- Angular velocity tensor
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 matrixwith zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements::
At any instant, , the angular velocity tensor is a linear map between the position vectors and their velocity vectors of a rigid body rotating around the origin:
where we omitted the parameter, and regard and as elements of the same 3-dimensional
Euclidean vector space.
The relation between this linear map and the angular velocity
pseudovectoris the following.
Because of "T" is the derivative of an
orthogonal transformation, the
bilinear formis skew-symmetric. (Here stands for the scalar product). So we can apply the fact of exterior algebrathat there is a unique linear formon that
where is the
wedge productof and .
dual vector"L"* of "L" we get
Introducing , as the
Hodge dualof "L"* , and apply further Hodge dual identities we arrive at
Because is an arbitrary vector, from nondegeneracy of
Rigid body dynamics
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