Variational principle, characteristic electric multipoles, and higher polarizing moments in field theory

Based on the variational principle, we introduce a new notion: the characteristic electric multipoles constituting a system of basic distributions of charge on the boundary of a spatial domain. Inside the domain, potentials of the characteristic multipoles are harmonic polynomials whose orders determine the minimum orders of nonzero spherical multipole moments of the characteristic multipoles. Using the characteristic multipole formalism, we solve the moment problem in electrostatics and construct the superconductor Lagrangian in an electrostatic field. We express the empty-space Green's function for the Laplace equation using the characteristic multipole potentials.