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The steep Nekhoroshev’s theorem. (English) Zbl 1337.37038

The authors revisit Nekhoroshev’s theorem on the exponential stability of nearly integrable real-analytic Hamiltonian systems [N. N. Nekhoroshev, Usp. Mat. Nauk 32, No. 6(198), 5–66 (1977; Zbl 0383.70023); Tr. Semin. Im. I. G. Petrovskogo 5, 5–50 (1979; Zbl 0473.34021)]. In general terms, exponential stability means stability of the action variables over times that are exponentially long with the inverse of the size of the perturbation. The Hamiltonian in the standard action-angle coordinates is \[ H(I,\varphi)= h(I)+\varepsilon f(I,\varphi) \] with \((I,\varphi)\in U\times\mathbb{R}^n\), where \(U\) is an open region in \(\mathbb{R}^n\), \(\mathbb{T}^n\) is the standard flat torus, and \(\varepsilon\) is a small parameter. The authors prove that the exponential stability exponent can be taken to be \(a=1/(2n\alpha_i,\dots, \alpha_{n-2})\), where the \(\alpha_i\) are Nekhoroshev’s steepness indices and \(n\geq 3\). Thus the authors improve the dependence of \(a^{-1}\) on \(n\), reducing it from Nekhoroshev’s quadratic to linear dependence. They conjecture that the new stability exponent is optimal, and they produce a heuristic argument in support.
The authors note that Nekhorovshev’s proof consists of geometric analytic and stability pieces, and that special attention to the geometric piece was crucial in achieving their new result.

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
70F15 Celestial mechanics
70E50 Stability problems in rigid body dynamics
70H05 Hamilton’s equations
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