Geodesics - Map Projections: Cartographic Information Systems - page 726

d q μ
d τ

= ∂H 2

1
λ 2 ( q 1 ,q 2 ) p μ , ∂p μ =

(E.27)

d p μ
d τ

( p 1 + p 2 )
2

∂
∂q μ

1

= −

λ 2 ( q 1 ,q 2 ) .

Note that the Hamilton equations as a system of four differential equations of first order in
the variable {q ( τ ) ,p ( τ ) } do not appear in the form we are used to from mechanics, a result
caused by the effect that no dynamical time has been introduced. Equation ( E.27 ) can also be
directly derived by ( E.21 ) as soon as the system of two differential equations of second order in
the variable q μ ( τ ) is transformed by means of ( E.22 ) into a system of four differential equations
of first order in the “state variable” {q ( τ ) ,p ( τ ) } . Furthermore, note that a condensed form of
the Hamilton equations is achieved as soon as we introduce the antisymmetric metric tensor
(symplectic tensor) ( E.28 ).

Ω :=

= 0 μν + δ μν

= 0+ 2 ,

ω ij := { }

−δ μν 0 μν − I 2 0

Ω − 1 :=

(E.28)

:= {ω ij } = 0 μν −δ μν

= 0 − I 2 .

+ δ μν 0 μν +I 2 0

In terms of the four-vector ( E.29 ) (state vector) being an element of the phase space, the Hamilton
equations in their contravariant and covariant form, respectively, are given by ( E.30 ). In particular,
the phase space is equipped with the metric ( E.31 ).

⎡ ⎤

q 1
q 2
p 1
p 2

⎣ ⎦

y := , (E.29)

dy i
d τ −

ω ij ∂H 2
∂y j

=0 ∀ i, j ∈{ 1 , 2 , 3 , 4 } ,

(E.30)

ω ij dy i

∂H 2
∂ y j d τ −

=0 ∀ i, j ∈{ 1 , 2 , 3 , 4 } ,