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

TL;DR

This paper analyzes the invariant motions of rigid body rotation over a fixed point using Euler angles, transforming solutions from rotating to fixed coordinate systems to provide a comprehensive analytical description.

Contribution

It introduces a method to represent the generalized Euler case in fixed Cartesian coordinates, offering an elegant analytical solution for rigid body rotations.

Findings

Derived an analytical case of general rotations

Represented the solution in fixed Cartesian coordinates

Provided a transformation from rotating to fixed frames

Abstract

The generalized Euler 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 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 angles.