# 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

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

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

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

$W = \int_{x_0}^{x_1} F dx$

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

$E_k = \frac{1}{2} m (\frac{dx}{dt})^2$

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

## 參考來源

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.