Mathematical Physics Vol 1

4.4 Generalized coordinates

109

Cartesian coordinates of the vector can now be expressed in the following form: x = x ( q i ) , y = y ( q i ) , z = z ( q i ) . It follows from the requirement for one-to-one correspondence between points in space and the coordinates q i that to each point with coordinates ( x , y , z ) three numbers q i must correspond , such that: q i = q i ( x , y , z ) , ∂ ( q 1 , q 2 , q 3 ) ∂ ( x , y , z )̸ = 0 . (4.128) Thus, equations (4.127) always satisfy the conditions necessary for solving them for q i . The equations (4.127) and (4.128) are equations of coordinate transformation. These trans formations are mutually reciprocal – inverse. Let us now define coordinate lines and coordinate surfaces. Definition Coordinate lines – represent the geometric location of points obtained when two coordinates are constant and the third changes. (4.127)

Coordinate lines can be straight or curved, and depending on that, we distinguish rectilinear and curvilinear coordinate systems (Fig. 4.16).

Figure 4.16: Coordinate lines.

Definition Coordinate surfaces – represent the geometric location of points obtained when two coordinates change and the third remains constant.

Figure 4.17: Coordinate surfaces