Lagrangian and Hamiltonian formulation of classical electrodynamics without potentials

摘要

In the standard Lagrangian and Hamiltonian approach to Maxwell's theory the potentials A(mu) are taken as the dynamical variables. In this paper I take the electric field (E) over right arrow and the magnetic field (B) over right arrow as the dynamical variables. I find a Lagrangian that gives the dynamical Maxwell equations and include the constraint equations by using Lagrange multipliers. In passing to the Hamiltonian one finds that the canonical momenta (Pi) over right arrow (E) and (Pi) over right arrow (B) are constrained giving 6 second class constraints at each point in space. Gauss's law and (Delta) over right arrow .(B) over right arrow = 0 can than be added in as additional constraints. There are now 8 second class constraints, leaving 4 phase space degrees of freedom. The Dirac bracket is then introduced and is calculated for the field variables and their conjugate momenta.