM
theory, whiskered tori, dimensional estimates, Celestial Mechanics,
D'Alembert Equinox Precession problem.
Fisica, Universita' di Roma La Sapienza,
P.le Moro 2, 00185, Roma, Italia.
e-mail giovanni@ipparco.roma1.infn.it
e-mail luigi@matrm3.mat.uniroma3.it
tel. 6-49914370, fax 6-4957697