Geodesic
The Maupertuis--Jacobi Principle for Hamiltonians of the Form~$F(x,|p|)$ in Two-Dimensional Stationary Semiclassical Problems
Matematičeskie zametki, Tome 97 (2015) no. 1, pp. 48-57.

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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.
Mots-clés : Maupertuis–Jacobi correspondence principle, Dirac equation
Keywords: Lagrangian manifold, Maslov canonical operator, Hamiltonian, Schrödinger equation, Hamiltonian flow, surface water wave, pseudodifferential equation. 
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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. Matematičeskie zametki, Tome 97 (2015) no. 1, pp. 48-57. http://geodesic.mathdoc.fr/item/MZM_2015_97_1_a4/

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