Lagrangian and Hamiltonian formulation of classical electrodynamics without potentials

Abstract

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 $ \overrightarrow{E}$ and the magnetic field $ \overrightarrow{B}$ 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 $ \overrightarrow{\Pi}_{E}$ and $ \overrightarrow{\Pi}_{B}$ are constrained giving 6 second class constraints at each point in space. Gauss's law and $ \overrightarrow{\nabla}\cdot\overrightarrow{B}=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.