Angular velocity tensor

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

:$$oldsymbolomega(t) imes mathbf{r}(t) = T(t) mathbf{r}(t)

It allows us to express the cross product:$$oldsymbolomega(t) imes mathbf{r}(t) as a matrix multiplication. It is, by definition, a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements::$$T(t) =egin{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 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