The variational principle and effective action for a spherical dust shell

The variational principle for a spherical configuration consisting of a thin spherical dust shell in a gravitational field is constructed. The principle is consistent with the boundary-value problem of the corresponding Euler-Lagrange equations, and leads to "natural boundary conditions." These conditions and the field equations following from the variational principle are used for performing of the reduction of this system. The equations of motion for the shell follow from the obtained reduced action. The transformation of the variational formula for the reduced action leads to two natural variants of the effective action. One of them describes the shell from a stationary interior observer's point of view, another from the exterior one. The conditions of isometry of the exterior and interior faces of the shell lead to the momentum and Hamiltonian constraints.