Angular velocity tensor

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

$$

boldsymbolomega(t) times mathbf{r}(t) = T(t) mathbf{r}(t)

It allows us to express the cross product

$boldsymbolomega(t)\; times\; mathbf\{r\}(t)$

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:

$$

T(t) = begin{pmatrix} 0 & -omega_z(t) & omega_y(t) omega_z(t) & 0 & -omega_x(t) -omega_y(t) & omega_x(t) & 0 end{pmatrix}

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

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

where

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

by definition.

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

$Tmathbf\{r\}\; =\; omega\; times\; mathbf\{r\}$

See also

• Angular velocity
• Rigid body dynamics