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An exponential estimate of the time of stability of nearly-integrable hamiltonian systems
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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 Referencesread more
Citations
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Recent progress and outstanding problems in Hamiltonian dynamics
TL;DR: In this article, progress in Hamiltonian dynamics over the last five years is reviewed, some outstanding problems are identified, and recent work with Aubry on "discrete breathers" is summarised.
The N-body problem
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Approximate invariant manifolds up to exponentially small terms
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Composition of Lie transforms with rigorous estimates and applications to Hamiltonian perturbation theory
TL;DR: In this article, a rigorous perturbation theory for nearly integrable Hamiltonian systems, based on the composition of Lie Transforms, is presented. But the perturbations are not considered in this paper, and the authors focus on a particular model-example, namely a weakly coupled harmonic oscillators having Diophantine frequencies.
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Optimal Stability and Instability for Near-Linear Hamiltonians
TL;DR: In this article, the authors prove a general result of stability for perturbations of linear integrable Hamiltonian systems, and construct an example of instability showing that both their result and their example are optimal.
References
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The applicability of the third integral of motion: Some numerical experiments
Michel Henon,Carl Heiles +1 more
TL;DR: In this paper, the existence of a third isolating integral of motion in an axisymmetric potential was investigated by numerical experiments and it was found that the third integral exists for only a limited rage of initial conditions.
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Small denominators and problems of stability of motion in classical and celestial mechanics
TL;DR: In this paper, the authors consider the problem of proper degeneracy and prove the existence of a non-degeneracy of diffeomorphisms with respect to a constant number of vertices.
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Proof of a theorem of a.?n.?kolmogorov on the invariance of quasi-periodic motions under small perturbations of the hamiltonian
TL;DR: In this paper, the rotatory motion of a heavy asymmetric rigid body is studied and the theorems of the rotational motion of such a rigid body are formulated and proved.