The new equation [1, 4 - 8] is obtained (1992), relating the left angular velocity with the derivative of the rotation vector. This equation is necessary to define the concept of a potential torque. Apart from that, it is very useful when solving numerically the problems of dynamics of rigid bodies, since then there is no need to introduce neither systems of angles, nor systems of parameters of the Klein-Hamilton type.
A new theorem [2, 3, 7, 8] on the representation of the rotation (turn) tensor in the form of a composition of turns about arbitrary fixed axes, is proved (1995). All previously known representations of the rotation (turn) tensors, (or, saying more precisely, of its matrix analogues) via Euler angles, Brayant angles, plane angles, ship angles etc., are particular cases of a general theorem, whose role, however, is not only a simple generalisation of these cases. The most important thing is that making a traditional choice of any system of angles, does not matter which one, we choose previously the axes. We describe the (unknown) rotation of a body under consideration in terms of turns about these axes. If this choice is made in an ineffectual way, and if it is difficult to make an appropriate choice, the chances to integrate or even to analyse qualitatively the resulting system of equations are very poor. Moreover, even in those cases when it is possible to integrate the system, often the obtained solution is not of big practical use, since this solution will contain poles or indeterminacy of the type zero divided by zero. As a result, the numerical solution, obtained with the help of computers, already after the first pole or indeterminacy becomes very distorted. The advantage and the purpose of the theorem under discussion is the fact that it allows to consider the axes of rotation as principal variables and to determine them in the process of the problem solution. As a result, one can obtain the simplest (among all possible forms) solutions.
An approach [4 - 6] is proposed (1997), which allows to analyse the stability of motion in the presence of spinor rotations described by the turn tensor. The method of perturbations for the group of proper orthogonal tensors is developed.