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.¹

In [5, 2] it is shown that the mean sun satisfies the functional equation

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 2) 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(, ) is the matrix

Then M(, )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 and latitude . This local coordinate frame is given by the unit vectors indicating the directions East, North, and to the zenith.

In the present paper we investigate a generalization of this equation for fixed . 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

(1)

which satisfies 1x = x and (₂₁)x = ₂(₁x) for all x X and ₁, ₂ G, where 1 is the identity element in G. In other words, a group action describes a homomorphism from G to the set of all bijections on X, which is a group together with the composition of functions as the multiplication in the group. Conversely, each homomorphism of this kind determines a group action of G on X.

A group action of G on X defines an equivalence relation on X. Two elements x₁, x₂ of x are called G-equivalent, x₁ ~ _Gx₂, if and only if there is some G such that x₂ = x₁. The equivalence classes G(x) with respect to ~ _G are called orbits of G on X, i.e.

For each x X the stabilizer G_x of x, which is the set

is a subgroup of G.

Two elements x and y of the same orbit have conjugate stabilizers. To be more precise, if y = x for G, then G_y = G_x^-1.

Coming back to the functional equation (1), we start with collecting 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

The properties of h are described in

Lemma 1. Assume that (g, x) is a solution of (1). Then the function h, defined above, satisfies h(0) = 1 G,

(2)

and

(3)

Proof.  The first statement is clear from the definition of h. Then h(s)x(u) = g(0)^-1g(s)x(u) = g(0)^-1g(0 + s)x(u) = g(0)^-1g(0)x(s + u) = x(s + u) for all s, u R. Finally h(s + t)x(u) = g(0)^-1g(s + t)x(u) = g(0)^-1g(s)x(t + u) = h(s)h(t)x(u) for all s, t, u R.       

It is also possible to determine g by h.

Proof.  g(s + t)x(u) = g₀h(s + t)x(u) = g₀x(s + t + u) = g₀h(s)x(t + u) = g(s)x(t + u).       

For the rest of the paper we will work with h instead of g. As was indicated earlier, for x X let G_x denote the stabilizer of x. From (3) we deduce that ^-1h(s + t) G _x(u) for all u R and all s, t R. In other words

Using this for t = -s, we see that there exists _s , such that h(s)^-1 = h(-s) _s. And for s = -t there is _t' , such that h(t)^-1 = _t'h(-t).

Let H := <h(R)> and := H, then the following lemma holds.

Proof.  It is clear that is a subgroup of H. We only have to prove that it is a normal subgroup. From the definition of H we know that

So it is enough to prove that h(r)h(r)^-1 < and h(r)^-1h(r) < for all r R. Let r, u R and , then there is a _r' , such that h(r)h(r)^-1x(u) = h(r)_r'h(-r)x(u) = h(r) _r'x(-r + u) = h(r)x(-r + u) = x(r - r + u) = x(u), since _r' stabilizes each element of the form x(t). This means, since was an arbitrary element of , that

so h(r)h(r)^-1 < H = . For the second part of the proof similar arguments can be used.       

This permits to define a function from R to the factor group H/ by

Proof.  For s, t R we know from (3) that h(s)h(t) h(s + t). So

In order to prove that is surjective let

Then

and _{i = 1}ⁿj_i ^. r_i R.       

Even the following result is true.

Proof.  It is enough to prove that N is a subgroup of G_x(u) for all u R, because then N is a subgroup of . By assumption N < H, so N < . From (2) it is clear that x(u) = h(u)x(0) for all u 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). Before describing sufficient conditions we prove a general result about group actions.

Lemma 6. Consider a group G acting on a set X. Let S be a subgroup of G, x₀ an arbitrary element of X, and N a normal subgroup of S, such that N is a subgroup of the stabilizer G_x0. Then the factor group S/N acts on the orbit S(x₀) in the following way:

(4)

Proof.  In order to prove that this action is well defined, consider an arbitrary N. Since N is a normal subgroup of S, for each S there exists ' N, such that = '. From

we derive that the action of on H(x₀) does not depend on the special choice of the representative of . Furthermore, it is clear that x₀ = 1x₀ = x₀, and ( _{1 2})x₀ = ₁₂x₀ = (₁₂)x₀ = ₁(₂)x₀ = ₁(₂x₀) = ₁( ₂x₀) for all ₁, ₂ S/N.       

Under the assumptions of the last lemma, N is a subgroup of each stabilizer G_x0 for S, since

Proof.  h(s)x(u) = (s)(u)x₀ = (s + u)x₀ = x(s + u) for all s, u R.       

Finally all these results are summarized in

Theorem 8. The functions x : R X and h : R G satisfy (2) if and only if there exist x₀ X, a subgroup S of G, a normal subgroup N of S, which is a subgroup of the stabilizer G_x0, and a homomorphism : R S/N, such that

where the natural action of the factor group S/R on the orbit S(x₀) is described by (4).

Proof.  If x and h satisfy (2), then choose x₀ = x(0). Moreover, S = H, N = , and = satisfy the above conditions by construction, Lemma 3 and Lemma 4. Finally, by construction it is clear that h(r) (r), and from (2) it follows that (r)x₀ = (r)x(0) = x(r) for all r R.

The rest of the present theorem follows immediately from Lemma 7.       

Acknowledgement: The author wants to express his thanks to Professor Jens Schwaiger for useful comments and hints while preparing this article.

References