Some applications of functional equations in
The mean sun is an artificial construction in astronomy which is used to define
civil time. It has the property that local mean time, dependent on the geographical
position on the earth’s surface, behaves regularly, which is not true for the (true)
local time, which depends on the real sun. We present a functional equation which
reflects this regularity property of the mean sun and show that the solution to this
equation essentially is the mean sun. Moreover we discuss the problem of satellite
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
In mathematical terms this means the following: Let e = (e1, e2, e3) be the canonical base
of 3 and think the origin to be positioned at the center of the earth. Then, choosing the
initial time and the length of a (mean) solar day appropriately, the law of motion of the mean
sun is given by
The position of the real or of the mean sun, or rather the vector pointing from the observer’s
position to the position of the sun then can be viewed as the hand of a sundial
indicating the local time. More exactly speaking this hand is given by the coordinate
vector of this vector when expressed in the observer’s local positional frame. So
observer’s position on the surface of the earth (r the radius of the earth, the geographical
longitude and the latitude of that point). Then the local coordinate frame is given by the
unit vectors indicating the directions East, North, and to the zenith. The transformation
matrix transforming the canonical base e to the local system, denoted by e(, ) is given by
Thus, if we dispose of the earth’s radius (which is very small compared to the distance of the
earth’s origin to the position of the sun), the local sundial is given by
We say that it is noon at P (, ), if the zenith component of C(, )(xm(s)) is
Local Mean time (or local civil time) has the convenient property, that,
if it is noon at Greenwich, it is 1 p.m. at a position 15o east of Greenwich, or
Thus xm satisfies the functional equation
The solution of (3) is quite simple.
- Theorem 1. Let y : 3 be given such that y(s) /= 0 for all real s and such that (3) holds
for all , , s and some ] - /2, /2[. Let furthermore
Then there is some constant vector c 3, c /= 0, such that
Conversely we have that every y given by (5) satisfies (3).
- Proof. (3) is equivalent to
But one easily can see that
Thus (3) is equivalent to
which for s = 0 and c = y(0) gives the desired result. The remaining part is obvious.
- Remark 1. The result of this theorem says, that every object behaving like the mean
sun with respect to (3) is given by a clockwise rotation with center at some point of the
earth’s axis and rotational velocity of one full rotation per day.
Now we come to a problem for which it is important to take the earth’s radius into account.
Satellite dishes point to positions given by geostationary satellites, and thus have the desired
property of fixed positions invariant with respect to time. Thus, if y(s) is the position of the
satellite at time s, the property required is that the direction vector belonging to ,
i.e., the unit vector corresponding to this vector, expressed in local coordinates be independent
Of course we may suppose that r, the radius of the earth, equals 1. Then we get the
- Theorem 2. If for y : 3, /= 1 for all s , the expression
is independent of s for at least three different points P (i, i), i = 1, 2, 3, then y itself
is constant. Moreover every constant vector of length different from 1 satisfies (6).
- Proof. (6) implies
for some unit vector u(, ). Thus
for i = 1, 2, 3, where the gi are straight lines in 3.
If, say, u(i, i) and u(j, j) are linearly independent, the intersection of gi and gj
contains at most one point. Since y(s) gi gj this implies that y(s) equals the point
of intersection of those two lines for all s. Thus y has to be constant.
The (only) other possibility left means that the two vectors u(1, 1), u(2, 2) as
well as the vectors u(1, 1) and u(3, 3) are linearly dependent. But then all three lines
g1, g2, g3 coincide, since they have a point in common and since they have proportional
direction vectors. Thus the three different points P (1, 1), P (2, 2), P (3, 3) of the
same length are collinear, which is impossible.
Now, as we have been successful with taking into account the radius of the earth in solving
this problem, we want to reconsider, in the same spirit, the original problem. We again
suppose this radius r to be of length 1 and we suppose for y that /= 0 for all s. The
functional equation then reads as
for all , , s and all -/2 < < /2.
- Theorem 3. Assume that (7) holds for all , , s and some -/2 < < /2,
where y : 3 is such, that /= 1 for all s. Then there is some constant vector
c 3, /= 1, such that
where U is given by (4). Moreover any y of the above form solves (7) for all , , s
and all -/2 < < /2.
- Proof. Fixing ] - /2, /2[ we put c(s, ) := and
u(s, ) := /c(s, ). This means
where c(s, ) > 0 and u(s, ) is a unit vector. Then (7) and the relation at the beginning of
the proof of the first theorem imply
Using (9) with = 0 gives
Moreover P (, ) = U()P (0, ) and U( + ) = U()U(). Thus (8) reads as
We also have, by (8) for = 0
For we consider the straight line determined by y(0) and P (, ).
that any may contain at most two unit vectors, one of them being P (, ).
Thus for any fixed real t there is some such that P (, )t. Then obviously
y(0) t; and it is the only point in that intersection of the two lines and t, since
P (, )t:
On the other hand we have by (8) with replaced by - t and s replaced by t
where we also used (9) with replaced by t, s by 0 and by - t. We also have
where we used (10) with s = 0 and = t.
desired. (The second part of the theorem is obvious.)
- Remark 2. A good description on the relation between the real and the mean sun may
be found in , one may also consult . Generalizations of the questions treated here,
especially conneted with the results of the first theorem, may be found in  and .
 Blatter, Ch., Von den Keplerschen Gesetzen zu einer minutengenauen Sonnenuhr
(From Kepler’s laws to a minute-exact sundial). Elem. Math. 49, No.4(1994), 155-165.
 Fripertinger, H., The functional equation of the mean sun written as a group action.
Grazer Math. Ber. ....
 Fripertinger, H. and Reich, L., Generalizations of the functional equation of the mean
sun. Grazer Math. Ber. ....
 Schwaiger, J., Die Ringsonnenuhr -- eine mathematische Analyse. To appear in Mitt.
Math. Ges. Hamburg.
Institut für Mathematik
Institut für Mathematik