Angular velocity tensor: Difference between revisions

Revision as of 10:37, 6 February 2008

Revision as of 08:48, 17 February 2008

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-\omega_y(t) & \omega_x(t) & 0 \\

\end{pmatrix}</math>

==Coordinate-free description==

At a given time instance <math>t</math>, the angular velocity tensor is a linear map between the postition vectors <math> \mathbf{r}(t) </math>

and their velocity vectors <math> \mathbf{v}(t) </math> of a rigid body rotating around the origo:

:<math> \mathbf{v} = T\mathbf{r} </math>

where we omitted the <math>t</math> parameter, and regard <math> \mathbf{v} </math> and <math> \mathbf{r} </math> as elements of the same 3-dimensional [[vectorspace]] <math>V</math>.

The relation between this linear map and the angular velocity [[pseudovector]] <math>\omega</math> is the following.

Because of ''T'' is the derivative of an [[orthogonal transformation]], the

:<math>B(\mathbf{r},\mathbf{s}) = (T\mathbf{r},\mathbf{s})</math>

[[bilinear form]] is [[skew-symmetric]]. So we can apply the fact of [[exterior algebra]] that there is a unique [[linear form]] <math>L</math> on <math>\Lambda^2 V </math> that

:<math>L(\mathbf{r}\wedge \mathbf{s}) = B(\mathbf{r},\mathbf{s})</math> ,

where <math>\mathbf{r}\wedge \mathbf{s} \in \Lambda^2 V </math> is the [[wedge product]] of <math>\mathbf{r}</math> and <math>\mathbf{s}</math>.

Taking the [[dual vector]] ''L''* of ''L'' we get

:<math> (T\mathbf{r},\mathbf{s}) = (L^*,\mathbf{r}\wedge \mathbf{s}) </math>

Introducing <math> \omega := *L^* </math>, as the [[Hodge dual]] of ''L''* , and apply further Hodge dual identities we arrive at

:<math> (T\mathbf{r},\mathbf{s}) = * (*L^* \wedge \mathbf{r} \wedge \mathbf{s}) = * (\omega \wedge \mathbf{r} \wedge \mathbf{s}) = (*(\omega \wedge \mathbf{r} ),\mathbf{s}) = (\omega \times \mathbf{r},\mathbf{s}) </math>

where

:<math>\omega \times \mathbf{r} := *(\omega \wedge \mathbf{r} )</math>

by definition.

Because <math>\mathbf{s}</math> is an arbitrary vector, from the nondegenerate property of scalar pruduct follows

:<math> T\mathbf{r} = \omega \times \mathbf{r}</math>

== See also ==