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13.15: Kinetic energy in terms of Euler angular velocities

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    The kinetic energy is a scalar quantity and thus is the same in both stationary and rotating frames of reference. It is much easier to evaluate the kinetic energy in the rotating Principal-axis frame since the inertia tensor is diagonal in the Principal-axis frame as given in equation \((13.12.14)\)

    \[T_{rot} = \frac{1}{2} \sum^3_i I_{ii} \omega^2_i\]

    Using equation \((13.14.1-13.14.3)\) for the body-fixed angular velocities gives the rotational kinetic energy in terms of the Euler angular velocities and principal-frame moments of inertia to be

    \[T_{rot}=\frac{1}{2}\left[I_{1}(\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi)^{2}+I_{2}(\dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi)^{2}+I_{3}(\dot{\phi} \cos \theta+\dot{\psi})^{2}\right]\]

    This page titled 13.15: Kinetic energy in terms of Euler angular velocities 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; a detailed edit history is available upon request.

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