1. Introduction
The center of mass (center of gravity or centroid) is a fundamental concept, and its geometrical and mechanical properties are for understanding many physical phenomena. Its definition for Euclidean spaces is elemental; nevertheless, in spaces with the non-zero curvature, it is rare. In [
1], the author gives an extensive explanation showing that the possibility of the concept can be correctly defined in more general spaces, and he signalizes the difficulties in defining spaces of non-zero curvature concerning the lack of the linear structure of ones. While it is true that the author synthesizes the basic properties of the center of mass, in his approach there are some entities without physical meaning, such as the non-conservation of the total mass of the system or the presence of unbounded speeds under normal conditions. In [
2], there is a definition of center of mass for two particles in hyperbolic space, in the same direction to the one presented here, but the authors do not give an expression for calculating it. In [
3], the author mentions the difficulty of defining the center of mass in curved spaces. He provides a class of orbits in the curved
n-body problem for which “no point that could play the role of the center of mass is fixed or moves uniformly along a geodesic”. This proves that the equations of motion lack center-of-mass and linear momentum integrals. Nevertheless, he does not provide a way to calculate or determine this element. In [
4], the center of the mass problem on two-point homogeneous spaces and the connection of existing mass center concepts with the two-body Hamiltonian functions are considered. We discussed different possibilities for defining a center of the mass in spaces of constant and non-zero curvature, and it was established that a natural way of defining a concept of center of mass for two particles on a Riemannian space as the point on the shortest geodesic interval joining these particles that divides the interval in the ratio of the masses of particles and this is denoted by
.
This last approach is followed in the present work.
In this article, the problem of finding a mathematical expression for computing the center of mass of a system of
n particles sited on the two-dimensional hyperbolic sphere
is considered. The stereographic projection of the upper sheet of
on the Poincaré disk
, endowed with the conformal metric (see [
5]).
Both
with the metric (
1) and
with the Euclidean metric have the same Gaussian curvature
, and for the Minding’s Theorem they belong to the isometric differentiable class (see [
6,
7], chapter 2). In [
8], we use the lever law, an explicit formula that allows us to calculate the center of mass of a system of
n particles with masses
located on the superior half plane of Lobachevsky
, endowed with a conformal metric which induces a constant and negative Gaussian curvature.
Following the basic geometry methods, we obtain the expression for the center of mass for a system of n particles sited in the hyperbolic sphere with arbitrary R.
We organized this article as follows: In
Section 1 we introduced some concepts relative to the center of mass in the Euclidean spaces. In
Section 2, some properties of stereographical projection are remembered, and we proceeded to deduce the expression for the center of mass, for two particles on the upper branch of hyperbola, from the “hyperbolic rule of the lever” (see [
1,
4]) extended to the surface of
. After obtaining the expression for the center of mass for two particles in
, we naturally extended to a system of
n particles in
, and in the same way, to a system of
n particles in
.
4. An Application to the Curved 2-Body Problem
In [
5], the curved
n-body problem in a two-dimensional space with constant negative curvature is studied, and the model
is considered, in which there are systems. Let
be the configuration of
n point particles with masses
where
is the conformal function of the Riemannian metric,
We consider functions of the form
where
is a solution of Equation (
10). Straightforward computations show that
For the configurations called relative equilibrium, concerning the 2-body problem, the next result is established.
Theorem 2. Consider two point particles of masses moving on the Poincaré disk , whose center is the origin, 0, of the coordinate system. Then is an elliptic relative equilibrium of system (10) with if and only if, for every circle centered at 0 of radius α, with , along which moves, there is a unique circle centered at 0 of radius r, which satisfies , along which moves, such that, at every time instant, and are on some diameter of , with 0 between them. Moreover, if and α are given, then ;
if and α are given, then ;
if and α are given, then .
This result was reformulated in a more precise form, using the expression for the hyperbolic center of mass taking into account that in a configuration corresponding to a relative equilibrium is invariant with the time, because the distance and angles between particles do not change. This is sufficient, considering the initial configuration on the
x-axis, and
corresponds to the length measure over the Poincaré disk of the projection of arc
over the hyperbolic sphere
, and
r is the projection length in disk one of the arc
over the hyperbolic sphere. Then we have the next relations:
and
Substituting in the hyperbolic rule
and the expression for the center of mass, we obtain
Thus it follows from Equation (
6) that
It follows that and so the center of mass is fixed for every time in the South Pole of the hyperbolic sphere , and Theorem 2 can be expressed in the following form.
Theorem 3. For every configuration of elliptic relative equilibrium for the 2-body problem with masses sited in the points and on the hyperbolic sphere of radius R. If and are the lengths of arcs measured from the South Pole to the points and , respectively, then it satisfies the relation , and the center of mass of the system is fixed in for every time.
The hyperbolic spaces are very special in relativity. A hyperbolic (i.e., Lobachevskian) space can be represented upon one sheet of a two-sheeted cylindrical hyperboloid in Minkowski space–time. According to works that recently appeared in literature (see [
9]), in hyperbolic spaces, the expression for the center of mass obtained by adopting the relativistic rule of lever reads
with
,
denoting the Riemannian distance of
to the center of mass and
k the (negative) Gaussian curvature, respectively. Using the stereographic projection of a hyperbolic sphere on the Poincaré disk.
For both the Euclidean and the hyperbolic spaces, the center of mass for the system particles plays a central role in the conserved momentum principle. Adoption of the conserved momentum principle for 2-body is expressed in spaces with negative Gaussian curvature is along the following lines.
Theorem 4. Consider two masses sited in the points , , respectively, and , the length or arc from to ; then, from the relation (hyperbolic rule of the lever) and using the stereographic projection of a hyperbolic sphere on the Poincaré disk the conserved momentum principle for 2-body expressed in spaces with negative Gaussian curvature is Proof. Following the ideas from [
1] on relativistic momentum, we have
where
p is the momentum,
v is the velocity of the particle with respect to a frame of reference and the velocity of the light is
We take
the distance of the particle with respect to the center of the reference frame. This solution with respect to
v yields
Evaluating this in the momentum gives
From the energy of the particle
and replacing the velocity of the particle
gets
From the hyperbolic identity
, the constant
can be obtained. Let
be the distance between the particle with mass
and the mass center, and
the distance between the particle with mass
and the mass center; then,
and
are the velocities of particles with mass
and
, respectively. In consequence, the relativistic momentum is constant,
□
The expression of the center of mass for a system of two particles in is presented in the following result.
Theorem 5. Consider two masses sited in the points , , respectively, and , the length or arc from to ; then, from the relation (hyperbolyc rule of the lever) and using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the center of mass for a system of two particles in is given bywhere k is the (negative) Gaussian curvature Proof. From the principle of conservation of relativistic momentum and total energy we obtain the desired result. Following the ideas from [
1],
with
□
One of the most interesting aspects concerning the determination of the center of mass of a particle system lies in its physical applications. As known, for Euclidean space, Equations (
2) may be derived from the Lever rule. If we suppose that the particles are under the influence of an attractive potential force, depending only on their mutual distance, this equation may be derived from the other two different characteristics of the center of mass: (a) Collision point and (b) center of steady rotation.
In situation (a), for a collisional point, if the particles are initially at rest they will collide at the center of mass; the expression of the centre of mass is along the following lines.
Theorem 6. Consider two masses sited in the points , , respectively, and , the length or arc from to ; then, from the relation (hyperbolyc rule of the lever) and using the stereographic projection of a hyperbolic sphere on the Poincaré disk the center of mass for a system of two particles in is given by Proof. Following the ideas from [
9] we obtain our result. □
In situation (b), for the center of steady rotation, if the particles rotate uniformly along with concentric circles, maintaining a constant distance over time, then the center of mass coincides with the circle’s center, the expression of the center of mass is given by the following.
Theorem 7. Consider two masses sited in the points , , respectively, and , the length or arc from to ; then, from the relation (hyperbolyc rule of the lever) and using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the center of mass for a system of two particles in is given by Proof. Following the ideas of [
2,
9], we can generalize, using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the center of mass for a system of two particles in
our idea and so obtain the result. □
Remark 2. It has been established that if the particles have distinct masses, then the above definitions of the center of mass are not equivalent for hyperbolic spaces. Similarly, using the stereographic projection of a hyperbolic sphere on the Poincaré disk, the three meanings for the center of mass (lever rule, collision point and center of steady rotation) are not equivalent. We consider that, from the physical point of view, the most appropriate definition is the definition present here, because it inherits two properties of the Euclidean center of mass (lever rule and collision point), while the relativistic definition only preserves one (conservation of angular momentum).