Hamiltonian mechanics of generalized eikonal waves

Abstract

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 $\Lambda$ in the ray phase space, (ii) a density $\mu$ on $\Lambda$, and (iii) an overall phase factor $\phi$. 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 $(\Lambda,\mu,\phi)$ are identitfied. Because we work in terms of the Keller-Maslov global WKB ansatz, as opposed to the more restrictive $\psi=a \exp(i S/\epsilon)$, 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.