# 经典场论

（重定向自古典場論

## 非相对论性场

### 牛顿重力

${\displaystyle {\vec {F}}={\frac {Gm_{1}m_{2}}{r^{3}}}{\vec {r}}}$

${\displaystyle {\vec {g}}={\frac {Gm}{r^{3}}}{\vec {r}}}$

### 电场

${\displaystyle {\vec {E}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {q}{r^{3}}}{\vec {r}}.}$

## 相对论场论

### 拉格朗日动力学

${\displaystyle S[\phi ]=\int {{\mathcal {L}}[\phi (x)]\,d^{4}x}.}$

${\displaystyle {\frac {\delta }{\delta \phi }}S={\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\right)=0.}$

## 相对论场

### 电磁场

${\displaystyle F_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}.}$

#### 拉格朗日函数

${\displaystyle {\mathcal {L}}={\frac {-1}{4\mu _{0}}}F^{ab}F_{ab}+j^{a}A_{a}.}$

#### 方程组

${\displaystyle \partial _{b}\left({\frac {\partial {\mathcal {L}}}{\partial \left(\partial _{b}A_{a}\right)}}\right)={\frac {\partial {\mathcal {L}}}{\partial A_{a}}}.}$

${\displaystyle \partial _{b}F^{ab}=\mu _{0}j^{a}.}$

${\displaystyle 6F_{[ab,c]}\,=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.}$

### 重力场

${\displaystyle {\mathcal {L}}=\,R{\sqrt {-g}}}$

${\displaystyle G_{ab}\,=0}$