7.5: Rotation Matrix of a Rigid Body

Let \(\boldsymbol{u}\) be an arbitrary vector attached to a rigid body moving with angular velocity \(\boldsymbol{\omega}(t)\). Let \(u(t)\) denote the column of coordinates of vector \(\boldsymbol{u}\) in the fixed system \(O x y z\). We have

\[u(t)=X(t) u_0, \quad u(0)=u_0, \notag \]

where \(X(t)\) is a \(3 \times 3\) orthogonal matrix with \(X(0)=E\).

Suppose that the frame \(Q \xi \eta \zeta\) attached to the rigid body coincides with \(O x y z\) at the initial instant.

In particular, this means that the coordinate column of vector \(\boldsymbol{u}\) in \(Q \xi \eta \zeta\) is equal to \(u_0\) at all times. Moreover, if \(v\) is the coordinate column of a vector \(\boldsymbol{v}\) in \(O x y z\), then \(X^{-1}(t) v\) is the coordinate column of vector \(\boldsymbol{v}\) in \(Q \xi \eta \zeta\).

The formula

\[\begin{equation*}
\dot{\boldsymbol{u}}=\omega \times u \tag{12}
\end{equation*}\]

is invariant and follows from formula (3).

Suppose the angular velocity vector \(\boldsymbol{\omega}\) is given by its coordinate column \(\omega\) in the frame \(Q \xi \eta \zeta\).
Then we can write formula (12) in \(Q \xi \eta \zeta\) coordinates:

\[X^{-1} \dot{u}(t)=\omega \times u_0=\Omega u_0, \notag\]

where

\[\Omega=\left(\begin{array}{ccc}
0 & -\omega_\zeta & \omega_\eta \\
\omega_\zeta & 0 & -\omega_{\xi} \\
-\omega_\eta & \omega_{\xi} & 0
\end{array}\right), \quad \omega=\left(\begin{array}{c}
\omega_{\xi} \\
\omega_\eta \\
\omega_\zeta
\end{array}\right) .\notag\]

Finally, we obtain:

\[X^{-1} \dot{X} u_0=\Omega u_0 \Longrightarrow \dot{X}=X \Omega .\notag \]

N. V. Efimov, E. R. Rozendorn, Linear Algebra and Multidimensional Geometry, Nauka, Moscow, 1970 (in Russian).

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