On Maxwell electrodynamics in multi-dimensional spaces

TL;DR

This paper derives Maxwell's equations in multi-dimensional spaces using variational principles, develops a Hamiltonian framework, and explores gauge choices and Einstein's equations in curved multi-dimensional geometries.

Contribution

It extends Maxwell electrodynamics to multi-dimensional spaces, introduces a Hamiltonian approach, and generalizes gauge conditions and Einstein's equations for these higher-dimensional contexts.

Findings

Derived Maxwell equations from variational principles in multi-dimensional spaces.

Developed a Hamiltonian formalism for electromagnetic fields in flat multi-dimensional spaces.

Rewritten Einstein's equations with electromagnetic fields in tensor form for multi-dimensional curved spaces.

Abstract

The governing equations of Maxwell electrodynamics in multi-dimensional spaces are derived from the variational principle of least action which is applied to the action function of the electromagnetic field. The Hamiltonian approach for the electromagnetic field in multi-dimensional pseudo-Euclidean (flat) spaces has also been developed and investigated. Based on the two arising first-class constraints we have generalized to multi-dimensional spaces a number of different gauges known for the three-dimensional electromagnetic field. For multi-dimensional spaces of non-zero curvature the governing equations for the multi-dimensional electromagnetic field are written in manifestly covariant form. Multi-dimensional Einstein's equations of metric gravity in the presence of electromagnetic field have been re-written in the true tensor form. Methods of scalar electrodynamics are applied to analyze Maxwell equations in the two- and one-dimensional spaces.