Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham 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 , , ..., 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 provides a method for  computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown. The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (for example, pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would  not be an identity, and the theorem would not hold. History. Although named for Edgar Buckingham, the theorem was first proved by French mathematician Joseph 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 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 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 "formula_1" for the dimensionless variables (or parameters), and this is the source of the theorem's name. Statement. More formally, the number formula_2 of  dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and formula_3 is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent. In mathematical terms, if we have a physically meaningful equation such as formula_4 where formula_5 are the formula_6 independent physical variables, and they are expressed in terms of formula_3 independent physical units, then the above equation can be restated as formula_8 where formula_9 are dimensionless parameters constructed from the formula_10 by formula_11 dimensionless equations — the so-called "Pi groups" — of the form  formula_12 where the exponents formula_13 are rational numbers. (They can always be taken to be integers by redefining formula_1 as being raised to a power that clears all denominators.) Significance. The Buckingham theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful". Two systems for which these parameters coincide are called "similar" (as with similar triangles, they differ  only in scale); they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions. Proof. Outline. It will be assumed that the space of fundamental and derived physical units forms a vector space over the rational numbers, with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" operation, and raising to powers as the "scalar multiplication" operation: represent a dimensional variable  as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). For instance, the standard gravity formula_15 has units of formula_16 (distance over time squared), so it is represented as the vector formula_17 with respect to the basis of fundamental units (distance, time). Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical-units vector space. Formal proof. Given a system of formula_6 dimensional variables formula_5 (with physical dimensions) in formula_3 fundamental (basis) dimensions, the "dimensional matrix