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

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.

DOI: https://doi.org/10.4213/mzm778

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

Bibliographic databases:

UDC: 517

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
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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
5. Gelfreich, VG, “The dynamical properties of a singularly perturbed Hamiltonian system near its slow manifold”, Doklady Mathematics, 66:3 (2002), 403
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
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
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
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
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