Euler's equations

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The equation of motion of a rigid body subject only to a torque $\boldsymbol{\tau}\,$ , is

$\frac{d\mathbf{L}}{dt} = \boldsymbol{\tau}\,$ ,

which is of course evaluated in a fixed frame $O\,$ . Instead, Euler's equations describe the motion of the body in its own, rotating frame $O'\,$ . This is usually more convenient, since, for a rigid body in the rotating frame, the moment of inertia tensor $\mathbf{I}\,$ is constant, and without loss of generality, diagonal.

We may instantaneously choose the axes of $O\,$ and $O'\,$ to coincide. Recall that

$\left(\frac{d}{dt}\right)_O = \left(\frac{d}{dt}\right)_{O'} + \boldsymbol{\omega}\times\,$ .

Thus, in the rotating frame $O'\,$

$\mathbf{I} \,\dot\boldsymbol{\omega} + \boldsymbol{\omega}\times\mathbf{L} = \boldsymbol{\tau}\,$ .

Having chosen our axes to be principal axes, the matrix $\mathbf{I}\,$ is diagonal, and this becomes

$\begin{matrix} I_1\dot{\omega}_{1}+(I_3-I_2)\omega_2\omega_3 &=& \tau_{1}\\ I_2\dot{\omega}_{2}+(I_1-I_3)\omega_3\omega_1 &=& \tau_{2}\\ I_3\dot{\omega}_{3}+(I_2-I_1)\omega_1\omega_2 &=& \tau_{3} \end{matrix}$ .

back to torque

back to Rigid body dynamics

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Euler's equations

From Physics wiki

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