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13.5: Matrix and Tensor Formulations of Rigid-Body Rotation

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  • The prior notation is clumsy and can be streamlined by use of matrix methods. Write the inertia tensor in a matrix form as

    \[\{\mathbb{I}\}= \begin{pmatrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{pmatrix}\]

    The angular velocity and angular momentum both can be written as a column vectors, that is

    \[\boldsymbol{\omega}=\begin{pmatrix} \omega_{1} \\ \omega_{2} \\ \omega_{3}
    \end{pmatrix} \quad \mathbf{L}=\begin{pmatrix} L_{1} \\ L_{2} \\ L_{3} \end{pmatrix}\]

    As discussed in appendix \(19.5.2\), Equation \(13.4.7\) now can be written in tensor notation as an inner product of the form

    \[L = \{\mathbb{I}\} \cdot \boldsymbol{\omega} \]

    Note that the above notation uses boldface for the inertia tensor \(\mathbb{I}\), implying a rank-2 tensor representation, while the angular velocity \(\boldsymbol{\omega}\) and the angular momentum \(\mathbf{L}\) are written as column vectors. The inertia tensor is a 9-component rank-2 tensor defined as the ratio of the angular momentum vector \(\mathbf{L}\) and the angular velocity \(\boldsymbol{\omega}\).

    \[\{\mathbb{I}\} = \frac{\mathbf{L}}{\boldsymbol{\omega}}\]

    Note that, as described in appendix \(19.5\), the inner product of a vector \(\boldsymbol{\omega}\), which is the rank 1 tensor, and a rank 2 tensor \(\{\mathbb{I}\}\), leads to the vector \(\mathbf{L}\). This compact notation exploits the fact that the matrix and tensor representation are completely equivalent, and are ideally suited to the description of rigid-body rotation.

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