Buckingham's ' Pi ' theorem

Buckingham's ' Pi ' theorem.

The Buckingham's ' Pi ' theorem is a key theorem in dimensional analysis. The theorem states that if we have a physically meaningful equation involving a certain number, n of physical variables and these variables are expressible in terms of k independent fundamental physical qualities, then the original expression is equivalent to an equation involving a set of p = n - k dimensionless variables constructed from the original variables. For the purpose of the experimenter, different systems which share the same description in terms of these dimensionless numbers are equivalent.

In mathematical terms, if we have a physically meaningful equation such as f ( q1, q2,……..qn ) = 0

where the qi are the n physical variables and they are expressed in terms of k independent physical units, the above equation can be restated as

F (?1, ?2………. ?p) = 0

Where, the ?i are the dimensionless parameters constructed from the qi by p = n-k equations of the form

?i  = q1^m, q2,^m……..qn^m

Where the exponents mi are the rational numbers. The use of the ?i the dimensionless parameters was introduced by Edge Buckingham in his original 1914 paper on the subject from which the theorem draws its name.

Most importantly, the Buckingham ? theorem provides a method for computing sets of dimensionless parameters is not unique: Buckingham's theorem only provides a way of generating sets of dimensionless parameters and will not choose the most 'physically meaningful'.

Proofs of the ? theorem often begin by considering the space of fundamental and derived physical unit's vector space, 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.

Making the physical units match across sets of physical equations can then the regarded as imposing linear constraints in the physical units vector space.

The theorem describes how every physical meaningful equation involving n variables can be equivalently rewritten as an equation of n - m dimensionless parameters, where m is the number of fundamental units used. Furthermore and most importantly it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown.

Two systems for which these parameters coincide are called similar; they are equivalent for the purposes of the experimental list who wants to determine the form of the equation can choose the most convenient one.

The ? theorem uses linear algebra: the space of all possible physical units can be seen as a vector space over the rational numbers if we represent a unit as the set of exponents needed for the fundamental units ( with a power of zero if the particular fundamental unit is not present) Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of the ?

theorem is essentially a Gauss - Jordan elimination carried out in this vector space.