Geodesics - Map Projections: Cartographic Information Systems - page 724

d s 2 =d q μ g μν d q ν g μν = λ 2 ( q 1 ,q 2 ) δ μν , ∀ (E.17)

2 ,
d s 2 = λ 2 ( q 1 ,q 2 )[(d q 1 ) 2 +(d q 2 ) 2 ] .

d q μ

2 , μν d q ν

∈ ∗ T q M

∈ T q M

(E.18)

E-311 Lagrange Portrait

The Lagrange portrait of the parameterized curve, called minimal geodesic q μ ( s ), a one-
dimensional submanifold in the two-dimensional Riemann manifold, is provided by ( E.19 )
and ( E.20 ). The functional ( E.19 ) subject to ( E.20 ) is minimal if the following hold: ( α ) zero
first variation (zero Frechet derivative) and ( β ) positive second variation (positive second Frechet
derivative).

L q, d q
d τ

d τ =min or

L 2 q, d q
d τ

d τ =min

(fixed boundary points) , (E.19)

2 L 2 q, d q
d τ

:= d s 2

d τ 2 = d q μ d τ g μν d q ν

g μν = λ 2 ( q 1 ,q 2 ) δ μν ,

d τ ∀

(E.20)

:= λ 2 ( q 1 ,q 2 ) d q 1
d τ

2 .

2 L 2 q, d q
d τ

2

+ d q 2
d τ

Under these necessary and sucient conditions, a minimal geodesic ( E.21 ) as a self-adjoint system
of two differential equations of second order with respect to the parameter arc length s and the
Christoffel symbols [ μν, λ ] of the first kind is being derived. (Only with respect to Christoffel
symbols of the first kind the system of differential equations ( E.21 ) is self-adjoint: the alternative
representation in terms of Christoffel symbols of the second kind with the second derivative
d 2 q μ / d s 2 as the leading term is not, thus cannot be derived from a variational principle.)

g μν d 2 q ν

d s 2 +[ μν λ ] d q ν

d q λ
d s

g μν = λ 2 ( q 1 ,q 2 ) δ μν , =0 ∀

d s

(E.21)

λ 2 d 2 q μ
d 2

+( ∂ ν λ 2 ) d q ν
d s

d q μ
d s −

1

2 λ 2 ∂ μ λ 2 =0 .

E-312 Hamilton Portrait

In contrast, the Hamilton portrait of a minimal geodesic q μ ( s )in { M

2 ,λ 2 δ μν } is based upon
the generalized momentum, the generalized velocity field g μν d q ν / d τ , being an element of the
cotangent space ∗ T q M

2 at point q ,namely