On the invariant motions of rigid body rotation over the fixed point, via Euler’s angles

Abstract

The generalized Euler’s case (rigid body rotation over the fixed point) is discussed here: The center of masses of non-symmetric rigid body is assumed to be located at the equatorial plane on axis Oy which is perpendicular to the main principal axis Ox of inertia at the fixed point. Such a case was presented in the rotating coordinate system, in a frame of reference fixed in the rotating body for the case of rotation over the fixed point (at given initial conditions). In our derivation, we have represented the generalized Euler’s case in the fixed Cartesian coordinate system; so, the motivation of our ansatz is to elegantly transform the proper components of the previously presented solution from one (rotating) coordinate system to another (fixed) Cartesian coordinates. Besides, we have obtained an elegantly analytical case of general type of rotations; also, we have presented it in the fixed Cartesian coordinate system via Euler’s angles.