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

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.