Infinitesimal rotation matrix: Difference between revisions

Revision as of 05:49, 6 September 2023

Revision as of 06:13, 6 September 2023

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===Associated quantities===

We can introduce here the associated ''[[infinitesimal rotation tensor]]'' <math>d\Phi(t) = A - I,</math> associated to this infinitesimal rotation matrix <math>A.</math> When it is divided by the time difference, this will yield the ''[[angular velocity tensor]]'':

: <math>

W = \frac{d\Phi(t)}{dt} = \begin{pmatrix}

0 & -d\phi_z(t) & d\phi_y(t) \\

0 d\phi_z(t) & 0 & -\omega_z(t) & & -d\phi_xomega_y(t) \\

-d\phi_yomega_z(t) & d0 & -\phi_xomega_x(t) & 0 \\

-\omega_y(t) & \omega_x(t) & 0 \\

\end{pmatrix}

</math>

Such that its associated rotation matrix is <math>A = I + d\Phi(t)</math>. When it is divided by the time difference, this will yield the ''[[angular velocity tensor]]'':

: <math>

W=\frac{d\Phi(t)}{dt} = \begin{pmatrix}

0 & -\omega_z(t) & \omega_y(t) \\

\omega_z(t) & 0 & -\omega_x(t) \\

-\omega_y(t) & \omega_x(t) & 0 \\

\end{pmatrix}

</math>