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In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem 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 of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π₁, π₂, ..., π_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 π 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 – 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 π theorem was first proved by the French mathematician Joseph Bertrand^[1] 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 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.^[8] It was Buckingham's article that introduced the use of the symbol "${\displaystyle \pi _{i}}$" for the dimensionless variables (or parameters), and this is the source of the theorem's name.

Statement

More formally, the number ${\displaystyle p}$ of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and ${\displaystyle k}$ 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 $${\displaystyle f(q_{1},q_{2},\ldots ,q_{n})=0,}$$ where ${\displaystyle q_{1},\ldots ,q_{n}}$ are any ${\displaystyle n}$ physical variables, and there is a maximal dimensionally independent subset of size ${\displaystyle k}$,^{[note 2]} then the above equation can be restated as $${\displaystyle F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,}$$ where ${\displaystyle \pi _{1},\ldots ,\pi _{p}}$ are dimensionless parameters constructed from the ${\displaystyle q_{i}}$ by ${\displaystyle p=n-k}$ dimensionless equations — the so-called Pi groups — of the form $${\displaystyle \pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}},}$$ where the exponents ${\displaystyle a_{i}}$ are rational numbers. (They can always be taken to be integers by redefining ${\displaystyle \pi _{i}}$ as being raised to a power that clears all denominators.) If there are ${\displaystyle \ell }$ fundamental units in play, then ${\displaystyle p\geq n-\ell }$.

Significance

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. 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 over the real 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 standard gravity ${\displaystyle g}$ has units of ${\displaystyle {\mathsf {L}}/{\mathsf {T}}^{2}={\mathsf {L}}^{1}{\mathsf {T}}^{-2}}$ (length over time squared), so it is represented as the vector ${\displaystyle (1,-2)}$ 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 ${\displaystyle q_{1},q_{2},\dots ,q_{n}}$, where the units of ${\displaystyle q_{i}}$ contain length raised to the power ${\displaystyle c_{i}}$. If we originally measure length in meters but later switch to centimeters, then the numerical value of ${\displaystyle q_{i}}$ would be rescaled by a factor of ${\displaystyle 100^{c_{i}}}$. 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 ${\displaystyle n}$ dimensional variables ${\displaystyle q_{1},\ldots ,q_{n}}$ in ${\displaystyle \ell }$ fundamental (basis) dimensions, the dimensional matrix is the ${\displaystyle \ell \times n}$ matrix ${\displaystyle M}$ whose ${\displaystyle \ell }$ rows correspond to the fundamental dimensions and whose ${\displaystyle n}$ columns are the dimensions of the variables: the ${\displaystyle (i,j)}$th entry (where ${\displaystyle 1\leq i\leq \ell }$ and ${\displaystyle 1\leq j\leq n}$) is the power of the ${\displaystyle i}$th fundamental dimension in the ${\displaystyle j}$Zdroj:https://en.wikipedia.org?pojem=Buckingham_π_theorem
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