Geodesics - Map Projections: Cartographic Information Systems - page 727

ω 2 :=

:= 1

2 ω ij dy i

dy j = ∧ (E.31)

d q μ =d p 1 ∧ d q 1 +d p 2 ∧ d q 2 . =d p μ ∧

E-32 The Maupertuis Gauge and the Newton Portrait of a Geodesic

How can we gauge dynamic time into the system of differential equations for a geodesic? According
to De Maupertuis ( 1744 ) elaborated by Jacobi ( 1866a ), let us represent the arc length d s according
to ( E.32 ) as the product of the factor of conformality λ 2 and the time differential d t or, equiv-
alently, according to ( E.33 ), identifying the factor of conformality as the kinetic energy ( E.34 ).
Therefore, we introduce dynamic time into the one-dimensional submanifold q ( τ ) → q ( t ), the

minimal geodesic in the two-dimensional Riemann manifold.

d s := λ 2 ( q 1 ,q 2 )d t, (E.32)

λ 4 ( q 1 ,q 2 ):=

:= d s 2

d t 2 = d q μ d t g μν d q ν

d t ∀g μν = λ 2 ( q 1 ,q 2 ) δ μν ,
λ 2 ( q 1 ,q 2 ) := (E.33)

d t = d q 1

2

+ d q 2
d t

2

:= d s ,

d t

2 T := d q 1
d t

2

+ d q 2
d t

2

. (E.34)

E-321 Lagrange Portrait

The Laqrangean L 2 ( q, d q/ d τ ) is transformed into the Lagrangean L 2 ( q, d q/ d t ) subject to the
metric g μν = λ 2 ( q 1 ,q 2 ) δ μν and λ 2 =( q 1 ) 2 +( q 2 ) 2 , where the dot differentiation is defined with
respect to the dynamic time t . We here note in passing that the prime differentiation is defined
with respect to the ane parameter arc length s .

L 2 ( q ( t ) , q ( t )) = 1

2 δ μν q μ q ν + 1 2 λ 2 ( q 1 ,q 2 ) ,

(E.35)