THE LAGRANGIAN FUNCTION AND SEPARATION OF VARIABLES FOR ELASTIC VIBRATIONS IN AXIALLY SYMMETRIC LAYERED MEDIAA. N. Kuznetsov
In frequently encountered cases of isotropic and, generally, inhomogeneous media, classical separation of variables in equations of linear elastodynamics is generally not applicable. It is therefore important to find special cases where variables can be separated. The separation of variables was carried out previously for axially symmetric motions in cases of flat, spherically, and cylindrically layered media. Sufficient conditions for such separation of variables were also suggested. It is also known that a restricted number of cases exists where separation of variables for a homogeneous body is compatible with its geometry. It is proved here that the classical separation of variables exists, even locally, in axially symmetric cases only for bodies with spherical, cylindrical, or flat layering. The statement of forward and inverse problems can be simplified in these cases, because the relevant equations of elastodynamics are reducible to sets of ordinary differential equations. The proof is based on methods of differential geometry and tensor calculus. Simple rules are deduced from basic properties of the Lagrangian function and Euler operator to separate variables in the Lagrangian function when it exists. In general, the result of the separation does not have a Lagrangian function. However, in the three cases mentioned above the result has a Lagrangian function, hence the separation results in a selfadjoint system of equations. That offers additional opportunities to investigate, among other items, variational symmetries and hence conservation laws. Also found are cases of nearly layered media where the separation of variables is possible, but the equations of motion are not selfadjoint, hence cannot be derived as the Euler-Lagrange equations for a Lagrangian function.