# Angular velocity tensor

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 $mathbf\left\{r\right\}\left(t\right)$and their velocity vectors $mathbf\left\{v\right\}\left(t\right)$ of a rigid body rotating around the origin:

:$mathbf\left\{v\right\} = Tmathbf\left\{r\right\}$

where we omitted the $t$ parameter, and regard $mathbf\left\{v\right\}$ and $mathbf\left\{r\right\}$ 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

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

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

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

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

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

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

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

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

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

by definition.

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

:$Tmathbf\left\{r\right\} = omega imes mathbf\left\{r\right\}$

* Angular velocity
* Rigid body dynamics

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