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2013
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Volume 22, Number 1
On the 70th Birthday of Nikolaí N. Nekhoroshev (Guest Editors: Pol Vanhaecke and Stephen Wiggins)

Marino M.
On the Classical and Quantum Integrability of Systems of Resonant Oscillators
Abstract
We study in this paper systems of harmonic oscillators with resonant frequencies. For these systems we present general procedures for the construction of sets of functionally independent constants of motion, which can be used for the definition of generalized actionangle variables, in accordance with the general description of degenerate integrable systems which was presented by Nekhoroshev in a seminal paper in 1972. We then apply to these classical integrable systems the procedure of quantization which has been proposed to the author by Nekhoroshev during his last years of activity at Milan University. This procedure is based on the construction of linear operators by means of the symmetrization of the classical constants of motion mentioned above.
For 3 oscillators with resonance 1 : 1 : 2, by using a computer program we have discovered an exceptional integrable system, which cannot be obtained with the standard methods based on the obvious symmetries of the Hamiltonian function. In this exceptional case, quantum integrability can be realized only by means of a modification of the symmetrization procedure.
Keywords: integrable systems, resonant harmonic oscillators, noncommutative integrability, quantization
Citation: Marino M., On the Classical and Quantum Integrability of Systems of Resonant Oscillators, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 1-17
DOI:10.1134/S1560354717010014
Bambusi D.,  Fusè A.
Nekhoroshev Theorem for Perturbations of the Central Motion
Abstract
In this paper we prove a Nekhoroshev type theorem for perturbations of Hamiltonians describing a particle subject to the force due to a central potential. Precisely, we prove that under an explicit condition on the potential, the Hamiltonian of the central motion is quasiconvex. Thus, when it is perturbed, two actions (the modulus of the total angular momentum and the action of the reduced radial system) are approximately conserved for times which are exponentially long with the inverse of the perturbation parameter.
Keywords: Nekhoroshev theorem, central motion, Hamiltonian dynamics
Citation: Bambusi D.,  Fusè A., Nekhoroshev Theorem for Perturbations of the Central Motion, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 18-26
DOI:10.1134/S1560354717010026
Bolotin S. V.
Degenerate Billiards in Celestial Mechanics
Abstract
In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincaré.
Keywords: Hamiltonian system, billiard, celestial mechanics, collision, regularization, shadowing, action functional
Citation: Bolotin S. V., Degenerate Billiards in Celestial Mechanics, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 27-53
DOI:10.1134/S1560354717010038
Giorgilli A.,  Locatelli U.,  Sansottera M.
Secular Dynamics of a Planar Model of the Sun-Jupiter-Saturn-Uranus System; Effective Stability in the Light of Kolmogorov and Nekhoroshev Theories
Abstract
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, which can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that underlies the analytic part of Nekhoroshev’s theorem to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.
Keywords: $n$-body planetary problem, KAM theory, Nekhoroshev theory, normal form methods, exponential stability, Hamiltonian systems, celestial mechanics
Citation: Giorgilli A.,  Locatelli U.,  Sansottera M., Secular Dynamics of a Planar Model of the Sun-Jupiter-Saturn-Uranus System; Effective Stability in the Light of Kolmogorov and Nekhoroshev Theories, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 54-77
DOI:10.1134/S156035471701004X
Delshams A.,  Schaefer R. G.
Arnold Diffusion for a Complete Family of Perturbations
Abstract
In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system  of $2+1/2$ degrees of freedom  $H(p,q,I,\varphi,s) = p^{2}/2+\cos q -1 +I^{2}/2 + h(q,\varphi,s;\varepsilon)$ — proving that for   any small periodic perturbation of the form $h(q,\varphi,s;\varepsilon) = \varepsilon\cos q\left( a_{00} + a_{10}\cos\varphi + a_{01}\cos s  \right)$  ($a_{10}a_{01} \neq 0$) there is global instability for the action.  For the proof we apply a geometrical mechanism based in the so-called Scattering map. This work has the following structure: In a first stage, for a more restricted case ($I^*\thicksim\pi/2\mu$, $\mu = a_{10}/a_{01}$), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of the instability for any $\mu$). The bifurcations of the scattering map are also studied as a function of $\mu$. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.
Keywords: Arnold diffusion, normally hyperbolic invariant manifolds, scattering maps
Citation: Delshams A.,  Schaefer R. G., Arnold Diffusion for a Complete Family of Perturbations, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 78-108
DOI:10.1134/S1560354717010051