A Ramsey-Theory-Based Approach to the Dynamics of Systems of Material Points

Abstract

We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles ${\overrightarrow{p}}_i$ serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are ${\overrightarrow{p}}_{cmi}.$ If ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))\ge 0$ is true, the vectors of momenta of the particles numbered i and j are connected with a red link; if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))<0$ takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed.

1. Introduction

In this paper, we expand the Ramsey theory to the analysis of dynamic systems built of point masses. In its most general meaning, the Ramsey theory refers to any set of objects interrelated by different kinds of distinguishable connections/interrelations [1,2,3,4,5,6,7,8,9,10,11,12]. The Ramsey theory was introduced by the British mathematician, logician, and thinker Frank Plumpton Ramsey [1]. Today, it is seen as a field of combinatorics/graph theory, which deals with the specific kind of mathematical structures, namely, complete graphs [3,4,5,6]. A graph is a mathematical structure comprising a set of objects in which pairs of the objects are in some sense “related” [3,4,5,6]. A complete graph, in turn, is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge [2,3]. In this paper, we propose a procedure to enable the treatment of a dynamics problem with the tools of the graph theory.

Much progress in the field of the Ramsey theory was achieved by Paul Erdős [7,8]. The Ramsey theorem states that a structure of a given kind is guaranteed to contain a well-defined substructure. The classical problem in the Ramsey theory is the so-called “party problem”, which asks what is the minimum number of guests, denoted R(m, n), that must be invited so that at least m will know each other, or at least n will not know each other (i.e., there exists an independent set of order n [1,2,3,4,5,6,7,8,9,10,11,12]). R(m, n) is called the Ramsey number. When the Ramsey theory is reshaped in the notions of the graph theory, it states that any structure will necessarily contain an interconnected substructure [2,3,4,5,6]. The Ramsey theorem, in its graph-theoretic forms, states that one will find monochromatic cliques in any edge color labelling of a sufficiently large complete graph [2,3,4,5,6]. In the discrete problems of physics, we deal with objects/particles interacting via various forces. Thus, it seems that the Ramsey theory is well suited for physical problems. However, papers addressing Ramsey-theory-based analysis of physical systems are still scarce [13,14,15].

2. Results

2.1. Isolated System of Point Masses and Its Analysis with the Ramsey Theory

Consider an isolated system built of six interacting non-relativistic point masses $m_i, i=1,\dots ,6$, as depicted in Figure 1. The momenta of the particles are described by ${\overrightarrow{p}}_i\left(t\right)=m_i{\overrightarrow{v}}_i\left(t\right)$.

The system is isolated; thus, the conservation of the total momentum of the system, denoted as ${\overrightarrow{P}}_{tot}$ and expressed by Equation (1), takes place:

$$\sum _{i=1}^6{\overrightarrow{p}}_i={\overrightarrow{P}}_{tot}=const.$$

(1)

There exists a frame in which the total momentum is zero, and this is a frame related to the center of masses of the system, which moves with velocity ${\overrightarrow{v}}_{cm}$ [16]:

$${\overrightarrow{v}}_{cm}={\displaystyle \frac{\sum _{i=1}^6{\overrightarrow{p}}_i}{\sum _{i=1}^6{m}_i}}.$$

(2)

It is convenient to consider the motion of the particles in the frame related to the center of masses of the system (“the center mass system”). The momenta of the particles in this system are denoted as ${\overrightarrow{p}}_{cmi}$. The total momentum of the set of the interacting particles in the center mass system is zero; $\sum _{i=1}^6{\overrightarrow{p}}_i={\overrightarrow{P}}_{tot}=0.$ Subsequently, the Ramsey-theory-based approach to the motion of the system is developed. The set of momenta ${\overrightarrow{p}}_{cmi}(t), i=(1,\dots , 6)$ are denoted as the set of vectors generating the complete bi-colored graph. Vectors ${\overrightarrow{p}}_{cmi}, i=(1,\dots , 6)$ are considered as the “generators” of the Ramsey graph. In other words, vectors ${\overrightarrow{p}}_{cmi}, i=(1\dots 6)$ serve as the vertices of the graph. The graph is built according to the following procedure. Vertices numbered i and j are connected with a red link when Equation (3) takes place:

$${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))=e_{ij}\ge 0.$$

(3)

Parentheses denote the scalar product of the vectors, and, correspondingly, the vertices numbered i and j are connected with a green link when Equation (4) occurs:

$${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))=e_{ij}(t)<0; i,j=1,\dots ,6$$

(4)

Within the terms of the Ramsey theory (recall the “party problem” addressed in the Section 1), the momenta of the particles are each “acquainted” with another and are connected with a red link when Equation (3) takes place, and the vectors are each “not acquainted” with another and are connected with a green link when Equation (4) takes place. Figure 2 exemplifies the suggested coloring procedure. Thus, a typical Ramsey problem arises, and the Ramsey numbers may be introduced [2,3,4,5,6].

According to the Ramsey theorem, two monochromatic triangles are recognized (namely, red triangles “135” and “356 in the graph shown in Figure 2). Indeed, R(3, 3) = 6; this means that at least one mono-colored triangle should inevitably appear in the bi-colored complete graph built of six vertices. Regrettably, the Ramsey theorem says nothing about what kind of monochromic triangles will necessarily be present in the graph.

Now, consider that the graph was built emerging from the momenta of the particles established in the frame of the center of mass. The scalar product of the momenta, defined by Equations (3) and (4), being time dependent, is invariant relative to the rotations and translations of the frames. Thus, the coloring procedure introduced at a given time is invariant relative to the aforementioned transformations of the frames. However, the coloring is sensitive to the Galilean/Lorenz transformations. Indeed, the sign of the scalar products defined by Equations (3) and (4) is sensitive to the Galilean/Lorenz transformations. It is noteworthy that it remains the same for the slow motions of the system of frames $S^{\prime }$ relative to the frame related to the center of masses, denoted as S. Consider the system of frames $S^{\prime }$ moving with velocity $\overrightarrow{u}$ relative to S (we restrict our treatment using the Galilean transformations). The scalar product of the momenta in $S^{\prime }$ is given by:

$$\left({\overrightarrow{p}}_i^{\prime }·{\overrightarrow{p}}_j^{\prime }\right)=m_i{m}_j\left({\overrightarrow{v}}_{cmi}-\overrightarrow{u}\right)·\left({\overrightarrow{v}}_{cmj}-\overrightarrow{u}\right).$$

(5)

Neglecting terms of a second-order smallness in speed u, we obtain:

$$\left({\overrightarrow{p}}_i^{\prime }·{\overrightarrow{p}}_j^{\prime }\right)\cong \left({\overrightarrow{p}}_{cmi}·{\overrightarrow{p}}_{cmj}\right)-m_i{m}_j\overrightarrow{u}\left({\overrightarrow{v}}_{cmi}+{\overrightarrow{v}}_{cmj}\right).$$

(6)

The scalar product $\left({\overrightarrow{p}}_i^{\prime }·{\overrightarrow{p}}_j^{\prime }\right)$ will be close to the scalar product $\left({\overrightarrow{p}}_{cmi}·{\overrightarrow{p}}_{cmj}\right)$ when Equation (7) takes place:

$$\left(\overrightarrow{u}·\left({\overrightarrow{v}}_{cmi}+{\overrightarrow{v}}_{cmj}\right)\right)\ll \left({\overrightarrow{v}}_{cmi}·{\overrightarrow{v}}_{cmj}\right).$$

(7)

Equation (7) defines when the motion of the frames may be considered as “slow”. Thus, we come to a very important and general conclusion: the mono-colored triangles appearing in the frame of the center of mass will remain mono-colored under rotations and translations of the frames, and they will remain monochromatic in any slowly moving inertial frame when Equation (7) is true. It again should be emphasized that the coloring of the graph changes with time. Thus, a somewhat surprising result is obtained: the monochromatic triangles, emerging from the coloring procedure defined by Equations (3) and (4), will remain monochromatic when the aforementioned conditions take place.

Consider the system built of the six interacting particles. The Ramsey complete graph generated by this system inevitably contains a “green” or “red” triangle. Assume that a red triangle built of vertices numbered i, k, and l is actually revealed. According to Equation (3), we have ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmk}(t))\ge 0$; ${(\overrightarrow{p}}_{cml}(t)·{\overrightarrow{p}}_{cmk}(t))\ge 0$; and ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cml}(t))=e_{ij}\ge 0$. Thus, for these “red” triangles, the following can be derived: $\left({\overrightarrow{p}}_{cmi}+{\overrightarrow{p}}_{cml}\right)·{\overrightarrow{p}}_{cmk}\ge 0$ and $\left({\overrightarrow{p}}_{cmi}+{\overrightarrow{p}}_{cmk}\right)·{\overrightarrow{p}}_{cml}\ge 0.$ The extension for the “green” triangles is trivial.

2.2. A Ring-like System of Momenta and Its Properties

The introduced Ramsey-theory-based approach enables an elegant geometric interpretation in the frame of the center of mass, in which $\sum _{i=1}^6{\overrightarrow{p}}_i={\overrightarrow{P}}_{tot}=0$ is true. The conservation of momentum may be interpreted as follows: the momenta of the particles form a ring-like system of vectors such as that depicted in Figure 3. Of course, this is true only in the frame of the center of mass. In general, the ring-like system of vectors shown in Figure 3 is a 3D one.

This set of momenta gives rise to a bi-colored complete graph, built according to the procedure described by Equations (3) and (4). It is instructive to build a graph for a system of particles in which $\left|{\overrightarrow{p}}_i\right|=const$ takes place, i.e., where the moduli of the momenta remain constant. It is taken that the vectors ${\overrightarrow{p}}_i$ form a hexagon (in principle, the hexagon may rotate), as shown in Figure 4A.

The 2D bi-colored complete Ramsey graph generated by the vectors of momenta ${\overrightarrow{p}}_i, i=1\dots 6$ ($\left|{\overrightarrow{p}}_i\right|=const$), according to Equations (3) and (4), is shown in Figure 4B. According to the Ramsey theorem, $R\left(3, 3\right)=6;$ this guarantees the presence of at least one mono-colored triangle in the graph, as shown in Figure 4B. Indeed, the triangles “135” and “246” that appear in Figure 4B are green ones. It should again be stressed that the suggested coloring of the Ramsey graph will remain untouched under rotations/translations of the frames, and ($\left|{\overrightarrow{p}}_i\right|=const$) the coloring will now not evolve with time. The suggested coloring procedure, given by Equations (3) and (4), is easily generalized for any arbitrary number of the momenta vectors. However, the calculation of large Ramsey numbers remains an unsolved problem [2,3,4].

The graph depicted in Figure 4A,B is correct for a system of interacting (say, gravitating) rotating particles. Thus, it is applicable for a description of the co-rotation observed within clusters of stars [17]. A cluster built of six gravitating stars will give rise to a graph containing green triangles.

Now, consider a graph emerging from five vectors generating momenta, ${\overrightarrow{p}}_i, i=1\dots 5$, $\left|{\overrightarrow{p}}_i\right|=const$, forming a pentagon. In this specific case, the conservation of momentum is also assumed, i.e., $\sum _{i=1}^5{\overrightarrow{p}}_i=0$ (the system is isolated, and the frame of the center of mass is considered). The 2D ring-like system of the momenta vectors is shown in Figure 5A. The bi-colored graph generated by the momenta vectors ${\overrightarrow{p}}_i, i=1,\dots , 5,$ defined by Equations (3) and (4), is shown in Figure 5B. The coloring of the graph is given by Equations (3) and (4). Five momenta vectors, ${\overrightarrow{p}}_i, i=1,\dots , 5$, now serve as the vertices of the complete Ramsey graph, as shown in Figure 5B.

The complete bi-colored graph, as shown in Figure 5B, does not contain any mono-colored triangle. Indeed, the Ramsey number $R\left(3, 3\right)=6.$ Moreover, we state that any graph emerging from the vectors ${\overrightarrow{p}}_i, i=1,\dots ,5$, $\left|{\overrightarrow{p}}_i\right|=const$ forming a pentagon, as shown in Figure 5A, will not contain a monochromatic triangle in any frame obtained by rotations or translations of the center of mass frame.

What do we have when the total moment of the system is not equal to zero? Consider a system containing five particles for which $\sum _{i=1}^5{\overrightarrow{p}}_i={\overrightarrow{P}}_{tot}\ne 0$ takes place. In this case, we have an open chain of momenta, such as that shown in Figure 6. The chain may be easily closed by vector $-{\overrightarrow{P}}_{tot}$, as depicted in Figure 6.

The further treatment of the Ramsey graph is trivially reduced to the aforementioned mathematical procedure when $-{\overrightarrow{P}}_{tot}$ is included in the set of generating vectors.

2.3. Extension of the Suggested Analysis for the Angular Momenta of the Particles

The suggested Ramsey-theory-based approach can easily be extended to the angular momenta of particles, denoted as ${\overrightarrow{L}}_i$. Consider an isolated system built of N interacting point masses, $m_i, i=1,\dots ,N$. For the sake of simplicity, assume that the total initial angular moment of the system is zero. The system is isolated; thus, the total angular moment of the system is conserved. Now, the set of angular momenta is defined as ${\overrightarrow{L}}_i(t), i=(1,\dots ,N)$, which is the set of vectors generating the complete bi-colored graph. The vectors ${\overrightarrow{L}}_i(t), i=(1,\dots ,N)$ serve now as the vertices of the graph. The graph is built according to the following procedure. Vertices numbered i and j are connected with a red link when Equation (8) takes place:

$${(\overrightarrow{L}}_i(t)·{\overrightarrow{L}}_j(t))\ge 0, i,j=1,\dots ,N.$$

(8)

Parentheses denote the scalar product of the vectors, and, correspondingly, the vertices numbered i and j are connected with a green link when Equation (9) is true:

$${(\overrightarrow{L}}_i(t)·{\overrightarrow{L}}_j(t))<0.$$

(9)

The further reasoning is similar to that discussed in Section 2.1 and Section 2. The bi-colored graphs arising from the coloring procedure defined by Equations (8) and (9) can be analyzed within the Ramsey theory. The number of mono-colored substructures remains invariant in all of the frames emerging from the original frame under its translations or rotations.

It is noteworthy that there exists a particular case when the coloring of the momenta graph coincides with that of the angular momenta. The scalar product of the angular momenta is calculated according to Equation (10):

$$\left({\overrightarrow{L}}_i·{\overrightarrow{L}}_j\right)=\left(\overrightarrow{r}\times {\overrightarrow{p}}_i·\overrightarrow{r}\times {\overrightarrow{p}}_j\right)$$

(10)

Considering the equation $\left(\overrightarrow{a}\times \overrightarrow{b}·\overrightarrow{c}\times \overrightarrow{d}\right)=\left(\overrightarrow{a}·\overrightarrow{c}\right)\left(\overrightarrow{b}·\overrightarrow{d}\right)-\left(\overrightarrow{a}·\overrightarrow{d}\right)\left(\overrightarrow{b}·\overrightarrow{c}\right)$ and Equation (10) yields:

$$\left({\overrightarrow{L}}_i·{\overrightarrow{L}}_j\right)=r^2\left({\overrightarrow{p}}_i·{\overrightarrow{p}}_j\right)-\left(\overrightarrow{r}·{\overrightarrow{p}}_i\right)\left(\overrightarrow{r}·{\overrightarrow{p}}_j\right)$$

(11)

We conclude from Equation (11) that when $\left(\overrightarrow{r}·{\overrightarrow{p}}_i\right)=0, i=1,\dots ,N$, the signs of the scalar products $\left({\overrightarrow{L}}_i·{\overrightarrow{L}}_j\right)$ and $\left({\overrightarrow{p}}_i·{\overrightarrow{p}}_j\right)$ coincide. This takes place when the rotational motion of the point masses occurs, as takes place in rotating clusters of stars [17]. In this case, the coloring of the momenta graph coincides with that of the angular momenta graph.

2.4. Graph Representation of the Motion of a Single Point Mass

The suggested approach can easily be extended to the motion of a single point mass, as shown in Figure 7.

.

The motion of the particle is seen in the momentum space, which is used broadly for solutions to physical problems [18,19]. A time sequence of six momenta of a point mass performing a curvilinear movement, ${\overrightarrow{p}}_i$, is presented in Figure 7. The graph representing the motion and its coloring were determined as outlined in detail in Section 2.1. The coloring procedure is defined by Equations (3) and (4). The emerging bi-colored graph, according to the Ramsey theorem, will inevitably contain at least one monocromatic triangle.

Consider one possible application of the suggested approach. We address a system called a dynamical billiard, in which a particle moves along a straight line and is reflected from the boundaries. Billiards are Hamiltonian idealizations of the known game billiards, in which the boundaries have a general geometric shape (rather than a rectangular shape). In Figure 8, a point particle m alternates between free motion (presupposed to be a straight line) and specular reflections from an elliptic boundary. This class of systems is called a dynamical billiard [20,21,22]. Dynamical billiards are described by mathematical models that appear in a broad diversity of physical phenomena [20,21,22]. The dynamical properties of such models are determined by the shape of the walls of the container, and they may vary from completely regular (integrable) to fully chaotic [20,21,22]. Thus, consider the simplest system in which the particles move in a 2D elliptic container and collide with its walls/boundary, as shown in Figure 8. The reflection points are marked as circles. Consider a set of six reflections from the boundary; the momenta after the reflections are denoted as ${\overrightarrow{p}}_i, i=1,\dots , 6.$

The reflections may be elastic or non-elastic. The motion of the particle may be regular or chaotic. The graph generated by the momenta and its coloring is determined as outlined in Section 2.1. The coloring procedure is defined by Equations (3) and (4). The emerging bi-colored, complete graph, according to the Ramsey theorem, will inevitably contain at least one monocromatic triangle, whatever the motion of particle m is.

3. Discussion

In this paper, we introduced a procedure that enables a Ramsey-theory-based analysis of isolated mechanical systems. The vectors of the momenta or angular momenta of the particles are taken as the vertices of the graph. The vertices are connected with colored links. The coloring procedure is defined by Equations (3) and (4) and Equations (8) and (9). It should be stressed that the coloring defined by Equations (3) and (4) and Equations (8) and (9) is non-transitive [23,24]. The transitive Ramsey numbers are different from the non-transitive ones [23,24]. The aforementioned coloring of the graphs, being time-dependent, is insensitive to the translations/rotations of frames. It should be emphasized that the conservation of the momentum and angular momentum of isolated mechanical systems emerges from the homogeneity and isotropy of space; in other words, the conservation of the total momentum and angular momentum arises from the insensitivity of the Lagrangian to the infinitesimal translations and rotations of the system [16].

However, the suggested coloring procedure is sensitive to the Galilean/Lorenz transformations. An isolated system containing six interacting particles will contain at least one mono-colored triangle.

In our future investigations, we plan to carry out the following:

(i)

Investigate a Hamiltonian interpretation of the introduced graphs emerging from the Hamiltonian of a given mechanical system.

(ii)

Study the relativistic generalization of the introduced graphs.

4. Conclusions

This paper introduces a Ramsey-theory-based analysis of mechanical systems. The Ramsey theory is a field of discrete mathematics that refers to any set of objects interrelated by different kinds of connections. Today, it is seen as a field of graph theory [25]. We propose considering the vectors of momenta, labeled ${\overrightarrow{p}}_i$, and angular momenta of the interacting point masses, denoted as ${\overrightarrow{L}}_i$, constituting an isolated physical system as the vertices of a graph [25]. We start from the analysis of the Ramsey-theory-based dynamics of an isolated system from the establishment of the momenta of the point masses in the frames of the center of masses, denoted as ${\overrightarrow{p}}_{cmi}$. The coloring of the links/edges of the graph is performed as follows: if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))\ge 0$ is true, the vectors of the momenta of the particles are connected with a red link; if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))<0$ is true, the vectors of the momenta are connected with a green link. Thus, the complete bi-colored Ramsey graph emerges. If the mechanical system contains six point masses, then the emerging graph contains at least one monochromatic triangle (which is totally red or green). The coloring of the graph may evolve with time; however, it is independent of the rotations/translations of the frames. Thus, the number of monochromatic triangles will be the same in all of the frames obtained by the rotation/translation of original frame at a given time. However, the coloring procedure is sensitive to Galilean transformations. On the other hand, the Ramsey theory predicts that an isolated physical system built of five interacting point masses may be represented by a bi-colored momenta graph that does not contain monochromatic triangles (if the coloring procedure remains untouched). In isolated systems, the total momentum is conserved. The conservation of momentum may be geometrically interpreted as follows: the momenta of the particles form a 3D ring-like system of vectors. Bi-colored, complete graphs emerging from 2D ring-like sets of momentum vectors are treated. Systems of particles in which $\left|{\overrightarrow{p}}_i\right|=const$ takes place, i.e., where the moduli of the momenta remain constant, are addressed. Ramsey graphs built for these systems, illustrating the Ramsey theorem and demonstrating the Ramsey numbers, are supplied in this paper. The presented approach can easily be extended to bi-colored complete graphs emerging from the angular momenta of interacting particles. An extension of the suggested approach to systems in which the momentum is not conserved is suggested. A generalization of the approach to the motion of a single particle seen in the momenta space is introduced.