Journal ArticleDOI
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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Polynomial normal forms with exponentially small remainder for analytic vector fields
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TL;DR: In this article, it was shown that for any vector field in R m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small.
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Frequency map analysis and particle accelerators
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Numerical computation of normal forms around some periodic orbits of the restricted three-body problem
Àngel Jorba,Jordi Villanueva +1 more
TL;DR: In this paper, the authors introduce a general methodology for computing the normal form around a periodic orbit of an autonomous real analytic Hamiltonian system, which is carried out up to some finite order and, neglecting the remainder, they obtain an accurate description of the dynamics in a small enough neighborhood of the orbit.
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Nekhoroshev stability at L4 or L5 in the elliptic‐restricted three‐body problem – application to Trojan asteroids
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Book ChapterDOI
Arnold Diffusion; a Compendium of Remarks and Questions
TL;DR: The problem of the instability of multidimensional, close to integrable Hamiltonian systems is a notoriously difficult one; the underlying phenomenon now usually goes under the name "Arnold diffusion" paying tribute to the oldest and still most significant paper in the subject.
References
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Journal ArticleDOI
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.