Finite Maxwell field and electric displacement Hamiltonians derived from a current dependent Lagrangian11 Jan 2018
In the common Ewald summation technique for the evaluation of electrostatic forces, the average electric field E is strictly zero. Finite uniform E can be accounted for by adding it as a new degree of freedom in an extended Lagrangian. Representing the uniform polarization P as the time integral of the internal current and E as the time derivative of a uniform vector field A, we define such an extended Lagrangian coupling A to the total current j_t(internal plus external) and hence derive a Hamiltonian resembling the minimal coupling Hamiltonian of electrodynamics. Next, applying a procedure borrowed from nonrelativistic molecular electrodynamics the j_t · A coupling is transformed to P · D form where D is the electric displacement acting as an electrostatic boundary condition. The resulting Hamiltonian is identical to the constant-D Hamiltonian obtained by Stengel, Spaldin and Vanderbilt (SSV) using thermodynamic arguments. The corresponding SSV constant E Hamiltonian is derived from an alternative extended Lagrangian.