Zhilin P.A. “A New Approach to the Analysis of Free Rotations of Rigid Bodies” // ZAMM. Z. angew. Math. Mech. 76 (1996). Vol. 4. P. 187-204.

Euler angles and the problem of free rotations of rigid bodies

Free rotation of rigid bodies was the first problem that was completely solved in dynamics of rigid bodies. Originally it was studied by Leonard Euler. Later Poinsot offered the famous geometrical interpretation to show real rotation of the body. In modern literature this problem is called the case of the integrability by Euler – Poinsot.
Ejler3 (18K)
L.Euler
Up to now no theoretical adaptations were made to the classical solution, the description of which is presented in all books on dynamics of rigid bodies. The classical solution allows perfectly to find the rotations, i.e. angular velocities, of the body. However, the determination of the turns, i.e. angles, does not impress so much. Moreover, it may be shown that the application of the Euler angles to this problem is not the best way because of several reasons. Firstly, the Euler angles, as a rule, give a representation which is rather difficult for interpretation. Secondly, this representation generates difficulties for the numerical realization on computers. By these reasons it seems to be useful to give an alternative approach to the analysis of the Euler – Poinsot problem, that is based on the concept of the tensor of turn called in the sequel turn-tensor. Some main facts concerning the turn-tensor are presented in the introduction, where the new theorem on the representation of the turn-tensor is given. The theorem allows to simplify the solution of problems of the dynamics of rigid bodies.

In Euler-Poinsot’s problem it is not difficult to find four first integrals of the basic equations. Three of them are well known. They express that the angular momentum vector of the body is constant. The fourth integral is that of energy which is directly expressed in terms of turns rather than of angular velocities. The energy integral in such a form allows to construct the most suitable representation of the turn-tensor to make the picture of turns of the body clear. It is found that there exist three and only three different types of rotations. Two of them give stable rotations, and the third type describes an unstable rotation. The type of rotation is determined for the given body by initial conditions only. In fact, the third type of rotation is the separatrix between two stable types of rotations. Under some conditions the stable rotations at certain moments of time can be very close to each other. Thus it is possible for the body to jump from one stable solution to another stable rotation. For example, the body can be rotating around the axis with minimal moment of inertia and there upon it can change the rotation to begin the rotation around the axis with maximal moment of inertia. Of course, small perturbations acting on the body are needed to provoke such a situation. As final result the problem is reduced to the integration of the simple differential equation of first order, the solution of which is a monotonically increasing function. All required quantities can be expressed in term of this function. It is shown how to see the turns of the body without integration of the equations if initial conditions are given.