# 因次分析

## 牛頓相似性原理

### 數學性質

1. 這群的運算方法是乘法，Ln×Lm = Ln+m。因此，這種運算方法符合閉包律
2. 單位元L0 = 1。量綱為L0的物理量是無量綱物理量。
3. 逆元是1/L or L−1
4. L提升至任意有理數冪pLp也是群的元素。其逆元是Lp或1/Lp

## 例子

${\displaystyle F=ma=m{\frac {d^{2}x}{dt^{2}}}=m{\frac {d}{dt}}{\frac {dx}{dt}}}$

${\displaystyle [F]=[M][L][T^{-2}]\,}$

${\displaystyle N=kg\cdot {\frac {m}{s^{2}}}}$ ，即公斤（kg）·（m）·秒（s）負二次方

${\displaystyle W=\int _{x_{0}}^{x_{1}}Fdx}$

${\displaystyle [W]=[F][L]=[M][L^{2}][T^{-2}]\,}$

${\displaystyle E_{k}={\frac {1}{2}}m\left({\frac {dx}{dt}}\right)^{2}}$

${\displaystyle [E_{k}]=[M]([L][T^{-1}])^{2}=[M][L^{2}][T^{-2}]\,}$

## 注释

1. ^ 例如，速度的量綱為長度每單位時間，而計量單位為公尺每秒、英里每小時或其它單位。量綱分析所根據的重要原理是，物理定律必需跟其計量物理量的單位無關。任何有意義的方程式，其左手邊與右手邊的量綱必需相同。檢查有否遵循這規則是做量綱分析最基本的步驟。

## 参考文献

1. ^ Price, Bartholomew, A treatise on infinitesimal calculus, containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics, Volume 4, University Press: pp. 119ff, 1862
2. ^ Stahl, Walter R, Dimensional Analysis In Mathematical Biology, Bulletin of Mathematical Biophysics, 1961, 23: 355
3. ^ Roche, John J, The Mathematics of Measurement: A Critical History, London: Springer: 203, 1998, ISBN 978-0387915814, Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.
4. ^ Mason, Stephen Finney, A history of the sciences, New York: Collier Books: 169, 1962, ISBN 0-02-093400-9
5. ^ M. J. Duff, L. B. Okun and G. Veneziano, Trialogue on the number of fundamental constants, JHEP 0203, 023 (2002) preprint页面存档备份，存于互联网档案馆）.