質量行列

二体一次元システム

${\displaystyle q={\begin{bmatrix}x_{1}&x_{2}\end{bmatrix}}^{\textsf {T}}}$ 。

${\displaystyle T=\sum _{i=1}^{2}{\frac {1}{2}}m_{i}{\dot {x_{i}}}^{2}}$

${\displaystyle T={\frac {1}{2}}{\dot {q}}^{\textsf {T}}M{\dot {q}}}$

${\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}}$

N体システム

${\displaystyle M=\operatorname {diag} \left[m_{1}I_{n_{1}},\,m_{2}I_{n_{2}},\,\ldots ,\,m_{N}I_{n_{N}}\right]}$

_niがI×N '_I nは単位行列、または：

${\displaystyle M={\begin{bmatrix}m_{1}&\cdots &0&0&\cdots &0&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &m_{1}&0&\cdots &0&\cdots &0&\cdots &0\\0&\cdots &0&m_{2}&\cdots &0&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &m_{2}&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &0&\cdots &m_{N}&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &0&\cdots &0&\cdots &m_{N}\\\end{bmatrix}}}$

回転ダンベル

${\displaystyle q={\begin{bmatrix}x&y&\alpha \end{bmatrix}}}$

${\displaystyle {\begin{aligned}x_{1}&=(x,y)+R(\cos \alpha ,\sin \alpha )&v_{1}&=\left({\dot {x}},{\dot {y}}\right)+R{\dot {\alpha }}(-\sin \alpha ,\cos \alpha )\\x_{2}&=(x,y)-R(\cos \alpha ,\sin \alpha )&v_{2}&=\left({\dot {x}},{\dot {y}}\right)-R{\dot {\alpha }}(-\sin \alpha ,\cos \alpha )\end{aligned}}}$

${\displaystyle 2T=m{\dot {x}}^{2}+m{\dot {y}}^{2}+mR^{2}{\dot {\alpha }}^{2}-2Rd\sin(\alpha ){\dot {x}}{\dot {\alpha }}+2Rd\cos(\alpha ){\dot {y}}{\dot {\alpha }}}$

${\displaystyle m=m_{1}+m_{2}}$と${\displaystyle d=m_{1}-m_{2}}$ 。この式は、次のように行列形式で記述でき

${\displaystyle T={\frac {1}{2}}{\dot {q}}^{\textsf {T}}M{\dot {q}}}$

${\displaystyle M={\begin{bmatrix}m&0&-Rd\sin \alpha \\0&m&Rd\cos \alpha \\-Rd\sin \alpha &Rd\cos \alpha &R^{2}m\end{bmatrix}}}$