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13.5: Matrix and Tensor Formulations of Rigid-Body Rotation

( \newcommand{\kernel}{\mathrm{null}\,}\)

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

{I}=(I11I12I13I21I22I23I31I32I33)

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

ω=(ω1ω2ω3)L=(L1L2L3)

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={I}ω

Note that the above notation uses boldface for the inertia tensor I, implying a rank-2 tensor representation, while the angular velocity ω and the angular momentum 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 L and the angular velocity ω.

{I}=Lω

Note that, as described in appendix 19.5, the inner product of a vector ω, which is the rank 1 tensor, and a rank 2 tensor {I}, leads to the vector 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.


This page titled 13.5: Matrix and Tensor Formulations of Rigid-Body Rotation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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