Angular velocity tensor

In physics, the angular velocity tensor is defined as a matrix T such that:

:

It allows us to express the cross product:as a matrix multiplication. It is, by definition, a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements::

Coordinate-free description

At any instant, $t$, the angular velocity tensor is a linear map between the position vectors $mathbf\{r\}(t)$and their velocity vectors $mathbf\{v\}(t)$ of a rigid body rotating around the origin:

:$mathbf\{v\}\; =\; Tmathbf\{r\}$

where we omitted the $t$ parameter, and regard $mathbf\{v\}$ and $mathbf\{r\}$ as elements of the same 3-dimensional Euclidean vector space $V$.

The relation between this linear map and the angular velocity pseudovector $omega$ is the following.

Because of "T" is the derivative of an orthogonal transformation, the

:$B(mathbf\{r\},mathbf\{s\})\; =\; (Tmathbf\{r\})\; cdot\; mathbf\{s\}$

bilinear form is skew-symmetric. (Here $cdot$ stands for the scalar product). So we can apply the fact of exterior algebra that there is a unique linear form $L$ on $Lambda^2\; V$ that

:$L(mathbf\{r\}wedge\; mathbf\{s\})\; =\; B(mathbf\{r\},mathbf\{s\})$ ,

where $mathbf\{r\}wedge\; mathbf\{s\}\; in\; Lambda^2\; V$ is the wedge product of $mathbf\{r\}$ and $mathbf\{s\}$.

Taking the dual vector "L"* of "L" we get

:$(Tmathbf\{r\})cdot\; mathbf\{s\}\; =\; L^*\; cdot\; (mathbf\{r\}wedge\; mathbf\{s\})$

Introducing $omega\; :=\; *L^*$, as the Hodge dual of "L"* , and apply further Hodge dual identities we arrive at

:$(Tmathbf\{r\})\; cdot\; mathbf\{s\}\; =\; *\; (\; *L^*\; wedge\; mathbf\{r\}\; wedge\; mathbf\{s\})\; =\; *\; (omega\; wedge\; mathbf\{r\}\; wedge\; mathbf\{s\})\; =\; *(omega\; wedge\; mathbf\{r\})\; cdot\; mathbf\{s\}\; =\; (omega\; imes\; mathbf\{r\})\; cdot\; mathbf\{s\}$

where :$omega\; imes\; mathbf\{r\}\; :=\; *(omega\; wedge\; mathbf\{r\})$

by definition.

Because $mathbf\{s\}$ is an arbitrary vector, from nondegeneracy of scalar product follows

:$Tmathbf\{r\}\; =\; omega\; imes\; mathbf\{r\}$

See also

* Angular velocity
* Rigid body dynamics