Buckingham π theorem

Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n 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, where 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 π theorem indicates that validity of the laws of physics does not depend on a specific unit system.

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.

Although named for Edgar Buckingham, the π theorem was first proved by the French mathematician Joseph Bertrand in 1878.

[1] 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[note 1] to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892,[2] a heuristic proof with the use of series expansions, to 1894.

[3] Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy [fr] in 1892,[4][5] then in 1911—apparently independently—by both A. Federman[6] and D. Riabouchinsky,[7] and again in 1914 by Buckingham.

For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.

physical variables, and there is a maximal dimensionally independent subset of size

The Buckingham π theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown.

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.

If we originally measure length in meters but later switch to centimeters, then the numerical value of

Any physically meaningful law should be invariant under an arbitrary rescaling of every fundamental unit; this is the fact that the pi theorem hinges on.

In order to convert this into a linear algebra problem, we take logarithms (the base is irrelevant), yielding

The elements of the matrix correspond to the powers to which the respective dimensions are to be raised.

In linear algebra, the set of vectors with this property is known as the kernel (or nullspace) of the dimensional matrix.

The dimensional matrix as written above is in reduced row echelon form, so one can read off a non-zero kernel vector to within a multiplicative constant:

Dimensional analysis has thus provided a general equation relating the three physical variables:

The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant:

Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel.

Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass

For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle.

To demonstrate the application of the π theorem, consider the power consumption of a stirrer with a given shape.

Those n = 5 variables are built up from k = 3 independent dimensions, e.g., length: L (SI units: m), time: T (s), and mass: M (kg).

According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = n − k = 5 − 3 = 2 independent dimensionless numbers.

, commonly named the Reynolds number which describes the fluid flow regime, and

An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc.