Rotations and proper-orthogonal tensors

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. 

Proper-orthogonal tensors

Recall that a proper-orthogonal second-order tensor {\bf R} is a tensor that has a unit determinant and whose inverse is its transpose:

(1)   \begin{eqnarray*} && \det({\bf R}) = 1, \\ \\ && {\bf R} {\bf R}^T = {\bf R}^T {\bf R} = {\bf I}. \end{eqnarray*}

The second of these equations implies that there are six restrictions on the nine components of {\bf R}. Consequently, only three components of {\bf R} are independent. In other words, any proper-orthogonal tensor can be parameterized by using three independent parameters. Because {\bf R} is a second-order tensor, it has the representation

(2)   \begin{equation*} {\bf R} = \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik} {\bf p}_i\otimes{\bf p}_k. \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}} \end{equation*}

Consider the transformation induced by {\bf R} on the orthonormal vectors {\bf p}_1, {\bf p}_2, and {\bf p}_3 that comprise a right-handed basis.  We define

(3)   \begin{eqnarray*} && {\bf t}_1 = {\bf R}{\bf p}_1 = \sum_{i \, \, = \, 1}^3 R_{i1}{\bf p}_i, \\ \\[0.10in] && {\bf t}_2 = {\bf R}{\bf p}_2 = \sum_{i \, \, = \, 1}^3 R_{i2}{\bf p}_i, \\ \\[0.10in] && {\bf t}_3 = {\bf R}{\bf p}_3 = \sum_{i \, \, = \, 1}^3 R_{i3}{\bf p}_i. \end{eqnarray*}

Notice that by using the vectors {\bf t}_i as defined in (3), {\bf R} may also be represented as

(4)   \begin{equation*} {\bf R} = {\bf t}_{1} \otimes {\bf p}_{1} + {\bf t}_{2} \otimes {\bf p}_{2} + {\bf t}_{3} \otimes {\bf p}_{3}. \end{equation*}

We now wish to show that \{{\bf t}_1, \, {\bf t}_2, \, {\bf t}_3 \} is also a right-handed orthonormal basis. First, let us verify orthonormality:

(5)   \begin{equation*} {\bf t}_i\cdot{\bf t}_k = {\bf R}{\bf p}_i \cdot {\bf R}{\bf p}_k = {\bf R}^T{\bf R} {\bf p}_i \cdot  {\bf p}_k = {\bf p}_i \cdot  {\bf p}_k = \delta_{ik}. \end{equation*}

Hence, the vectors {\bf t}_i are orthonormal. To establish right-handedness, we use the definition of the determinant that features the scalar triple product of three vectors:

(6)   \begin{equation*} [{\bf t}_1, \, {\bf t}_2, \, {\bf t}_3 ] = [{\bf R}{\bf p}_1, \, {\bf R}{\bf p}_2, \, {\bf R}{\bf p}_3 ] = \det({\bf R}) [{\bf p}_1, \, {\bf p}_2, \, {\bf p}_3 ] = (1)(1) = 1. \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}} \end{equation*}

Therefore, \{{\bf t}_1, \, {\bf t}_2, \, {\bf t}_3 \} 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 {\bf R}:

(7)   \begin{eqnarray*} {\bf R} \!\!\!\!\! &=& \!\!\!\!\! {\bf R}{\bf R}{\bf R}^T = {\bf R} \left(\sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik} {\bf p}_i\otimes{\bf p}_k \right) {\bf R}^T = \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik} {\bf R}\left({\bf p}_i\otimes{\bf p}_k\right){\bf R}^T = \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik} \left({\bf R}{\bf p}_i\otimes{\bf R}{\bf p}_k \right) \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}} \\ \\[0.05in] &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik}  {\bf t}_i\otimes{\bf t}_k. \end{eqnarray*}

In summary, we have the following representations for {\bf R}:

(8)   \begin{eqnarray*} {\bf R} \!\!\!\!\! &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik} {\bf p}_i\otimes{\bf p}_k \\ \\[0.05in] &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 R_{ik} {\bf t}_i\otimes{\bf t}_k \\ \\[0.05in] &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3 {\bf t}_i\otimes{\bf p}_i. \end{eqnarray*}

Notice that the components of {\bf R} for the first two representations are identical, and the handedness of \{{\bf p}_1, \, {\bf p}_2, \, {\bf p}_3 \} is transferred without change by {\bf R} to \{{\bf t}_1, \, {\bf t}_2, \, {\bf t}_3 \}. Also observe that the components R_{ik} of {\bf R} are equal to {\bf t}_k\cdot{\bf p}_i. As this product is equal to the cosine of the angle between {\bf t}_k and {\bf p}_i, each R_{ik} is often referred to as a direction cosine. Consequently, the matrix \left[R_{ik}\right] is known as the direction cosine matrix. Clearly, the nine angles whose cosines are {\bf t}_k\cdot{\bf p}_i are not all independent, for if they were, then \left[R_{ik}\right] would have nine independent components, which would contradict the requirement {\bf R}^T{\bf R} = {\bf I}. Indeed, as we shall see shortly, it is possible to arrive at three independent angles to parameterize {\bf R}, but these angles are not all easily related to the angles between {\bf t}_k and {\bf p}_i.

Derivatives of a proper-orthogonal tensor and angular velocity vectors

Consider a proper-orthogonal tensor {\bf R} that is a function of time: {\bf R} = {\bf R}(t). By the product rule, the time derivative of {\bf R}{\bf R}^T is

(9)   \begin{equation*} \dot{ \overline{ {\bf R}{\bf R}^T } } = \dot{\bf R}{\bf R}^T + {\bf R}\dot{\bf R}^T. \end{equation*}

Because \dot{\bf I} = {\bf O}, the right-hand side of (9) is zero, and thus

(10)   \begin{equation*} \dot{\bf R}{\bf R}^T = - {\bf R}\dot{\bf R}^T = - \left(\dot{\bf R}{\bf R}^T\right)^T. \end{equation*}

In other words, the second-order tensor \dot{\bf R}{\bf R}^T is skew-symmetric. For convenience, we define

(11)   \begin{equation*} {\bOmega}_{\bf R} = \dot{\bf R}{\bf R}^T, \end{equation*}

in part because this tensor, known as the angular velocity tensor of {\bf R}, appears in numerous places later on. The skew-symmetry of \dot{\bf R}{\bf R}^T allows us to define an associated angular velocity vector {\bomega}_{\bf R}:

(12)   \begin{equation*} {\bomega}_{\bf R} = - \frac{1}{2} {\bepsilon}\left[\dot{\bf R}{\bf R}^T\right] , \end{equation*}

where {\bomega}_{\bf R}\times{\bf a} = (\dot{\bf R}{\bf R}^T){\bf a} for any vector {\bf a}. A common example of the calculation of an axial vector arises when we consider the motion of a rigid body rotating about the {\bf E}_3 direction. In this case, the skew-symmetric angular velocity tensor

(13)   \begin{equation*} \bOmega_{\bf R} = \Omega \left({\bf E}_2\otimes{\bf E}_1 - {\bf E}_1\otimes{\bf E}_2\right). \end{equation*}

Consequently,

(14)   \begin{equation*} \bepsilon\left[  \bOmega_{\bf R} \right] = - 2 \Omega {\bf E}_3, \end{equation*}

and hence we conclude that the axial vector of \bOmega_{\bf R} is the angular velocity vector \Omega{\bf E}_3. It is also useful to verify that

(15)   \begin{equation*} \left(  \Omega \left({\bf E}_2\otimes{\bf E}_1 - {\bf E}_1\otimes{\bf E}_2\right) \right) {\bf a} = \Omega{\bf E}_3 \times {\bf a} \end{equation*}

for any vector {\bf a}.

In a similar manner, we can also show that {\bf R}^T\dot{\bf R} is a skew-symmetric tensor and define an angular velocity tensor {\bOmega}_{0_{\bf R}} with corresponding angular velocity vector {\bomega}_{0_{\bf R}}:

(16)   \begin{eqnarray*} && {\bOmega}_{0_{\bf R}} = {\bf R}^T\dot{\bf R}, \\ \\ && {\bomega}_{0_{\bf R}} = - \frac{1}{2} {\bepsilon}\left[{\bf R}^T \dot{\bf R}\right]. \end{eqnarray*}

With the help of the identity

(17)   \begin{equation*} \bepsilon \left[ {\bf Q} {\bf B} {\bf Q}^T \right] = \mbox{det}\left({\bf Q}\right) {\bf Q} \left(\bepsilon \left[ {\bf B} \right]\right) , \end{equation*}

which holds for all orthogonal {\bf Q} and all second-order tensors {\bf B}, it is possible to show that

(18)   \begin{eqnarray*} && {\bf R}{\bomega}_{0_{\bf R}} = {\bomega}_{\bf R}, \\ \\ && {\bf R}{\bOmega}_{0_{\bf R}}{\bf R}^T = {\bOmega}_{{\bf R}}. \end{eqnarray*}

Notice how identity (17) simplifies when {\bf Q} is a proper-orthogonal tensor.

On a final note, if we utilize the representation

(19)   \begin{equation*} {\bf R} = \sum_{i \, \, = \, 1}^3 {\bf t}_i\otimes{\bf p}_i \end{equation*}

and take \{ {\bf p}_i \} to be a fixed basis, then

(20)   \begin{eqnarray*} {\bOmega}_{{\bf R}} \!\!\!\!\! &=& \!\!\!\!\! \dot{\bf R}{\bf R}^T = \left(\sum_{i \, \, = \, 1}^3 \dot{\bf t}_i\otimes{\bf p}_i + \sum_{i \, \, = \, 1}^3 {\bf t}_i\otimes{\dot{\bf p}_i} \right){\bf R}^T = \left(\sum_{i \, \, = \, 1}^3 \dot{\bf t}_i\otimes{\bf p}_i\right){\bf R}^T \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}} \\ \\[0.05in] &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3 \dot{\bf t}_i\otimes{\bf t}_i. \end{eqnarray*}

Consequently, we find a familiar result:

(21)   \begin{equation*} \dot{\bf t}_i = {\bOmega}_{{\bf R}}{\bf t}_i = {\bomega}_{{\bf R}}\times{\bf t}_i. \end{equation*}

That is, if {\bf t}_i are defined by use of a proper-orthogonal tensor {\bf R} and a fixed basis \{ {\bf p}_i \}, then their time derivatives can be expressed in terms of the angular velocity vector of the rotation tensor and the basis vectors {\bf t}_i.

Corotational derivatives

Consider the following representations of a vector {\bf a} and a second-order tensor {\bf A}:

(22)   \begin{eqnarray*} && {\bf a} = \sum_{i \, \, = \, 1}^3 a_i {\bf t}_i, \\ \\ \\ && {\bf A} = \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 A_{ik} {\bf t}_i\otimes{\bf t}_k. \end{eqnarray*}

If we assume that {\bf a} and {\bf A} are functions of time, then

(23)   \begin{eqnarray*} \dot{\bf a} \!\!\!\!\! &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3 \dot{a}_i {\bf t}_i + \sum_{i \, \, = \, 1}^3 a_i \dot{\bf t}_i = \sum_{i \, \, = \, 1}^3 \dot{a}_i {\bf t}_i + \sum_{i \, \, = \, 1}^3 a_i ({\bomega}_{{\bf R}}\times{\bf t}_i) = \sum_{i \, \, = \, 1}^3 \dot{a}_i {\bf t}_i + {\bomega}_{{\bf R}}\times \left ( \sum_{i \, \, = \, 1}^3 a_i {\bf t}_i \right ) \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}} \\ \\[0.05in] \!\!\!\!\! &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3 \dot{a}_i {\bf t}_i +  {\bomega}_{{\bf R}}\times{\bf a} \end{eqnarray*}

and

(24)   \begin{eqnarray*} \dot{\bf A} \!\!\!\!\! &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 \dot{A}_{ik} {\bf t}_i\otimes{\bf t}_k + \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 A_{ik} \dot{\bf t}_i\otimes{\bf t}_k + \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 A_{ik} {\bf t}_i\otimes\dot{\bf t}_k \\ \\[0.05in] &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 \dot{A}_{ik} {\bf t}_i\otimes{\bf t}_k + \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 A_{ik} ({\bOmega}_{{\bf R}}{\bf t}_i)\otimes{\bf t}_k + \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 A_{ik} {\bf t}_i\otimes({\bOmega}_{{\bf R}}{\bf t}_k) \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}} \\ \\[0.05in] &=& \!\!\!\!\! \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 \dot{A}_{ik} {\bf t}_i\otimes{\bf t}_k + {\bOmega}_{{\bf R}}{\bf A} - {\bf A}{\bOmega}_{{\bf R}}. \end{eqnarray*}

Let \corot{\bf a} and \corot{\bf A} denote the corotational derivatives (with respect to {\bf R}) of {\bf a} and {\bf A}, respectively:

(25)   \begin{eqnarray*} && \corot{\bf a} = \sum_{i \, \, = \, 1}^3 \dot{a}_{i} {\bf t}_i , \\ \\ \\ && \corot{\bf A} = \sum_{i \, \, = \, 1}^3\sum_{k \, \, = \, 1}^3 \dot{A}_{ik} {\bf t}_i\otimes{\bf t}_k . \end{eqnarray*}

Simply put, \corot{\bf a} and \corot{\bf A} are the respective derivatives of {\bf a} and {\bf A} if the vectors {\bf t}_i are constant. With this notation, (23) and (24) simplify to

(26)   \begin{eqnarray*} && \dot{\bf a} = \corot{\bf a} + {\bomega}_{{\bf R}}\times{\bf a} , \\ \\ && \dot{\bf A} = \corot{\bf A} + {\bOmega}_{{\bf R}}{\bf A} - {\bf A}{\bOmega}_{{\bf R}}. \end{eqnarray*}

The terms in these expressions involving the angular velocity vector {\bomega}_{{\bf R}} and the angular velocity tensor {\bOmega}_{{\bf R}} are the result of the orthonormal vectors {\bf t}_i changing with time. Corotational derivatives of vectors and tensors feature prominently in our discussions of relative rotations and the kinematics of rigid bodies.