L. NIEDERMAN
Exponential stability for small perturbations of steep integrable Hamiltonian systems
(15K, Postscript)
ABSTRACT. In the 70's, Nekhorochev proved that for an analytic
nearly integrable Hamiltonian system with a perturbation of size $\varepsilon$,
the actions linked to the unperturbed Hamiltonian vary only of the order of
$\varepsilon^b$ over a time of the order of ${\rm exp}\left(
c\varepsilon^{-a}\right)$ for some positive constants $a$, $b$ and $c$,
provided that the unperturbed Hamiltonian meets certain {\it generic}
transversality conditions known as {\it steepness}. Among steep systems, convex
or quasi-convex systems are easier to analyze since the use of energy
conservation allows to shorten the proof of exponential estimates of stability.
In this case, Lochak and P${\ddot{\rm o}}$schel have obtained independently the
stability exponents $a=b=1/2n$ for a $n$ degrees of freedom system, especially
the time exponent ($a$) is expected to be optimal (see [LMS]). Moreover, the
study of Lochak relies on simultaneous Diophantine approximation which gives a
very transparent proof.
On the other hand, the proof in the steep case has almost not been resumed
since Nekhorochev original work despite various physical examples where the
model Hamiltonian is only steep. Here, we combine the original scheme with
simultaneous Diophantine approximation as in Lochak's proof. This yields
significant simplifications with respect to Nekhorochev's reasoning, it also
allows to obtain the exponents $a=b=\left( 2n p_1\ldots p_n\right)^{-1}$ where
$\left( p_1\ldots p_n\right)$ are the steepness indices of the considered
Hamiltonian. In the quasi-convex case the steepness indices are all equal to
one, hence we recover $a=b=1/2n$ and our result generalize in the steep case
those of Lochak and P${\ddot{\rm o}}$schel.