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Angular velocity tensor

 

 
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Angular velocity tensor
 
 
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In physicsPhysics

Physics , the most fundamental physical science, is concerned with the underlying principles of the natural world....
, the angular velocity tensor is defined as a matrix T such that:

It allows us to express the cross productCross product

In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space....

as a matrix multiplication. It is, by definition, a skew-symmetric matrixFacts About Skew-symmetric matrix

In linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it ...
 with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements:

Coordinate-free description

At a given time instance , the angular velocity tensor is a linear map between the position vectors
and their velocity vectors of a rigid body rotating around the origo:

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 pseudovectorFacts About Pseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an...
  is the following.

Because of T is the derivative of an orthogonal transformation, the

bilinear formBilinear form

In mathematics, a bilinear form on a vector space V over a field F is a mapping V V ? F which is linear...
 is skew-symmetric. (Here stands for the scalar product). So we can apply the fact of exterior algebraExterior algebra

In mathematics, the exterior algebra of a given vector space V over a field K is a certain unital associative algebr...
 that there is a unique linear form  on that

,

where is the wedge product of and .

Taking the dual vector L* of L we get

Introducing , as the Hodge dualHodge dual

In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which est...
 of L* , and apply further Hodge dual identities we arrive at

where

by definition.

Because is an arbitrary vector, from nondegeneracy of scalar product follows

See also

  • Angular velocityAngular velocity

    In physics angular velocity is the speed at which something rotates together with the direction it rotates in....
  • Rigid body dynamicsRigid body dynamics Summary

    In physics, Rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate, which introdu...