Angular velocity tensor

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

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