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Mat. Zametki, 2015, Volume 97, Issue 1, Pages 48–57 (Mi mz10569)  

This article is cited in 7 scientific papers (total in 7 papers)

The Maupertuis–Jacobi Principle for Hamiltonians of the Form $F(x,|p|)$ in Two-Dimensional Stationary Semiclassical Problems

S. Yu. Dobrokhotovab, D. S. Minenkovab, M. Rouleuxcd

a Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
b A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow
c Université du Sud Toulon-Var, France
d Centre de Physique Théorique, France

Abstract: We consider two-dimensional asymptotic formulas based on the Maslov canonical operator arising in stationary problems for differential and pseudodifferential equations. In the case of Lagrangian manifolds invariant with respect to Hamiltonian flow with Hamiltonians of the form $F(x,|p|)$, we show how asymptotic formulas can be simplified by using the well-known (in classical mechanics) Maupertuis–Jacobi correspondence principle to replace the Hamiltonians $F(x,|p|)$ by Hamiltonians of the form $C(x)|p|$ arising, in particular, in geometric optics and related to the Finsler metric. As examples, we consider Hamiltonians corresponding to the Schrödinger equation, the two-dimensional Dirac equation, and the pseudodifferential equations for surface water waves.

Keywords: Maupertuis–Jacobi correspondence principle, Lagrangian manifold, Maslov canonical operator, Hamiltonian, Schrödinger equation, Dirac equation, Hamiltonian flow, surface water wave, pseudodifferential equation.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00521
Ministry of Education and Science of the Russian Federation МК-1017.2013.1
This work was supported by the Russian Foundation for Basic Research (grant no. 14-01-00521) and by the Grant of the President of the Russian Federation (grant no. MK-1017.2013.1).


DOI: https://doi.org/10.4213/mzm10569

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English version:
Mathematical Notes, 2015, 97:1, 42–49

Bibliographic databases:

Document Type: Article
UDC: 517.9
Received: 29.08.2014

Citation: S. Yu. Dobrokhotov, D. S. Minenkov, M. Rouleux, “The Maupertuis–Jacobi Principle for Hamiltonians of the Form $F(x,|p|)$ in Two-Dimensional Stationary Semiclassical Problems”, Mat. Zametki, 97:1 (2015), 48–57; Math. Notes, 97:1 (2015), 42–49

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    This publication is cited in the following articles:
    1. S. Yu. Dobrokhotov, A. Cardinali, A. I. Klevin, B. Tirozzi, “Maslov complex germ and high-frequency Gaussian beams for cold plasma in a toroidal domain”, Dokl. Math., 94:1 (2016), 480–485  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “New integral representations of the Maslov canonical operator in singular charts”, Izv. Math., 81:2 (2017), 286–328  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. Tirozzi, “Scalarization of stationary semiclassical problems for systems of equations and its application in plasma physics”, Theoret. and Math. Phys., 193:3 (2017), 1761–1782  mathnet  crossref  crossref  adsnasa  isi  elib
    4. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rouleux, “The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides”, Dokl. Math., 96:1 (2017), 406–410  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    5. Anikin A.Yu. Dobrokhotov S.Yu. Nazaikinskii V.E. Rouleux M., “Asymptotics of Green Function For the Linear Waves Equations in a Domain With a Non-Uniform Bottom”, Proceedings of the International Conference Days on Diffraction (Dd) 2017, ed. Motygin O. Kiselev A. Goray L. Suslina T. Kazakov A. Kirpichnikova A., IEEE, 2017, 18–23  isi
    6. A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. Tirozzi, “Gausian packets and beams with focal points in vector problems of plasma physics”, Theoret. and Math. Phys., 196:1 (2018), 1059–1081  mathnet  crossref  crossref  adsnasa  isi  elib
    7. Reijnders K.J.A., Minenkov D.S., Katsnelson I M., Dobrokhotov S.Yu., “Electronic Optics in Graphene in the Semiclassical Approximation”, Ann. Phys., 397 (2018), 65–135  crossref  mathscinet  zmath  isi  scopus
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