E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) 547
Important!
{M
2 , g µν }: orientable, star-shaped, second order H¨ older continuous, compact.
Thus, we have excluded corners, edges, and self-intersections. {M
2 , g µν } is totally covered by a set of
charts which form by their union a complete atlas. Any chart is an open subset of a two-dimensional
Euclidean manifold E
2 := {R
2 , δ µν } with standard canonical metric I = {δ µν }, the unit matrix.
Indeed, we shall deal only with topographic surfaces which are assumed to be topologically similar to
the sphere. Thus, a minimal atlas of {M
2 , g µν } is established by two charts, for example, as described
by J. Engels and E. Grafarend (1995), based upon the following coordinates.
Important!
Quasi-spherical coordinates. Meta-quasi-spherical coordinates.
According to a standard theorem, for example, S. S. Chern (1955) applied to two-dimensional Riemann manifolds, conformal (isothermal, isometric) coordinates {q
1 , q
2
} always exist. They establish
a conformal diffeomorphism {M
2 , g µν } → {R
2 , δ µν } which is angle preserving. In the following, we
shall adopt {M
2 , g µν } to be conformally flat, in particular g µν = λ
2 (q
1 , q
2 )δ µν , where λ
2 (q
1 , q
2 ) is
the factor of conformality. The infinitesimal distance between two points (q, q + dq) in {M
2 , λ
2 δ µν }
can correspondingly be represented by (E.17) and (E.18). Obviously, dq
µ is an element of the tangent
space T q M
2 at q, while g µν dq
ν is an element of the cotangent space
∗ T q M
2 , which is the dual space
of T q M
2 := {R
2 , G q }.
ds
2 = dq
µ g µν dq
ν
∀ g µν = λ
2 (q
1 , q
2 )δ µν ,
dq
µ
∈ T q M
2 , g µν dq
ν
∈
∗ T q M
2 ,
(E.17)
ds
2 = λ
2 (q
1 , q
2 )
(dq
1 )
2 + (dq
2 )
2
.
(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 Fr´ echet
derivative) and (β) positive second variation (positive second Fr´ echet derivative).
L
q,
dq
dτ
dτ = min or
L
2
q,
dq
dτ
dτ = min (fixed boundary points) , (E.19)
2L
2
q,
dq
dτ
:=
ds
2
dτ 2 =
dq
µ
dτ
g µν
dq
ν
dτ
∀ g µν = λ
2 (q
1 , q
2 )δ µν ,
2L
2
q,
dq
dτ
:= λ
2 (q
1 , q
2 )
dq
1
dτ
2
+
dq
2
dτ
2
.
(E.20)
Important!
{M
2 , g µν }: orientable, star-shaped, second order H¨ older continuous, compact.
Thus, we have excluded corners, edges, and self-intersections. {M
2 , g µν } is totally covered by a set of
charts which form by their union a complete atlas. Any chart is an open subset of a two-dimensional
Euclidean manifold E
2 := {R
2 , δ µν } with standard canonical metric I = {δ µν }, the unit matrix.
Indeed, we shall deal only with topographic surfaces which are assumed to be topologically similar to
the sphere. Thus, a minimal atlas of {M
2 , g µν } is established by two charts, for example, as described
by J. Engels and E. Grafarend (1995), based upon the following coordinates.
Important!
Quasi-spherical coordinates. Meta-quasi-spherical coordinates.
According to a standard theorem, for example, S. S. Chern (1955) applied to two-dimensional Riemann manifolds, conformal (isothermal, isometric) coordinates {q
1 , q
2
} always exist. They establish
a conformal diffeomorphism {M
2 , g µν } → {R
2 , δ µν } which is angle preserving. In the following, we
shall adopt {M
2 , g µν } to be conformally flat, in particular g µν = λ
2 (q
1 , q
2 )δ µν , where λ
2 (q
1 , q
2 ) is
the factor of conformality. The infinitesimal distance between two points (q, q + dq) in {M
2 , λ
2 δ µν }
can correspondingly be represented by (E.17) and (E.18). Obviously, dq
µ is an element of the tangent
space T q M
2 at q, while g µν dq
ν is an element of the cotangent space
∗ T q M
2 , which is the dual space
of T q M
2 := {R
2 , G q }.
ds
2 = dq
µ g µν dq
ν
∀ g µν = λ
2 (q
1 , q
2 )δ µν ,
dq
µ
∈ T q M
2 , g µν dq
ν
∈
∗ T q M
2 ,
(E.17)
ds
2 = λ
2 (q
1 , q
2 )
(dq
1 )
2 + (dq
2 )
2
.
(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 Fr´ echet
derivative) and (β) positive second variation (positive second Fr´ echet derivative).
L
q,
dq
dτ
dτ = min or
L
2
q,
dq
dτ
dτ = min (fixed boundary points) , (E.19)
2L
2
q,
dq
dτ
:=
ds
2
dτ 2 =
dq
µ
dτ
g µν
dq
ν
dτ
∀ g µν = λ
2 (q
1 , q
2 )δ µν ,
2L
2
q,
dq
dτ
:= λ
2 (q
1 , q
2 )
dq
1
dτ
2
+
dq
2
dτ
2
.
(E.20)