Rigid Bodies; Euler's Eq of Motion derivation

[logic smogic]

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?

 

[Jhero]

Hello,

It seems like you are on the right track! Your approach of using the kinetic energy and angular velocity in terms of Euler angles is correct. However, there is one more step that you need to take to derive Euler's equations of motion.

In the Lagrange formulation, the generalized coordinates q_j are used to describe the configuration of the system. In this case, we can use the Euler angles as our generalized coordinates. This means that we need to express the kinetic energy in terms of the Euler angles and their derivatives, rather than the angular velocity.

To do this, we can use the relationship between the angular velocity and the Euler angles that you have already provided. We can then use the chain rule to express the kinetic energy in terms of the Euler angles and their derivatives.

Once you have the expression for the kinetic energy in terms of the Euler angles and their derivatives, you can plug it into the Lagrange equation for \psi and reduce it to the desired form. This will give you the first Euler's equation of motion.

You will need to repeat this process for the other two Euler's equations, using the Lagrange equation for \theta and \phi. It may seem like a lot of work, but this approach is the most systematic and will give you the desired result.

I hope this helps! Let me know if you have any further questions.