Mass in the Hyperbolic Plane

Archimedes computed the center of mass of several regions and bodies [Dijksterhuis], and this fundamental physical notion may very well be due to him. He based his investigations of this concept on the notion of moment as it is used in his Law of the Lever. A hyperbolic version of this law was formulated in the nineteenth century leading to the notion of a hyperbolic center of mass of two point-masses [Andrade, Bonola]. In 1987 Galperin proposed an axiomatic definition of the center of mass of finite systems of point-masses in Euclidean, hyperbolic and elliptic n-dimensional spaces and proved its uniqueness. His proof is based on Minkowskian, or relativistic, models and evades the issue of moment. A surprising aspect of this work is that hyperbolic mass is not additive. Ungar [2004] used the theory of gyrogroups to show that in hyperbolic geometry the center of mass of three point-masses of equal mass coincides with the point of intersection of the medians. Some information regarding the centroids of finite point sets in spherical spaces can be found in [Fog, Fabricius-Bjerre].

In this article we offer a physical motivation for the hyperbolic Law of the Lever and go on to provide a model-free definition and development of the notions of center of mass, moment, balance and mass of finite point-mass systems in hyperbolic geometry. All these notions are then extended to linear sets and laminae. Not surprisingly, the center of mass of the uniformly dense hyperbolic triangle coincides with the intersection of the triangle’s medians. However, it is pleasing that a hyperbolic analog of Archimedes’s mechanical method can be brought to bear on this problem. The masses of uniform disks and regular polygons are computed in the Gauss model and these formulas are very surprising. Other configurations are examined as well.

For general information regarding the hyperbolic plane the reader is referred to [Greenberg, Stahl] 2 THE HYPERBOLIC LAW OF THE LEVER Many hyperbolic formulas can be obtained from their Euclidean analogs by the mere replacement of a length d by sinh d. The Law of Sines and the Theorems of Menelaus and Ceva (see Appendix) are cases in point. It therefore would make sense that for a lever in the hyperbolic plane a suitable definition of the moment of a force w acting perpendicularly at distance d from the fulcrum is w sinh d

Nevertheless, a more physical motivation is in order. We begin with an examination of the balanced weightless lever of Figure 1. This lever is pivoted at E and has masses of weights w 1 and w 2 at A and B respectively. By this is meant that there is a mass D, off the lever, which exerts attractive forces w 1 and w 2 along the straight lines AD and BD. Since this system is assumed to be in equilibrium, it follows that the resultant of the forces w 1 and w 2 acts along the straight line ED. Neither the direction nor the intensity of the resultant are affected by the addition of a pair of equal but opposite forces f 1 and f 2 at A and B. (Here and below we employ the convention that the magnitude of the vector v is denoted by v.) We assume that the common magnitude of f 1 and f 2 is large enough so that the lines of direction of the partial resultants r i = f i + w i , i = 1,2, intersect in some point, say C. Note that the quadrilateral ACBD lies in the hyperbolic plane whereas the parallelograms of forces at A and B lie in the respective Euclidean tangent planes. This is the standard operating procedure in mathematical physics.

It is now demonstrated that such a system in equilibrium must satisfy the equation

where each F i is the component of w i in the direction orthogonal to AB. Indeed, it follows from several applications of both the Euclidean and the hyperbolic Laws of Sines that

and Eq’n (1) follows by cross-multiplication. If we take the mass at D out of the picture and stipulate that F 1 and F 2 are simply two forces that act perpendicularly to the lever AB (Fig. 2) then it is makes sense to regard the quantities

as the respective moments of the forces F 1 and F 2 with respect to the pivot point E. This facilitates the derivation of the resultant of F 1 and F 2 . Suppose c 1 , c 2 and F 3 ⊥ AB are such that

Then the moments of F 3 with respect to A and B are, respectively

Since the right hand sides of these two equations, are, respectively, the moments of F 2 with respect to A and the moment of F 1 with respect to B, it follows that the equations of (2) do indeed imply equilibrium. Consequently, the reverse of F 3 is indeed the resultant of F 1 and F 2 .

The physical considerations of the previous section motivate the following formal definitions. A point-mass is an ordered pair (X, x) where its location X is a point of the hyperbolic plane and its weight x is a positive real number. The (unsigned) moment of the point-mass (X, x) with respect to the point N or the straight line n is, respectively,

where d(X, N ) and d(X, n) are the respective hyperbolic distances fro

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