Angular Momentum Vector in Matrix Form
The two cross-products in Eq.(B.19) can be written out with the help
of the vector analysis identityB.23
![$\displaystyle \underline{x}\times (\underline{y}\times\underline{z}) \eqsp \und...
...underline{z}^T\underline{x})-\underline{z}\cdot(\underline{x}^T\underline{y}).
$](http://www.dsprelated.com/josimages_new/pasp/img2867.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
where
![$\displaystyle \mathbf{I}\underline{\omega}\eqsp
\left[\begin{array}{ccc}
I_{1...
...begin{array}{c} \omega_1 \\ [2pt] \omega_2 \\ [2pt] \omega_3\end{array}\right]
$](http://www.dsprelated.com/josimages_new/pasp/img2872.png)
![$ I_{ii}=m\left(\sum_{j=1}^3x_j^2 - x_i^2\right)$](http://www.dsprelated.com/josimages_new/pasp/img2873.png)
![$ I_{ij}=-mx_ix_j$](http://www.dsprelated.com/josimages_new/pasp/img2874.png)
![$ i\ne j$](http://www.dsprelated.com/josimages_new/pasp/img2875.png)
The matrix
![$ \mathbf{I}$](http://www.dsprelated.com/josimages_new/pasp/img558.png)
The vector angular momentum of a rigid body is obtained by summing the angular momentum of its constituent mass particles. Thus,
![$\displaystyle \underline{L}\eqsp \sum_i m_i \left(\left\Vert\,\underline{x}_i\,...
...e{x}_i^T\right)\underline{\omega}
\,\isdefs \, \mathbf{I}\,\underline{\omega}.
$](http://www.dsprelated.com/josimages_new/pasp/img2877.png)
![$ \underline{\omega}$](http://www.dsprelated.com/josimages_new/pasp/img2799.png)
In summary, the angular momentum vector
is given by the mass
moment of inertia tensor
times the angular-velocity vector
representing the axis of rotation.
Note that the angular momentum vector
does not in general
point in the same direction as the angular-velocity vector
. We
saw above that it does in the special case of a point mass traveling
orthogonal to its position vector. In general,
and
point
in the same direction whenever
is an eigenvector of
, as will be discussed further below (§B.4.16). In this
case, the rigid body is said to be dynamically balanced.B.24
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Simple Example
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Relation of Angular to Linear Momentum