To discuss some elementary features of rotation tensors, it is convenient to first discuss proper-orthogonal tensors. We then appeal to a theorem by Euler who showed that all rotation tensors are, by definition, proper-orthogonal tensors. The forthcoming discussion will lay the foundations for the possibility of three-parameter representations of rotations and, subsequently, the existence of angular velocity vectors. It is also a starting point for several investigations on experimental measurements of rotations.
Recall that a proper-orthogonal second-order tensor is a tensor that has a unit determinant and whose inverse is its transpose:
The second of these equations implies that there are six restrictions on the nine components of . Consequently, only three components of are independent. In other words, any proper-orthogonal tensor can be parameterized by using three independent parameters. Because is a second-order tensor, it has the representation
Consider the transformation induced by on the orthonormal vectors , , and that comprise a right-handed basis. We define
Notice that by using the vectors as defined in (3), may also be represented as
We now wish to show that is also a right-handed orthonormal basis. First, let us verify orthonormality:
Hence, the vectors are orthonormal. To establish right-handedness, we use the definition of the determinant that features the scalar triple product of three vectors:
Therefore, is a right-handed orthonormal basis. Lastly, after a series of manipulations, we can arrive at another, but rather unusual, representation of the proper-orthogonal tensor :
In summary, we have the following representations for :
Notice that the components of for the first two representations are identical, and the handedness of is transferred without change by to . Also observe that the components of are equal to . As this product is equal to the cosine of the angle between and , each is often referred to as a direction cosine. Consequently, the matrix is known as the direction cosine matrix. Clearly, the nine angles whose cosines are are not all independent, for if they were, then would have nine independent components, which would contradict the requirement . Indeed, as we shall see shortly, it is possible to arrive at three independent angles to parameterize , but these angles are not all easily related to the angles between and .
Derivatives of a proper-orthogonal tensor and angular velocity vectors
Consider a proper-orthogonal tensor that is a function of time: . By the product rule, the time derivative of is
Because , the right-hand side of (9) is zero, and thus
In other words, the second-order tensor is skew-symmetric. For convenience, we define
in part because this tensor, known as the angular velocity tensor of , appears in numerous places later on. The skew-symmetry of allows us to define an associated angular velocity vector :
where for any vector . A common example of the calculation of an axial vector arises when we consider the motion of a rigid body rotating about the direction. In this case, the skew-symmetric angular velocity tensor
and hence we conclude that the axial vector of is the angular velocity vector . It is also useful to verify that
for any vector .
In a similar manner, we can also show that is a skew-symmetric tensor and define an angular velocity tensor with corresponding angular velocity vector :
With the help of the identity
which holds for all orthogonal and all second-order tensors , it is possible to show that
Notice how identity (17) simplifies when is a proper-orthogonal tensor.
On a final note, if we utilize the representation
and take to be a fixed basis, then
Consequently, we find a familiar result:
That is, if are defined by use of a proper-orthogonal tensor and a fixed basis , then their time derivatives can be expressed in terms of the angular velocity vector of the rotation tensor and the basis vectors .
Consider the following representations of a vector and a second-order tensor :
If we assume that and are functions of time, then
Let and denote the corotational derivatives (with respect to ) of and , respectively:
Simply put, and are the respective derivatives of and if the vectors are constant. With this notation, (23) and (24) simplify to
The terms in these expressions involving the angular velocity vector and the angular velocity tensor are the result of the orthonormal vectors changing with time. Corotational derivatives of vectors and tensors feature prominently in our discussions of relative rotations and the kinematics of rigid bodies.