Quasijets in anisotropic media, Finsler geometry, and Fermi coordinates

The Hamilton–Jacobi equations for the phase function of the quasijet solutions in the case of Finsler geometry are considered in the paper. This case corresponds to the physical probleom of waves propagation in anisotropic media. The wave field corresponding to the quasijet solution propagate along a geodesic. Due to that all considerations of the paper are provided in the Fermi coordinates close to the geodesic. The quadratic term of the phase function after extracting the frequency factor satisfy the covariant Riccati equation. Especially simple form for the equation is obtained for the case of Riemannian geometry. Nontrivial coefficients of the Riccati equation coincide with elements of the curvature tensor. In the case of Findsler geometry all considerations are more complicated. Nevertheless, the main role in the Riccati equations play elements of the third Kartan curvature tensor computen on the tangential elements to the geodesic.