1. The Foundations of Dynamics of a Rigid Body with a Fixed Point

Consider a rigid body moving about a fixed point O. The position of the rigid body in space is determined at each moment of time by the position of the moving coordinate system Oxyz connected with the body relative to the fixed coordinate system Ox ₁ y ₁ z ₁. To determine the body position, three independent (among themselves) parameters corresponding to the number of degrees of freedom of the body are introduced. Let us consider one of the most common ways of determining orientation of a rigid body using the Euler angles [1–4]. The plane Oxy intersects with planeOx ₁ y ₁ along the line ON, called the line of nodes. Figure 1.1 shows the angle ψ of precession as the angle between the axis Ox ₁ and the line of nodes ON, the angle of nutation θ as the angle between the axes Oz and Oz ₁, and the angle φ of proper rotation as the angle between the line of nodes ON and the axis Ox.