Mathematical Preliminaries

Abstract

In classical mechanics the motion of bodies takes place in Euclidean space. To describe geometrical and mechanical relations a frame of axes spanning the space is required. It is common practice to use the Cartesian frame as shown in Fig.1. It is characterized by the location of its origin O and by the orientation of three perpendicular axes, each of which has a scale of distance established by making a correspondence between the real numbers and the Euclidean distance of each point on the axis from the origin O. Orientation and scale may be visualized by three directed line segments e_α, α = 1, 2, 3¹ extending from the origin O to the points on the axes whose distance from O is 1. The set {e_α} is sometimes called an axis triad, and is denoted by symbol e (see also Eq.2.1.4-1 later). The complete frame, consisting of an origin and a triad, is designated by the notation {O, e} in which its location and orientation are explicit. When referring to both location and orientation, we use the term position with this generalized meaning. As regards the numbering of the triad axes, in this book we use only right-handed or dextral frames. A dextral frame is one in which a rotation about the 3-axis in a positive sense given by the right-hand rule moves the 1-axis toward the 2-axis, as shown by the bold curved arrow in Fig.1.