- #1
sergiokapone
- 302
- 17
Lets consider the angular momentum tensor (here ##m=1##)
\begin{equation}
L^{ij} = x^iv^j - x^jv^i
\end{equation}
and rortational velocity of particle (expressed via angular momentum tensor)
\begin{equation}
v^j = \omega^{jm}x_m.
\end{equation}
Then
\begin{equation}
L^{ij} = x^ix_m\omega^{jm} - x^jx_m\omega^{im}
\end{equation}
Now we can lower indices near ##\omega## with metric tensor:
\begin{align}
\omega^{jm} = g^{jn}g^{mr}\omega_{nr} \\
\omega^{im} = g^{in}g^{mr}\omega_{nr}
\end{align}
So, we get
\begin{equation}
L^{ij} = \left( x^ix_mg^{jn}g^{mr} - x^jx_m g^{in}g^{mr}\right) \omega_{nr}
\end{equation}
So, we can conclude ##L^{ij} = I^{ijnr} \omega_{nr}##, the inertia tensor is
\begin{equation}
I^{ijnr} = x^ix_mg^{jn}g^{mr} - x^jx_m g^{in}g^{mr}
\end{equation}Common form of inertia tensor
\begin{equation}
I^i_j = \delta_j^i x^2 - x^ix_j
\end{equation}
So, my question, how can I get common form of inertia tensor ##I^i_j ## in case 3D-space based on my ##I^{ijnr} ## (if it correct, of course)?
\begin{equation}
L^{ij} = x^iv^j - x^jv^i
\end{equation}
and rortational velocity of particle (expressed via angular momentum tensor)
\begin{equation}
v^j = \omega^{jm}x_m.
\end{equation}
Then
\begin{equation}
L^{ij} = x^ix_m\omega^{jm} - x^jx_m\omega^{im}
\end{equation}
Now we can lower indices near ##\omega## with metric tensor:
\begin{align}
\omega^{jm} = g^{jn}g^{mr}\omega_{nr} \\
\omega^{im} = g^{in}g^{mr}\omega_{nr}
\end{align}
So, we get
\begin{equation}
L^{ij} = \left( x^ix_mg^{jn}g^{mr} - x^jx_m g^{in}g^{mr}\right) \omega_{nr}
\end{equation}
So, we can conclude ##L^{ij} = I^{ijnr} \omega_{nr}##, the inertia tensor is
\begin{equation}
I^{ijnr} = x^ix_mg^{jn}g^{mr} - x^jx_m g^{in}g^{mr}
\end{equation}Common form of inertia tensor
\begin{equation}
I^i_j = \delta_j^i x^2 - x^ix_j
\end{equation}
So, my question, how can I get common form of inertia tensor ##I^i_j ## in case 3D-space based on my ##I^{ijnr} ## (if it correct, of course)?