The Hamiltonian description of the motion of arbitrary discontinuity surfaces is proposed on the basis of the variational principle, taking into account the conservation laws in terms of consecutively introduced volume potentials (of the Clebsch type) as constraints. Such a method of introduction of Hamiltonian variables makes it possible to generalize the known canonical variables for the interface between two media to the cases of shock waves and slip surfaces. The results are compared with the introduction of surface Hamiltonian variables through the canonical transformations of bulk Hamiltonian variables. The results permit a direct generalization to the case of magnetohydrodynamics, plasmas, superfluid liquids, and other media for which the bulk Hamiltonian equations are known. ©1997 American Institute of Physics.