On Kinematic Description of the Motion of a Rigid Body

A system of ordinary differential equations has been derived for a vector of finite rotation corresponding to Euler's theorem: the vector of finite rotation is directed along the axis of finite rotation of a solid, and its length is equal to the angle of plane rotation around this axis. The system of equations is explicitly resolved with respect to the time derivative of the rotation vector components. The right part of the system depends on the rotation vector and the angular velocity vector in the principle axes. The obtained system of equations is shown to be equivalent to the system of equations for quaternions. The coordinates of the orts of the principle axes of a rigid body in fixed axes are expressed in terms of finite rotation angles and the components of angular velocity using simple analytical formulas.