弹性杆动力学中的速度牵连关系
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当仅存在沿着杆方向的运动时，动量和动量矩是如何通过速度以及曲率张量进行表达的？
杆的动力学方程表示为：
\[\begin{cases} \mathbf{n}_s+\mathbf{q}_n=\rho A\mathbf{r}_{tt}\\ \mathbf{m}_s+\mathbf{r}_s\times \mathbf{n}+\mathbf{q}_m=\rho \left( I_1\mathbf{d}_2\times \left( \mathbf{d}_2 \right) _{tt}+I_2\mathbf{d}_1\times \left( \mathbf{d}_1 \right) _{tt} \right)\\ \end{cases}\]考虑左边的项，动量可表示为：
\[\mathbf{r}_{tt}=\frac{\partial}{\partial t}\left( \frac{\partial \mathbf{r}}{\partial t} \right) =\frac{\partial}{\partial t}\left( \frac{\partial \mathbf{r}}{\partial s}*v \right) =\frac{\partial \mathbf{d}_3}{\partial s}v^2+v_t\mathbf{d}_3=\left( u_2\mathbf{d}_1-u_1\mathbf{d}_2 \right) v^2+v_t\mathbf{d}_3\]动量矩表示：
\[\mathbf{d}_2\times \frac{\partial}{\partial t}\left( \frac{\partial \mathbf{d}_2}{\partial t} \right) \\ =\mathbf{d}_2\times \frac{\partial}{\partial t}\left( \frac{\partial \mathbf{d}_2}{\partial s}*v \right)\\ =\mathbf{d}_2\times \left( \frac{\partial}{\partial s}\left( \frac{\partial \mathbf{d}_2}{\partial s} \right) v^2+v_t\frac{\partial \mathbf{d}_2}{\partial s} \right) =\mathbf{d}_2\times \left( \left( \mathbf{u}_s\times \mathbf{d}_2+\mathbf{u}\times \left( \mathbf{u}\times \mathbf{d}_2 \right) \right) v^2+v_t\left( \mathbf{u}\times \mathbf{d}_2 \right) \right)\\ =\mathbf{d}_2\times \left( \mathbf{u}_s\times \mathbf{d}_2-u_2\mathbf{u} \right) v^2+v_t\left( \mathbf{u}-u_2\mathbf{d}_2 \right)\]现在考虑Darboux矢量对空间的导数：
\[\frac{\partial \mathbf{u}}{\partial s}=\left( u_1,u_2,u_3 \right) _s\cdot \left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{d}_3\\ \end{array} \right) +\left( u_1,u_2,u_3 \right) \left( \begin{matrix} 0& u_3& -u_2\\ -u_3& 0& u_1\\ u_2& -u_1& 0\\ \end{matrix} \right) \left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{d}_3\\ \end{array} \right) =\left( u_1,u_2,u_3 \right) _s\cdot \left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{d}_3\\ \end{array} \right)\]Darboux矢量（特征向量）本征性质：对Darboux矢量求导相当于只对曲率张量系数求导。
\[\mathbf{d}_1\times \frac{\partial}{\partial t}\left( \frac{\partial \mathbf{d}_1}{\partial t} \right) =\left( \left( \left( u_2 \right) _s-u_1u_3 \right) \mathbf{d}_2+\left( \left( u_3 \right) _s+u_1u_2 \right) \mathbf{d}_3 \right) v^2+v_t\left( u_2\mathbf{d}_2+u_3\mathbf{d}_3 \right)\] \[\mathbf{d}_2\times \frac{\partial}{\partial t}\left( \frac{\partial \mathbf{d}_2}{\partial t} \right) =\left( \left( \left( u_1 \right) _s+u_2u_3 \right) \mathbf{d}_1+\left( \left( u_3 \right) _s-u_1u_2 \right) \mathbf{d}_3 \right) v^2+v_t\left( u_1\mathbf{d}_1+u_3\mathbf{d}_3 \right)\]