# 白金漢π定理

（重定向自Π定理

f1(u, ρ, ν, A, FD)= 0，

f2(CD, Re)= 0，

## 定理描述

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

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

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

## 例子

### 速度

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

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

${\displaystyle \pi =D^{0}T^{0}V^{0}=D^{-1}T^{1}(D^{1}T^{-1})=D^{-1}T^{1}V^{1}=TV/D}$ .

${\displaystyle f(\pi )=0,}$

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

${\displaystyle f(\pi )=\pi -1=VT/D-1=0.}$

### 單擺

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

（此處寫成四個物理量之間的關係，不是直接將週期T寫成另M, Lg的函數）

${\displaystyle f(\pi )=0,}$

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

a1, ..., a4為待確定的數值

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

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

（行表示基本因次tml，而列表示物理量T, M, Lg。例如第四行為(−2, 0, 1)表示變數g的因次為${\displaystyle t^{-2}m^{0}\ell ^{1}}$

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

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

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

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

## 参考文献

1. ^ Bertrand, J. Sur l'homogénéité dans les formules de physique. Comptes rendus. 1878, 86 (15): 916–920.
2. ^ When in applying the pi–theorem there arises an arbitrary function of dimensionless numbers.
3. ^ Rayleigh. On the question of the stability of the flow of liquids. Philosophical magazine. 1892, 34: 59–70. doi:10.1080/14786449208620167.
4. ^ Second edition of The Theory of Sound’’(Strutt, John William. The Theory of Sound 2. Macmillan. 1896.).
5. ^ Quotes from Vaschy’s article with his statement of the pi–theorem can be found in: Macagno, E. O. Historico-critical review of dimensional analysis. Journal of the Franklin Institute. 1971, 292 (6): 391–402. doi:10.1016/0016-0032(71)90160-8.
6. ^ Федерман, А. О некоторых общих методах интегрирования уравнений с частными производными первого порядка. Известия Санкт-Петербургского политехнического института императора Петра Великого. Отдел техники, естествознания и математики. 1911, 16 (1): 97–155. (Federman A., On some general methods of integration of first-order partial differential equations, Proceedings of the Saint-Petersburg polytechnic institute. Section of technics, natural science, and mathematics)
7. ^ Riabouchinsky, D. Мéthode des variables de dimension zéro et son application en aérodynamique. L'Aérophile. 1911: 407–408.
8. ^ Original text of Buckingham’s article in Physical Review