Mathematical Physics Vol 1

4.4 Generalized coordinates

113

Note that for an orthogonal coordinate system g i j = h 2

i δ i j . Namely, in that case we have

e i · e j = δ i j ,

(4.144)

or, in the expanded format

e 1 · e 2 = e 2 · e 3 = e 3 · e 1 = 0 , e 1 · e 1 = e 2 · e 2 = e 3 · e 3 = 1 ,

(4.145)

and thus in an orthogonal coordinate system, for the arc element we obtain d s 2 = d r · d r = = 3 ∑ i , j = 1 h i dq i h j dq j e i · e j = 3 ∑ i , j = 1 δ i j h i dq i h j dq j =

(4.146)

3 ∑ i = 1

i 2

h i dq

=

.

Finally, let us express the volume element (Fig. 4.21) by generalized orthogonal coordinates d V = h 1 dq 1 e 1 · h 2 dq 2 e 2 × h 3 dq 3 e 3 = (4.147) = h 1 h 2 h 3 dq 1 dq 2 dq 3 ( e 1 · e 2 × e 3 )= h 1 h 2 h 3 dq 1 dq 2 dq 3 .

Figure 4.21: Volume element.

4.4.2 Gradient, divergence, rotor and Laplacian - expressed by generalized coordi nates The gradient of an arbitrary function U can be expressed in generalized coordinates as grad U = ∇ U =

∂ U ∂ q 1

∂ U ∂ q 2

∂ U ∂ q 3

1 h 1 ·

1 h 2 ·

1 h 3 ·

e 1 +

e 2 +

e 3 =

(4.148)

=

3 ∑ i = 1

∂ U ∂ q i

1 h i ·

e i .

=

Divergence is given by the expression div A = ∇ · A =

(4.149)

1 h 1 h 2 h 3

( h 1 h 2 A 3 ) .

∂ ∂ q 1

∂ ∂ q 2

∂ ∂ q 3

( h 2 h 3 A 1 )+

( h 3 h 1 A 2 )+

=

Rotor is given by the expression

h 1 e 1 h 2 e 2 h 3 e 3 ∂ ∂ q 1 ∂ ∂ q 2 ∂ ∂ q 3 h 1 A 1 h 2 A 2 h 3 A 3

1 h 1 h 2 h 3 ·

rot A = ∇ × A =

(4.150)

.