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The Maupertuis-Jacobi principle for Hamiltonians of the form F(x, |p|) in two-dimensional stationary semiclassical problems

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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.

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Authors and Affiliations

  1. Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

    S. Yu. Dobrokhotov & D. S. Minenkov

  2. Moscow Institute of Physics and Technology, Moscow, Russia

    S. Yu. Dobrokhotov & D. S. Minenkov

  3. CNRS, CPT, UMR 7332, Aix Marseille Université, 13288, Marseille, France

    M. Rouleux

  4. CNRS, CPT, UMR 7332, Université de Toulon, 83957, La Garde, France

    M. Rouleux

Authors
  1. S. Yu. Dobrokhotov
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  2. D. S. Minenkov
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  3. M. Rouleux
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Correspondence to S. Yu. Dobrokhotov.

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Original Russian Text © S. Yu. Dobrokhotov, D. S. Minenkov, M. Rouleux, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 1, pp. 48–57.

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Dobrokhotov, S.Y., Minenkov, D.S. & Rouleux, M. The Maupertuis-Jacobi principle for Hamiltonians of the form F(x, |p|) in two-dimensional stationary semiclassical problems. Math Notes 97, 42–49 (2015). https://doi.org/10.1134/S0001434615010058

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  • DOI: https://doi.org/10.1134/S0001434615010058

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