Angular velocity tensor

The angular velocity tensor is a skew-symmetric matrix defined by: The scalar elements above correspond to the angular velocity vector components

The linear mapping Ω acts as a cross product

When multiplied by a time difference, it results in the angular displacement tensor.

undergoing uniform circular motion around a fixed axis satisfies: Let

Therefore, the angular velocity tensor is: since the inverse of an orthogonal matrix

In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor, which is a second rank skew-symmetric tensor.

Since the angular velocity tensor Ω = Ω(t) is a skew-symmetric matrix: its Hodge dual is a vector, which is precisely the previous angular velocity vector

If we know an initial frame A(0) and we are given a constant angular velocity tensor Ω, we can obtain A(t) for any given t. Recall the matrix differential equation: This equation can be integrated to give: which shows a connection with the Lie group of rotations.

We prove that angular velocity tensor is skew symmetric, i.e.

a frame matrix, taking the time derivative of the equation gives: Applying the formula

, Thus, Ω is the negative of its transpose, which implies it is skew symmetric.

, the angular velocity tensor represents a linear map between the position vector

of a point on a rigid body rotating around the origin: The relation between this linear map and the angular velocity pseudovector

Because Ω is the derivative of an orthogonal transformation, the bilinear form is skew-symmetric.

Thus we can apply the fact of exterior algebra that there is a unique linear form

is an arbitrary vector, from nondegeneracy of scalar product follows Since the spin angular velocity tensor of a rigid body (in its rest frame) is a linear transformation that maps positions to velocities (within the rigid body), it can be regarded as a constant vector field.

In particular, the spin angular velocity is a Killing vector field belonging to an element of the Lie algebra SO(3) of the 3-dimensional rotation group SO(3).

Also, it can be shown that the spin angular velocity vector field is exactly half of the curl of the linear velocity vector field v(r) of the rigid body.

In symbols, The same equations for the angular speed can be obtained reasoning over a rotating rigid body.

Instead, it can be supposed rotating around an arbitrary point that is moving with a linear velocity V(t) in each instant.

To obtain the equations, it is convenient to imagine a rigid body attached to the frames and consider a coordinate system that is fixed with respect to the rigid body.

As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O′ and the vector from O to O′ is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is Ri in the lab frame, and at position ri in the body frame.

It is seen that the position of the particle can be written: The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time.

It can be proved that this is a skew symmetric matrix, so we can take its dual to get a 3 dimensional pseudovector that is precisely the previous angular velocity vector

: Substituting ω for Ω into the above velocity expression, and replacing matrix multiplication by an equivalent cross product: It can be seen that the velocity of a point in a rigid body can be divided into two terms – the velocity of a reference point fixed in the rigid body plus the cross product term involving the orbital angular velocity of the particle with respect to the reference point.

We have supposed that the rigid body rotates around an arbitrary point.

(Note the marked contrast of this with the orbital angular velocity of a point particle, which certainly does depend on the choice of origin.)

See the graph to the right: The origin of lab frame is O, while O1 and O2 are two fixed points on the rigid body, whose velocity is

) is arbitrary, it follows that If the reference point is the instantaneous axis of rotation the expression of the velocity of a point in the rigid body will have just the angular velocity term.

Another example is the point of contact of a purely rolling spherical (or, more generally, convex) rigid body.