Hamiltonian mechanics of generalized eikonal waves

الأصل · EN

In accordance with the Keller-Maslov global WKB theory, a semiclassical scalar wave field is best encoded as a triple consisting of (i) a Lagrangian submanifold Λ in the ray phase space, (ii) a density μ on Λ, and (iii) an overall phase factor ϕ. We present the Hamiltonian structure of the Cauchy problem for such a "geometric semiclassical state" in the special case where the wave operator is Hermetian. Variational, symplectic, and Poisson formulations of the time evolution equations for (Λ,μ,ϕ) are identitfied. Because we work in terms of the Keller-Maslov global WKB ansatz, as opposed to the more restrictive ψ=a (i S/ε), all of our results are insensitive to the presence of caustics. In particular, because the variational principle is insensitive to caustics, the latter may be used to construct structure-perserving numerical integrators for scalar wave equations.