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.