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Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable hamiltonian systems. A review.
"... : Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the invariant tori of the rotators system at fixed rotation numbers: ..."
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Cited by 51 (12 self)
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: Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the invariant tori of the rotators system at fixed rotation numbers: we exhibit cancellations, to all orders of perturbation theory, that allow proving the stability and analyticity of the dipohantine tori. We find in this way a proof of the KAM theorem by direct bounds of the kth order coefficient of the perturbation expansion of the parametric equations of the tori in terms of their average anomalies: this extends Siegel's approach, from the linearization of analytic maps to the KAM theory; the convergence radius does not depend, in this case, on the twist strength, which could even vanish ("twistless KAM tori"). The same ideas apply to the case in which the potential couples the pendulum and the rotators: in this case the invariant tori with diophantine...
Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics  A review with some applications
, 1995
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Stability of motions near resonances in quasiintegrable Hamiltonian systems
 J. Stat. Phys
, 1986
"... Nekhoroshev's theorem on the stability of motions in quasiintegrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonan ..."
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Cited by 22 (4 self)
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Nekhoroshev's theorem on the stability of motions in quasiintegrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.
A Proof of Existence of Whiskered Tori With Quasi Flat Homoclinic Intersections in a Class of Almost Integrable Hamiltonian Systems
, 1995
"... : Rotators interacting with a pendulum via small, velocity independent, potentials are considered: the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds ("whiskers"), whose intersections define a set of homoclinic points. The homoclinic splitting ca ..."
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Cited by 18 (6 self)
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: Rotators interacting with a pendulum via small, velocity independent, potentials are considered: the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds ("whiskers"), whose intersections define a set of homoclinic points. The homoclinic splitting can be introduced as a measure of the splitting of the stable and unstable manifolds near to any homoclinic point. In a previous paper, [G1], cancellation mechanisms in the perturbative series of the homoclinic splitting have been investigated. This led to the result that, under suitable conditions, if the frequencies of the quasi periodic motion on the tori are large, the homoclinic splitting is smaller than any power in the frequency of the forcing ("quasi flat homoclinic intersections"). In the case l = 2 the result was uniform in the twist size: for l ? 2 the discussion relied on a recursive proof, of KAM type, of the whiskers existence, (so loosing the uniformity in the twist size). Here ...
Whiskered tori with prefixed frequencies and Lyapunov spectrum
, 1995
"... : A classical mechanics problem, as the existence of whiskered tori for an almost integrable hamiltonian system, is analyzed with techniques reminiscent of the quantum field theory, following the strategy developed in recent works about the matter. The system consists in a collection of rotators int ..."
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Cited by 16 (14 self)
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: A classical mechanics problem, as the existence of whiskered tori for an almost integrable hamiltonian system, is analyzed with techniques reminiscent of the quantum field theory, following the strategy developed in recent works about the matter. The system consists in a collection of rotators interacting with a pendulum via a small potential depending only on the angle variables. The proof of the existence of the stable and unstable manifolds ("whiskers") of the rotators invariant tori corresponding to diophantine rotation numbers is simplified by setting the Lyapunov spectrum to prefixed values via the introduction, in the hamiltonian function, of "counterterms" depending on the strength of the interaction; this is a feature usual in quantum field theory, and emphasizes the analogy between the the field theory and the KAM framework pointed out already in the mentioned works. Key words: KAM, perturbation theory, classical mechanics, quantum field theory, renormalization group 1. In...
Forni: Transport properties of kicked and quasiperiodic
, 1998
"... to appear in the Journal of Statistical Physics We study transport properties of Schrödinger operators depending on one or more parameters. Examples include the kicked rotor and operators with quasiperiodic potentials. We show that the mean growth exponent of the kinetic energy in the kicked rotor ..."
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Cited by 12 (0 self)
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to appear in the Journal of Statistical Physics We study transport properties of Schrödinger operators depending on one or more parameters. Examples include the kicked rotor and operators with quasiperiodic potentials. We show that the mean growth exponent of the kinetic energy in the kicked rotor and of the mean square displacement in quasiperiodic potentials is generically equal to 2: this means that the motion remains ballistic, at least in a weak sense, even away from the resonances of the models. Stronger results are obtained for a class of tightbinding Hamiltonians with an electric field E(t) = E0 + E1 cosωt. For H = ∑ an−k(  n − k>< n  +  n>< n − k ) + E(t)  n>< n  with an ∼  n  −ν (ν> 3/2) we show that the mean square displacement satisfies < ψt, N2ψt> ≥ Cǫt2/(ν+1/2)−ǫ for suitable choices of ω, E0 and E1. We relate this behaviour to the spectral properties of the Floquet operator of the problem.
Arnold’s diffusion in isochronous systems
 Math. Phys. Anal. Geom
, 1998
"... Abstract: an illustration of a mechanism for Arnold’s diffusion following a nonvariational approach and finding explicit estimates for the diffusion time. Keywords: Arnold’s diffusion, homoclinic splitting, KAM ..."
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Cited by 12 (0 self)
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Abstract: an illustration of a mechanism for Arnold’s diffusion following a nonvariational approach and finding explicit estimates for the diffusion time. Keywords: Arnold’s diffusion, homoclinic splitting, KAM
Between classical and quantum
, 2005
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
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Cited by 12 (3 self)
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The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
Twistless KAM tori.
, 1994
"... : A selfcontained proof of the KAM theorem in the Thirring model is discussed. Keywords: KAM, invariant tori, classical mechanics, perturbation theory, chaos I shall particularize the Eliasson method, [E], for KAM tori to a special model, of great interest, whose relevance for the KAM problem was p ..."
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Cited by 9 (5 self)
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: A selfcontained proof of the KAM theorem in the Thirring model is discussed. Keywords: KAM, invariant tori, classical mechanics, perturbation theory, chaos I shall particularize the Eliasson method, [E], for KAM tori to a special model, of great interest, whose relevance for the KAM problem was pointed out by Thirring, [T] (see [G] for a short discussion of the model). The idea of exposing Eliasson's method through simple particular cases appears in [V], where results of the type of the ones discussed here, and more general ones, are announced. The connection between the methods of [E] and the tree expansions in the renormalization group approaches to quantum field theory and many body theory can be found also in [G[. The connection between the tree expansions and the breakdown of invariant tori is discussed in [PV]. The Thirring model is a system of rotators interacting via a potential. It is described by the hamiltonian (see [G] for a motivation of the name): 1 2 J \Gamma1 ~ A ...