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Erratum for ``Drift and diffusion in phase space''
No full textStatements c),d) on p. 71 and the reference to them in the
following remark are not correct: this is due to a computational error
(claiming that is exponentially small) in the corresponding
proof in Appendix A13 referred to on p. 76 (-th paragraph)
On the measure of kam tori in two degrees of freedom
No full textA conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the “non-torus” set in analytic systems with two degrees of freedom is discussed
Global properties of generic real–analytic nearly–integrable Hamiltonian systems
No full textWe introduce a new class Gns of generic real analytic potentials on Tn and study global analytic properties of natural nearly-integrable Hamiltonians 12|y|2 + epsilon f (x), with potential f is an element of Gns, on the phase space M = B x Tn with B a given ball in Rn. The phase space M can be covered by three sets: a 'non-resonant' set, which is filled up to an exponentially small set of measure e-cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a 'simply resonant set' of measure root epsilon Ka and a third set of measure epsilon Kb which is 'non perturbative', in the sense that the H-dynamics on it can be described by a natural system which is not nearly-integrable. We then focus on the simply resonant set - the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) - and show that on such a set the secular (averaged) 1 degree-of-freedom Hamiltonians (labeled by the resonance index k is an element of Zn) can be put into a universal form (which we call 'Generic Standard Form'), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k. (c) 2023 Elsevier Inc. All rights reserved
On the topology of nearly-integrable Hamiltonians at simple resonances
No full textWe show that, in general, averaging at simple resonances a real-analytic, nearly-integrable Hamiltonian, one obtains a one-dimensional system with a cosine-like potential; 'in general' means for a generic class of holomorphic perturbations and apart from a finite number of simple resonances with small Fourier modes; 'cosine-like' means that the potential depends only on the resonant angle, with respect to which it is a Morse function with one maximum and one minimum. Furthermore, the (full) transformed Hamiltonian is the sum of an effective one-dimensional Hamiltonian (which is, in turn, the sum of the unperturbed Hamiltonian plus the cosine-like potential) and a perturbation, which is uniformly exponentially small. As a corollary, under the above hypotheses, if the unperturbed Hamiltonian is also strictly convex, the effective Hamiltonian at any simple resonance (apart a finite number of low-mode resonances) has the phase portrait of a pendulum. The results presented in this paper are an essential step in the proof (in the 'mechanical' case) of a conjecture by Arnold-Kozlov-Neishdadt [Arnold V I, Kozlov V V and Neishtadt A I 2006 Mathematical aspects of classical and celestial mechanics Encyclopaedia of Mathematical Sciences 3rd edn vol 3 (Berlin: Springer), remark 6.8, p 285], claiming that the measure of the 'non-torus set' in general nearly-integrable Hamiltonian systems has the same size of the perturbation; compare [Biasco L and Chierchia L 2015 On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems Rendiconti Lincei. Mat. Appl. 26 1-10 and Biasco L and Chierchia L KAM Theory for Secondary Tori (arXiv:1702.06480v1 [math.DS])]
V. I. Arnold’s “Pointwise” KAM Theorem
No full textWe review V. I. Arnold’s 1963 celebrated paper [1] Proof of A. N. Kolmogorov’s Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp” asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed
Measures of basins of attraction in spin-orbit dynamics
No full textWe consider a dissipative spin-orbit model where it is assumed that the orbit of the satellite is Keplerian, the obliquity is zero, and the dissipative effects depend linearly on the relative angular velocity. The measure of the basins of attraction associated to periodic and quasi-periodic attractors is numerically investigated. The results depend on the interaction among the physically relevant parameters, namely, the orbital eccentricity, the equatorial oblateness and the dissipative constant. In particular, it appears that, for astronomically relevant parameter values, for low eccentricities (as in the Moon's case) about 96% of the initial data belong to the basin of attraction of the 1/1 spin-orbit resonance; for larger values of the eccentricities higher order spin-orbit resonances and quasi-periodic attractors become dominant providing a mechanism for explaining the observed state of Mercury into the 3/2 resonance
KAM Stability and Celestial Mechanics
No full textKAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to "physical systems" for "observable" values of the perturbation parameters.
We consider the Restricted, Circular, Planar, Three-Body Problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000.
We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCP3BP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points "close" to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion.
The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system.
The paper is self-contained but does not include the (similar to 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors
Properly-degenerate KAM theory (following V.I. Arnold)
No full textArnold’s “Fundamental Theorem” on properly–degenerate systems
[3, Chapter IV] is revisited and improved with particular attention to the relation
between the perturbative parameters and to the measure of the Kolmogorov
set. Relations with the planetary many–body problem are shortly
discussed
Planetary Birkhoff normal forms
Full text linkBirkhoff normal forms for the (secular) planetary problem are investigated. Existence and uniqueness is discussed and it is shown the classical Poincare variables and the RPS-variables (introduced in [6]), after a trivial lift, lead to the same Birkhoff normal form; as a corollary the Birkhoff normal form (in Poincare variables) is degenerate at all orders (answering a question of M. Herman). Non-degenerate Birkhoff normal forms for partially and totally reduced cases are provided and an application
to long-time stability of secular action variables (eccentricities and inclinations) is discussed
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