 9292 Chierchia L., Gallavotti G.
 Drift and diffusion in phase space
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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, 581585 (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 socalled 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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