Geodesics - Map Projections: Cartographic Information Systems - page 746

leads to ( E.115 ).

d , g 1

:= cos α, (E.112)

d || g 1

d , g 2
d || g 2

:= sin α,

(i)
d , g 1 = g 11 q 1 = d g 1 cos α = √ g 11 cos α,
(ii)
d , g 2 = g 22 q 2 = d g 2 sin α = √ g 22 sin α,

(E.113)

(iii)

cos α,

∂ √ g 22
∂q 1

sin α + ∂ √ g 11
∂q 2

d α 1

√ g 11 g 22

d s = α ( s )=

−

q 1 = cos α
λ

,

q 2 = sin α
λ

, (E.114)

λ 2 ∂q 2 cos α ,

α = 1

∂q 1 sin α + ∂λ

∂λ

−

d q 1
d t

= d q 1
d s

d s

d t = λ cos α,

d q 2
d t

= d q 2
d s

d s
d t = λ sin α,

(E.115)

dα

d t = d α

d s

d t = ∂λ

∂λ
∂q 1 sin α. ∂q 2 cos α

−

d s

Similarly, the generalized momenta p μ ∈ ∗ T q M

2 are represented by ( E.116 ), leading to the Hamil-
ton equations of a geodesic in terms of polar coordinates, namely to ( E.117 ), and these are solved
by means of Lie series in Sect. E-36 .

p 1 = λ cos α, p 2 = λ sin α, (E.116)

q 1 = λ cos α, q 2 = λ sin α,

∂λ 2
∂q 1 ,

∂λ 2
∂q 2 .

p 1 = 1
2

p 2 = 1
2

(E.117)

Finally, as local polar/normal coordinates, we take advantage of Definition E.3 .