本文仅供学习使用，总结很多本现有讲述运动学或动力学书籍后的总结，从矢量的角度进行分析，方法比较传统，但更易理解，并且现有的看似抽象方法，两者本质上并无不同。
2024年底本人学位论文发表后方可摘抄
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本文参考：
黎 旭,陈 强 洪,甄 文 强 等.惯 性 张 量 平 移 和 旋 转 复 合 变 换 的 一 般 形 式 及 其 应 用[J].工 程 数 学 学 报,2022,39(06):1005-1011.
食用方法质量点的动量与角动量
刚体的动量与角动量------力与力矩的关系
惯性矩阵的表达与推导------在刚体运动过程中的作用
惯性矩阵在不同坐标系下的表达
务必自己推导全部公式，并理解每个符号的含义
机构运动学与动力学分析与建模 Ch00-2质量刚体的在坐标系下运动Part3
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- [2.2.3 欧拉方程 Euler equation - 2](#2.2.3 欧拉方程 Euler equation - 2)
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2.2.3 欧拉方程 Euler equation - 2
- 进而分析 H ⃗ Σ M F = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) ω ⃗ M F d m i − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) R ⃗ G P i F d m i \vec{H}{\Sigma {\mathrm{M}}}^{F}=m{\mathrm{total}}\cdot \vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}+\int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \vec{\omega}{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \right) \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m{\mathrm{i}} H ΣMF=mtotal⋅R GF×V GF+∫(R GPiF⋅R GPiF)ω MFdmi−∫(R GPiF⋅ω MF)R GPiFdmi，有：
H ⃗ Σ M F = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + [ I ] Σ M / G F ⋅ ω ⃗ M F H ⃗ Σ M / G F = H ⃗ Σ M F − m t o t a l ⋅ R ⃗ G F × V ⃗ G F = [ I ] Σ M / G F ⋅ ω ⃗ M F \begin{split} &\vec{H}{\Sigma {\mathrm{M}}}^{F}=m{\mathrm{total}}\cdot \vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}+\int{\left( {\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m{\mathrm{i}}\cdot \vec{\omega}{\mathrm{M}}^{F} =m{\mathrm{total}}\cdot \vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}+\left[ I \right] {\Sigma {\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \\ &\vec{H}{\Sigma {\mathrm{M}}/\mathrm{G}}^{F}=\vec{H}{\Sigma {\mathrm{M}}}^{F}-m{\mathrm{total}}\cdot \vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}=\left[ I \right] {\Sigma {\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \end{split} H ΣMF=mtotal⋅R GF×V GF+∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi⋅ω MF=mtotal⋅R GF×V GF+[I]ΣM/GF⋅ω MFH ΣM/GF=H ΣMF−mtotal⋅R GF×V GF=[I]ΣM/GF⋅ω MF
则相对于质心点 G G G 存在：
{ M ⃗ Σ M / G F = [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) [ I ] Σ M / G F = ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i F ⃗ G F = m t o t a l a ⃗ G F \begin{cases} \vec{M}{\Sigma {\mathrm{M}}/\mathrm{G}}^{F}=\left[ I \right] {\Sigma {\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}{\mathrm{M}}^{F}+\vec{\omega}{\mathrm{M}}^{F}\times \left( \left[ I \right] {\Sigma {\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \right)\\ \left[ I \right] {\Sigma {\mathrm{M}}/\mathrm{G}}^{F}=\int{\left( {\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\\ \vec{F}{\mathrm{G}}^{F}=m{\mathrm{total}}\vec{a}_{\mathrm{G}}^{F}\\ \end{cases} ⎩ ⎨ ⎧M ΣM/GF=[I]ΣM/GFα MF+ω MF×([I]ΣM/GF⋅ω MF)[I]ΣM/GF=∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmiF GF=mtotala GF - 对 H ⃗ Σ M / O F \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF进一步推导（分析 [ I ] Σ M / G F \left[ I \right] {\Sigma {\mathrm{M}}/\mathrm{G}}^{F} [I]ΣM/GF），可得：
H ⃗ Σ M / O F = ∑ i N R ⃗ O P i F × P ⃗ P i F = ∑ i N m P i ⋅ R ⃗ O P i F × ( V ⃗ O F + ω ⃗ F × R ⃗ O P i F ) = ∑ i N m P i ⋅ R ⃗ ~ O P i F ⋅ ( ω ⃗ ~ F ⋅ R ⃗ O P i F ) + ∑ i N m P i ⋅ R ⃗ ~ O P i F ⋅ V ⃗ O F = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ 0 − z O P i F y O P i F z O P i F 0 − x O P i F − y O P i F x O P i F 0 ] ⋅ [ I ^ J ^ K ^ ] T ( [ 0 − w z P i F w y P i F w z P i F 0 − w x P i F − w y P i F w x P i F 0 ] ⋅ [ x O P i F y O P i F z O P i F ] ) + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ [ ( y O P i F ) 2 + ( z O P i F ) 2 ] w x P i F − ( x O P i F y O P i F ) w y P i F − ( x O P i F z O P i F ) w z P i F − ( y O P i F x O P i F ) w x P i F + [ ( x O P i F ) 2 + ( z O P i F ) 2 ] w y P i F − ( y O P i F z O P i F ) w z P i F − ( z O P i F x O P i F ) w x P i F − ( z O P i F y O P i F ) w y P i F + [ ( x O P i F ) 2 + ( y O P i F ) 2 ] w z P i F ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ ( y O P i F ) 2 + ( z O P i F ) 2 − x O P i F y O P i F − x O P i F z O P i F − y O P i F x O P i F ( x O P i F ) 2 + ( z O P i F ) 2 − y O P i F z O P i F − z O P i F x O P i F − z O P i F y O P i F ( x O P i F ) 2 + ( y O P i F ) 2 ] [ w x P i F w y P i F w z P i F ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F = [ I ^ J ^ K ^ ] T [ ∑ i N m P i ⋅ [ ( y O P i F ) 2 + ( z O P i F ) 2 ] − ∑ i N m P i ⋅ x O P i F y O P i F − ∑ i N m P i ⋅ ( x O P i F z O P i F ) − ∑ i N m P i ⋅ ( y O P i F x O P i F ) ∑ i N m P i ⋅ [ ( x O P i F ) 2 + ( z O P i F ) 2 ] − ∑ i N m P i ⋅ ( y O P i F z O P i F ) − ∑ i N m P i ⋅ ( z O P i F x O P i F ) − ∑ i N m P i ⋅ ( z O P i F y O P i F ) ∑ i N m P i ⋅ [ ( x O P i F ) 2 + ( y O P i F ) 2 ] ] [ w x P i F w y P i F w z P i F ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F = [ I ^ J ^ K ^ ] T [ I x x I x y I x z I y x I y y I y z I z x I z y I z z ] [ w x P i F w y P i F w z P i F ] = [ I ^ J ^ K ^ ] T [ I x x w x P i F + I x y w y P i F + I x z w z P i F I y x w x P i F + I y y w y P i F + I y z w z P i F I z x w x P i F + I z y w y P i F + I z z w z P i F ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F = [ I ^ J ^ K ^ ] T [ H x H y H z ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F \begin{aligned} \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F}&=\sum_i^N{\vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \vec{P}{\mathrm{P}{\mathrm{i}}}^{F}}=\sum_i^N{m_{\mathrm{P}{\mathrm{i}}}\cdot \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \left( \vec{V}{\mathrm{O}}^{F}+\vec{\omega}^F\times \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F} \right)}=\sum_i^N{m_{\mathrm{P}{\mathrm{i}}}\cdot \tilde{\vec{R}}{\mathrm{OP}{\mathrm{i}}}^{F}\cdot \left( \tilde{\vec{\omega}}^F\cdot \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F} \right)}+\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \tilde{\vec{R}}{\mathrm{OP}{\mathrm{i}}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}}\\ &=\sum_i^N{m_{\mathrm{P}{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} 0& -z{\mathrm{OP}{\mathrm{i}}}^{F}& y{\mathrm{OP}{\mathrm{i}}}^{F}\\ z{\mathrm{OP}{\mathrm{i}}}^{F}& 0& -x{\mathrm{OP}{\mathrm{i}}}^{F}\\ -y{\mathrm{OP}{\mathrm{i}}}^{F}& x{\mathrm{OP}{\mathrm{i}}}^{F}& 0\\ \end{matrix} \right] \cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left( \left[ \begin{matrix} 0& -w{\mathrm{z}{\mathrm{Pi}}}^{F}& w{\mathrm{y}{\mathrm{Pi}}}^{F}\\ w{\mathrm{z}{\mathrm{Pi}}}^{F}& 0& -w{\mathrm{x}{\mathrm{Pi}}}^{F}\\ -w{\mathrm{y}{\mathrm{Pi}}}^{F}& w{\mathrm{x}{\mathrm{Pi}}}^{F}& 0\\ \end{matrix} \right] \cdot \left[ \begin{array}{c} x{\mathrm{OP}{\mathrm{i}}}^{F}\\ y{\mathrm{OP}{\mathrm{i}}}^{F}\\ z{\mathrm{OP}{\mathrm{i}}}^{F}\\ \end{array} \right] \right) +m{\mathrm{total}}\cdot \tilde{\vec{R}}{\mathrm{OG}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}}\\ &=\sum_i^N{m_{\mathrm{P}{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \left[ \left( y{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( z{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2 \right] w{\mathrm{x}{\mathrm{Pi}}}^{F}-\left( x{\mathrm{OP}{\mathrm{i}}}^{F}y{\mathrm{OP}{\mathrm{i}}}^{F} \right) w{\mathrm{y}{\mathrm{Pi}}}^{F}-\left( x{\mathrm{OP}{\mathrm{i}}}^{F}z{\mathrm{OP}{\mathrm{i}}}^{F} \right) w{\mathrm{z}{\mathrm{Pi}}}^{F}\\ -\left( y{\mathrm{OP}{\mathrm{i}}}^{F}x{\mathrm{OP}{\mathrm{i}}}^{F} \right) w{\mathrm{x}{\mathrm{Pi}}}^{F}+\left[ \left( x{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( z{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2 \right] w{\mathrm{y}{\mathrm{Pi}}}^{F}-\left( y{\mathrm{OP}{\mathrm{i}}}^{F}z{\mathrm{OP}{\mathrm{i}}}^{F} \right) w{\mathrm{z}{\mathrm{Pi}}}^{F}\\ -\left( z{\mathrm{OP}{\mathrm{i}}}^{F}x{\mathrm{OP}{\mathrm{i}}}^{F} \right) w{\mathrm{x}{\mathrm{Pi}}}^{F}-\left( z{\mathrm{OP}{\mathrm{i}}}^{F}y{\mathrm{OP}{\mathrm{i}}}^{F} \right) w{\mathrm{y}{\mathrm{Pi}}}^{F}+\left[ \left( x{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( y{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2 \right] w{\mathrm{z}{\mathrm{Pi}}}^{F}\\ \end{array} \right] +m{\mathrm{total}}\cdot \tilde{\vec{R}}{\mathrm{OG}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}}\\ &=\sum_i^N{m_{\mathrm{P}{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} \left( y{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( z{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2& -x{\mathrm{OP}{\mathrm{i}}}^{F}y{\mathrm{OP}{\mathrm{i}}}^{F}& -x{\mathrm{OP}{\mathrm{i}}}^{F}z{\mathrm{OP}{\mathrm{i}}}^{F}\\ -y{\mathrm{OP}{\mathrm{i}}}^{F}x{\mathrm{OP}{\mathrm{i}}}^{F}& \left( x{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( z{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2& -y{\mathrm{OP}{\mathrm{i}}}^{F}z{\mathrm{OP}{\mathrm{i}}}^{F}\\ -z{\mathrm{OP}{\mathrm{i}}}^{F}x{\mathrm{OP}{\mathrm{i}}}^{F}& -z{\mathrm{OP}{\mathrm{i}}}^{F}y{\mathrm{OP}{\mathrm{i}}}^{F}& \left( x{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( y{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2\\ \end{matrix} \right] \left[ \begin{array}{c} w{\mathrm{x}{\mathrm{Pi}}}^{F}\\ w{\mathrm{y}{\mathrm{Pi}}}^{F}\\ w{\mathrm{z}{\mathrm{Pi}}}^{F}\\ \end{array} \right] +m{\mathrm{total}}\cdot \tilde{\vec{R}}{\mathrm{OG}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}}\\ &=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} \sum_i^N{m_{\mathrm{P}{\mathrm{i}}}\cdot \left[ \left( y{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( z{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2 \right]}& -\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot x{\mathrm{OP}{\mathrm{i}}}^{F}y{\mathrm{OP}{\mathrm{i}}}^{F}}& -\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left( x{\mathrm{OP}{\mathrm{i}}}^{F}z{\mathrm{OP}{\mathrm{i}}}^{F} \right)}\\ -\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left( y{\mathrm{OP}{\mathrm{i}}}^{F}x{\mathrm{OP}{\mathrm{i}}}^{F} \right)}& \sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left[ \left( x{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( z{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2 \right]}& -\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left( y{\mathrm{OP}{\mathrm{i}}}^{F}z{\mathrm{OP}{\mathrm{i}}}^{F} \right)}\\ -\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left( z{\mathrm{OP}{\mathrm{i}}}^{F}x{\mathrm{OP}{\mathrm{i}}}^{F} \right)}& -\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left( z{\mathrm{OP}{\mathrm{i}}}^{F}y{\mathrm{OP}{\mathrm{i}}}^{F} \right)}& \sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left[ \left( x{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2+\left( y{\mathrm{OP}{\mathrm{i}}}^{F} \right) ^2 \right]}\\ \end{matrix} \right] \left[ \begin{array}{c} w{\mathrm{x}{\mathrm{Pi}}}^{F}\\ w{\mathrm{y}{\mathrm{Pi}}}^{F}\\ w{\mathrm{z}{\mathrm{Pi}}}^{F}\\ \end{array} \right] +m{\mathrm{total}}\cdot \tilde{\vec{R}}{\mathrm{OG}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}\,\,\\ &=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} I_{\mathrm{xx}}& I_{\mathrm{xy}}& I_{\mathrm{xz}}\\ I_{\mathrm{yx}}& I_{\mathrm{yy}}& I_{\mathrm{yz}}\\ I_{\mathrm{zx}}& I_{\mathrm{zy}}& I_{\mathrm{zz}}\\ \end{matrix} \right] \left[ \begin{array}{c} w_{\mathrm{x}{\mathrm{Pi}}}^{F}\\ w{\mathrm{y}{\mathrm{Pi}}}^{F}\\ w{\mathrm{z}{\mathrm{Pi}}}^{F}\\ \end{array} \right] =\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} I{\mathrm{xx}}w_{\mathrm{x}{\mathrm{Pi}}}^{F}+I{\mathrm{xy}}w_{\mathrm{y}{\mathrm{Pi}}}^{F}+I{\mathrm{xz}}w_{\mathrm{z}{\mathrm{Pi}}}^{F}\\ I{\mathrm{yx}}w_{\mathrm{x}{\mathrm{Pi}}}^{F}+I{\mathrm{yy}}w_{\mathrm{y}{\mathrm{Pi}}}^{F}+I{\mathrm{yz}}w_{\mathrm{z}{\mathrm{Pi}}}^{F}\\ I{\mathrm{zx}}w_{\mathrm{x}{\mathrm{Pi}}}^{F}+I{\mathrm{zy}}w_{\mathrm{y}{\mathrm{Pi}}}^{F}+I{\mathrm{zz}}w_{\mathrm{z}{\mathrm{Pi}}}^{F}\\ \end{array} \right] +m{\mathrm{total}}\cdot \tilde{\vec{R}}{\mathrm{OG}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}\\ &=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} H_{\mathrm{x}}\\ H_{\mathrm{y}}\\ H_{\mathrm{z}}\\ \end{array} \right] +m_{\mathrm{total}}\cdot \tilde{\vec{R}}{\mathrm{OG}}^{F}\cdot \vec{V}{\mathrm{O}}^{F}\\ \end{aligned} H ΣM/OF=i∑NR OPiF×P PiF=i∑NmPi⋅R OPiF×(V OF+ω F×R OPiF)=i∑NmPi⋅R ~OPiF⋅(ω ~F⋅R OPiF)+i∑NmPi⋅R ~OPiF⋅V OF=i∑NmPi⋅ I^J^K^ T 0zOPiF−yOPiF−zOPiF0xOPiFyOPiF−xOPiF0 ⋅ I^J^K^ T 0wzPiF−wyPiF−wzPiF0wxPiFwyPiF−wxPiF0 ⋅ xOPiFyOPiFzOPiF +mtotal⋅R ~OGF⋅V OF=i∑NmPi⋅ I^J^K^ T [(yOPiF)2+(zOPiF)2]wxPiF−(xOPiFyOPiF)wyPiF−(xOPiFzOPiF)wzPiF−(yOPiFxOPiF)wxPiF+[(xOPiF)2+(zOPiF)2]wyPiF−(yOPiFzOPiF)wzPiF−(zOPiFxOPiF)wxPiF−(zOPiFyOPiF)wyPiF+[(xOPiF)2+(yOPiF)2]wzPiF +mtotal⋅R ~OGF⋅V OF=i∑NmPi⋅ I^J^K^ T (yOPiF)2+(zOPiF)2−yOPiFxOPiF−zOPiFxOPiF−xOPiFyOPiF(xOPiF)2+(zOPiF)2−zOPiFyOPiF−xOPiFzOPiF−yOPiFzOPiF(xOPiF)2+(yOPiF)2 wxPiFwyPiFwzPiF +mtotal⋅R ~OGF⋅V OF= I^J^K^ T ∑iNmPi⋅[(yOPiF)2+(zOPiF)2]−∑iNmPi⋅(yOPiFxOPiF)−∑iNmPi⋅(zOPiFxOPiF)−∑iNmPi⋅xOPiFyOPiF∑iNmPi⋅[(xOPiF)2+(zOPiF)2]−∑iNmPi⋅(zOPiFyOPiF)−∑iNmPi⋅(xOPiFzOPiF)−∑iNmPi⋅(yOPiFzOPiF)∑iNmPi⋅[(xOPiF)2+(yOPiF)2] wxPiFwyPiFwzPiF +mtotal⋅R ~OGF⋅V OF= I^J^K^ T IxxIyxIzxIxyIyyIzyIxzIyzIzz wxPiFwyPiFwzPiF = I^J^K^ T IxxwxPiF+IxywyPiF+IxzwzPiFIyxwxPiF+IyywyPiF+IyzwzPiFIzxwxPiF+IzywyPiF+IzzwzPiF +mtotal⋅R ~OGF⋅V OF= I^J^K^ T HxHyHz +mtotal⋅R ~OGF⋅V OF
其中：
- 若有： ω ⃗ = [ I ^ J ^ K ^ ] T [ ω 1 ω 2 ω 3 ] , R ⃗ = [ I ^ J ^ K ^ ] T [ r 1 r 2 r 3 ] \vec{\omega}=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \omega _1\\ \omega _2\\ \omega _3\\ \end{array} \right] ,\vec{R}=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} r_1\\ r_2\\ r_3\\ \end{array} \right] ω = I^J^K^ T ω1ω2ω3 ,R = I^J^K^ T r1r2r3 ，则有如下叉乘的计算：
ω ⃗ × R ⃗ = ω ⃗ ~ ⋅ R ⃗ = [ I ^ J ^ K ^ ] T ( [ 0 − ω 3 ω 2 ω 3 0 − ω 1 − ω 2 ω 1 0 ] ⋅ [ r 1 r 2 r 3 ] ) \vec{\omega}\times \vec{R}=\tilde{\vec{\omega}}\cdot \vec{R}=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left( \left[ \begin{matrix} 0& -\omega _3& \omega _2\\ \omega _3& 0& -\omega _1\\ -\omega _2& \omega _1& 0\\ \end{matrix} \right] \cdot \left[ \begin{array}{c} r_1\\ r_2\\ r_3\\ \end{array} \right] \right) ω ×R =ω ~⋅R = I^J^K^ T 0ω3−ω2−ω30ω1ω2−ω10 ⋅ r1r2r3- H ⃗ Σ M / O F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF表示刚体 Σ M \Sigma _{\mathrm{M}} ΣM
相对于(with respect to/W.R.T)点 O O O 的角动量在固定坐标系 { F } \left\{ F \right\} {F}的表达。其投影分量满足：H x H y H z \] = \[ I x x w x P i F + I x y w y P i F + I x z w z P i F I y x w x P i F + I y y w y P i F + I y z w z P i F I z x w x P i F + I z y w y P i F + I z z w z P i F \] = \[ I x x I x y I x z I y x I y y I y z I z x I z y I z z \] \[ w x P i F w y P i F w z P i F \] = \[ I \] \[ w x P i F w y P i F w z P i F \] \\left\[ \\begin{array}{c} H_{\\mathrm{x}}\\\\ H_{\\mathrm{y}}\\\\ H_{\\mathrm{z}}\\\\ \\end{array} \\right\] =\\left\[ \\begin{array}{c} I_{\\mathrm{xx}}w_{\\mathrm{x}_{\\mathrm{Pi}}}\^{F}+I_{\\mathrm{xy}}w_{\\mathrm{y}_{\\mathrm{Pi}}}\^{F}+I_{\\mathrm{xz}}w_{\\mathrm{z}_{\\mathrm{Pi}}}\^{F}\\\\ I_{\\mathrm{yx}}w_{\\mathrm{x}_{\\mathrm{Pi}}}\^{F}+I_{\\mathrm{yy}}w_{\\mathrm{y}_{\\mathrm{Pi}}}\^{F}+I_{\\mathrm{yz}}w_{\\mathrm{z}_{\\mathrm{Pi}}}\^{F}\\\\ I_{\\mathrm{zx}}w_{\\mathrm{x}_{\\mathrm{Pi}}}\^{F}+I_{\\mathrm{zy}}w_{\\mathrm{y}_{\\mathrm{Pi}}}\^{F}+I_{\\mathrm{zz}}w_{\\mathrm{z}_{\\mathrm{Pi}}}\^{F}\\\\ \\end{array} \\right\] =\\left\[ \\begin{matrix} I_{\\mathrm{xx}}\& I_{\\mathrm{xy}}\& I_{\\mathrm{xz}}\\\\ I_{\\mathrm{yx}}\& I_{\\mathrm{yy}}\& I_{\\mathrm{yz}}\\\\ I_{\\mathrm{zx}}\& I_{\\mathrm{zy}}\& I_{\\mathrm{zz}}\\\\ \\end{matrix} \\right\] \\left\[ \\begin{array}{c} w_{\\mathrm{x}_{\\mathrm{Pi}}}\^{F}\\\\ w_{\\mathrm{y}_{\\mathrm{Pi}}}\^{F}\\\\ w_{\\mathrm{z}_{\\mathrm{Pi}}}\^{F}\\\\ \\end{array} \\right\] =\\left\[ I \\right\] \\left\[ \\begin{array}{c} w_{\\mathrm{x}_{\\mathrm{Pi}}}\^{F}\\\\ w_{\\mathrm{y}_{\\mathrm{Pi}}}\^{F}\\\\ w_{\\mathrm{z}_{\\mathrm{Pi}}}\^{F}\\\\ \\end{array} \\right\] HxHyHz = IxxwxPiF+IxywyPiF+IxzwzPiFIyxwxPiF+IyywyPiF+IyzwzPiFIzxwxPiF+IzywyPiF+IzzwzPiF = IxxIyxIzxIxyIyyIzyIxzIyzIzz wxPiFwyPiFwzPiF =\[I\] wxPiFwyPiFwzPiF
I \] Σ M / O F = \[ I x x Σ M / O F I x y Σ M / O F I x z Σ M / O F I y x Σ M / O F I y y Σ M / O F I y z Σ M / O F I z x Σ M / O F I z y Σ M / O F I z z Σ M / O F \] = \[ ∑ i N m P i ⋅ \[ ( y O P i F ) 2 + ( z O P i F ) 2 \] − ∑ i N m P i ⋅ x O P i F y O P i F − ∑ i N m P i ⋅ ( x O P i F z O P i F ) − ∑ i N m P i ⋅ ( y O P i F x O P i F ) ∑ i N m P i ⋅ \[ ( x O P i F ) 2 + ( z O P i F ) 2 \] − ∑ i N m P i ⋅ ( y O P i F z O P i F ) − ∑ i N m P i ⋅ ( z O P i F x O P i F ) − ∑ i N m P i ⋅ ( z O P i F y O P i F ) ∑ i N m P i ⋅ \[ ( x O P i F ) 2 + ( y O P i F ) 2 \] \] \\begin{aligned} \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\&=\\left\[ \\begin{matrix} {I_{\\mathrm{xx}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{xy}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{xz}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\\\\ {I_{\\mathrm{yx}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{yy}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{yz}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\\\\ {I_{\\mathrm{zx}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{zy}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{zz}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\\\\ \\end{matrix} \\right\]\\\\ \&=\\left\[ \\begin{matrix} \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( y_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right) \^2+\\left( z_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right) \^2 \\right\]}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot x_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}y_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( x_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}z_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right)}\\\\ -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( y_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}x_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right)}\& \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( x_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right) \^2+\\left( z_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right) \^2 \\right\]}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( y_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}z_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right)}\\\\ -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( z_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}x_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right)}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( z_{\\mathrm{OP}_{\\mathrm{i}}}\^{F}y_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right)}\& \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( x_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right) \^2+\\left( y_{\\mathrm{OP}_{\\mathrm{i}}}\^{F} \\right) \^2 \\right\]}\\\\ \\end{matrix} \\right\]\\\\ \\end{aligned} \[I\]ΣM/OF= IxxΣM/OFIyxΣM/OFIzxΣM/OFIxyΣM/OFIyyΣM/OFIzyΣM/OFIxzΣM/OFIyzΣM/OFIzzΣM/OF = ∑iNmPi⋅\[(yOPiF)2+(zOPiF)2\]−∑iNmPi⋅(yOPiFxOPiF)−∑iNmPi⋅(zOPiFxOPiF)−∑iNmPi⋅xOPiFyOPiF∑iNmPi⋅\[(xOPiF)2+(zOPiF)2\]−∑iNmPi⋅(zOPiFyOPiF)−∑iNmPi⋅(xOPiFzOPiF)−∑iNmPi⋅(yOPiFzOPiF)∑iNmPi⋅\[(xOPiF)2+(yOPiF)2
上式的实际推导过程，是进行两次转置变化，在实际过程中可以理解成，适用于矩阵与矢量相乘的张量Tensor乘法，因此也可将惯性矩阵 [ I ] \left[ I \right] [I]称为惯性张量Inertia Tensor。而采用基于拉格朗日恒等式证明的三个向量的双重矢积公式，可能更利于理解：（为方便运算，忽略点 O O O的运动）
- 三个向量的双重矢积公式： ( r ⃗ 1 × r ⃗ 2 ) × r ⃗ 3 = ( r ⃗ 1 ⋅ r ⃗ 3 ) r ⃗ 2 − ( r ⃗ 2 ⋅ r ⃗ 3 ) r ⃗ 1 \left( \vec{r}1\times \vec{r}2 \right) \times \vec{r}3=\left( \vec{r}1\cdot \vec{r}3 \right) \vec{r}2-\left( \vec{r}2\cdot \vec{r}3 \right) \vec{r}1 (r 1×r 2)×r 3=(r 1⋅r 3)r 2−(r 2⋅r 3)r 1
H ⃗ Σ M / O F = ∑ i N R ⃗ O P i F × P ⃗ P i F = ∑ i N m P i ⋅ R ⃗ O P i F × ( ω ⃗ F × R ⃗ O P i F ) = ∑ i N m P i ⋅ [ ( R ⃗ O P i F ⋅ R ⃗ O P i F ) ω ⃗ F − ( ω ⃗ F ⋅ R ⃗ O P i F ) R ⃗ O P i F ] \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F}=\sum_i^N{\vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \vec{P}{\mathrm{P}{\mathrm{i}}}^{F}}=\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \left( \vec{\omega}^F\times \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F} \right)}=\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \left[ \left( \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\cdot \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F} \right) \vec{\omega}^F-\left( \vec{\omega}^F\cdot \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F} \right) \vec{R}{\mathrm{OP}_{\mathrm{i}}}^{F} \right]} H ΣM/OF=i∑NR OPiF×P PiF=i∑NmPi⋅R OPiF×(ω F×R OPiF)=i∑NmPi⋅[(R OPiF⋅R OPiF)ω F−(ω F⋅R OPiF)R OPiF]