The Maupertuis–Jacobi Principle and the Fermat Variational Principle in the Problem of Short-Wavelength Asymptotics of the Solution of the Helmholtz Equation with a Localized Source

The paper considers the problem of short-wavelength asymptotics of the Helmholtz equation with a localized right-hand side in the form of a rapidly decreasing function. An algorithm is presented for calculating rays by the variational method and the wave field by the Maslov canonical operator method for given boundary conditions. This approach is used for model examples, including those with a logarithmic singularity of the ray family. In addition, we consider applications of the variational method for calculating rays in the illuminated region and in the caustic shadow region.