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engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
,
applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, and
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Buckingham theorem is a key
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. 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 1, 2, ..., ''p'' 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 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 Boyle's law, also referred to as the Boyle–Mariotte law or Mariotte's law (especially in France), is an empirical gas laws, gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: ...
– 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 the 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''When in applying the –theorem there arises an ''arbitrary function'' of dimensionless numbers. 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 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 "\pi_i" for the dimensionless variables (or parameters), and this is the source of the theorem's name.


Statement

More formally, the number p of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and k 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 f(q_1,q_2,\ldots,q_n)=0, where q_1, \ldots, q_n are any n physical variables, and there is a maximal dimensionally independent subset of size k,A dimensionally independent set of variables is one for which the only exponents q_1^ \, q_2^ \cdots q_k^ yielding a dimensionless quantity are a_1 = a_2 = \cdots = 0. This is precisely the notion of
linear independence In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then th ...
.
then the above equation can be restated as F(\pi_1,\pi_2,\ldots,\pi_p)=0, where \pi_1, \ldots, \pi_p are dimensionless parameters constructed from the q_i by p = n - k dimensionless equations — the so-called ''Pi groups'' — of the form \pi_i=q_1^\,q_2^ \cdots q_n^, where the exponents a_i are rational numbers. (They can always be taken to be integers by redefining \pi_i as being raised to a power that clears all denominators.) If there are \ell fundamental units in play, then p \geq n - \ell.


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

For simplicity, it will be assumed that the space of fundamental and derived physical units forms a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, 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 g has units of \mathsf / \mathsf^2 = \mathsf^1 \mathsf^ (length over time squared), so it is represented as the vector (1, -2) with respect to the basis of fundamental units (length, time). We could also require that exponents of the fundamental units be rational numbers and modify the proof accordingly, in which case the exponents in the pi groups can always be taken as rational numbers or even integers.


Rescaling units

Suppose we have quantities q_1, q_2, \dots, q_n, where the units of q_i contain length raised to the power c_i. If we originally measure length in meters but later switch to centimeters, then the numerical value of q_i would be rescaled by a factor of 100^. 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.


Formal proof

Given a system of n dimensional variables q_1, \ldots, q_n in \ell fundamental (basis) dimensions, the ''dimensional matrix'' is the \ell \times n matrix M whose \ell rows correspond to the fundamental dimensions and whose n columns are the dimensions of the variables: the (i, j)th entry (where 1 \leq i \leq \ell and 1 \leq j \leq n) is the power of the ith fundamental dimension in the jth variable. The matrix can be interpreted as taking in a combination of the variable quantities and giving out the dimensions of the combination in terms of the fundamental dimensions. So the \ell \times 1 (column) vector that results from the multiplication M\begina_1\\ \vdots \\ a_n\end consists of the units of q_1^\,q_2^\cdots q_n^ in terms of the \ell fundamental independent (basis) units. If we rescale the ith fundamental unit by a factor of \alpha_i, then q_j gets rescaled by \alpha_1^\, \alpha_2^ \cdots \alpha_\ell^, where m_ is the (i, j)th entry of the dimensional matrix. In order to convert this into a linear algebra problem, we take logarithms (the base is irrelevant), yielding \begin \log \\ \vdots \\ \log \end \mapsto \begin \log \\ \vdots \\ \log \end - M^\operatorname \begin \log \\ \vdots \\ \log \end, which is an action of \mathbb^\ell on \mathbb^n. We define a physical law to be an arbitrary function f \colon (\mathbb^+)^n \to \mathbb such that (q_1, q_2, \dots, q_n) is a permissible set of values for the physical system when f(q_1, q_2, \dots, q_n) = 0. We further require f to be invariant under this action. Hence it descends to a function F \colon \mathbb^n / \operatorname \to \mathbb. All that remains is to exhibit an isomorphism between \mathbb^n/\operatorname and \mathbb^p, the (log) space of pi groups (\log, \log, \dots, \log). We construct an n \times p matrix K whose columns are a basis for \ker. It tells us how to embed \mathbb^p into \mathbb^n as the kernel of M. That is, we have an exact sequence : 0 \to \mathbb^p \xrightarrow \mathbb^n \xrightarrow \mathbb^\ell. Taking tranposes yields another exact sequence :\mathbb^\ell \xrightarrow \mathbb^n \xrightarrow \mathbb^p \to 0. The first isomorphism theorem produces the desired isomorphism, which sends the
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
v + M^\operatorname \mathbb^\ell to K^\operatorname v. This corresponds to rewriting the tuple (\log q_1, \log q_2, \dots, \log q_n) into the pi groups (\log\pi_1, \log\pi_2, \dots, \log\pi_p) coming from the columns of K. The
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
defines seven base units, which are the
ampere The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
, kelvin,
second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...
,
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
,
kilogram The kilogram (also spelled kilogramme) is the base unit of mass in the International System of Units (SI), equal to one thousand grams. It has the unit symbol kg. The word "kilogram" is formed from the combination of the metric prefix kilo- (m ...
,
candela The candela (symbol: cd) is the unit of luminous intensity in the International System of Units (SI). It measures luminous power per unit solid angle emitted by a light source in a particular direction. Luminous intensity is analogous to radi ...
and mole. It is sometimes advantageous to introduce additional base units and techniques to refine the technique of dimensional analysis. (See orientational analysis and reference.)


Examples


Speed

This example is elementary but serves to demonstrate the procedure. Suppose a car is driving at 100 km/h; how long does it take to go 200 km? This question considers n = 3 dimensioned variables: distance d, time t, and speed v, and we are seeking some law of the form t = \operatorname(v, d). Any two of these variables are dimensionally independent, but the three taken together are not. Thus there is p = n - k = 3 - 2 = 1 dimensionless quantity. The dimensional matrix is M = \begin 1 & 0 & \;\;\;1\\ 0 & 1 & -1 \end in which the rows correspond to the basis dimensions L and T, and the columns to the considered dimensions L, T, \text V, where the latter stands for the speed dimension. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised. For instance, the third column (1, -1), states that V = L^0 T^0 V^1, represented by the column vector \mathbf= ,0,1 is expressible in terms of the basis dimensions as V = L^1 T^ = L/T, since M\mathbf = ,-1 For a dimensionless constant \pi=L^T^V^, we are looking for vectors \mathbf= _1,a_2,a_3/math> such that the matrix-vector product M \mathbf equals the zero vector , 0 In linear algebra, the set of vectors with this property is known as the kernel (or nullspace) of the dimensional matrix. In this particular case its kernel is one-dimensional. 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: \mathbf = \begin -1\\ \;\;\;1\\ \;\;\;1\\ \end. If the dimensional matrix were not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant, replacing the dimensions by the corresponding dimensioned variables, may be written: \pi = d^t^1v^1 = tv/d. Since the kernel is only defined to within a multiplicative constant, the above dimensionless constant raised to any arbitrary power yields another (equivalent) dimensionless constant. Dimensional analysis has thus provided a general equation relating the three physical variables: F(\pi)=0, or, letting C denote a zero of function F, \pi=C, which can be written in the desired form (which recall was t = \operatorname(v, d)) as t = C\frac. The actual relationship between the three variables is simply d = vt. In other words, in this case F has one physically relevant root, and it is unity. The fact that only a single value of C will do and that it is equal to 1 is not revealed 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 dimensions 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 is not written here as a function of M, L, \text g.) Period, mass, and length are dimensionally independent, but acceleration can be expressed in terms of time and length, which means the four variables taken together are not dimensionally independent. Thus we need only p = n - k = 4 - 3 = 1 dimensionless parameter, denoted by \pi, and the model can be re-expressed as F(\pi) = 0, where \pi is given by \pi = T^M^L^g^ for some values of a_1, a_2, a_3, a_4. The dimensions of the dimensional quantities are: T = t, M = m, L = \ell, g = \ell/t^2. The dimensional matrix is: \mathbf = \begin 1 & 0 & 0 & -2\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1 \end. (The rows correspond to the dimensions t, m, and \ell, and the columns to the dimensional variables T, M, L, \text g. For instance, the 4th column, (-2, 0, 1), states that the g variable has dimensions of t^m^0 \ell^1.) We are looking for a kernel vector a = \left _1, a_2, a_3, a_4\right/math> such that the matrix product of \mathbf on a yields the zero vector ,0,0 The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant: a = \begin2\\ 0 \\ -1 \\ 1\end. Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant may be written: \begin \pi &= T^2M^0L^g^1\\ &= gT^2/L \end. In fundamental terms: \pi = (t)^2 (m)^0 (\ell)^ \left(\ell/t^2\right)^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. In this example, three of the four dimensional quantities are fundamental units, so the last (which is g) must be a combination of the previous. Note that if a_2 (the coefficient of M) had been non-zero then there would be no way to cancel the M value; therefore a_2 be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass M. (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\left(gT^2/L\right) = 0. Then this implies that gT^2/L = C_i for some zero C_i of the function F. If there is only one zero, call it C, then gT^2/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 = 4\pi^2. 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.


Electric power

To demonstrate the application of the theorem, consider the power consumption of a stirrer with a given shape. The power, ''P'', in dimensions · L2/T3 is a function of the
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
, ''ρ'' /L3 and the
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
of the fluid to be stirred, ''μ'' /(L · T) as well as the size of the stirrer given by its
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
, ''D'' and the angular speed of the stirrer, ''n'' /T Therefore, we have a total of ''n'' = 5 variables representing our example. 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 -theorem, the ''n'' = 5 variables can be reduced by the ''k'' = 3 dimensions to form ''p'' = ''n'' − ''k'' = 5 − 3 = 2 independent dimensionless numbers. Usually, these quantities are chosen as \mathrm = , commonly named the
Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
which describes the fluid flow regime, and N_\mathrm = \frac, the power number, which is the dimensionless description of the stirrer. Note that the two dimensionless quantities are not unique and depend on which of the ''n'' = 5 variables are chosen as the ''k'' = 3 dimensionally independent basis variables, which, in this example, appear in both dimensionless quantities. The Reynolds number and power number fall from the above analysis if \rho, ''n'', and ''D'' are chosen to be the basis variables. If, instead, \mu, ''n'', and ''D'' are selected, the Reynolds number is recovered while the second dimensionless quantity becomes N_\mathrm = \frac. We note that N_\mathrm is the product of the Reynolds number and the power number.


Other examples

An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method. The theorem has also been used in fields other than physics, for instance in
sports science Sports science is a discipline that studies how the healthy human body works during exercise, and how sports and physical activity promote health and performance from cellular to whole body perspectives. The study of sports science traditionally i ...
.


See also

* Blast wave *
Dimensionless quantity Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
* Natural units * Similitude (model) *
Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...


References


Notes


Citations


Bibliography

* * * * * * * * * *


Original sources

* * * * * * *


External links


Some reviews and original sources on the history of pi theorem and the theory of similarity (in Russian)
{{DEFAULTSORT:Buckingham Pi Theorem Articles containing proofs Dimensional analysis Eponymous theorems of physics