TITLE:
Effective stability and KAM theory
AUTHORS:
Amadeu Delshams(1) and Pere Gutierrez(2)
(1) Dept. de Matematica Aplicada I (ETSEIB), Universitat Politecnica de
Catalunya, Diagonal 647, 08028 Barcelona (Spain).
E-mail: amadeu@ma1.upc.es
(2) Dept. de Matematica Aplicada II, Universitat Politecnica de
Catalunya, Pau Gargallo 5, 08071 Barcelona (Spain).
E-mail: gutierrez@ma2.upc.es
ABSTRACT:
The two main stability results for nearly-integrable
Hamiltonian systems are revisited:
Nekhoroshev theorem, concerning exponential lower
bounds for the stability time (effective stability),
and KAM theorem, concerning the preservation of a
majority of the nonresonant invariant tori
(perpetual stability).
To stress the relationship between both theorems,
a common approach is given to their proof, consisting
of bringing the system to a normal form constructed
through the Lie series method.
The estimates obtained for the size of the remainder
rely on bounds of the associated vectorfields,
allowing to get the ``optimal'' stability exponent
in Nekhoroshev theorem for quasiconvex systems.
On the other hand, a direct and complete proof of
the isoenergetic KAM theorem is obtained.
Moreover, a modification of the proof leads to
the notion of nearly-invariant torus, which constitutes
a bridge between KAM and Nekhoroshev theorems.
JOURNAL:
J. Differential Equations, 128:415-490, 1996.