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空间转动坐标系中质点的运动

平面转动系统中质点的运动中,我们研究了平面转动系统里质点的运动.现在,我们用类似的方法来研究三维空间转动系统里质点的运动.

设$\{O,\mathbf{E}_1,\mathbf{E}_2,\mathbf{E}_3\}$是三维空间中的一个标架,其中$\alpha=(\mathbf{E}_1,\mathbf{E}_2,\mathbf{E}_3)$是$\mathbf{R}^3$中的一组有序标准正交基.$\{o(t),\mathbf{e}_1(t),\mathbf{e}_2(t),\mathbf{e}_3(t)\}$是三维空间中的另外一个运动标架,其中$\beta=(\mathbf{e}_1(t),\mathbf{e}_2(t),\mathbf{e}_3(t))$是$\mathbf{R}^3$中的另外一组有序标准正交基.易得存在过渡矩阵
$$
[I]_{\beta}^{\alpha}=
\begin{pmatrix}
a_{11}(t)&a_{12}(t)&a_{13}(t)\\
a_{21}(t)&a_{22}(t)&a_{23}(t)\\
a_{31}(t)&a_{32}(t)&a_{33}(t)
\end{pmatrix},
$$
其中$[I]_{\beta}^{\alpha}$是一个正交矩阵.设三维空间中有一个质点,该质点在标架$\beta$下的坐标为$(a(t),b(t),c(t))_{x’y’z’}$,则质点在标架$\alpha$下的坐标应该为
\begin{align*}
o(t)_{xyz}&+\begin{pmatrix}
a_{11}(t)&a_{12}(t)&a_{13}(t)\\
a_{21}(t)&a_{22}(t)&a_{23}(t)\\
a_{31}(t)&a_{32}(t)&a_{33}(t)
\end{pmatrix}
\begin{pmatrix}
a(t)\\
b(t)\\
c(t)\\
\end{pmatrix}_{xyz}=(o_{x}(t),o_y(t),o_z(t))_{xyz}\\&+\begin{pmatrix}
a_{11}(t)&a_{12}(t)&a_{13}(t)\\
a_{21}(t)&a_{22}(t)&a_{23}(t)\\
a_{31}(t)&a_{32}(t)&a_{33}(t)
\end{pmatrix}
\begin{pmatrix}
a(t)\\
b(t)\\
c(t)\\
\end{pmatrix}_{xyz}.
\end{align*}
于是,在标架$\alpha$下,质点的速度应该为
$$
(o_{x}’(t),o_y’(t),o_z’(t))_{xyz}+\begin{pmatrix}
a_{11}’(t)&a_{12}’(t)&a_{13}’(t)\\
a_{21}’(t)&a_{22}’(t)&a_{23}’(t)\\
a_{31}’(t)&a_{32}’(t)&a_{33}’(t)
\end{pmatrix}
\begin{pmatrix}
a(t)\\
b(t)\\
c(t)\\
\end{pmatrix}_{xyz}+\begin{pmatrix}
a_{11}(t)&a_{12}(t)&a_{13}(t)\\
a_{21}(t)&a_{22}(t)&a_{23}(t)\\
a_{31}(t)&a_{32}(t)&a_{33}(t)
\end{pmatrix}
\begin{pmatrix}
a’(t)\\
b’(t)\\
c’(t)\\
\end{pmatrix}_{xyz}.
$$
从而,在标架$\alpha$下,质点的加速度为
\begin{align*}
(o_x’’(t),o_y’’(t),o_z’’(t))_{xyz}&+
\begin{pmatrix}
a_{11}’’(t)&a_{12}’’(t)&a_{13}’’(t)\\
a_{21}’’(t)&a_{22}’’(t)&a_{23}’’(t)\\
a_{31}’’(t)&a_{32}’’(t)&a_{33}’’(t)
\end{pmatrix}
\begin{pmatrix}
a(t)\\
b(t)\\
c(t)
\end{pmatrix}_{xyz}+
2\begin{pmatrix}
a_{11}’(t)&a_{12}’(t)&a_{13}’(t)\\
a_{21}’(t)&a_{22}’(t)&a_{23}’(t)\\
a_{31}’(t)&a_{32}’(t)&a_{33}’(t)
\end{pmatrix}
\begin{pmatrix}
a’(t)\\
b’(t)\\
c’(t)
\end{pmatrix}_{xyz}\\&+
\begin{pmatrix}
a_{11}(t)&a_{12}(t)&a_{13}(t)\\
a_{21}(t)&a_{22}(t)&a_{23}(t)\\
a_{31}(t)&a_{32}(t)&a_{33}(t)
\end{pmatrix}
\begin{pmatrix}
a’’(t)\\
b’’(t)\\
c’’(t)
\end{pmatrix}_{xyz}.
\end{align*}