 9568 Delshams A., Guti\'errez P.
 Effective stability and KAM theory
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Feb 15, 95

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Abstract. The two main stability results for nearlyintegrable 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 nearlyinvariant
torus, which constitutes a bridge between KAM and Nekhoroshev theorems.
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