Mathematical Physics Vol 1

4.6 Examples

195

Differentiating once again by time, we obtain the acceleration

d 2 r d t 2

d d t

( ˙ ρ e ρ + ρ ˙ ϕ e ϕ + ˙ z e z )

a =

=

d e ρ d t

d e ϕ d t

= ˙ ρ + ρ ¨ ϕ e ϕ + ˙ ρ ˙ ϕ e ϕ + ¨ z e z = ˙ ρ ˙ ϕ e ρ + ¨ ρ e ρ + ρ ˙ ϕ ( − ˙ ϕ e ρ )+ ρ ¨ ϕ e ϕ + ˙ ρ ˙ ϕ e ϕ + ¨ z e z =( ¨ ρ − ρ ¨ ϕ 2 ) e ρ +( ρ ¨ ϕ + 2˙ ρ ˙ ϕ ) e ϕ + ¨ z e z . + ¨ ρ e ρ + ρ ˙ ϕ

Exercise 148 Find the infinitesimal part of an arc in cylindrical coordinates and determine the corresponding Lame’s coefficients.

Solution Given that

x = ρ cos ϕ y = ρ sin ϕ z = z ,

and

d x = − ρ sin ϕ d ϕ + cos ϕ d ρ , d y = ρ cos ϕ d ϕ + sin ϕ d ρ , d z = d z ,

it follows that

( d s ) 2 = d x 2 + d y 2 + d z 2 =( − ρ sin ϕ d ϕ + cos ϕ d ρ ) 2 +( ρ cos ϕ d ϕ + sin ϕ d ρ ) 2 +( d z ) 2 =( d ρ ) 2 + ρ 2 ( d ϕ ) 2 +( d z ) 2 = h 2 1 ( d ρ ) 2 + h 2 2 ( d ϕ ) 2 + h 2 3 ( d z ) 2 . From here, by comparison with (4.146) on p. 113, we obtain Lame’s coefficients h 1 = h ρ = 1, h 2 = h ϕ = ρ and h 3 = h z = 1.

Exercise 149 Find the arc element in a) spherical and

b) parabolic coordinates and determine the corresponding Lame’s coefficients.