Zhilin P.A. Symmetries and Orthogonal Invariants in Oriented Space // Proceedings of XXXII Summer School - Conference Advanced Problems in Mechanics. St. Petersburg, Russia. 2005. P. 470-483.
The necessity of orientation of reference system arises due to our desire to take into account the moment interaction in mechanics. In the nature there are two principally different kinds of motion: the translation motion and the spinor (rotational) motion. Under translational motion a body is changing the position in the space. Under spinor motion a body is changing an orientation in the space without changing of position. The changing of translational motion is determined by forces. The changing of spinor motion is determined by moments. Note that in general moments can not be reduced to the concept of the force moment. In order to describe the spinor movements and the moment interactions we must orient the reference system and to introduce some new objects called axial objects in addition to the conventional objects called polar. There are many different but mathematically almost equivalent ways to introduce the space orientation. We prefer a way with clear physical sense. The physical (and mathematical) image of the spinor movement is given by so-called spin-vector whose introduction does not require the space orientation. Let there be given some system of reference (SR). Polar vector is represented in SR as an arrow. In addition to polar vector let us introduce a new object called spin-vector. For this it is necessary to take a strait line in SR called axis of a spin-vector. After that a circular arrow around the axis of a spin-vector must be drown in the plain orthogonal to the axis. Now we have a visual image of the spin-vector see Figure at the left.
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1) a is placed on the axis of spin-vector s,
2) modulus of a is equal to the modulus of s,
3) the vector a is directed as shown at the Figure at the center
(in such a case we have the right-oriented SR) or as shown at the Figure at the right
(in such a case we have the left-oriented SR).
The concept of an axial vector introduced above is the exact expression of a physical idea about angular velocity, moment and so on. The introduction of axial vector does not require any system of coordinates.