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.

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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.