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A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems. (English) Zbl 0602.58022

Let \(n\) be an integer \(>1\), \(\mathcal G\) a bounded, open and convex domain of \(\mathbb R^ n\) and \(\mathbb T^ n\) the \(n\)-dimensional real torus. For vectors \(v\in\mathbb C^ n\) we set \(\| v\| =\max_ i | v_ i|\). If \(\rho\) and \(\sigma\) are positive constants, let \(\mathcal D\) be the complex thickening of \(\mathcal G\times\mathbb T^ n\) relative to \(\rho\) and \(\sigma\). The norm \(\| f\|_{\mathcal D}\) of an analytic function \(f\) on \(\mathcal D\) is defined by \[ \| f\|_{{\mathcal D}}=\sup_{(p,q)\in {\mathcal D}}| f(p,q)|. \] If \(\delta\) is a positive constant, \(\mathcal G\)-\(\delta\) denotes the set of points contained, together with a \(\delta\)-neighborhood, in \(\mathcal G\). The following notations and assumptions are also needed for the precise statement of the main result of this paper:
(i) An analytic function \(H(p,q)\), which may be written in the form \(H(p,q)= h(p)+ f(p,q)\), is defined on \(\mathcal D\). There exist constants \(E\) and \(\varepsilon\) such that \(\| H\|_{\mathcal D}\leq E\) and \(\| f\|_{\mathcal D}\leq \varepsilon E\).
(ii) A matrix \(A(p)\) is defined by \(A(p)_{ij}=\partial^ 2h/\partial p_ i\partial p_ j\). There exists a constant \(M\) such that \(\| A(p)v\|_{{\mathcal D}}\leq M \| v\|\) for all \(v\in\mathbb C^ n.\)
(iii) For each pair \(\{v,w\}\) of vectors in \(\mathbb R^ n\) we set \(v\cdot w=\sum_{i}v_ iw_ i\). There exists a constant \(m\) such that \(| A(p)v\cdot v| \geq mv\cdot v\) for all \(p\in\mathcal G\) and all \(v\in\mathbb R^ n.\)
(iv) Assume that \(\sigma\leq 1\) and \(\rho\leq \sqrt{E/M}\). Set \(c=4(n^ 2+2n+2)\) and \[ \varepsilon_ 0=\min\left((M\rho^ 2/E)^ 2,\;(\sigma /8n)^{4c}(m/M)^{8n}\right). \]
Define positive dimensional constants \(\mathcal P\) and \(\mathcal T\) by \(\mathcal P=\sqrt{EM}/m\) and \(\mathcal T=1/\sqrt{EM}\). Let \(\varepsilon\) be such that \(0<\varepsilon <\varepsilon_ 0\), and define further constants \(\Delta\) and \(T\) by \(\Delta =2(n+1)\mathcal P\varepsilon^{1/c}\) and \(T=(\mathcal T/\sqrt{\varepsilon}) \exp (\varepsilon^{- 1/4c})\).
Solutions of Hamiltonian equations with Hamiltonian \(H\) are called “real motions”.
With these notations, and under these assumptions, the authors’ main conclusion may be stated as follows:
For all real motions \((p(t),q(t))\) with \(p(0)\in\mathcal G\text{-}2\Delta\), and for any \(t\in [0,T]\), the inequality \(\| p(t)-p(0)\| \leq \Delta\) holds.
Stability estimates of nearly integrable Hamiltonian systems for finite but “large” times have been given by N. N. Nekhoroshev [Usp. Mat. Nauk 32, No. 6(198), 5–66 (1977; Zbl 0383.70023)]. In that work, the geometrical assumption of steepness on the “unperturbed Hamiltonian” \(h\), which is similar to a condition introduced by J. Glimm [Commun. Pure Appl. Math. 17, 509–526 (1964; Zbl 0125.40804)] (cf. esp. p. 521) was used. Hypothesis (iii) of the present paper amounts to a uniform convexity property for \(h\) more restrictive than the steepness condition, but it permits substantial simplifications in the so-called “analytic part” of Nekhoroshev’s proof [N. N. Nekhoroshev, Tr. Semin. Im. I. G. Petrovskogo 5, 5–50 (1979; Zbl 0473.34021)], a paper that the authors indicate does not seem to have been translated from the Russian (meanwhile see [Topics in modern mathematics, Petrovskii Semin. 5, 1–58 (1985; Zbl 0668.34046)] (the editor)).
The physical relevance of analogues of the Kolmogorov-Arnold-Moser stability theorem valid on “good” domains for finite but large times has been described by the authors elsewhere [Nature 311, 444–446 (1984); cf. also Nuovo Cim. B 89, 89–102, 103–119 (1985)].

MSC:

37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
70H08 Nearly integrable Hamiltonian systems, KAM theory
70F15 Celestial mechanics
37C75 Stability theory for smooth dynamical systems
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References:

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