\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

Kinetics: Time derivative of the Inertia tensor

In a previous section we showed that

\[\begin{align} \bs{J}_t &= (J_{0})_{ij}\uv{e}_i \otimes \uv{e}_j, \label{eq:Jt} \end{align}\]

where \(\left(J_{0}\right)_{ij}\) are the components of \(\bs{J}_0\) w.r.t \(\uv{E}_i\). Specifically, note that \(\left(J_{0}\right)_{ij}\) are constant w.r.t time.

Differentiating $\eqref{eq:Jt}$ w.r.t time we get

\[\begin{align} \dot{\bs{J}}_t &= (J_{0})_{ij}\left(\dot{\uv{e}}_i\otimes \uv{e}_j+\uv{e}_i\otimes \dot{\uv{e}}_j\right), \end{align}\]

In a previous section we showed that \(\dot{\uv{e}}_i =\bs{\omega}_t \times \uv{e}_i\), or equivalently, \(\dot{\uv{e}}_i =\bs{\Omega}_t \uv{e}_i\). The angular velocity tensor is sometimes denoted as \(\bs{\omega}_{\times}\) or \(\tilde{\bs{\omega}}\). Using these results, we get,

\[\begin{align} \dot{\bs{J}}_t &= (J_{0})_{ij}\left(\left(\bs{\Omega}_t \uv{e}_i\right)\otimes \uv{e}_j+\uv{e}_i\otimes \left(\bs{\Omega}_t \uv{e}_j\right)\right),\\ &=(J_{0})_{ij}\left(\bs{\Omega}_t\left( \uv{e}_i\otimes \uv{e}_j\right)+\left(\uv{e}_i\otimes \uv{e}_j\right) \bs{\Omega}^{\textsf{T}}_t\right),\\ &=(J_{0})_{ij}\left(\bs{\Omega}_t\left( \uv{e}_i\otimes \uv{e}_j\right)+\left(\uv{e}_i\otimes \uv{e}_j\right) \bs{\Omega}^{\textsf{T}}_t\right),\\ &=\bs{\Omega}_t\left((J_{0})_{ij} \uv{e}_i\otimes \uv{e}_j\right)+\left((J_{0})_{ij}\uv{e}_i\otimes \uv{e}_j\right) \bs{\Omega}^{\textsf{T}}_t,\\ &=\bs{\Omega}_t\bs{J}_t+\bs{J}_t\bs{\Omega}^{\textsf{T}}_t,\\ \end{align}\]

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