Lagrangian and Hamiltonian formulation of classical electrodynamics without potentials
Irving K. Barber School of Arts and Sciences, University of British Columbia Okanagan, 3333 University Way, V1V 1V7, Kelowna, B.C., Canada
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Accepted: 22 September 2017
Published online: 30 October 2017
In the standard Lagrangian and Hamiltonian approach to Maxwell's theory the potentials are taken as the dynamical variables. In this paper I take the electric field and the magnetic field 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 and are constrained giving 6 second class constraints at each point in space. Gauss's law and 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.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2017