# 二體問題

## 約化為兩個獨立的單體問題

${\displaystyle \mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{1}{\ddot {\mathbf {x} }}_{1}\,\!}$ —— (1)、
${\displaystyle \mathbf {F} _{21}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{2}{\ddot {\mathbf {x} }}_{2}\,\!}$ —— (2)；

### 質心運動（第一個單體問題）

${\displaystyle \mathbf {x} _{cm}\ {\stackrel {def}{=}}\ (m_{1}\mathbf {x} _{1}+m_{2}\mathbf {x} _{2})/M\,\!}$

${\displaystyle {\ddot {\mathbf {x} }}_{cm}=(m_{1}{\ddot {\mathbf {x} }}_{1}+m_{2}{\ddot {\mathbf {x} }}_{2})/M\,\!}$

${\displaystyle M{\ddot {\mathbf {x} }}_{cm}=\mathbf {F} _{12}+\mathbf {F} _{21}=0\,\!}$

${\displaystyle \mathbf {v} _{cm}={\dot {\mathbf {x} }}_{cm}=(m_{1}\mathbf {v} _{10}+m_{2}\mathbf {v} _{20})/M\,\!}$

${\displaystyle m_{1}\mathbf {v} _{1}+m_{2}\mathbf {v} _{2}=M\mathbf {v} _{cm}=m_{1}\mathbf {v} _{10}+m_{2}\mathbf {v} _{20}\,\!}$

${\displaystyle \mathbf {x} _{cm}=\mathbf {v} _{cm}t+(m_{1}\mathbf {x} _{10}+m_{2}\mathbf {x} _{20})/M\,\!}$

### 位移向量運動（第二個單體問題）

${\displaystyle {\ddot {\mathbf {r} }}={\ddot {\mathbf {x} }}_{1}-{\ddot {\mathbf {x} }}_{2}=\left({\frac {\mathbf {F} _{12}}{m_{1}}}-{\frac {\mathbf {F} _{21}}{m_{2}}}\right)\,\!}$

${\displaystyle {\ddot {\mathbf {r} }}=\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\mathbf {F} _{12}\,\!}$

${\displaystyle \mu {\ddot {\mathbf {r} }}=\mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=\mathbf {F} (\mathbf {r} )\,\!}$

${\displaystyle \mathbf {x} _{1}(t)=\mathbf {x} _{cm}(t)+m_{2}\mathbf {r} (t)/M\,\!}$
${\displaystyle \mathbf {x} _{2}(t)=\mathbf {x} _{cm}(t)-m_{1}\mathbf {r} (t)/M\,\!}$

## 角動量

{\displaystyle {\begin{aligned}\mathbf {L} _{tot}&=\mathbf {x} _{1}\times (m_{1}{\dot {\mathbf {x} }}_{1})+\mathbf {x} _{2}\times (m_{2}{\dot {\mathbf {x} }}_{2})=\mathbf {x} _{cm}\times M{\dot {\mathbf {x} }}_{cm}+\mathbf {r} \times \mu {\dot {\mathbf {r} }}\\&=\mathbf {L} _{cm}+\mathbf {L} _{rel}\\\end{aligned}}\,\!}

${\displaystyle \mathbf {x} _{cm}=\mathbf {v} _{cm}t+(m_{1}\mathbf {x} _{10}+m_{2}\mathbf {x} _{20})/M\,\!}$

${\displaystyle \mathbf {L} _{cm}=\mathbf {v} _{cm}t\times M\mathbf {v} _{cm}=0\,\!}$
${\displaystyle \mathbf {L} _{tot}=\mathbf {L} _{rel}\,\!}$

${\displaystyle (Lu')'+Lu=1/L\!}$

### 角動量守恆與連心力

${\displaystyle {\boldsymbol {\tau }}_{tot}=\mathbf {x} _{1}\times \mathbf {F} _{12}+\mathbf {x} _{2}\times \mathbf {F} _{21}=\mathbf {r} \times \mathbf {F} _{12}\,\!}$

${\displaystyle {\boldsymbol {\tau }}_{tot}={\frac {d\mathbf {L} _{tot}}{dt}}\,\!}$

${\displaystyle \mathbf {F} (\mathbf {r} )=F(r){\hat {\mathbf {r} }}}$

${\displaystyle \mu {\ddot {\mathbf {r} }}={F}(r){\hat {\mathbf {r} }}\,\!}$

### 平面運動與角動量守恆

${\displaystyle \mathbf {r} \cdot \mathbf {L} _{tot}=\mathbf {r} \cdot (\mathbf {r} \times (\mu {\dot {\mathbf {r} }}))=0\,\!}$

## 參考文獻

### 引用

1. ^ David Betounes. Differential Equations. Springer. 2001: 58; Figure 2.15.
2. ^ Luo, Siwei. The Sturm-Liouville problem of two-body system. Journal of Physics Communications. 22 June 2020, 4 (6): 061001. Bibcode:2020JPhCo...4f1001L. .
3. ^ Goldstein, Herbert. Classical Mechanics 3rd. United States of America: Addison Wesley. 1980: pp. 7–8. ISBN 0201657023 （英语）.