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Part2
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- [2.2.3 欧拉方程 Euler equation](#2.2.3 欧拉方程 Euler equation)
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2.2.3 欧拉方程 Euler equation
对式 H ⃗ Σ M / O F \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF进一步分析，有：
H ⃗ Σ M / O F = ∫ R ⃗ O P i F × ( d m i ⋅ d R ⃗ P i F d t ) = ∫ ( ( R ⃗ P i F − R ⃗ O F ) × V ⃗ P i F ) d m i = ∫ ( R ⃗ P i F × V ⃗ P i F ) d m i − ∫ ( R ⃗ O F × V ⃗ P i F ) d m i = H ⃗ Σ M F − R ⃗ O F × P ⃗ G F \begin{split} \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F}&=\int{\vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \left( \mathrm{d}m_i\cdot \frac{\mathrm{d}\vec{R}{\mathrm{P}{\mathrm{i}}}^{F}}{\mathrm{d}t} \right)}=\int{\left( \left( \vec{R}{\mathrm{P}{\mathrm{i}}}^{F}-\vec{R}{\mathrm{O}}^{F} \right) \times \vec{V}{\mathrm{P}{\mathrm{i}}}^{F} \right) \mathrm{d}m_i} \\ &=\int{\left( \vec{R}{\mathrm{P}{\mathrm{i}}}^{F}\times \vec{V}{\mathrm{P}{\mathrm{i}}}^{F} \right) \mathrm{d}m_i}-\int{\left( \vec{R}{\mathrm{O}}^{F}\times \vec{V}{\mathrm{P}{\mathrm{i}}}^{F} \right) \mathrm{d}m_i} \\ &=\vec{H}{\Sigma {\mathrm{M}}}^{F}-\vec{R}{\mathrm{O}}^{F}\times \vec{P}{\mathrm{G}}^{F} \end{split} H ΣM/OF=∫R OPiF×(dmi⋅dtdR PiF)=∫((R PiF−R OF)×V PiF)dmi=∫(R PiF×V PiF)dmi−∫(R OF×V PiF)dmi=H ΣMF−R OF×P GF
对上式进一步求导，则有：
d H ⃗ Σ M / O F d t = d H ⃗ Σ M F d t − d ( R ⃗ O F × P ⃗ G F ) d t = d H ⃗ Σ M F d t − V ⃗ O F × P ⃗ G F − m t o t a l ⋅ R ⃗ O F × a ⃗ G F \frac{\mathrm{d}\vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F}}{\mathrm{d}t}=\frac{\mathrm{d}\vec{H}{\Sigma {\mathrm{M}}}^{F}}{\mathrm{d}t}-\frac{\mathrm{d}\left( \vec{R}{\mathrm{O}}^{F}\times \vec{P}{\mathrm{G}}^{F} \right)}{\mathrm{d}t}=\frac{\mathrm{d}\vec{H}{\Sigma {\mathrm{M}}}^{F}}{\mathrm{d}t}-\vec{V}{\mathrm{O}}^{F}\times \vec{P}{\mathrm{G}}^{F}-m_{\mathrm{total}}\cdot \vec{R}{\mathrm{O}}^{F}\times \vec{a}{\mathrm{G}}^{F} dtdH ΣM/OF=dtdH ΣMF−dtd(R OF×P GF)=dtdH ΣMF−V OF×P GF−mtotal⋅R OF×a GF
其中：
H ⃗ Σ M F = ∫ R ⃗ P i F × p ⃗ P i F = ∫ ( R ⃗ G F + R ⃗ G P i F ) × ( d m i ⋅ ( V ⃗ G F + V ⃗ G P i F ) ) = ∫ R ⃗ G F × V ⃗ G F d m i ⏟ m t o t a l ⋅ R ⃗ G F × V ⃗ G F + ∫ R ⃗ G F × V ⃗ G P i F d m i ⏟ 0 + ∫ R ⃗ G P i F × V ⃗ G F d m i ⏟ 0 + ∫ R ⃗ G P i F × V ⃗ G P i F d m i ⏟ ∫ R ⃗ G P i F × ( ω ⃗ M F × R ⃗ G P i F ) d m i = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + ∫ R ⃗ G P i F × ( ω ⃗ M F × R ⃗ G P i F ) d m i = 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 \begin{split} \vec{H}{\Sigma {\mathrm{M}}}^{F}&=\int{\vec{R}{\mathrm{P}{\mathrm{i}}}^{F}\times \vec{p}{\mathrm{P}{\mathrm{i}}}^{F}}=\int{\left( \vec{R}{\mathrm{G}}^{F}+\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \times \left( \mathrm{d}m_i\cdot \left( \vec{V}{\mathrm{G}}^{F}+\vec{V}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \right)} \\ &=\begin{array}{c} \underbrace{\int{\vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}}\mathrm{d}m_i}\\ m_{\mathrm{total}}\cdot \vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}\\ \end{array}+\begin{array}{c} \underbrace{\int{\vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m_i}\\ 0\\ \end{array}+\begin{array}{c} \underbrace{\int{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\times \vec{V}{\mathrm{G}}^{F}}\mathrm{d}m_i}\\ 0\\ \end{array}+\begin{array}{c} \underbrace{\int{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\times \vec{V}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m_i}\\ \int{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\times \left( \vec{\omega}{\mathrm{M}}^{F}\times \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right)}\mathrm{d}m_i\\ \end{array} \\ &=m{\mathrm{total}}\cdot \vec{R}{\mathrm{G}}^{F}\times \vec{V}{\mathrm{G}}^{F}+\int{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\times \left( \vec{\omega}{\mathrm{M}}^{F}\times \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right)}\mathrm{d}m_i \\ &=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_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_i \end{split} H ΣMF=∫R PiF×p PiF=∫(R GF+R GPiF)×(dmi⋅(V GF+V GPiF))= ∫R GF×V GFdmimtotal⋅R GF×V GF+ ∫R GF×V GPiFdmi0+ ∫R GPiF×V GFdmi0+ ∫R GPiF×V GPiFdmi∫R GPiF×(ω MF×R GPiF)dmi=mtotal⋅R GF×V GF+∫R GPiF×(ω MF×R GPiF)dmi=mtotal⋅R GF×V GF+∫(R GPiF⋅R GPiF)ω MFdmi−∫(R GPiF⋅ω MF)R GPiFdmi
将 H ⃗ Σ M F \vec{H}{\Sigma {\mathrm{M}}}^{F} H ΣMF进一步求导，则有：
d H ⃗ Σ M F d t = { R ⃗ G F × m t o t a l ⋅ a ⃗ G F + 2 ∫ ( V ⃗ P i F ⋅ R ⃗ G P i F ) ω ⃗ M F d m i + ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) α ⃗ M F d m i − ∫ ( V ⃗ G P i F ⋅ ω ⃗ M F ) R ⃗ G P i F d m i − ∫ ( R ⃗ G P i F ⋅ α ⃗ M F ) R ⃗ G P i F d m i − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) V ⃗ G P i F d m i = { R ⃗ G F × m t o t a l ⋅ a ⃗ G F + ( ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) α ⃗ M F d m i − ∫ ( R ⃗ G P i F ⋅ α ⃗ M F ) R ⃗ G P i F d m i ) − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) ( ω ⃗ M F × R ⃗ G P i F ) d m i = { R ⃗ G F × m t o t a l ⋅ a ⃗ G F + ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ) ⋅ E 3 × 3 α ⃗ M F d m i − ∫ ( R ⃗ G P i F T α ⃗ M F ) R ⃗ G P i F d m i ) − ∫ ( R ⃗ G P i F T ω ⃗ M F ) ( ω ⃗ M F × R ⃗ G P i F ) d m i = { R ⃗ G F × m t o t a l ⋅ a ⃗ G F + α ⃗ M 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 × ( ∫ ( R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F ) \begin{split} \frac{\mathrm{d}\vec{H}{\Sigma {\mathrm{M}}}^{F}}{\mathrm{d}t}&=\begin{cases} \vec{R}{\mathrm{G}}^{F}\times m{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F}+2\int{\left( \vec{V}{\mathrm{P}{\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{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \vec{\alpha}{\mathrm{M}}^{F}}\mathrm{d}m{\mathrm{i}}\\ -\int{\left( \vec{V}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \right) \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m{\mathrm{i}}-\int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{\alpha}{\mathrm{M}}^{F} \right) \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m{\mathrm{i}}-\int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \right) \vec{V}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m{\mathrm{i}}\\ \end{cases} \\ &=\begin{cases} \vec{R}{\mathrm{G}}^{F}\times m{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F}+\left( \int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \vec{\alpha}{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{\alpha}{\mathrm{M}}^{F} \right) \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m{\mathrm{i}} \right)\\ -\int{\left( \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot \vec{\omega}{\mathrm{M}}^{F} \right) \left( \vec{\omega}{\mathrm{M}}^{F}\times \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \mathrm{d}m_{\mathrm{i}}}\\ \end{cases} \\ &=\begin{cases} \vec{R}{\mathrm{G}}^{F}\times m{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F}+\left( \int{\left( {\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \cdot E^{3\times 3}\vec{\alpha}{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( {\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{\alpha}{\mathrm{M}}^{F} \right) \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}\mathrm{d}m{\mathrm{i}} \right)\\ -\int{\left( {\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{\omega}{\mathrm{M}}^{F} \right) \left( \vec{\omega}{\mathrm{M}}^{F}\times \vec{R}{\mathrm{GP}{\mathrm{i}}}^{F} \right) \mathrm{d}m_{\mathrm{i}}}\\ \end{cases} \\ &=\begin{cases} \vec{R}{\mathrm{G}}^{F}\times m{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F}+\vec{\alpha}{\mathrm{M}}^{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{\omega}{\mathrm{M}}^{F}\times \left( \int{\left( \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} \right)\\ \end{cases} \end{split} dtdH ΣMF=⎩ ⎨ ⎧R GF×mtotal⋅a GF+2∫(V PiF⋅R GPiF)ω MFdmi+∫(R GPiF⋅R GPiF)α MFdmi−∫(V GPiF⋅ω MF)R GPiFdmi−∫(R GPiF⋅α MF)R GPiFdmi−∫(R GPiF⋅ω MF)V GPiFdmi=⎩ ⎨ ⎧R GF×mtotal⋅a GF+(∫(R GPiF⋅R GPiF)α MFdmi−∫(R GPiF⋅α MF)R GPiFdmi)−∫(R GPiF⋅ω MF)(ω MF×R GPiF)dmi=⎩ ⎨ ⎧R GF×mtotal⋅a GF+(∫(R GPiFTR GPiF)⋅E3×3α MFdmi−∫(R GPiFTα MF)R GPiFdmi)−∫(R GPiFTω MF)(ω MF×R GPiF)dmi=⎩ ⎨ ⎧R GF×mtotal⋅a GF+α MF∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi−ω MF×(∫(R GPiFR GPiFT)dmi⋅ω MF)
其中：
⇒ − ω ⃗ M F × ∫ ( R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F = ω ⃗ M 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 − R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 ) d m i ⋅ ω ⃗ M F ) = ω ⃗ M 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 ) − ω ⃗ M F × ( ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 ) d m i ⋅ ω ⃗ M F ) ⏟ 0 \begin{split} \Rightarrow &-\vec{\omega}{\mathrm{M}}^{F}\times \int{\left( \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} \\ &=\vec{\omega}{\mathrm{M}}^{F}\times \left( \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}}-{\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot E^{3\times 3} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}{\mathrm{M}}^{F} \right) \\ &=\vec{\omega}{\mathrm{M}}^{F}\times \left( \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} \right) -\begin{array}{c} \underbrace{\vec{\omega}{\mathrm{M}}^{F}\times \left( \int{\left( {\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}{\mathrm{GP}{\mathrm{i}}}^{F}\cdot E^{3\times 3} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) }\\ 0\\ \end{array} \end{split} ⇒−ω MF×∫(R GPiFR GPiFT)dmi⋅ω MF=ω MF×(∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT−R GPiFTR GPiF⋅E3×3)dmi⋅ω MF)=ω MF×(∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi⋅ω MF)− ω MF×(∫(R GPiFTR GPiF⋅E3×3)dmi⋅ω MF)0
将上两式进行汇总，可得：
⇒ d H ⃗ Σ M F d t = { R ⃗ G F × m t o t a l ⋅ a ⃗ 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 α ⃗ M F + ω ⃗ M 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 ) = R ⃗ G F × m t o t a l ⋅ a ⃗ G F + [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \begin{split} \Rightarrow \frac{\mathrm{d}\vec{H}{\Sigma {\mathrm{M}}}^{F}}{\mathrm{d}t}&=\begin{cases} \vec{R}{\mathrm{G}}^{F}\times m{\mathrm{total}}\cdot \vec{a}{\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{\alpha}{\mathrm{M}}^{F}\\ +\vec{\omega}{\mathrm{M}}^{F}\times \left( \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} \right)\\ \end{cases} \\ &=\vec{R}{\mathrm{G}}^{F}\times m_{\mathrm{total}}\cdot \vec{a}_{\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) \end{split} ⇒dtdH ΣMF=⎩ ⎨ ⎧R GF×mtotal⋅a GF+∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmiα MF+ω MF×(∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi⋅ω MF)=R GF×mtotal⋅a GF+[I]ΣM/GFα MF+ω MF×([I]ΣM/GF⋅ω MF)
其中：
[ 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 \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_i [I]ΣM/GF=∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi
[ I ] Σ M / G F \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F} [I]ΣM/GF被称为惯性矩阵inertia matrix（或称为惯量矩阵 ），为该物体在固定坐标系下相对于质心点 G G G的惯性张量。
进而可知：
d H ⃗ Σ M F d t = M ⃗ Σ M F = ∫ R ⃗ P i F × d F ⃗ P i F = R ⃗ G F × m t o t a l ⋅ a ⃗ G F + [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \frac{\mathrm{d}\vec{H}{\Sigma {\mathrm{M}}}^{F}}{\mathrm{d}t}=\vec{M}{\Sigma {\mathrm{M}}}^{F}=\int{\vec{R}{\mathrm{P}{\mathrm{i}}}^{F}\times \mathrm{d}\vec{F}{\mathrm{P}{\mathrm{i}}}^{F}}=\vec{R}{\mathrm{G}}^{F}\times m{\mathrm{total}}\cdot \vec{a}_{\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) dtdH ΣMF=M ΣMF=∫R PiF×dF PiF=R GF×mtotal⋅a GF+[I]ΣM/GFα MF+ω MF×([I]ΣM/GF⋅ω MF)
上式被称为：欧拉方程在惯性坐标系下相对固定点的表达式 ；当固定点与质心点重合时(此时G点为固定点)，则有：
M ⃗ Σ M / G F = M ⃗ Σ M F − R ⃗ G F × ( m t o t a l ⋅ a ⃗ G F ) = R ⃗ G F × ( m t o t a l ⋅ a ⃗ G F ) + [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) − R ⃗ G F × ( m t o t a l ⋅ a ⃗ G F ) = [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \begin{split} \vec{M}{\Sigma {\mathrm{M}}/\mathrm{G}}^{F}&=\vec{M}{\Sigma {\mathrm{M}}}^{F}-\vec{R}{\mathrm{G}}^{F}\times \left( m{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F} \right) \\ &=\vec{R}{\mathrm{G}}^{F}\times \left( m_{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F} \right) +\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) -\vec{R}{\mathrm{G}}^{F}\times \left( m{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F} \right) \\ &=\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) \end{split} M ΣM/GF=M ΣMF−R GF×(mtotal⋅a GF)=R GF×(mtotal⋅a GF)+[I]ΣM/GFα MF+ω MF×([I]ΣM/GF⋅ω MF)−R GF×(mtotal⋅a GF)=[I]ΣM/GFα MF+ω MF×([I]ΣM/GF⋅ω MF)
此时为固定坐标系下相对固定点质心 G G G求解的欧拉方程。