Euler Equations of Motion

Imagine a rigid body that is rotating at an angular velocity of ω within an inertial frame of reference. Along with this, picture a second rotating frame that is attached to the body itself. This frame moves along with the body and possesses an angular velocity of Ω. The total moment about the center of mass is calculated by adding the rate of change of angular momentum about the center of mass in relation to the rotating frame and the cross-product of the body's angular velocity and its angular momentum.

Now, consider a situation where the angular velocity of these rotating axes equals the angular velocity of the body itself. In such a scenario, the moments and product of inertia concerning the rotating axes will remain constant. Recalling the scalar components of the angular momentum and using these, one can express the equation for the total moment in terms of scalar components.

If one chooses the rotating axes as the principle axes of inertia, the product of the inertia term disappears. This simplification results in a more manageable scalar form of the total moment equation. These principles and equations constitute Euler's equations of motion for rotating bodies. These equations provide valuable insights into the dynamics of rotating rigid bodies, enabling us to understand and predict their behavior under various conditions.