Nekhoroshev estimates for quasi-convex hamiltonian systems
About: This article is published in Mathematische Zeitschrift.The article was published on 1993-05-01. It has received 247 citations till now. The article focuses on the topics: Covariant Hamiltonian field theory & Superintegrable Hamiltonian system.
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01 Jan 2002
TL;DR: In this paper, the authors describe elementary celestial and Hamiltonian mechanics and a quasi-integrable Hamiltonian system for the detection of chaos in the universe. But they do not describe the physical structure of the world.
Abstract: 1. Elementary Celestial and Hamiltonian Mechanics 2. Quasi-Integrable Hamiltonian System 3. Kam Tori 4. Single Resonance Dyanmics 5. Numerical Tools for the Detection of Chaos 6. Interactions Among Resonances 7. Secular Dynamics of the Planets 8. Secular Dynamics of Small Bodies 9. Mean Motion Resonances 10. Three Body Resonances 11. Secular Dynamics Inside Mean Motion Resonances 12. Global Dynamical Structure of the Belts of Small Bodies.
401 citations
Book•
01 Dec 2003
TL;DR: This is a tutorial on some of the main ideas in KAM theory, an expanded version of the lectures given by the author in the Summer Research Institute on Smooth Ergodic Theory Seattle, 1999.
Abstract: This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the Summer Research Institute on Smooth Ergodic Theory Seattle, 1999. The style is pedagogical and expository and it only aims to be an introduction to the primary literature. It does not aim to be a systematic survey nor to present full proofs.
173 citations
Cites background from "Nekhoroshev estimates for quasi-con..."
...Let us just mention some of the developments very quickly: An elegant proof of the theorem based on approximation by periodic orbits [Loc92], the proof of what are conjectured to be the optimal exponents [LN92], [Pös93], [DG96] – the later paper contains a unified point of view for KAM and Nekhoroshev theorems – and the proof of Nekhoroshev estimates in a neighborhood of an elliptic fixed point [GFB98], [FGB98], [Nie98], [Pös]....
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..., [LN92], and in the proof of estimates in a neighborhood of KAM torus [FdlL92b], [PW94], [Pös93], See also [Val00] for general estimates in problems without small divisors....
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...Other proofs of Nekhoroshev theorems are covered in [Pös93], [Loc92]....
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TL;DR: In this paper, the authors studied the dynamics in the neighborhood of an invariant torus of a nearly integrable system and provided an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/σ)], σ being the distance from the invariant Torus.
Abstract: We study the dynamics in the neighborhood of an invariant torus of a nearly integrable system. We provide an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/σ)], σ being the distance from the invariant torus. We also discuss the connection of this result with the existence of many invariant tori close to the considered one.
155 citations
TL;DR: This work represents graphically the evolution of the set of resonances of a quasi-integrable dynamical system, the so-called Arnold web, whose structure is crucial for the stability properties of the system.
Abstract: We represent graphically the evolution of the set of resonances of a quasi-integrable dynamical system, the so-called Arnold web, whose structure is crucial for the stability properties of the system. The basis of our representation is the use of an original numerical method, whose definition is directly related to the dynamics of orbits, and the careful choice of a model system. We also show the transition from the Nekhoroshev stability regime to the Chirikov diffusive one, which is a generic nontrivial phenomenon occurring in many physical processes, such as slow chaotic transport in the asteroid belt and beam-beam interaction.
154 citations
TL;DR: In this article, the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems was studied, and it was shown that 1/2nα is the optimal exponent for the time of stability and b = 1 2n as an exponent for radius of confinement of the action variables.
Abstract: . – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.
119 citations
Cites methods from "Nekhoroshev estimates for quasi-con..."
...Finally, the Nekhoroshev Theorem holds with a = b = 1/2n if h is assumed to be quasi-convex, as proved independently by Lochak and Neishtadt [LN92,LNN93] and by Poschel [ Po93 ]....
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References
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TL;DR: The main ideas of the proof of the exponential estimate were discussed in this paper, including steepness conditions and forbidden motions of the discs of fast drift on the steepness of the unperturbed Hamiltonian.
Abstract: CONTENTS § 1 Introduction § 2 Unsolved problems Conjectures Generalizations § 3 The main ideas of the proof of the exponential estimate § 4 Steepness conditions Precise statement of the main theorem § 5 Forbidden motions § 6 Resonances Resonance zones and blocks § 7 Dependence of the diameters of the discs of fast drift on the steepness of the unperturbed Hamiltonian § 8 Condition for the non-overlapping of resonances § 9 Traps in frequency systems Completion of the proof of the main theorem § 10 Statement of the lemma on the elimination of non-resonance harmonics, and of the technical lemmas used in the proof of the main theorem § 11 Remarks on the proof of the main theorem § 12 Application of the main theorem to the many-body problem References
726 citations
"Nekhoroshev estimates for quasi-con..." refers background in this paper
...In his original paper [ 20 ] Nekhoroshev was mainly concerned with the generic steep case, so his estimates are not particularly sharp....
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...for all orbits, provided the hamiltonian H is real analytic, and the unperturbed hamiltonian h meets certain generic transversality conditions known as steepness [19, 20 , 21, 11]....
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TL;DR: In this paper, it was shown that in analytic systems with rapidly rotating phase this separation can be achieved so that the error is exponentially small and that the remaining small error is theoretically impossible to eliminate in any version of the averaging method.
360 citations
"Nekhoroshev estimates for quasi-con..." refers background in this paper
...Such a scheme first appeared in [ 18 ] in a related averaging problem, and the author is indebted to Anatolij Neishtadt for communicating this idea....
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348 citations