Dynamical systems admitting normal shift and wave equations

Abstract

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton equations being special case for Newton's equations. In simplest cases (e. g. light propagation in isotropic non-homogeneous refracting media) the dynamics of points preserves orthogonality of wave front and rays. This property was generalized in the theory of dynamical systems admitting normal shift of hypersurfaces. In present paper inverse problem is studied, i. e. the problem of reconstructing wave equation corresponding wave front dynamics for which is described by Newtonian dynamical system admitting normal shift of hypersurfaces in Riemannian manifold.