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Mat. Zametki, 2001, Volume 70, Issue 5, Pages 660–669 (Mi mz778)  

This article is cited in 13 scientific papers (total in 13 papers)

Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes

J. Brüninga, S. Yu. Dobrokhotovb, M. A. Poteryakhinc

a Humboldt University
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
c Russian Research Centre "Kurchatov Institute"

Abstract: In this paper we consider an analytic Hamiltonian system differing from an integrable system by a small perturbation of order $\varepsilon$. The corresponding unperturbed integrable system is degenerate with proper and limit degeneracy: all variables, except two, are at rest and there is an elliptic singular point in the plane of these two variables. It is shown that by an analytic symplectic change of the variable, which is $O(\varepsilon)$-close to the identity substitution, the Hamiltonian can be reduced to a form differing only by exponentially small ($O(e^{-\operatorname{const}/\varepsilon})$) terms from the Hamiltonian possessing the following properties: all variables, except two, change slowly at a rate of order $\varepsilon$ and for the two remaining variables the origin is the point of equilibrium; moreover, the Hamiltonian depends only on the action of the system linearized about this equilibrium.


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English version:
Mathematical Notes, 2001, 70:5, 599–607

Bibliographic databases:

UDC: 517
Received: 04.04.2001

Citation: J. Brüning, S. Yu. Dobrokhotov, M. A. Poteryakhin, “Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes”, Mat. Zametki, 70:5 (2001), 660–669; Math. Notes, 70:5 (2001), 599–607

Citation in format AMSBIB
\by J.~Br\"uning, S.~Yu.~Dobrokhotov, M.~A.~Poteryakhin
\paper Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes
\jour Mat. Zametki
\yr 2001
\vol 70
\issue 5
\pages 660--669
\jour Math. Notes
\yr 2001
\vol 70
\issue 5
\pages 599--607

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    This publication is cited in the following articles:
    1. J. Brüning, S. Yu. Dobrokhotov, M. A. Poteryakhin, “Integral Representation of Analytical Solutions of the Equation $yf_x'-xf_y'=g(x,y)$”, Math. Notes, 72:4 (2002), 583–585  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. J. Brüning, S. Yu. Dobrokhotov, K. V. Pankrashin, “The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field”, Theoret. and Math. Phys., 131:2 (2002), 704–728  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Bruning, J, “The spectral asymptotics of the two-dimensional Schrodinger operator with a strong magnetic field. II”, Russian Journal of Mathematical Physics, 9:4 (2002), 400  crossref  mathscinet  zmath  isi  scopus
    4. Bruning, J, “The spectral asymptotics of the two-dimensional Schrodinger operator with a strong magnetic field. I”, Russian Journal of Mathematical Physics, 9:1 (2002), 14  mathscinet  zmath  isi
    5. Gelfreich, VG, “The dynamical properties of a singularly perturbed Hamiltonian system near its slow manifold”, Doklady Mathematics, 66:3 (2002), 403  zmath  isi
    6. Gelfreich V, Lerman L, “Long-periodic orbits and invariant tori in a singularly perturbed Hamiltonian system”, Physica D-Nonlinear Phenomena, 176:3–4 (2003), 125–146  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Dobrokhotov S.Yu., Minenkov D.S., “On Various Averaging Methods for a Nonlinear Oscillator with Slow Time-Dependent Potential and a Nonconservative Perturbation”, Regul. Chaotic Dyn., 15:2-3 (2010), 285–299  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. L. A. Kalyakin, “Analysis of the Bloch equations for the nuclear magnetization model”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 64–81  mathnet  crossref  isi  elib
    9. A. Yu. Anikin, J. Brüning, S. Yu. Dobrokhotov, “Averaging and trajectories of a Hamiltonian system appearing in graphene placed in a strong magnetic field and a periodic potential”, J. Math. Sci., 223:6 (2017), 656–666  mathnet  crossref  mathscinet  elib
    10. L. A. Kalyakin, “Adiabatic approximation for a Model of Cyclotron Motion”, Math. Notes, 101:5 (2017), 850–862  mathnet  crossref  crossref  mathscinet  isi  elib
    11. L. A. Kalyakin, “Adiabatic approximation in a resonance capture problem”, Ufa Math. J., 9:3 (2017), 61–75  mathnet  crossref  isi  elib
    12. L. A. Kalyakin, “Resonance capture in a system of two oscillators near equilibrium”, Theoret. and Math. Phys., 194:3 (2018), 331–346  mathnet  crossref  crossref  isi  elib
    13. Kalyakin L.A., “Capture and Keeping of a Resonance Near Equilibrium”, Russ. J. Math. Phys., 26:2 (2019), 152–167  crossref  isi
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