91影视

The kinetic energy,$T=\frac{1}{2}m{v}^2=\frac{1}{2}m\left(蝇脳r\right)鈻�\left(蝇脳r\right)$.

The velocity is, $v=蝇脳r$.

Kinetic energy is the energy that a particle has as a result of its movement.

$T=\underset{n=1}{\overset{n}{鈭�}}\frac{1}{2}{m}_n{v}_n^2=\underset{n=1}{\overset{N}{鈭�}}\frac{1}{2}{m}_n{\left(蝇脳r_n\right)}^2$

Find the square of velocity vector.

$$\begin{gathered} v^2=\left(蝇脳r\right).\left(蝇脳r\right) \\ =\left|{\left(蝇脳r\right)}_x{蝇}_{x}x \\ {\left(蝇脳r\right)}_y{蝇}_y{y}^{} \\ {\left(蝇脳r\right)}_z{蝇}_{z}z\right| \\ =\left|{蝇}_{y}z-{蝇}_z{y}_{}{蝇}_{x}x \\ {蝇}_{z}x-{蝇}_x{z}^{}{蝇}_{y}y \\ {蝇}_{x}y{蝇}_{y}x{蝇}_{z}z\right| \\ =\left|{蝇}_{y}z{蝇}_{x}x \\ {蝇}_{zx}{蝇}_y{y}^{} \\ {蝇}_{x}y{蝇}_{z}z\right|-=\left|{蝇}_{z}y{蝇}_{x}x \\ {蝇}_{x}z{蝇}_y{y}^{} \\ {蝇}_{y}x{蝇}_{z}z\right| \end{gathered}$$

Simplify further,

$$\begin{gathered} =-{蝇}_x\left|{蝇}_{z}xy \\ {蝇}_{x}yz\right|+{蝇}_y\left|{蝇}_{y}zx \\ {蝇}_{x}yz\right|-{蝇}_z\left|{蝇}_{y}zx \\ {蝇}_{z}yz\right|+{蝇}_x\left|{蝇}_{z}yx \\ {蝇}_{y}xz\right|-{蝇}_y\left|{蝇}_{z}yx \\ {蝇}_{y}xz\right|+{蝇}_z\left|{蝇}_{z}yx \\ {蝇}_{x}zz\right| \\ ={蝇}_x^2{y}^2+{蝇}_y^2+{蝇}_z^2{x}^2+{蝇}_x^2{z}^2+{蝇}_z^2{y}^2-2\left({蝇}_x{蝇}_{z}xz+{蝇}_x{蝇}_{y}xy+{蝇}_y{蝇}_{z}yz\right)卤\left({蝇}_x{x}^2+{蝇}_y{y}^2+{蝇}_z{z}^2\right) \\ =\left({蝇}_x^2+{蝇}_y^2+{蝇}_z^2\right)\left(x^2+y^2+z^2\right)-{\left({蝇}_{x}x+{蝇}_{y}y+{蝇}_{z}z\right)}^2 \\ ={蝇}^2{r}^2-{\left(蝇路r\right)}^2 \end{gathered}$$

So Kinetic energy is

$T=\underset{n=1}{\overset{N}{鈭�}}\frac{1}{2}{m}_i\left({蝇}^2{r}_n^2-{\left(蝇路r_n\right)}^2\right)$

Calculate $I蝇$.

$$\begin{gathered} {\left(I蝇\right)}_i=I_k{蝇}_k \\ =\underset{n=1}{\overset{N}{鈭�}}{m}_n\left(r_n^2{\mathcal{S}}_{ik}-r_{n,i}{r}_{n,k}\right){蝇}_k \\ =\underset{n=1}{\overset{N}{鈭�}}{m}_n\left(r_n^2{蝇}_i-r_{n,i}\left(r_{n,k}{蝇}_k\right)\right) \\ =\underset{n=1}{\overset{N}{鈭�}}{m}_n\left(r_n^2{蝇}_i-r_{n,i}\left(蝇.r_n\right)\right) \end{gathered}$$

Since, ${蝇}^{T}I蝇=蝇.\left(I蝇\right)$.

$$\begin{gathered} \frac{1}{2}{蝇}^{T}I蝇=\frac{1}{2}{蝇}_i{\left(I蝇\right)}_i \\ =\frac{1}{2}\underset{n=1}{\overset{N}{鈭�}}{m}_n\left(r_n^2{蝇}_i{蝇}_i-\left(r_{n,i}{蝇}_i\right)\left(蝇路r_n\right)\right) \\ =\frac{1}{2}\underset{n=1}{\overset{N}{鈭�}}{m}_n\left(r_n^2{蝇}^2-{\left(蝇路r_n\right)}^2\right) \end{gathered}$$

Equate kinetic energy with matrix of tensor of inertia.

$$\begin{gathered} T=\underset{n=1}{\overset{N}{鈭�}}\frac{1}{2}{m}_i\left({蝇}^2{r}_n^2-{\left(蝇路r_n\right)}^2\right) \\ =\frac{1}{2}{蝇}^{T}I蝇 \end{gathered}$$

Hence, Kinetic energy is $T=\frac{1}{2}{蝇}^T蝇$.