E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) 551
E-322 Hamilton portrait
It should not be too surprising that by introducing the Maupertuis gauge, in particular dynamic time
t, we are led to familiar Hamiltonian equations. The Hamiltonian H
2 (q(τ ), p(τ )) is transformed into
the Hamiltonian H
2 (q(t), p(t)) subject to the metric g µν = λ
2 (q
1 , q
2 )δ µν and λ
2 = ( ˙
q
1 )
2 + ( ˙
q
2 )
2 .
H
2 (q(t), p(t)) =
=
1
2
δ
µν p µ p ν −
1
2
λ
2 (q
1 , q
2 ) .
(E.38)
(E.38) used as the input to minimize the functional leads to Corollary E.2.
Corollary E.2 (Newton portrait of a Maupertuis gauged geodesic in phase space).
dq
µ
dt
=
∂H
2
∂p µ
= δ
µν p ν ,
(E.39)
dp µ
dt
= −
∂H
2
∂q µ =
1
2
∂
∂q µ λ
2 (q
1 , q
2 ) .
(E.40)
End of Corollary.
E-33 A geodesic as a submanifold of the sphere (conformal coordinates)
Let us represent the differential equations of a geodesic as a submanifold of the sphere M
2 := S
2
R as
functions q
µ (t) as well as the Lagrangean L
2 (q(t), ˙
q(t)) and the Hamiltonian H
2 (q(t), p(t)) with respect
to conformal coordinates (isothermal, isometric) of the most important applicable map projections
listed below. Compare with Figs. E.1–E.4. The related computer-graphical illustrations of Maupertuis
gauged geodesics document the elegance of those submanifolds in S
2
R parameterized in dynamic time.
Important!
(i) The Universal Polar Stereographic Projection (UPS) (central perspective projection from
the South Pole to a tangent plane T np M
2 at the North Pole), (ii) the Universal Mercator
Projection (UM) (conformal diffeomorphism of the sphere onto a circular cylinder S
1
R × R
where the circle S
1
R is chosen as the equator of the sphere), (iii) the Universal Transverse
Mercator Projection (UTM) (conformal diffeomorphism of the sphere onto a circular cylinder
S
1
R × R where the circle S
1
R is chosen as a definite meridian of the sphere), and (iv) the
Universal Conic Lambert Projection (UC) (conformal diffeomorphism of the sphere onto a
circular cone {x ∈ R
3 (x
2 + y
2 )/a
2
− z
2 /b
2 = 0, a
2 + b
2 = R
2
} where the circle S
1
a is chosen
as a definite parallel circle (line-of-contact) and b as the definite distance of the circle S
1
a
from the equatorial plane).
Boxes E.1–E.3 contain (i) the conformal coordinates as a function of spherical longitude and spherical latitude, especially the factor of conformality, (ii) the Lagrangean version versus the Hamiltonian
version of a geodesic in S
2
R in terms of conformal coordinates (isometric coordinates) and Maupertuis gauge, and (iii) the differential equations of a geodesic in S
2
R in terms of conformal coordinates
(isometric coordinates) and Maupertuis gauge: Lagrange portrait, two differential equations of second
order, Hamilton portrait, four differential equations of first order.
E-322 Hamilton portrait
It should not be too surprising that by introducing the Maupertuis gauge, in particular dynamic time
t, we are led to familiar Hamiltonian equations. The Hamiltonian H
2 (q(τ ), p(τ )) is transformed into
the Hamiltonian H
2 (q(t), p(t)) subject to the metric g µν = λ
2 (q
1 , q
2 )δ µν and λ
2 = ( ˙
q
1 )
2 + ( ˙
q
2 )
2 .
H
2 (q(t), p(t)) =
=
1
2
δ
µν p µ p ν −
1
2
λ
2 (q
1 , q
2 ) .
(E.38)
(E.38) used as the input to minimize the functional leads to Corollary E.2.
Corollary E.2 (Newton portrait of a Maupertuis gauged geodesic in phase space).
dq
µ
dt
=
∂H
2
∂p µ
= δ
µν p ν ,
(E.39)
dp µ
dt
= −
∂H
2
∂q µ =
1
2
∂
∂q µ λ
2 (q
1 , q
2 ) .
(E.40)
End of Corollary.
E-33 A geodesic as a submanifold of the sphere (conformal coordinates)
Let us represent the differential equations of a geodesic as a submanifold of the sphere M
2 := S
2
R as
functions q
µ (t) as well as the Lagrangean L
2 (q(t), ˙
q(t)) and the Hamiltonian H
2 (q(t), p(t)) with respect
to conformal coordinates (isothermal, isometric) of the most important applicable map projections
listed below. Compare with Figs. E.1–E.4. The related computer-graphical illustrations of Maupertuis
gauged geodesics document the elegance of those submanifolds in S
2
R parameterized in dynamic time.
Important!
(i) The Universal Polar Stereographic Projection (UPS) (central perspective projection from
the South Pole to a tangent plane T np M
2 at the North Pole), (ii) the Universal Mercator
Projection (UM) (conformal diffeomorphism of the sphere onto a circular cylinder S
1
R × R
where the circle S
1
R is chosen as the equator of the sphere), (iii) the Universal Transverse
Mercator Projection (UTM) (conformal diffeomorphism of the sphere onto a circular cylinder
S
1
R × R where the circle S
1
R is chosen as a definite meridian of the sphere), and (iv) the
Universal Conic Lambert Projection (UC) (conformal diffeomorphism of the sphere onto a
circular cone {x ∈ R
3 (x
2 + y
2 )/a
2
− z
2 /b
2 = 0, a
2 + b
2 = R
2
} where the circle S
1
a is chosen
as a definite parallel circle (line-of-contact) and b as the definite distance of the circle S
1
a
from the equatorial plane).
Boxes E.1–E.3 contain (i) the conformal coordinates as a function of spherical longitude and spherical latitude, especially the factor of conformality, (ii) the Lagrangean version versus the Hamiltonian
version of a geodesic in S
2
R in terms of conformal coordinates (isometric coordinates) and Maupertuis gauge, and (iii) the differential equations of a geodesic in S
2
R in terms of conformal coordinates
(isometric coordinates) and Maupertuis gauge: Lagrange portrait, two differential equations of second
order, Hamilton portrait, four differential equations of first order.