The torque in Euler's equations for rigid body rotations

Question

$$\large{I_1\dot{\omega}_1+(I_3-I_2)\omega_2\omega_3=M_1}$$ $$\large{I_2\dot{\omega}_2+(I_1-I_3)\omega_3\omega_1=M_2}$$ $$\large{I_3\dot{\omega}_3+(I_2-I_1)\omega_1\omega_2=M_3}$$

In Taylor's Classical Mechanics, it is said that the equations are generally difficult to use because the components $M_1,M_2$ and $M_3$ of the applied torque as seen in the rotating body frame are complicated functions of time. My question is why would the torques $M_1, M_2$ and $M_3$ be in the rotating body reference frame? The Euler equations are derived using: $$ \frac{dL}{dt}_{lab} = M$$ and then putting in: $$ \frac{dL}{dt}_{lab} = \frac{dL}{dt}_{body} + \omega \times L$$ so shouldn't the torque be with respect to the lab frame? What am I missing here?

Answer 1

My question is why would the torques $M_1, M_2$ and $M_3$ and be in the rotating body reference frame?

They are not. As you noted further, they are the applied torque with respect to the fixed space/lab frame.

These set of equations describe the dynamics as seen in frame fixed in the body or the rotating frame. Unless you have a symmetrical object (such as sphere), in general, it is difficult to keep track principal axes as they vary with time. The principal axis, will not remain principal axis,and the inertia ”seen” in this coordinate system will vary with time.

Since the axes are fixed to the body, we are committed to follow the body as it rotates in order to use these equations and for obtaining the solutions. We need to come up with equations of motion relative to a nonrotating space frame, and before we can do that we need a set of coordinates that specifiy the orientation of our body relative to such a frame -- Euler angles.