Group actions and the functional equation of the mean sun
Harald Fripertinger^{1}
^{2}
Abstract
A generalization of the functional equation
g(s+t)x(u) = g(s)x(t+u) ("s,t,u Î R) of the mean sun is studied,
where a group G acts on a set X, (R,+) is a not necessarily commutative
group and both
x: R® X and g: R® G
are unknown functions, which will be determined by the equation.
Une généralisation de l'équation fonctionnelle g(s+t)x(u) = g(s)x(t+u)
("s,t,u Î R) du soleil moyen est examinée, où une groupe agit
sur un ensemble X, (R,+) est un groupe, mais pas nécessairement
commutatif, et x: R® X et aussi g: R® G sont des fonctions
inconnues lesquelles seront déterminées par l'équation fonctionnelle.
Local solar time is measured by a sundial. When the center of the sun is on
an observer's meridian, the observer's local solar time is zero hours (noon).
Because the earth moves with varying speed in its orbit at different times of
the year and because the plane of the earth's equator is inclined to its
orbital plane, the length of the solar day is different depending on the time
of year. It is more convenient to define time in terms of the average of local
solar time. Such time, called mean solar time, may be thought of as being
measured relative to an imaginary sun (the mean sun) that lies in the earth's
equatorial plane and about which the earth orbits with constant speed.
Every mean solar day is of the same
length.^{3}
In [5,2] it is shown
that the mean sun satisfies the functional equation
M(l+t,f)^{T} y(s) = M(l,f)^{T} y(s+t) "s,t,l Î \mathbbR, p/2 < f < p/2 

where y(s) is a vector of length 1 which is the direction from the center
of the earth to the sun at the time s (one day corresponds to 2p)
expressed in a geocentric coordinate system. As a basis of this system we
can choose two orthogonal vectors in the equatorial plane and one vector along
the axis of the earth.
M(l,f) is the matrix
Then M(l,f)y(s) is the direction from the earth to the sun
expressed in a local coordinate system on the surface of the earth in the
point of longitude l and latitude f.
In the present paper we investigate a generalization of this equation
for fixed f. To be more precise we will deal with the following problem:
Let (G,·) be a group acting on a set X (cf.
[1,3,4,6]) and let
(R,+) be a not necessarily commutative group.
Find all functions x: R® X and g: R® G which satisfy
g(s+t)x(u) = g(s)x(t+u), "s,t,u Î R. 
 (1) 
The group R is a generalization of \mathbbR, the matrices expressing the
change of the coordinate system are now elements of the group G and X
represents a generalization of the set of all vectors in \mathbbR^{3}.
To begin with we will collect some properties of the functions g and x.
Later we determine all solutions g and x of (1).
In a first step we replace g by another function h: R® G
defined by
It is easy to prove that h(0) = 1 Î G and
h(s)x(u) = x(s+u) "s,u Î R. 
 (2) 
Lemma 1
If the functions x: R® X and h: R® G satisfy
(2) then for arbitrary g_{0} Î G the function g: R®G defined by g(r): = g_{0}h(r) satisfies (1).
Proof.
g(s+t)x(u) = g_{0}h(s+t)x(u) = g_{0}x(s+t+u) = g_{0}h(s)x(t+u) = g(s)x(t+u).
^{[¯]}
Furthermore the function h satisfies
h(s+t)x(u) = h(s)h(t)x(u) "s,t,u Î R, 
 (3) 
since h(s+t)x(u) = g(0)^{1}g(s+t)x(u) = g(0)^{1}g(s)x(t+u) = h(s)h(t)x(u).
For the rest of the paper we will work with h instead of g.
For x Î X let G_{x} denote the stabilizer of x, i.e.
G_{x}: = { g Î G  gx = x} , 

which is a subgroup of G.
From (3) we deduce that
(h(s)h(t))^{1}h(s+t) Î G_{x(u)} for all u Î R and all
s,t Î R. In other words
(h(s)h(t))^{1}h(s+t) Î 
Ç
u Î R

G_{x(u)} = : 
^ G

. 

Using this for t = s we see that there exists g_{s} Î [^G] such that
h(s)^{1} = h(s)g_{s} and for s = t there is g_{t}¢ Î [^G] such that
h(t)^{1} = g_{t}¢h(t).
Let H: = áh(R)ñ and [G\tilde]: = [^G]ÇH then the following lemma
holds.
Lemma 2
The subgroup [G\tilde] of H is normal.
Proof.
It is clear that [G\tilde] is a subgroup of H. We only have to prove that
it is a normal subgroup. From the definition of H we know that
H = 
ì í
î


n Õ
i = 1

h(r_{i})^{ji}  n Î N, r_{i} Î R, j_{i} Î { 1,1} 
ü ý
þ

. 

So it is enough to prove that h(r)[G\tilde]h(r)^{1} £ [G\tilde] and
h(r)^{1}[G\tilde]h(r) £ [G\tilde] for all r Î R.
Let r,u Î R and g Î [G\tilde] then there is a g_{r}¢ Î [^G] such that
h(r)gh(r)^{1}x(u) = h(r)gg_{r}¢h(r)x(u) = h(r)gg_{r}¢x(r+u) = h(r)x(r+u) = x(rr+u) = x(u)
since gg_{r}¢ Î [^G] stabilizes each element of the form x(t). This
means, since g was an arbitrary element of [G\tilde], that
h(r) 
~ G

h(r)^{1} £ G_{x(u)} "u Î R 

so h(r)[G\tilde]h(r)^{1} £ [^G]ÇH = [G\tilde].
For the second part of the proof similar arguments can be used.
^{[¯]}
This permits to define a function j from R to the factor group
H/[G\tilde] by
j(r): = h(r) 
~ G

= : 
h(r)

. 

Lemma 3
The mapping j is a surjective group homomorphism.
Proof.
For s,t Î R we know from (3)
that h(s)h(t) Î h(s+t)[G\tilde]. So
j(s+t) = h(s+t) 
~ G

= h(s)h(t) 
~ G

= h(s) 
~ G

h(t) 
~ G

= j(s)j(t). 

In order to prove that j is surjective let
y: = 
æ è

n Õ
i = 1

h(r_{i})^{ji} 
ö ø


~ G

= 

Î H/ 
~ G

. 

Then
y = 
n Õ
i = 1


h(r_{i})^{ji}

= 
n Õ
i = 1


h(r_{i})

j_{i}

= 
n Õ
i = 1

j(r_{i})^{ji} = 
n Õ
i = 1

j(j_{i}·r_{i}) = j 
æ è

n å
i = 1

j_{i}·r_{i} 
ö ø



and å_{i = 1}^{n} j_{i}·r_{i} Î R.
^{[¯]}
Even the following result is true.
Lemma 4
If a subgroup N of G_{x(0)} is a normal subgroup of H then N is a
subgroup of [G\tilde].
Proof.
It is enough to prove that N is a subgroup of G_{x(u)} for all u Î R,
because
then N is a subgroup of [^G]. By assumption N £ H, so N £ [G\tilde].
From (2) it is clear that x(u) = h(u)x(0) for all u Î R, so
G_{x(u)} = G_{h(u)x(0)} = h(u)G_{x(0)}h(u)^{1}. Since N is a normal subgroup
of H it is obvious that N = h(u)Nh(u)^{1} £ h(u)G_{x(0)}h(u)^{1} = G_{x(u)}.
^{[¯]}
So far we derived necessary conditions for solutions of
(2).
Conversely consider a group G acting on a set X.
Let H be a subgroup of G, x_{0} an arbitrary element of X and
[G\tilde] a normal subgroup of H such that [G\tilde] is a subgroup of the
stabilizer G_{x0}. Then the factor group H/[G\tilde] acts on the orbit
H(x_{0}): = { hx_{0}  h Î H} in the following way:
H/ 
~ G

×H(x_{0})® H(x_{0}) ( 
_ h

,kx_{0})® (hk)x_{0}. 
 (4) 
In order to prove that this action is well defined consider an arbitrary
g Î [G\tilde]. Since [G\tilde] is a normal subgroup of H there exists
g¢ Î [G\tilde] such that gk = kg¢. From
(hg)kx_{0} = h(gk)x_{0} = h(kg¢)x_{0} = (hk)g¢x_{0} = (hk)x_{0} 

we derive that the action of [`h] on H(x_{0}) does not depend on the
special choice of the representative of [`h].
Furthermore it is clear that [`1]kx_{0} = 1kx_{0} = kx_{0} and ([`h]_{1} [`h]_{2})kx_{0} = [`(h_{1}h_{2})]kx_{0} = (h_{1}h_{2})kx_{0} = h_{1}(h_{2}k)x_{0} = [`h]_{1}(h_{2}kx_{0}) = [`h]_{1}([`h]_{2} kx_{0}) for all [`h]_{1},[`h]_{2} Î H/[G\tilde].
Moreover [G\tilde] is a subgroup of all the stabilizers G_{hx0} for all
h Î H since

~ G

= h 
~ G

h^{1} £ hG_{x0}h^{1} = G_{hx0}. 

Lemma 5
Let j: R® H/[G\tilde] be a homomorphism. When defining the two
functions x and h by
x(r): = j(r)x_{0}, and h(r) being an arbitrary element in
the coset j(r) for r Î R then h and x satisfy
(2).
Proof.
h(s)x(u) = j(s)j(u)x_{0} = j(s+u)x_{0} = x(s+u) for all s,u Î R.
^{[¯]}
These results are summarized in the following
Theorem 1
The functions x: R® X and h: R® G satisfy (2) if and
only if there exist x_{0} Î X, a subgroup H of G, a normal subgroup
[G\tilde] of H which is a subgroup of the stabilizer G_{x0} and a
homomorphism j: R® H/[G\tilde] such that
x(r) = j(r) x_{0} and h(r) Î j(r) "r Î R 

where the natural action of the factor group H/[G\tilde] the orbit H(x_{0})
is described by (4).
Acknowledgement: The author wants to express his thanks to Professor
Jens Schwaiger for useful comments and hints while preparing this article.
References
 [1]

P.M. Cohn.
Algebra, volume 3.
J. Wiley & Sons, Chichester etc., 2nd edition, 1989.
ISBN 0471101699.
 [2]

H. Fripertinger and J. Schwaiger.
Some applications of functional equations in astronomy.
Grazer Mathematische Berichte, 344 (2001), 16.
 [3]

S. Lang.
Algebra.
Addison Wesley, Reading, Massachusetts, 3rd edition, 1993.
ISBN 0201555409.
 [4]

K. Meyberg.
Algebra. Teil 1.
Carl Hanser Verlag, München, Wien, 2nd edition, 1980.
ISBN 3446119655.
 [5]

J. Schwaiger.
Some applications of functional equations in astronomy.
Aequationes Mathematicae, 60 (2000), p. 185.
In Report of the meeting: The Thirtyseventh International Symposium
on Functional Equations, May 1623, 1999, Huntington, WV.
 [6]

M. Suzuki.
Group Theory I.
Grundlehren der mathematischen Wissenschaften 247. Springer Verlag,
Berlin, Heidelberg, New York, 1982.
ISBN 3540109153.
 
Institut für Mathematik
Karl Franzens Universität Graz
Heinrichstr. 36/4
A8010 Graz, AUSTRIA
harald.fripertinger@kfunigraz.ac.at

 
Footnotes:
^{1}Supported by the Fonds zur Förderung
der wissenschaftlichen Forschung P14342MAT.
^{2}Mathematics Subject Classification 2000: 3902, 20A05
^{3}http://www.infoplease.com/ce6/society/A0845838.html
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