# 極移

## 成因

### 剛體的旋轉

#### 歐拉動力學方程

${\displaystyle {\partial {\vec {H}} \over \partial t}+{\vec {\omega }}\times {\vec {H}}={\vec {L}}}$

${\displaystyle {\begin{cases}A{\dot {\omega }}_{1}+(C-B)\omega _{2}\omega _{3}=L_{1}\\B{\dot {\omega }}_{2}+(A-C)\omega _{1}\omega _{3}=L_{2}\\C{\dot {\omega }}_{3}+(B-A)\omega _{1}\omega _{2}=L_{3}\end{cases}}}$

${\displaystyle {\vec {H}}=I\times {\vec {\omega }}={\begin{pmatrix}A&0&0\\0&B&0\\0&0&C\end{pmatrix}}\times {\begin{pmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{pmatrix}}}$

#### 旋轉地球體

${\displaystyle {\begin{cases}A{\dot {\omega }}_{1}+(C-A)\omega _{2}\omega _{3}=L_{1}\\A{\dot {\omega }}_{2}+(A-C)\omega _{1}\omega _{3}=L_{2}\\C{\dot {\omega }}_{3}=L_{3}\end{cases}}}$

${\displaystyle {\begin{cases}A{\dot {\omega }}_{1}+(C-A)\omega _{2}\omega _{3}=0\\A{\dot {\omega }}_{2}+(A-C)\omega _{1}\omega _{3}=0\\C{\dot {\omega }}_{3}=0\end{cases}}\Longrightarrow {\begin{cases}A{\dot {\omega }}_{1}+(C-A)\omega _{2}\Omega =0\\A{\dot {\omega }}_{2}+(A-C)\omega _{1}\Omega =0\\\omega _{3}={\text{const}}=\Omega \end{cases}}}$

${\displaystyle {\begin{cases}A{\dot {\omega }}_{1}+(C-A)\Omega \cdot \omega _{2}=0\\A{\dot {\omega }}_{2}+(A-C)\Omega \cdot \omega _{1}=0\\\end{cases}}\Longrightarrow {\begin{cases}A{\ddot {\omega }}_{1}+(C-A)\Omega \cdot {\dot {\omega }}_{2}=0\\A{\ddot {\omega }}_{2}+(A-C)\Omega \cdot {\dot {\omega }}_{1}=0\\\end{cases}}}$

${\displaystyle {\begin{cases}{\ddot {\omega }}_{1}+{\left[{\frac {C-A}{A}}\Omega \right]}^{2}\omega _{1}=0\\{\ddot {\omega }}_{2}+{\left[{\frac {C-A}{A}}\Omega \right]}^{2}\omega _{2}=0\\\end{cases}}\Longrightarrow {\begin{cases}\omega _{1}=p\cos {\left[({\frac {C-A}{A}}\Omega )t+\varphi _{0}\right]}\\\omega _{2}=p\sin {\left[({\frac {C-A}{A}}\Omega )t+\varphi _{0}\right]}\\\end{cases}}}$

${\displaystyle {\begin{cases}\lVert {\vec {\omega }}\rVert ={\sqrt {\Omega ^{2}+p^{2}}}\\{\vec {e_{\omega }}}={\left[{\frac {\omega _{1}}{\omega }},{\frac {\omega _{2}}{\omega }},{\frac {\Omega }{\omega }}\right]}^{\text{T}}\\\end{cases}}}$

## 參考文獻

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