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 TMF, 2003, Volume 135, Number 3, Pages 378–408 (Mi tmf196)

Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

V. V. Belova, S. Yu. Dobrokhotovb, V. A. Maksimova

a Moscow State Institute of Electronics and Mathematics
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.

Keywords: isotropic tori, action–angle variables, semiclassical asymptotic approximations

DOI: https://doi.org/10.4213/tmf196

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English version:
Theoretical and Mathematical Physics, 2003, 135:3, 765–791

Bibliographic databases:

Citation: V. V. Belov, S. Yu. Dobrokhotov, V. A. Maksimov, “Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application”, TMF, 135:3 (2003), 378–408; Theoret. and Math. Phys., 135:3 (2003), 765–791

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. S. Yu. Dobrokhotov, M. A. Poteryakhin, “Normal Forms near Two-Dimensional Resonance Tori for the Multidimensional Anharmonic Oscillator”, Math. Notes, 76:5 (2004), 653–664
2. M. A. Poteryakhin, “Normal forms near an invariant torus and the asymptotic eigenvalues of the operator $\langle V,\nabla\rangle-\epsilon\Delta$”, Math. Notes, 77:1 (2005), 140–145
3. Albeverio S, Dobrokhotov S, Poteryakhin M, “On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods”, Asymptotic Analysis, 43:3 (2005), 171–203
4. Bruning J, Dobrokhotov SY, Semenov ES, “Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem”, Regular & Chaotic Dynamics, 11:2 (2006), 167–180
5. V. V. Belov, V. A. Maksimov, “Semiclassical quantization of Bohr orbits in the helium atom”, Theoret. and Math. Phys., 151:2 (2007), 659–680
6. Davila-Rascon G., Vorobiev Yu., “The First Step Normalization for Hamiltonian Systems With Two Degrees of Freedom Over Orbit Cylinders”, Electronic Journal of Differential Equations, 2009, 54
7. S. Yu. Dobrokhotov, M. Rouleux, “The Semiclassical Maupertuis–Jacobi Correspondence and Applications to Linear Water Waves Theory”, Math. Notes, 87:3 (2010), 430–435
8. Dobrokhotov S., Rouleux M., “The semi-classical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory”, Asymptot Anal, 74:1–2 (2011), 33–73
9. Bruening J., Dobrokhotov S.Yu., Sekerzh-Zen'kovich S.Ya., Tudorovskiy T.Ya., “Spectral Series of the Schrodinger Operator in a Thin Waveguide with a Periodic Structure. 2. Closed Three-Dimensional Waveguide in a Magnetic Field”, Russian Journal of Mathematical Physics, 18:1 (2011), 33–53
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