Mathematical Physics Vol 1

Chapter 4. Field theory

114

Finally, let us represent the Laplacian ( ∆ = ∇ 2 ) with respect to generalized coordinates ∆ U = (4.151)

1 h 1 h 2 h 3

∂ ∂ q 1

∂ U ∂ q 1

∂ ∂ q 2

∂ U ∂ q 2

∂ ∂ q 3

∂ U ∂ q 3

h 2 h 3

h 3 h 1

h 1 h 2

h 1 ·

h 2 ·

h 3 ·

=

+

+

.

4.5 Special coordinate systems 1. CYLINDRICAL ( ρ , ϕ , z )

The relation between Cartesian and cylindrical coordinates is given by (Fig. 4.22):

Figure 4.22: Cylindrical coordinate system.

x = ρ cos ϕ , y = ρ sin ϕ , z = z , where ρ > 0 , 0 ≤ ϕ < 2 π , − ∞ < z < + ∞ . Let us now show on the example of these coordinates how the Lamé coefficients are deter mined. Let us first express the differentials d x , d y , d z by the new coordinates dx = cos ϕ d ρ +( − sin ϕ ) ρ d ϕ , dy = sin ϕ d ρ + cos ϕρ d ϕ , dz = dz . Let us then express the arc element d s 2 = d x 2 + d y 2 + d z 2 = = d ρ 2 + ρ 2 d ϕ 2 + d z 2 , (4.152) and then, comparing with (4.141), we can conclude that h ρ = 1 h ϕ = ρ h z = 1 .

2. SPHERICAL ( r , θ , ϕ )