Mathematical Physics Vol 1

4.4 Generalized coordinates

111

x 2

x 2

P

x 2

P

P

O

O

x 1

x 1

x 1

O

(a)

(b)

(c)

Figure 4.18: Coordinate lines forming an acute angle.

In addition to this set of unit vectors, we can also observe a set of vectors grad q i , which are in the directions of the normals to the coordinate surfaces, and are defined by

∇ q i | ∇ q i |

E i =

(4.134)

.

Vectors E i are therefore unit vectors of normals to the coordinate surfaces. Thus, at each point in space there are two sets of unit vectors: e i – unit vectors in the directions of the tangents, and E i – unit vectors of the normals to the coordinate surfaces (Fig. 4.19). If the coordinate system is orthogonal these two sets coincide. Let us observe an arbitrary vector v and represent it by these two systems v = = v 1 e 1 + v 2 e 2 + v 3 e 3 = 3 ∑ i = 1 v i e i = = V 1 E 1 + V 2 E 2 + V 3 E 3 = 3 ∑ i = 1 V i E i . (4.135)

This vector can also be represented by the base vectors g i = ∂ r q i : v =

3 ∑ i = 1

3 ∑ i = 1

∂ r ∂ q 1

∂ r ∂ q 2

∂ r ∂ q 3

∂ r ∂ q i

= c 1

+ c 2

+ c 3

c i g

c i

=

i =

(4.136)

3 ∑ i = 1

i = C

1 + C

2 + C

3 .

C i ∇ q

1 ∇ q

2 ∇ q

3 ∇ q

=

Figure 4.19: Contravariant and covariant base.

The coordinates obtained by decomposing this vector are called c i – contravariant and C i – covariant , respectively.

4.4.1 Arc and volume elements

The length of an arc element (Fig. 4.20) is determined by the relation d s 2 = d r · d r .

(4.137)