A reduction of a three-dimensional initial-boundary value problem of mechanics of solids to a two-dimensional initial-boundary value problem of N -th order shell theory is performed on the groundwork of variational principles of the analytical continuum mechanics. The shell model is formulated as a two-dimensional continuum (a material base surface) with a set of generalized coordinates (field variables), and the boundary conditions on the shell faces are translated to the base surface, therefore they become supplementary constraints for the field variables of the first kind. The dynamic equations are constructed as Lagrange equations of the second kind of the continuum mechanical system using the Lagrange multipliers method. The obtained equations are invariant with respect to the base functions of the thickness coordinate and similar to the ones of the simplified N -th order shell theory. The new generalized forces defined on the basis of the variational formulation of the initial-boundary value problem contain the additive components with Lagrange multipliers. The constructed problem’s statement for the N -th order shell theory secures the accurate definition of the physical constants for lowest order shell models.