Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem can be seen as a scheme for nondimensionalization because it provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown.
Although named for Edgar Buckingham, the π theorem was first proved by French mathematician J. Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains in distinct terms all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem (“the method of dimensions”) became widely known due to the works of Rayleigh (the first application of the π theorem in the general case to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansions, to 1894).
Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892, then in 1911—apparently independently—by both A. Federman and D. Riabouchinsky, and again in 1914 by Buckingham. It was Buckingham's article that introduced the use of the symbol "πi" for the dimensionless variables (or parameters), and this is the source of the theorem's name.
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