Dynamic systems admitting normal shift and wave equations (Q1417372)

Let \(M\) be an \(n\)-dimensional Riemannian manifold. For smooth symmetric tensors \(a_{r}\), \(r =0, \dots, m\) with components \(a^{k_{1}\cdots k_{r}} ( x^{1}, \dots , x^{n})\) (in a local coordinate system) consider the operator \(H(p, D)= \sum_{r=0 }^{m }a_{r}D^{r } \), where \(a_{r} D^{ r }= \sum_{k_{1}=1 }^{ n }\cdots \sum_{k_{r}=1 }^{ n } a^{k _{1}\cdots k_{r}} ( x^{1}, \dots , x ^{n}) \nabla_{k_{1}} \cdots \nabla_{k _{r}}\) and \(\nabla\) denotes the operator of covariant differentiation with respect to the metric connection in \(M\). Also consider the operator \(H(p, \lambda^{-1}D)= \sum_{r=0 }^{ m }\lambda^{ -r } a_{r}D^{ r }\), where \( \lambda \) is a large parameter. The main argument of the paper is a formal study of asymptotic solutions of the form \( u = \sum_{\alpha =0 } ^{ \infty} \varphi_{\alpha }/ (i \lambda ) ^{\alpha } \exp [ i \lambda S]\) where \( \varphi_{\alpha }\) and \(S\) are smooth scalar fields in \(M\). (This is thus the standard WKB method.) The ``phase'' \(S\) will then satisfy the associated Hamilton-Jacobi equation, etc. After a number of tedious calculations, the author establishes a relation between the phenomenon of wave motion and the theory of Newtonian dynamic systems admitting a normal shift of hypersurfaces, which (theory) had been previously developed, apparently mainly by himself and A. Yu. Boldin, in a series of papers.