Chierchia L.
On the stability problem for nearly--integrable Hamiltonian systems
(31K, Plain TeX)
ABSTRACT. {\bf Abstract:} {\it The problem of stability of the action variables in
nearly--integrable (real--analytic) Hamiltonian systems is considered.
Several results (fully described in {\rm [Chierchia-Gallavotti deposited in
mp_arc (1992)]}) are discussed; in particular:
(i) a generalization of Arnold's method ({\rm [Arnold (1966)]})
allowing to prove instability
(i.e. drift of action variables by an amount of order $1$, often called
``Arnold's diffusion") for general perturbations of ``a--priori unstable
integrable systems" (i.e. systems for which the integrable structure
carries separatrices); (ii) Examples of perturbations of ``a--priori stable
sytems" (i.e. systems whose integrable part can be completely described by
regular
action--angle variables) exhibiting instability. In such examples, inspired
by the ``D'Alembert problem" in Celestial Mechanics (treated, in full
details, in {\rm [Chierchia-Gallavotti (1992)]}),
the splitting of the asymptotic manifolds is not exponentially
small in the perturbation parameter.}