# 歐拉運動定律

## 剛體

### 歐拉第一運動定律

${\displaystyle \mathbf {F} ^{(ext)}=m\mathbf {a} _{cm}}$

#### 導引

${\displaystyle \mathbf {r} _{cm}{\stackrel {def}{=}}{\frac {\sum _{i=1}^{n}m_{i}\mathbf {r} _{i}}{m}}}$ ;

${\displaystyle \mathbf {v} _{cm}={\frac {\mathrm {d} \mathbf {r} _{cm}}{\mathrm {d} t}}={\cfrac {\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}}{m}}}$

${\displaystyle \mathbf {a} _{cm}={\frac {\mathrm {d} \mathbf {v} _{cm}}{\mathrm {d} t}}={\cfrac {\sum _{i=1}^{n}m_{i}\mathbf {a} _{i}}{m}}}$

${\displaystyle i}$ 個粒子感受到的力 ${\displaystyle \mathbf {F} _{i}}$

${\displaystyle \mathbf {F} _{i}=\mathbf {F} _{i}^{(ext)}+\sum _{j=1,j\neq i}^{n}\mathbf {F} _{ji}}$

${\displaystyle \mathbf {F} =\sum _{i=1}^{n}\mathbf {F} _{i}=\sum _{i=1}^{n}\mathbf {F} _{i}^{(ext)}+\sum _{i=1}^{n}\sum _{j=1,j\neq i}^{n}\mathbf {F} _{ji}}$

${\displaystyle \mathbf {F} _{ji}=-\mathbf {F} _{ji}}$

${\displaystyle \mathbf {F} =\sum _{i=1}^{n}\mathbf {F} _{i}^{(ext)}=\mathbf {F} ^{(ext)}}$

${\displaystyle \mathbf {F} _{i}=m_{i}\mathbf {a} _{i}}$

${\displaystyle \mathbf {F} =\sum _{i=1}^{n}\mathbf {F} _{i}=\sum _{i=1}m_{i}\mathbf {a} _{i}=m\mathbf {a} _{cm}}$

${\displaystyle \mathbf {F} ^{(ext)}=m\mathbf {a} _{cm}}$

#### 動量守恆定律

${\displaystyle \mathbf {p} =\sum _{i=1}^{n}\mathbf {p} _{i}=\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}=m\mathbf {v} _{cm}}$

${\displaystyle \mathbf {F} ^{(ext)}={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}}$

### 歐拉第二運動定律

${\displaystyle {\boldsymbol {\tau }}_{O}^{(ext)}={\frac {\mathrm {d} \mathbf {L} _{O}}{\mathrm {d} t}}}$

#### 導引

${\displaystyle \mathbf {L} _{i}=\mathbf {r} _{i}\times \mathbf {p} _{i}=\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}}$

${\displaystyle \mathbf {L} _{i}}$ 對於時間的導數為

${\displaystyle {\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\frac {\mathrm {d} (\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i})}{\mathrm {d} t}}=\mathbf {v} _{i}\times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {a} _{i}=\mathbf {r} _{i}\times m_{i}\mathbf {a} _{i}}$

${\displaystyle {\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}=\mathbf {r} _{i}\times \mathbf {F} _{i}}$

${\displaystyle i}$ 個粒子所感受到的淨力矩 ${\displaystyle {\boldsymbol {\tau }}_{i}}$${\displaystyle {\boldsymbol {\tau }}_{i}=\mathbf {r} _{i}\times \mathbf {F} _{i}}$ 。所以，${\displaystyle {\boldsymbol {\tau }}_{i}}$${\displaystyle \mathbf {L} _{i}}$ 的關係為

${\displaystyle {\boldsymbol {\tau }}_{i}={\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}}$

${\displaystyle {\boldsymbol {\tau }}_{O}=\sum _{i=1}^{n}{\boldsymbol {\tau }}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}\sum _{i=1}^{n}\mathbf {L} _{i}={\frac {\mathrm {d} \mathbf {L} _{O}}{\mathrm {d} t}}}$

${\displaystyle i}$ 個粒子所感受到的淨力 ${\displaystyle \mathbf {F} _{i}}$

${\displaystyle \mathbf {F} _{i}=\mathbf {F} _{i}^{(ext)}+\sum _{j=1,j\neq i}^{n}\mathbf {F} _{ji}}$

${\displaystyle i}$ 個粒子所感受到的淨力矩 ${\displaystyle {\boldsymbol {\tau }}_{i}}$

${\displaystyle {\boldsymbol {\tau }}_{i}=\mathbf {r} _{i}\times \mathbf {F} _{i}=\mathbf {r} _{i}\times \mathbf {F} _{i}^{(ext)}+\sum _{j=1,j\neq i}^{n}\mathbf {r} _{i}\times \mathbf {F} _{ji}}$

${\displaystyle {\boldsymbol {\tau }}_{O}=\sum _{i=1}^{n}{\boldsymbol {\tau }}_{i}=\sum _{i=1}^{n}\mathbf {r} _{i}\times \mathbf {F} _{i}^{(ext)}+\sum _{i=1}^{n}\sum _{j=1,j\neq i}^{n}\mathbf {r} _{i}\times \mathbf {F} _{ji}}$

${\displaystyle \mathbf {r} _{i}\times \mathbf {F} _{ji}+\mathbf {r} _{j}\times \mathbf {F} _{ji}=\mathbf {r} _{i}\times \mathbf {F} _{ji}-\mathbf {r} _{j}\times \mathbf {F} _{ji}=(\mathbf {r} _{i}-\mathbf {r} _{j})\times \mathbf {F} _{ji}=\mathbf {r} _{ij}\times \mathbf {F} _{ji}}$

${\displaystyle {\boldsymbol {\tau }}_{O}=\sum _{i=1}^{n}\mathbf {r} _{i}\times \mathbf {F} _{i}^{(ext)}={\boldsymbol {\tau }}_{O}^{(ext)}}$

${\displaystyle {\boldsymbol {\tau }}_{O}^{(ext)}={\frac {\mathrm {d} \mathbf {L} _{O}}{\mathrm {d} t}}}$

#### 相對於質心的歐拉第二運動定律

${\displaystyle {\boldsymbol {\tau }}_{O}=\sum _{i=1}^{n}{\boldsymbol {\tau }}_{i}=\sum _{i=1}^{n}\mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})=\sum _{i=1}^{n}(\mathbf {r} _{cm}+\mathbf {r} '_{i})\times (m_{i}(\mathbf {a} _{cm}+\mathbf {a} '_{i}))}$

${\displaystyle {\boldsymbol {\tau }}_{O}=\mathbf {r} _{cm}\times m\mathbf {a} _{cm}+\sum _{i=1}^{n}\mathbf {r} '_{i}\times m_{i}\mathbf {a} '_{i}}$

${\displaystyle {\boldsymbol {\tau }}_{cm}=\sum _{i=1}^{n}\mathbf {r} '_{i}\times m_{i}\mathbf {a} '_{i}}$

${\displaystyle \mathbf {L} _{cm}=\sum _{i=1}^{n}\mathbf {r} '_{i}\times m_{i}\mathbf {v} '_{i}}$

${\displaystyle {\boldsymbol {\tau }}_{cm}={\frac {\mathrm {d} \mathbf {L} _{cm}}{\mathrm {d} t}}}$

## 可變形體

${\displaystyle \mathbf {F} _{b}=\int _{\mathbb {V} }\mathbf {b} \,\mathrm {d} m=\int _{\mathbb {V} }\rho \mathbf {b} \,\mathrm {d} V}$
${\displaystyle \mathbf {F} _{t}=\int _{\mathbb {S} }\mathbf {t} \,\mathrm {d} S}$

${\displaystyle \mathbf {L} _{b}=\int _{\mathbb {V} }\mathbf {r} \times \rho \mathbf {b} \,\mathrm {d} V}$
${\displaystyle \mathbf {L} _{t}=\int _{\mathbb {S} }\mathbf {r} \times \mathbf {t} \,\mathrm {d} S}$

${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}}$

${\displaystyle \int _{\mathbb {V} }\rho \mathbf {b} \,\mathrm {d} V+\int _{\mathbb {S} }\mathbf {t} \,\mathrm {d} S={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\mathbb {V} }\rho \mathbf {v} \,\mathrm {d} V}$

${\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}$

${\displaystyle \int _{\mathbb {V} }\mathbf {r} \times \rho \mathbf {b} \,\mathrm {d} V+\int _{\mathbb {S} }\mathbf {r} \times \mathbf {t} \,\mathrm {d} S={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\mathbb {V} }\mathbf {r} \times \rho \mathbf {v} \,\mathrm {d} V}$

## 參考文獻

1. ^ Beatty, Millard F. Principles of engineering mechanics Volume 2 of Principles of Engineering Mechanics: Dynamics-The Analysis of Motion,. Springer. 2006: pp. 405. ISBN 0387237046.
2. ^ Bradley, Robert E., Sandifer, Charles. Leonhard Euler: life, work and legacy Volume 5 of Studies in the history and philosophy of mathematics. Elsevier. 2007: pp. 196. ISBN 9780444527288.
3. ^ Rao, Anil Vithala. Dynamics of particles and rigid bodies. Cambridge University Press. 2006: 355. ISBN 978-0-521-85811-3.
4. ^ Lubliner, Jacob. Plasticity Theory (Revised Edition) (PDF). Dover Publications. 2008: pp. 27–28. ISBN 0486462900. （原始内容 (PDF)存档于2010-03-31）.