Rigid Bodies; Euler's Eq of Motion derivation

by logic smogic
Tags: bodies, derivation, euler, motion, rigid
 P: 56 Problem Derive Euler's equations of motion for a rigid body from the Lagrange formulation (for generalized coordinate $$\psi$$, the third Euler angle). Applicable Formulae Euler's equations of motion (what we are trying to derive) are: $$I_{1} \dot{\omega_{1}} - \omega_{2} \omega_{3} (I_{2}-I_{3}) = N_{1}$$ $$I_{2} \dot{\omega_{2}} - \omega_{3} \omega_{1} (I_{3}-I_{1}) = N_{2}$$ $$I_{3} \dot{\omega_{3}} - \omega_{1} \omega_{2} (I_{1}-I_{2}) = N_{3}$$ Lagrange Formulation: $$\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q_{j}}} \right) - \frac{\partial T}{\partial q_{j}} = Q_{j}$$ Attempt at a solution It seems I need an expression for the kinetic energy T of the rigid body in terms of the Euler angles. I would then plug that into the Lagrange equation above for $$\psi$$, and reduce it to the desired form. The kinetic energy is given by: $$T = \frac{\vec{\omega}\cdot\bar{I}\cdot\vec{\omega}}{2}$$ and the angular velocity in terms of Euler angles is: $$\vec{\omega_{x,y,z}}=\left( \begin{array}{c} \dot{\phi} sin\theta sin\psi + \dot{\theta} cos\psi \\ \dot{\phi} sin\theta cos\psi - \dot{\theta} sins\psi \\ \dot{\phi}cos\theta + \dot{\psi} \end{array} \right) \cdot \left( \begin{array}{c} \hat{x} \\ \hat{y} \\ \hat{z} \end{array} \right)$$ I've already started working on it, but it seems like a lot of work. Am I on the right track, or is there something I'm missing here?

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