Natural microstructures 9
for any t . Since the Heisenberg algebra bundle evolves from G
n
we may ask how α
n
evolves along σ , in particular for α
1
. The evolution of α
n
can be expressed in terms of ˙α
n
defined by
˙α
n
σ t
; h := d
dt α
n
σ t
; h − α
n
˙σ t, h =
σ t
× σ
n +1
t , h ∀ h ∈ F
n σ
t
. A slightly more informative form for
˙α
1
is ˙α
1
σ t
; h = ω
2
σ t
; σ t, h ∀ h ∈ F
1 σ
t
. Thus the evolution of α
1
along σ is governed by the Heisenberg algebras G
2
, yielding in particular
α
1
= const. iff
σ × ¨σ = 0,
meaning i
σ
ω
2
= 0. Hence α
1
= const. is the analogue of Keppler’s second law. In this case the quaternion b := σ
× ˙σ is constant and hence σ is in the plane F ⊂ E perpendicular to b. Thus R · b × F
b
is a Heisenberg algebra with
ω
b
h
1
, h
2
: = h
1
× b, h
2
∀ h
1
, h
2
∈ F
b
as symplectic form on F
a
. Hence the planetary motion can be described in only one Heisenberg algebra, namely in G
b
, which is caused by the angular momentum b, of course. We have ¨σ =
f · σ for some smooth real-valued function f defined along a planetary motion σ , implying
ω
2
= f ·
|σ |
2
¯ m
· ω
a
. In case σ is a circle, f is identical with the constant map with value
¯ m
|σ |
2
, due to the third Kepplerian law cf. equation 7. This motivates us to set
G
2 σ
t
= G
a σ
t
∀t 8
along any closed planetary motion σ which hence implies ω
2
= ω
a
along σ . In turn one obtains
¨σ t = grad V
sol
σ t
∀ t, 9
a well-known equation from Newton implying Keppler’s laws. Equation 9 is derived from a holographic principle in the sense that equation 8 states that the oriented circle of P
2 σ
t
matches the oriented circle of P
a σ
t
at any t .
6. Horizontal and periodic lifts of β
Since, in general,
a
6= 0, the horizontal distribution in T P
a
does not need to be integrable along level surfaces. However,
a
vanishes along field lines and thus the horizontal distribution is integrable along these curves. Let us look at P
a
|
β
where β is a field line of the singularity free vector field X .
A horizontal lift of ˙ β
is a curve ˙ β
hor
in H or
β
= ker α
a
which satisfies T pr
a
˙ β
hor
= ˙ β
and obeys an initial condition in T P
a
|
β
. Hence there is a unique curve β
hor
passing through v
β t
∈ P
a β
t
, say, called horizontal lift of β. In the case of a constant vector field or in the
10 E. Binz - S. Pods - W. Schempp
case of the solar field this is nothing else but a meridian of the cylinder respectively the cone P
a
|
β
containing v
β t
. Let βt = x for a fixed x ∈ O.
Obviously, a horizontal lift is a geodesic on P
a
|
β
equipped with the metric g
H or
β
, say, induced by the scalar product , on E .
At first let a be a non-vanishing constant. A curve γ on P
a
|
β
here is called a periodic lift of β through v
x
iff it is of the form γ
s = β
hor
s · e
p ·s·
a |a|
∈ P
a β
s
∀ s where p is a fixed real.
Clearly, γ is a horizontal lift through v
x
iff γ = β
hor
, i.e. iff p = 0. In fact any periodic
lift γ of β is a geodesic on P
a
|
β
. Hence ¨γ is perpendicular to P
a
|
β
. Due to the U 1-symmetry of P
a
|
β
, a geodesic σ on P
a
|
β
is of the form σ
s = β
hor
θ · s · e
p ·θ·s·
a |a|
∀ s as it is easily verified. Here p and θ denote reals. θ determines the speed of the geodesic. Thus
σ and β have accordant speeds if θ
= 1 which will be assumed from now on, as can be easily seen from
˙γ 0 = p · v
x
· a
|a| + ˙
β
hor
for t = 0. The real number p determines the spatial frequency of the periodic lift γ due to
2 ·π
T
=
p |v
x
|
. The spatial frequency of γ counts the number of revolutions around P
a
|
β
per unit time and is determined by the F
a x
-component p of the initial velocity due to the U 1-symmetry of the cylinder P
a
|
β
. We refer to p as a momentum. For the solar field X x
= x,
−
x |x|
3
with x ∈ O, let |x
| = 1 and let a parametrization of the body of revolution P
a
|
β
be given in Clairaut coordinates via the map x : U → E defined by
xu, v :
= − ¯ m
· 3v − 2
1 3
· r e
u ·
a |a|
· v
x
+ a
|a| on an open set U
⊂ R
2
. Here r is the representation of U
a |a|
1 onto S O F
a |a|
for any x ∈ O.
Then a geodesic γ on P
a
|
β
takes the form γ
s = xus, vs = − ¯
m · 3vs − 2
1 3
· r e
us ·
a |a|
· v
x
+ a
|a| where the functions u and v are determined by
us =
√ 2
· arctan s
√ 2 d
+ c
1
2 d + c
2
10 and
v s
= ± 1
3 1
√ 2
s + c
1 2
+ d
2
3 2
+ 2
3 11
cf. [12] with s in an open interval I ⊂ R containing 1. Here c
1
and c
2
are integration constants determining the initial conditions. Since we are concerned with a forward movement along the
Natural microstructures 11
channel R ·
a |a|
, only the positive sign in 11 is of interest. The constant d fixes the slope of the geodesic via
cos ϑ =
d r
1 √
2
s + c
1 2
+ d
2
where ϑ is the constant angle between the geodesic γ , called periodic lift, again, and the parallels given in Clairaut coordinates. This means that d vanishes precisely for a meridian. A periodic
lift γ is a horizontal lift of β iff γ is a meridian. Thus the parametrization of a meridian as a horizontal lift β
hor
of an integral curve β parametrized as in 5 has the form β
hor
t = − ¯
m · 3t − 2
1 3
· v
x
with β
hor
1 = − ¯
m · v
x
as well as β1 = − ¯
m · x for
2 3
≤ t 1 and any initial v
x
∈ P
a β
1
. For the constant vector field from above, any periodic lift γ of β through v
x
is uniquely determined by the U
a
1-valued map s
7→ e
p ·s·
a |a|
, while for the solar field a periodic lift is characterized by
s 7→ e
us ·
a |a|
with us as in 10. These two maps here are called an elementary periodic function respectively an elementary Clairaut map. Therefore, we can state:
P
ROPOSITION
2. Let x = β0. Under the hypothesis that a is a non-zero constant, there
is a one-to-one correspondence between all elementary periodic U
a
1-valued functions and all periodic lifts of β passing through a given v
x
∈ P
a x
. In case X is the solar field there is a one-to-one correspondence between all periodic lifts passing through a given v
x
∈ P
a x
and all elementary Clairaut maps.
An internal variable can be interpreted as a piece of information. Thus the fibres F
a x
and P
a x
can be regarded as a collection of pieces of information at x. The periodic lifts of β on P
a
|
β
describe the evolution of information of P
a
|
β
along β. This evolution can be further realized by a circular polarized wave: Let the lift rotate with frequency ν
6= 0. Then a point ws; t, say, on this rotating lift is described by
w s
; t = |v
x
| · β
hor v
x
s |β
hor v
x
s |
· e
2π νt −p·s·
a |a|
∀s, t ∈ R, s
6= 0 12
a circular polarized wave on the cylinder with
1 |p|
as speed of the phase and |v
x
| as amplitude. w
travels along R ·
a |a|
, the channel of information. Clearly, P
a
|
im β
is in O × E and not in E.
However, w could be coupled to the space E and could be a wave in E traveling along β, e.g. as an electric or magnetic field. More types of waves can be obtained by using the complex line
bundle F
a
instead of the principal bundle P
a
, of course.
12 E. Binz - S. Pods - W. Schempp
7. Representation of the Heisenberg group associated with periodic lifts of β on P
