An exponential estimate of the time of stability of nearly-integrable hamiltonian systems
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
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TL;DR: In this article, a general framework for introducing nonlinear corrections into ordinary quantum mechanics, that can serve as a guide to experiments that would be sensitive to such corrections, is presented, including the precession of spinning particles in external fields, experiments of Stern-Gerlach type, and the broadening and de-tuning of absorption lines.
446 citations
Book•
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
Cites background or result from "An exponential estimate of the time..."
...Arnold (1963c) and Nekhoroshev (1977) already worked in this direction, and, more recently, Guzzo and Morbidelli (1997) reformulated both theorems taking into account most features of small body dynamics....
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...A detailed description of how a Nekhoroshev result can be established goes beyond the scope of this section, and we report just a brief guideline. The interested reader can refer to Guzzo and Morbidelli (1997). In the Nekhoroshev domain, one first considers the subdomains where there are no mean motion resonances of Fourier order smaller than 1/ε....
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...An analogous study has been recently done by Nesvorný and Roig (2001) for the Kuiper belt....
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...2, the correct attitude for approaching the investigation of a given dynamical system is to search for an answer to the following questions: has the system the Nekhoroshev structure? In the negative case, at which order do resonances overlap? Unfortunately, it is not an easy task to answer these questions in an analytic way. This can be understood with the following qualitative argument, inspired by the work of Arnold (1963b). The width of a resonant domain scales as the square root of the coefficient of the corresponding resonant harmonic, i....
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TL;DR: In this paper, the authors reinterpreted the Landau damping phenomenon in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism.
Abstract: Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.
367 citations
TL;DR: In this paper, a brief review of the Fermi-pasta-Ulam paradox is given, together with its suggested resolutions and its relation to other physical problems, focusing on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective.
Abstract: A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Bose-particles are also considered.
280 citations
TL;DR: In this article, a review is devoted to an examination of quantum systems possessing stochasticity in the classical limit, and the problem of quantization of systems fulfilling the stochasticallyity condition is connected both with a wide class of problems which are physical in principle (destruction of quantum numbers due to interactions, statistics of the energy spectrum, kinetic description and so on) and with applications (molecular dynamics, interaction of atoms and molecules with a strong radiation field and such on).
269 citations
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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.
Abstract: The problem of the existence of a third isolating integral of motion in an axisymmetric potential is investigated by numerical experiments. It is found that the third integral exists for only a limited rage of initial conditions.
1,728 citations
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.
Abstract: CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systemsChapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional caseChapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincare variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of § 5. The many-body problemChapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences
1,057 citations
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.
Abstract: CONTENTS § 1. Introduction § 2. Formulation of the theorems § 3. Proofs § 4. Technical lemmas § 5. Appendix. The rotatory motion of a heavy asymmetric rigid bodyReferences
845 citations
Journal Article•
756 citations