Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

TL;DR

This paper demonstrates how the electromagnetic Lagrangian can be reconstructed from a set of Poisson brackets, linking Hamiltonian formulations with fundamental equations of electromagnetism.

Contribution

It introduces a method to derive the electromagnetic Lagrangian directly from Poisson brackets, bridging Hamiltonian and Lagrangian frameworks.

Findings

Maxwell equations derived from Hamiltonian and Poisson brackets

Lorentz force law obtained from Poisson brackets

Reconstruction of the electromagnetic Lagrangian from brackets

Abstract

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.