92-92 Chierchia L., Gallavotti G.

Drift and diffusion in phase space (365K, Plain TeX) Jul 24, 92

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Abstract. \noindent {\bf Abstract:} {\it The problem of stability of the action variables (i.e. 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 [Arnold, Sov. Math. Dokl., 5, 581-585 (1966)], we work out a general quantitative theory, from the point of view of {\sl dimensional analysis}, for {\sl a priori unstable systems} (i.e. 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 $O(1)$. \noindent The above theory can be extended so as to cover cases of {\sl a priori stable systems} (i.e. 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). }

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