Abstract
The Hamiltonian formalism for nonrelativistic charged particles in an electromagnetic field is shown to be manifestly gauge invariant. The Hamiltonian for the subsystem consisting of the interacting particles is shown to be form invariant under gauge transformation, but numerically dependent on the choice of gauge. The energy, which is numerically the same in all gauges, is obtained by subtracting the scalar potential of the time-dependent radiation field from the Hamiltonian. The time rate of change of the energy of the particles is equal to the Joule power, which is the power lost by the field, so that energy of the entire isolated system consisting of particles and radiation field is conserved. In the quasistatic approximation, the energy of the longitudinal electric field gives the instantaneous Coulomb interaction energy, which can be associated with the particles.
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