Buckingham π theorem

In mathematical physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. The theorem loosely 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 quantities, then the original expression is equivalent to an equation involving a set of p = n − k  dimensionless parameters constructed from the original variables: it is a scheme for nondimensionalization. This 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'.

Statement

More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. 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

$f(q_{1},q_{2},\ldots ,q_{n})=0\,$

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

$F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0\,$

where the π_i are dimensionless parameters constructed from the q_i  by p = n − k  equations of the form

$\pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}}\,$

where the exponents a_i are rational numbers (they can always be taken to be integers: just raise it to a power to clear denominators).

The use of the π_i as the dimensionless parameters was introduced by Edgar Buckingham in his original 1914 paper on the subject from which the theorem draws its name.

Significance

The Vaschy–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'.

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.

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 gravitational constant g has units of $\ell /t^{2}=\ell ^{1}t^{-2}$ (distance over time squared), so it is represented as the vector $(1,-2)$ 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 unit vector space.

Formal proof

Given a system of n-dimensional variables (physical variables), in k (physical) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the (i, j)th entry is the power of the ith unit in the jth variable. The matrix can be interpreted as taking in a combination of the dimensional quantities and giving out the dimensions of this product. So

$M{\begin{bmatrix}a_{1}\\\vdots \\a_{n}\end{bmatrix}}$

is the units of

$q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}}.\,$

A dimensionless variable is a combination whose units are all zero (hence, dimensionless), which is equivalent to the kernel of this matrix; a dimensionless variable is a linear relation between units of dimensional variables.

By the rank-nullity theorem, a system of n vectors in k dimensions (where all dimensions are necessary) satisfies a (p = n − k)-dimensional space of relations. Any choice of basis will have p elements, which are the dimensionless variables.

The dimensionless variables can always be taken to be integer combinations of the dimensional variables (by clearing denominators). There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used.

Examples

Speed

This example is elementary, but demonstrates the general procedure: Suppose a car is driving at 100 km/hour; how long does it take it to go 200 km?

This question has two fundamental physical units: time t and length $\ell$, and three dimensional variables: distance D, time taken T, and velocity V. Thus there is 3 − 2 = 1 dimensionless quantity. The units of the dimensional quantities are:

$D\sim \ell ,\ T\sim t,\ V\sim \ell /t.$

The dimensional matrix is:

$M={\begin{bmatrix}1&0&1\\0&1&-1\end{bmatrix}}$

The rows correspond to the dimensions $\ell$, and t, and the columns to the dimensional variables D, T, V. For instance, the 3rd column, (1, −1), states that the V (velocity) variable has units of $\ell ^{1}t^{-1}=\ell /t$.

For a dimensionless constant $\pi =D^{a_{1}}T^{a_{2}}V^{a_{3}}$ we are looking for a vector $a=[a_{1},a_{2},a_{3}]$ such that the matrix product of M on a yields the zero vector [0,0]. In linear algebra, this vector is known as the kernel of the dimensional matrix, and it spans the nullspace of the dimensional matrix, which in this particular case is one dimensional. The dimensional matrix as written above is in reduced row echelon form, so one can read off that a kernel vector may be written (to within a multiplicative constant) by:

$a={\begin{bmatrix}-1\\1\\1\end{bmatrix}}.$

If the dimensional matrix were not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix in order to more easily determine the kernel. It follows that the dimensionless constant may be written:

${\begin{aligned}\pi &=D^{-1}T^{1}V^{1}\\&=TV/D\end{aligned}}$

or, in dimensional terms:

$\pi \sim (\ell )^{-1}(t)^{1}(\ell /t)^{1}\sim 1$

Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.

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

$f(\pi )=0\,$

which may be written:

$T={\frac {CD}{V}}$

where C is one of a set of constants, such that $C=f^{-1}(0)$. The actual relationship between the three variables is simply $D=VT$ so that the actual dimensionless equation ($f(\pi )=0$) is written:

$f(\pi )=\pi -1=VT/D-1=0\,$

In other words, there is only one value of C and it is unity. The fact that there is only a single value of C and that it is equal to unity is a level of detail not provided by the technique of dimensional analysis.

The simple pendulum

We wish to determine the period T  of small oscillations in 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

$f(T,M,L,g)=0.\,$

(Note that it is written as a relation, not as a function: T isn't here written as a function of M, L, and g.)

There are 3 fundamental physical units in this equation: time t, mass m, and length l, and 4 dimensional variables, T, M, L, and g. Thus we need only 4 − 3 = 1 dimensionless parameter, denoted π, and the model can be re-expressed as

$f(\pi )=0\,$

where π is given by

$\pi =T^{a_{1}}M^{a_{2}}L^{a_{3}}g^{a_{4}}\,$

for some values of a₁, ..., a₄.

The units of the dimensional quantities are:

$T=t,M=m,L=\ell ,g=\ell /t^{2}.\,$

The dimensional matrix is:

$M={\begin{bmatrix}1&0&0&-2\\0&1&0&0\\0&0&1&1\end{bmatrix}}$

(The rows correspond to the dimensions t, m, and l, and the columns to the dimensional variables T, M, L and g. For instance, the 4th column, (−2, 0, 1), states that the g variable has units of $t^{-2}m^{0}\ell ^{1}$.)

We are looking for a kernel vector a = [a1,a2,a3,a4] such that the matrix product of M on a yields the zero vector [0,0,0]. The dimensional matrix as written above is in reduced row echelon form, so one can read off that a kernel vector may be written (to within a multiplicative constant) by:

$a={\begin{bmatrix}2\\0\\-1\\1\end{bmatrix}}.$

Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix in order to more easily determine the kernel. It follows that the dimensionless constant may be written:

${\begin{aligned}\pi &=T^{2}M^{0}L^{-1}g^{1}\\&=gT^{2}/L\end{aligned}}$

In dimensional terms:

$\pi =(t)^{2}(m)^{0}(\ell )^{-1}(\ell /t^{2})^{1}=1\,$

which is dimensionless. Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.

This example is easy because 3 of the dimensional quantities are fundamental units, so the last (g) is a combination of the previous. Note that if a₂ were non-zero there would be no way to cancel the M value—therefore a₂ must be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass. (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor, ${\vec {g}}-2{\vec {T}}-{\vec {L}}$ is the only nontrivial way to construct a vector of a dimensionless parameter.)

The model can now be expressed as:

$f(gT^{2}/L)=0.\$

Assuming the zeroes of f  are discrete, we can say gT²/L = C_n  where C_n  is the nth zero. If there is only one zero, then gT²/L = C . 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 C = 4π².

For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the angle approaches zero.

See also

• Blast wave
• Dimensional analysis
• Dimensionless number
• Natural units
• Nondimensionalization
• Similitude (model)
• Rayleigh's method of dimensional analysis

References

Exposition

• Hanche-Olsen, Harald (2004). "Buckingham's pi-theorem" (PDF). NTNU. Retrieved April 9, 2007.
• Hart, George W. (March 1, 1995). Multidimensional Analysis: Algebras and Systems for Science and Engineering. Springer-Verlag. ISBN 0387944176.
• Kline, Stephen J. (1986). Similitude and Approximation Theory. Springer-Verlag, New York. ISBN 0387165185.
• Wan, Frederic Y.M. (1989). Mathematical Models and their Analysis. Harper & Row Publishers, New York. ISBN 0060469021.
• Vignaux, G.A. (1991). "Dimensional analysis in data modelling" (PDF). Victoria University of Wellington. Retrieved December 15, 2005.
• Mike Sheppard, 2007 Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants
• Gibbings, J.C. (2011). Dimensional Analysis. Springer. ISBN 1849963169.

Original sources

• Vaschy, A. (1892). "Sur les lois de similitude en physique". Annales Télégraphiques. 19: 25–28.
• Buckingham, E. (1914). "On physically similar systems; illustrations of the use of dimensional equations". Physical Review. 4 (4): 345–376. Bibcode:1914PhRv....4..345B. doi:10.1103/PhysRev.4.345.
• Buckingham, E. (1915). "The principle of similitude". Nature. 96 (2406): 396–397. Bibcode:1915Natur..96..396B. doi:10.1038/096396d0.
• Buckingham, E. (1915). "Model experiments and the forms of empirical equations". Transactions of the American Society of Mechanical Engineers. 37: 263–296.
• Taylor, Sir G. (1950). "The Formation of a Blast Wave by a Very Intense Explosion. I. Theoretical Discussion". Proceedings of the Royal Society A. 201 (1065): 159–174. Bibcode:1950RSPSA.201..159T. doi:10.1098/rspa.1950.0049.
• Taylor, Sir G. (1950). "The Formation of a Blast Wave by a Very Intense Explosion. II. The Atomic Explosion of 1945". Proceedings of the Royal Society A. 201 (1065): 175–186. Bibcode:1950RSPSA.201..175T. doi:10.1098/rspa.1950.0050.

External links

• Some reviews and original sources on the history of pi theorem and the theory of similarity (in Russian)