Lagrangian and Hamiltonian Formulation of Classical Electrodynamics without Potentials

TL;DR

This paper reformulates classical electrodynamics using electric and magnetic fields as dynamical variables, deriving a Lagrangian and Hamiltonian framework with constraints, providing a new perspective on the theory's foundational structure.

Contribution

It introduces a novel Lagrangian and Hamiltonian formulation of Maxwell's equations without potentials, incorporating constraints via Lagrange multipliers and Dirac brackets.

Findings

Derived a Lagrangian yielding Maxwell's equations with constraints.

Identified second class constraints reducing phase space degrees of freedom.

Calculated the Dirac brackets for the field variables and conjugate momenta.

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 $\vec{E}$ and the magnetic field $\vec{B}$ as the 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 $\vec{\Pi}_E$ and $\vec{\Pi}_B$ are constrained giving 6 second class constraints at each point in space. Gauss's law and $\vec{\nabla}\cdot\vec{B}=0$ 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.