Potential theory

Potential theory may be defined as the study of harmonic functions.

Definition and comments

The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from potentials which satisfied Laplace's equation. Hence, potential theory was the study of functions which could serve as potentials. Nowadays, we know that nature is more complicated -- the equations which describe forces are systems of non-linear partial differential equations such as the Einstein equations and the Yang-Mills equations and that the Laplace equation is only valid as a limiting case. Nevertheless, the term "potential theory" has remained as a convenient term for describing the study of functions which satisfy the Laplace equation.

Related Topics:
Physics - Force - Potential - Laplace's equation - Functions - Partial differential equation - Einstein equations - Yang-Mills equations

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Obviously, there is considerable overlap between potential theory and the theory of the Laplace equation. To the extent that it is possible to draw a distinction between these two fields, the difference is more one of emphasis than subject matter and rests on the following distinction -- potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. Of course, this is not a hard and fast distinction and, in practice there is considerable overlap between the two fields, with methods and results from one being used in the other.

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