Electrostatic Origins of the Dirichlet Principle
Steven Deckelman
TL;DR
This paper investigates the electrostatic origins of the Dirichlet principle for solving the Dirichlet problem via energy minimization. It reconstructs the classical electrostatic argument using historical accounts and fills in missing physics and mathematics, showing how a Dirichlet energy integral arises from assembling charge distributions and how the minimum corresponds to a harmonic extension. The work clarifies the connection between physical intuition and the calculus of variations, situating Dirichlet's principle within Green's identities and Hilbert's variational framework. It provides a historically grounded, technically explicit account of how electrostatic reasoning motivated the Dirichlet principle and its mathematical justification.
Abstract
The Dirichlet Principle is an approach to solving the Dirichlet problem by means of a Dirichlet energy integral. It is part of the folklore of mathematics that the genesis of this argument was motivated by physical analogy involving electrostatic fields. The story goes something like this: If an electrostatic potential is prescribed on the boundary of a region, it will extend to a potential in the interior of the region which is harmonic when the electric field is in stable equilibrium, and that electrostatic field has minimum Dirichlet energy. The details of this argument are seldom given and where they are, they are typically scant, redacted, and speculative while often omitting either physics details or mathematics details. The purpose of this article is to give a detailed reconstruction of the electrostatic argument by combining accounts in several contemporary and historical disparate sources. Particular attention is given to explaining the frequently omitted physics and mathematical details and how they fit together to give the physical motivation.