91Ó°ÊÓ

Physics definitions are given.

Tensors, like scalars and vectors, are mathematical constructs that can be

used to describe physical qualities. Tensors are just a combination of scalars

and vectors, with a scalar being a zero rank tensor and a vector being a first

rank tensor.

Define angular momentum of a point mass.

$L=\mathrm{Ï–}\left(r^2\mathrm{Ï–}\left(r.\text{'}\right)r\right)$

Express the definition along the other axes.

$$\begin{gathered} L=m\left({\mathrm{Ó\neg }}^2{y}_v-\left(ii\right)\right) \\ L=m\left({\mathrm{Ó\neg }}^2{}_z-\left(ii\right)z\right) \end{gathered}$$

Compare the above equations with $L_i=l_{ij}{\mathrm{Ó\neg }}_j$.

$$\begin{gathered} l_{yx}=-myx \\ {l}_{yy}=m\left(x^2+y^2\right) \\ {l}_{yz}=-myz \end{gathered}$$

Continue the comparison.

$$\begin{gathered} l_{zx}=-mzx \\ {l}_{zy}=mzy \\ {l}_{zz}=m\left(x^2+y^2\right) \end{gathered}$$

Express the components of the inertia tensor based on the earlier equations.

$\mathrm{δ}=m\left(r^2{}_{ij}-ij\right)$

Extend the previous equations to a multi-particle system.

$$\begin{gathered} L=\underset{k}{∑}{m}_k{r}_k×\left(\mathrm{Ó\neg }×r_k\right) \\ =\underset{k}{∑}{m}_k\left(r_k^2\mathrm{Ó\neg }-\left(r_k.\mathrm{Ó\neg }\right)r_k\right) \\ {\mathrm{Ó\neg }}_1=\underset{k}{∑}{m}_k^j\left(r{r}_k^2{}_i-{}_{k.jj{k}_i;}\right) \\ =\underset{k}{∑}{\mathrm{δ}}_k^{}\left(r{r}_k^2{T}_{ij}-\underset{kj}{Ó\neg }k.j\right) \end{gathered}$$

Continue the evaluation.

${\mathrm{δ}}_{ij}^{}=\underset{k}{∑}m{T}_k\left(r_k^2{}_{ij}-kjk.j\right)$

Replace summation with integral to evaluate over a continuous system.

${\mathrm{δ}}_{ij}^{}=∫r\left(r{r}_{}^2{m}_i{}_{ij}\right)d$

Thus, the solution is${\mathrm{δ}}_{ij}^{}=\underset{k}{∑}{m}_k\left(r_k^2{}_{ij}{-}_{k,}{}_{kj}\right)$