Buckingham Pi theorem
From Academic Kids

The Buckingham π theorem is a key theorem in dimensional analysis.
The theorem states that if we have a physically meaningful equation involving a certain number (e.g.: n ) of physical variables, and these variables are expressible in terms of k independent fundamental physical quantities, 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 purposes 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:
 <math>f(q_1,q_2,\ldots,q_n)=0<math>
where the q_{i} are the n physical variables, and they are expressed in terms of k independent physical units, then the above equation can be restated as
 <math>F(\pi_1,\pi_2,\ldots,\pi_p)=0<math>
where the π_{i} are dimensionless parameters constructed from the q_{i} by p = n − k equations of the form
 <math>\pi_i=q_1^{m_1}\,q_2^{m_2}\ldots q_n^{m_n}<math>
where the exponents m_{i} are constants. The use of the π_{i} as the dimensionless parameters was introduced by E. Buckingham in his original 1914 paper on the subject from which the theorem draws its name.
Most importantly, the Buckingham π theorem provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown. However, the choice 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'.
Contents 
Proving the π theorem
Proofs of the π theorem often begin by considering the space of fundamental and derived physical units as a 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 be regarded as imposing linear constraints in the physical unit vector space.
The πtheorem theorem describes how every physically 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 equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.
The πtheorem 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 πtheorem is essentially a GaussJordan elimination carried out in this vector space.
Examples
The simple pendulum
We wish to determine the period T of a simple pendulum. It will be assumed that it is a function of the length L , the mass M , and the acceleration due to gravity on the surface of the Earth g, which has units of length divided by time squared. The model is of the form
 <math>f(T,M,L,g) = 0\,<math>
There are only three fundamental physical units in this equation: mass, time, and length. Thus we need only 43=1 dimensionless parameter and it is obviously
 <math>\pi=gT^2/L\,.<math>
This means that the model can be reexpressed as:
 <math>f(gT^2/L) = 0\,<math>
Assuming the zeroes of f are discrete, we can say gT^{2}/L = K_{n} where K_{n} is the nth zero. If there is only one zero, then gT^{2}/L=K . It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by K=4π^{2} . Note that the dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass.
The Atomic bomb
In 1941, Sir Geoffrey I. Taylor used dimensional analysis to estimate the energy released in an atomic bomb explosion. The first atomic bomb was detonated near Alamogordo, New Mexico on July 16, 1945. In 1947, movies of the explosion were declassified, allowing Sir Taylor to complete the analysis and estimate the energy released in the explosion, even though the energy release was still classified! The actual energy released was later declassified and its value was remarkably close to Taylor's estimate.
Taylor supposed that the description of the process was adequately described by five physical quantities, the time t since the detonation, the energy E which is released at a single point in space at detonation, the radius R of the shock wave at time t , the atmospheric pressure p and the ambient density ρ. There are only three fundamental physical units in this equation: mass, time, and length. Thus we need only 53=2 dimensionless parameters, which can be found to be:
 <math>\pi_0=R\,\left(\frac{\rho}{Et^2}\right)^{1/5}<math>
and
 <math>\pi_1=p\,\left(\frac{t^6}{E^2\rho^3}\right)^{1/5}<math>
The process can now be described by an equation of the form:
 <math>f(\pi_0,\pi_1)=0\,<math>
or, equivalently:
 <math>R=\left(\frac{Et^2}{\rho}\right)^{1/5}g(\pi_1)<math>
where g(π_{1}) is some function of π_{1}. The energy in the explosion is expected to be huge, so that for times of the order of a second after the explosion, we can estimate π_{1} to be approximately zero, and experiments using light explosives can be conducted to determine that g(0) is on the order of unity so that:
 <math>R\approx\left(\frac{Et^2}{\rho}\right)^{1/5}.<math>
This is Taylor's equation which, once he knew the radius of the explosion as a function of the time, allowed him to calculate the energy of the explosion. (See book by F.Y.M. Wan.)
See also
External links
 http://www.math.ntnu.no/~hanche/notes/buckingham/
 http://www.isd.unistuttgart.de/~rudolph/
 http://scienceworld.wolfram.com/physics/BuckinghamsPiTheorem.html
References
 Buckingham, E., On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4, 345376 (1914). Abstract (http://prola.aps.org/abstract/PR/v4/i4/p345_1)
 Buckingham, E., The principle of similitude. Nature 96, 396397 (1915).
 Buckingham, E., Model experiments and the forms of empirical equations. Trans. A.S.M.E. 37, 263296 (1915).
 Hart, George W., "Multidimensional Analysis: Algebras and Systems for Science and Engineering", SpringerVerlag (March 1, 1995). ISBN 0387944176 (Web page description) (http://www.georgehart.com/research/multanal.html)
 Kline, Stephen J., "Similitude and Approximation Theory", SpringerVerlag, New York, 1986. ISBN 0387165185
 Wan, Frederic Y.M., "Mathematical Models and their Analysis", Harper & Row Publishers, New York, 1989. ISBN 0060469021
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