A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics

TL;DR

This paper introduces Colombeau generalized functions, explaining their core concepts and demonstrating their application to nonlinear problems in classical physics, especially in electrodynamics, including detailed calculations of point charge self-energy.

Contribution

It provides an elementary introduction to Colombeau algebras and applies them to classical electrodynamics, including detailed treatment of point charge singularities.

Findings

Defined Coulomb potential as a Colombeau generalized function

Calculated nonlinear integrals involving products of distributions

Confirmed methods for handling point singularities in electrodynamics

Abstract

The objective of this introduction to Colombeau algebras of generalized-functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic non-linear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the self-energy of a point electric charge is worked out in detail: The Coulomb potential and field are defined as Colombeau generalized-functions, and integrals of nonlinear expressions corresponding to products of distributions (such as the square of the Coulomb field and the square of the delta-function) are calculated. Finally, the methods introduced in Eur. J. Phys. /28/ (2007) 267-275, 1021-1042, and 1241, to deal with point-like singularities in classical electrodynamics are confirmed.