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Nekhoroshev estimates for quasi-convex Hamiltonian systems. (English) Zbl 0857.70009

The paper is concerned with the stability of motions in nearly integrable autonomous Hamiltonian systems with \(n\) degrees of freedom, with Hamiltonian \(H=h(I)+ f_\varepsilon(I,\theta)\) in \(n\)-dimensional action-angle variables \(I\), \(\theta\) depending on a small parameter \(\varepsilon\) such that \(f_\varepsilon\sim\varepsilon\). Provided that the system is real analytic and that the unperturbed Hamiltonian \(h\) fulfills generic transversality conditions known as steepness, N. Nekhoroshev obtained certain stability bounds, called the “stability radius” and “stability time”, in order to prove that, generically, the variations of the actions of all orbits remain small over a finite but exponentially long time interval.
In this paper, the author gives simple and explicit expressions for these stability bounds in the case, when the unperturbed Hamiltonian \(h\) is assumed to be convex on all of its domain, or on each of its level sets, respectively. In order to obtain these results, he constructs normal forms on various subdomains of phase space known as “resonance blocks”, which are described as neighbourhoods of “admissible resonances”. Then, these normal forms lead to stability estimates, which turn out to be straightforward in the convex cases. Finally, a geometric argument is given how to cover the whole phase space by such resonance blocks. This last geometric construction is precisely a new one: resonance zones are described not through small divisor conditions, but they are given as tubular neighbourhoods of exact resonances, for which the required small divisor estimates are subsequently verified. This leads to a simpler and essentially optimal covering of action space by resonance blocks. In turn, this results in a substantial improvement of the stability exponent.

MSC:

70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C75 Stability theory for smooth dynamical systems
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References:

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