## Comments about the book "Exploring Complexity" by Grgoire Nicolis and Ilya Prigogineo

This document contains comments about the book: "Exploring Complexity" by Grgoire Nicolis and Ilya Prigogine. Munich 1989
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### page 2

Two great revolutions in physics at the beginning of this century were quantum mechanics and relativity.
Okay
The concerns of these two areas - elementary particles and cosmology - correspond to extreme conditions; they are part of high energy physics.
Okay
There were moments when the program of classical science seemed near completion: a fundamental level, which would be the carrier of deterministic and reversible laws, seemed in sight.
What are and why do we need deterministic and reversible laws?
A more deeper question is: are there reversible processes. What is the definition.
IMO there are three possibilities:
• When you start from A and B you will get C and D
• When you start from C and D you will get A and B
• When you start from A and B (or C and D) you will get a mixture of all.
However I don't think, this is all. I expect there is always an extra parameter involved. For example: Temperature.
That means as a function of the temperature the reaction goes in a certain direction. That means the reaction is not trully reversible.
We have long known that we are living in a pluralistic world in which we find deterministic as well as stochastic phenomena, reversible as well as irreversible.
What makes a phenomena or process deterministic or stochastic? What makes a phenomena or process reversible or irreversible?
This distinction is not explained, but this is a very important physical question
Moreover, we know that many phenomena - the frictionless pendulum, for one - are also reversible, for future and past play the same role in the equations describing the motion or dynamics involved.
Pendulums are not frictionless.. There is always friction involved. That means at any point in time the equations should not be symmetric. At one point or an other, without a continuous inflow of energy, the pendulum will stop ticking.
But other processes, such as diffussion or chemical reactions are irreversible.
Okay. Maybe all processes are like that?
In such processes, there is a priviliged direction of time: a system that originally is nonuniform becomes homogeneous in the course of time.
A very simple chemical reaction is: (1) 2H2 + O2 => 2(H2O). But you can also have: (2) 2(H2O) => 2H2 + O2. Both reactions are identical but in different directions. The direction depents about external conditions. What is important there is nothing like a direction of time involved. It is all chemistry.
In addition if we want to avoid the paradox of referring the variety of natural phenomena to a program printed at the moment of the Big Bang, we are forced to acknowledge the existance of stochastic processes, those whose dynamics is non deterministic, probabilistic, even completely random and unpredictable.
It makes sense to declare all processes as unpredictable, but that has nothing to with any computer program or the Big Bang.

### page 3

At the beginning of this century, continuing the tradition of the classical research program, physicists were almost unanimous in agreeing that the fundamental laws of the universe were deterministic and reversible.
The order in this sentence is wrong. First you must identify the processes in the Universe that you want to study. Next you must try to describe these processes from a physical or chemical point of view. And finally from a mathematical point of view.
Many chemical processes are identical caused by the same chemical reactions.
If you can declare these processes deterministic or reversible requires a clear definition what is meant.
Now at the end of this century, more and more scientists have come to think, as we do, that many fundamental processes shaping nature are irreversible and stochastic; that the deterministic and reversible laws describing the elementary interactions may not be telling the whole story.
I think that the scientist around 1900 and earlier were only interesting in the elementary interactions and in the reactions performed in a laboratory. I doubt if they cared much about deterministic and reversible processes except in the case of Isaac Newton. At the same time I expect that the time frame he studied and the accuracy he used was too small to conclude that his laws in fact were inaccurate.
In classical physics, the investigator is outside the system that he observes.
That is the correct way. Any human activity in principle should be excluded as part of the process under study.
He is the one who can make independent decisions, while the system itself is subject to deterministic laws.
Humans can be involved in the setup of any experiment. Any process is normally a mixture of many physical, mechanical biological or mechanical changes in time. Often the process can be subdivided in smaller parts. These smaller parts each involve changes, at the same time these smaller parts can influence each other. A typical case is the human body.
In other terms, there is a "decider" who is free, and the members of the system, be they individuals or organizations, who are not "free" but must conform to some master plan.
This sentence is difficult to compare with the reality.
Not only in human sciences, but in physics as well we know that we are both actors and spectators - to use a well-known expression by Niels Bohr.
There is a hugh difference between physics and human sciences. In physics humans are not a part of the physical process studied. Humans should not influence the process studied and not influence the evolution of the process. In human sciences certain people are part of the process studied and other people not. The people of both groups should not influence each other. For example a doctor in a medical study, in general, should not tell a patient that he looks bad. Its the patient who should inform him.

### Chapter 1. Complexity in Nature - page 1

What is the difference, if any, between the swinging of a pendulum and the beating of the hart, or between a crystal of ice and a snowflake?
All these four are completely different physical (chemical, biological, mechanical) systems.
Is the world of physical and chemical phenomena basically a simple and predictable world where all observed facts can be interpreted adequately by appealing to a few fundamental interactions.
The phenomena or events of our world are in general not simple and not predicatable.
An examination of experimental data will lead us to the conclusion that the distinction between physico-chemical and biological phenomena, between simple and complex behavior, is not as sharp as we might intuitively think.
Our intuition is a very poor judge to make any physical distinction. In fact all phenomena are complex.

### 1.1 What is Complexity - page 6

This is about 10^19 molecules crowded into a space of etc, moving in all possible directions and continuously colliding with one another.
This is almost all what can be written considering all molecules. For a single molecule this is slightly different.
Is this sheer vastness of number enough to qualify this system as complex?
That depends on the definition of complex. Such definition only makes sense if there are also other types of systems.
Intuition continues to tells us "No", because we do not perceive any coordinated activity or form or dynamics.
When we do not observe any particular pattern or behavior over a long period does not mean that the trajectory of each particle is simple. In actual fact the behavior of any particle can be very complex.
This incoherence of molecular movement, represented in Figure 1(a), results from the basic property of the intermolecular forces of interaction in this and many other systems encountered in nature, namely their short range character.
It is not Figure1(a) which is the issue. The question is which are the "intermolecular forces of interaction" used to calculate the trajectory of each molecul. A second question is, is such a trajectory correct with the actual observations. I think such a question is easier to handle with an individual dust particle.
This example teaches us an important lesson: It is more natural, or at least less ambiguous, to speak of complex behavior rather than complex systems.
Why this tricky distinquish? Complex systems show complex behaviour, compared with simple systems show simple behavior. The real issue is what are the differences between both.

### Figure 1a Visualization of molecular chaos.

The figure summarizes the results of a computer simulation of the equations of motion of 400 elastic hard disks in a rectangular box measuring about 120 x 60 (molecular diameter)^2
There is nothing against this simulation, but what is does, it shows a visualization of mathematical equations which are suppossed to be the same as a visualization of the physical reality. That is the question.
The boundary conditions are temperature on the two horizontal boundaries, fixed at the same value (T=1 in units in which kb/m = 1/2, where kB is is Boltzmann's constant and m the molecular mass), and periodic conditions on the vertical boundaries.
This seems to indicate that all particles have the same mass.

### 1.2 Self-organization in physico-chemical systems: The birth of complexity - page 8

Since the 1960s a revolution in mathematical and physical sciences has imposed a new attitude in the description of nature.
This sentence is difficult to understand, it is vaque.
It is important to agree upon that there are two different ways to describe a process:
• In a more physical, biological,mechanical and chemical sense. That means maily in words and concepts. A clock as such is described how all the parts work together that finally the system shows the time of the day.
• In a mathematical way. That means for example by means of mathematical equations. For example our solar system by means of Newton's Law, MOND or General Relativity ie. Einsteins equations.
The question is: which is the best strategy to follow, to understand something.
Such ordinary systems as a layer of fluid or a mixture of chemical products can generate, under certain conditions, self organization phenomena at a macroscopic scale in the form of spatial patterns or temporal rhythms.
You can call these process cyclic. The most famous example are the tides?
It is invading the physical sciences and appears to be rooted in the laws of nature.
These cyclic patterns are deeply rooted in the behaviour of certain physical process as described what is commonly called: "the laws of nature". Anyway such a discussion makes only sense when there exists a clear definition which these laws of nature are.

### 1.4 Self-organization phenomena in chemistry - page 15

This process is symbolized
 A + B --k--> C + D (1.2a)
in which k is the rate constant, generally a function of temperature and pressure.
When the process is a function of temperature there should be a heating element (controller) available as a part of the reactor.
That means when the temperature is too low no reaction takes place.
When the temperature increases slowly the reaction will start and C + D will be produced. This reaction, once started, can also create heat. That means that heating should be deminished. This can also be done automaticaly.

### page 16

More precisely, after a sufficient reaction time the amounts of coexisting constituents A,B,C and D attain a fixed value of the ratio cCcD/cAcB wher c denotes the concentration of each constituent.
Okay
When this fixed value is attained we say that the system is in chemical equilibrium and the value is of the ratio is equilibrium constant.
I doubt if this is simple. Assume that both C and D are zero. In Eq. (1.2)a the number of molecules C and D produced will be the same and the number of molecules A and B used. That means the reaction will be finished when A or B is depleted.
However that is maybe not always the case. In that case the reaction stops at a certain point. What that point is, is the great question.
Experiment shows the existence of the reverse transformation of equation (1.2a)
 C + D --k'--> A + B (1.2b)
I expect that experiment shows assuming that both A and B are zero that the reaction will be finished when C or D is depleted
However this maybe not the case. Anyway this reaction will also stop at a certain point.
We represent a reversible reaction as
 A + B --k--> C + D --k'--> A + B (1.2c)
That means the reaction can be both ways, as demonstrated by experiment. The point is that this has nothing to do with the concept of time.

### page 17

Moreover, if some parts of the chemical mechanism could enable the system to capture and further amplify the enhancement, we could have a potentially unstable situation, similar to the one depicted for the Bénard problem in Figure 4.
It is the purpose of the field of process control to take care that these situations don't arise.
Such mechanisms are known to exist in chemistry, and their most striking manifestation is autocatalysis.
For instance, the presence of a product may enhance the rate of its own production.
A typical case is a nuclear reaction. See https://en.wikipedia.org/wiki/Nuclear_chain_reaction

### page 19

This oscillation which measures time through an internally generated dynamics, constitutes a chemical clock.
It behaves to some sort of chemical clock. The accuracy (frequency) most probably is not high.
To derive the mathematical equations is also not simple.
Figure 8 illustrates the differences between the two kinds of oscillations. The upper left diagram shows a frictionless pendulum swinging with a maximum angle to the vertical opening théta1 [function theta(t)] and a period T1
To consider the performance frictionless creates an pendulum which operates ad infinitum, contrary to the reality.

### Figure 8

Comparison of sustained oscillations. Left, conservative system, the pendulum; right, dissipative system, the BZ reagent.
Both systems show oscillations.
From a physical point of view, both systems are rather easy to understand.
From a practical point of view, to actual simulate a real pendulum is not easy. To simulate the Belousof-Zhabotski reaction is even more complex.

### page 20

The upper right diagram shows the variation in time of the concentration of a chemical in the BZ reagent, irregular at first, and then characterized by regular oscillations of amplitude A and period T when the clocklike state has been attained.
To call the oscillations regular is for practical purpose okay.
In reality nor the oscillations, nor the amplitude, nor the period T are constant. This makes a real accurate simulation of such a system extremely difficult.

### page 44

Let us try to give the flavor of the relation, that may exist between gravitation, general relativity and irreversible processes, referring for more detail to Appendix 5. Primordial Irreversible Processes - page 283
As is well known, the world, as described in newtonian theory, presented an intrinsic duality: on one side, space-time; on the other, matter moving under the influence of forces.
This picture is wrong.
• What exist is space which contains objects (matter) which move through space. However there is one more important aspect: each and every object influences and is influenced by all the other objects. The result is that there at almost all levels groups of objects which move more or less autonomous.
• There exists nothing what we can call time.
• May be the most interesting type of movement are oscillators, like a pendulum.
Oscillators can be used to measure other types of movements.
Oscillators demonstrate what we humans consider: time.
At the borderline: There exists no duality.
The great idea of Einstein was to go beyond this duality, and to relate space-time to matter. General relativity is the basis of all cosmological evolution.
The idea of Einstein was to define a new parameter s, which is a function of x,y,z,t and c. This is also a distance, but does not exist in reality (cannot be measured) and can only be calculated.
However, there are some difficulties. Einstein's equations describe an evolution in which matter and entropy are conserved.
?
Therefore, if we go back to the far distant past, we arrive at a singularity for which all matter and entropy are concentrated in a single point.
The question is if this is true for entropy.
A more important question is: What is a point? Does it have a size?
Indeed, beyond it, the laws of physics have no meaning.
Maybe even before this moment.

### 2.1 Conservative systems - page 46

Indeed, Newton's second Law realting force to acceleration, and third law relating "action" to "reaction" imply that in a system of interacting massive points free of any external influence three quantities remain invariant in time:
1. Total energy
2. Total translational momentum
3. Total angular momentum
It is in the practical details of these three quantities that lie the problem. For example how is the mass measured or calculated?

### page 48

One of the most familiar conservative systems is the pendulum. If we denote the pendulum length by l,the accelaration gravity by g,and the angle of the downward vertical by theta and agree to measure energies using the position of the mass on this vertical as a reference, we can write the total energy - the hamilton - in the form:
 H = Kinetic energy + potential = 1/2 m*v^2 + m*g*l (1 - cos(Theta))
Okay
Sinc v is tangent to the trajectory of mass, v = l (dTeta/dt), and thus
 H =ml [ 1/2l (dtheta/dt)^2 + g (1 - cos(theta))] = constant (2.6a)
The question is what is the solution of the equation: 1/2l (dtheta/dt)^2 + g (1 - cos(theta)) = constant

### page 49

As discussed in more detail in Chapter 3 , it is very useful to visualize the evolution of a system in an abstract space known as phase space.
It is doubtfull to use the concept of abstract space to explain something that exists in physical space.
In phase space the coordinates are the positions and momenta appearing in Hamilton's equations, Eqs(2.5b).
Okay.
In the case of the pendulum q=0, p=m*l*(dtheta/dt) the phase space is the plane (dtheta/dt,theta).
This is a mathematical plane, not a physical plane.
These trajectories and velocities are represented in Figure 20. They should not be confused with the more familiar trajectory and velocity of a body moving in physical space, which, despite their usefulness, describes a less detailed description of the motion.
That is the question . When you want to perform measurements you have to perform physical measurements.
These measurements should be used to simulate a real pendulum, based on observations. If observations show that under certain situations a pendulum can make a complete revolution, than the simulation should also show that behaviour. If simulations show that when the pendulum shows dramatic movement when at almost its highest point, than the simulation should show the same.

### page 50

A still different class of conservative systems arises in the two-body problem , the motion of two bodies interacting by gravitational or any other force that depends only on the absolute value of the distance between the boies.
Such a system is does not physical exist, consisting of just 2 objects. It is too simple. See also: Reflection 3 - Physics versus Mathematics
However complications arise quickly in mechanics.
Seems logical
With two pendulums coupled through a spring, a single pendulum forced periodically from outside, or three interacting masses, we arrive at the frontier of present knowledge.
Strange. A three body problem IMO seems also a conservative system. Why is it not possible to handle such a system.

### 2.2 Dissipative System - page 50

In addition to conservative systems as described by classical mechanics, we must consider systems that give rise to irreversible processes. A very simple example is provided by systems exhibiting friction.
IMO almost all processes are irreversible. A comet, part of a binary system, also losses mass.
In contrast, the classical principle of inertia, which states that acceleration, not velocity, is the basic mechanical quantity, corresponds to a model is which friction is neglected.
Neither acceleration nor velocity are the most important, because both are calculated. The actual observations of the positions of the objects considered and the model used, are the most important.
But in the nineteeth century a conflict appeared. In physics, irreversibility and dissipation were interpreted as degradation, while among natural scientists biological evolution, which is obvious an irreversible process, was associated with increasing complexity.
What's in the name?

### page 51

When the equations of evolution of these variables are studied an important general feature emerges, namely, that the structure of the equations is not invariant under time reversal, contrary to Eqs(2.4) and (2.5b).
In this text the importance of equations is advocated. The importance of the physical processes should be amplified.
We therefore expect an irreversible course of events.

### 2.5 The many facets of the second law - page 61

Now it is time to raise the of how these solutions of the evolution laws, Eqs (2.12) are attained in the course of time.
I'm more interesting in the practical question: How are these solutions found in some more practical situations.
For conservative systems the answer to this question is remarkably simple at least in principle: Evolution is entirely dictated by the initial conditions (as in figures 8 and 20)
That is definite not true. The initial conditions are part of the evolution of any process. The physical (chemical, mechanical etc) details of the process are the most important. Those details can result in mathematical equations.
which fix the various constants of motion and hence the trajectories themselves through such relations as Eqs (2.6).
In practice all the parameters of these equations have to be measured or calculated, which can be complex.

### 3.3 Integrable Conservative systems - page 88

Consider a hamilton system described by the Eqs(2.5) or (3.7).
Are both equations a correct description of a physical system?
As Liouville demonstrated, if the latter are sufficiently regular in the mathematical sense of the term, the system can be integrated by simple quadratures.
This sentence is not clear.
Anyway, in general this is true in mathematical sense, but is it also true in physical sense? That means are both the differential equations and the solutions a correct description of a physical system?

### page 89

Equations (3.8) define the particular class of integrable systems
That ofcourse is true in mathematical sense. The problem is which type of systems can be accuratable described by the equations (3.8)
A hamiltonian system with two degrees of freedom is integrable if there exists a sufficiently regular first integral independent of the hamiltonian H
IMO this sentence explains something completely in mathematical sense, without any link to the physical 'world' i.e. reality.
So too is the system in a two-body problem in the presence of central interaction forces, whose importance stems from its relation to the motion of celestial bodies.
The main problem with this approach that this central interaction force is supposed to act instantaneous
More generally, all systems that can be separated into uncoupled systems of one degree of freedom are integratable.
How do you know that this can be done for any arbitrary system? I expect strict rules apply.

### page 90

If these periods are commensurate, that is, if there exists a set of nonvanishing integers k1, k2, .. such that
 k1w1 + k2w2 + .... = 0
then the motion is clearly periodic and is represented by a closed curve.

### 3.8 Nonintegratable Conservative Systems: The new mechanics - page 115

Let us think for a moment about the movements of the planets around the Sun.
Okay, but one moment is too short.

### page 116

Clearly, as long as one is interested in time scales of the order of the inverse of the strength of the perturbation (these would be in the 1000-year range for the planetary system), the effects induced by the latter can be handled straight forward.
This can be done by standard methods
Okay.
But if we look for solutions valid for very long times and, in particular, if we want to understand the qualitative behaviour of the exact solutions, we are immediately faced with a number of formidable difficulties.
 Google: In a nutshell, qualitative research generates “textual data” (non-numerical). Quantitative research, on the contrary, produces “numerical data” or information that can be converted into numbers.
That is correct. The central problem is accuracy. Specific in physics in the accuracy of the observations, in the detailed descriptions of how these observations are made.
In our example of the planetary system these would arise in a time scale in the range of 1 billion years, and the question at issue could be whether the planets will escape from the system, crash on the sun, or hit each other.
The central answer is that this question cannot be answered for our solar system. We can answer the question for a system that mimics our solar system, but not the system of which we are a part.
The principle manifestation of such difficulties is the presence of divergencies.
The reason that we cannot answer these questions 'exactly' lies first of all in physical reasons.
Secondly in mathematical issues.
Part of the problem is that our sun is influenced by all the 'stars' in our galaxy.
An other very important issue is that gravity forces involved between all the objects don't act instantaneous, but take time. What that means is that Newton's Law is too simple to be used.
At the same time I also have my doubts if GR is not too complex to be practical.

An important point is: to call the movements of the stars chaotic, is a misnomer.

The most ubiquitous such condition is the occurence of resonance, that is, the case where the frequencies of the integrable reference system are commensurate in the sense of k1w1 + k2w2 + ... = 0 (See Section 3.3 - page 90 )

### 4.3 Markovian processes and irreversibility. Information entropy and Physical entropy - page 160

We begin by introducing the entropy of a markovian process. The need for such a quantity arose quite early in the probability theory.
Okay
It was essentialy motivated by the development of exacty soluble probablistic models aiming at reconcile the irreversible trend to equilibrium predicted by Boltzmann's kinetic theory and the reversibility of the laws of motion of conservative systems.
Any closed process without any communication to its environment leads to an equilibrium state, which is by definition irreversible. That means such a process does not return back to its initial state. The laws of motion which describe the evolution of certain process, identified as objects, forward in time. They predict the future.
From a mathematical point these laws are (time) reversible, but from a physical point not.
Let us simply observe here that in both physicochemical and information theory contexts, it is customary to seek a quantity that satisfies a number of requirements imposed by intuition and by analogy with thermodynamics;
Intuition kan never be the reason and the why of thermodynamics has to be explained.
For example:
1. For a given number of states, N, and for
 P(Q)>=0, Sum, 1 over n, P(Q)=1
the entrophy function S takes its largest value Pq = 1/N.
That means the probability for each state is mathematical identical.
The question is if all physical states, with for example 7 coins and N = 2^7 = 128 states are visited.

### page 161

It is known from probability theory that this property is an outstanding feature of stochastic processes whose state are ergodic; in other words, all states are visited with probability one and the mean time of returning to any particular state from which the system started (known as recurrence time) is finite.
But that does not make it a reversible process. See also page 152

### page 163

These equations describe respectively, an irreversible decay to a final asymptotically stable state y=0 in a dissipative system (See e.g, Section 2.2 - page 55 ) and a typically time-reversible evolution such as the harmonic oscillator (see Section 3.3 - page 88 )

### A1.1 Basic Equations - page 243

This appendix is addressed to the mathematically more sophisticated reader; here the formal method for computing the stability of the solutions of a dynamical system is outlined.
FRom a mathematical point of view this is okay.
From a more realistic approach this raises a problem because no real problem is discussed.

### Appendix 5. Primordial Irreversible Processes - page 243

Indeed, the newtonian view was based on a duality: on one side, space-time; on the other matter (in fact, space and time were considered to be independent from each otheras well as from the matter content).
My understanding is that space, time and objects i.e. matter are independent from each other, but at the same time they belong together.

### page 284

The basic novelty of Einstein's general relativity was to suppress this duality and to establish a connection between space-time on one side and matter on the other.
You could also claim that Newton established a connection between space, time versus matter.

### A5.4 The role of irreversibility - page 287

We now come to the question of whether the origin of our universe is a singularity or an instability.
This question can not be answered. Examples of questions that can be answered are: Who has painted this painting? Who has written or composed this musical master piece. Nothing in that sense can be said about the Universe, partly because the definition of the word universe is not clear.
An idea often presented is that our universe would be a 'free lunch', that it derives from the fact that all available energy is ultimately present under two fundamental forms, which compensate each other: mass related energy (which is positive) and gravitation-related energy (which is negative).
It is never clever to explain something that is not clear: a 'free lunch', by two concepts which are also not clear: mass related energy and gravitation-related energy. The most important issue is: in order to create something out of nothing, don't you need something else? A world completely filled with only 'virtueel particles' is not the answer

### Reflection 1 - General

There is a hugh difference of the behaviour of the physical behavior of a system versus a simulated behaviour. The simulated behavior starts from a set of mathematical equations, which are described in this book. The solution of these equations are shown on a computer screen. Starting point of such a mathematical approach are measurements based on physical parameters, like position and time, in practice a clock reading. The end point of a simulation are also positions but at a different position and clock reading. To tune the mathematical equations require more measurements at different positions and clock readings.

To understand the physical behavior of a system requires the understanding of the physical subsystems and the connections between those subsystems. Most often to understand these subsystems require experiments.

### Reflection 2 - The history of simple versus complex

The study of complexity i.e. the evolution of a complex process makes only sense if there are also simple processes.
During my study at the university the subject was mainly about how to control a process. That means when you change the target, what else had to be done, such that the process reached this new target in a controlled fashion. In practice that means that the temperature has to be changed, or certain products have to be added. One strategy to use is to control the process manualy. That means every parameter is slightly changed one by one and changes in the process are monitored. When you do that carefully you get more insights about the details of the process, specific how the different parts influence each other. A second step is to quantify these changes.
The general lesson is that mixing processes are the most simple. Processes where reactions take place are complex. The most complex processes are when people are part of the process.

There are many types of processes. The first distinction is if as part of the process any living activity is involved or not. When the answer is no than the second distintion is: does the process takes place here on earth or not.
When the answer is no than you are typical studying the behavior of the stars and planets. My point is that all these processes are complex. A very important point is that each process studied should be realistic. For example: The simplest system studied is a binary system of two stars. In principle such a system can exist. If you are an observer on one star you have a 50% chance that you can observe the other star, otherwise you should change position. You could observe that the magtitude of the star changes, but it is impossible to explain.
A much more interesting stellar system is when three stars are involved. In that case it could be established that for example two stars form a binary system and that the third stars rotate at a large distance around these two.
What that means is how more complex the stellar system, how more interesting the situation is.
What can also happen when the system consists of three stars that one is ejected or one collides or merges with a star.
The same can also happen whan all the three stars are black holes
A related question is: Was the system stable before the collision and is the system stable after the collision? IMO opinion this question is rather academic.

A whole different question is: Is it possible to predict the outcome of a stellar system with three stars and to what extend is it possible to predict the outcome of any system. Generally speaking this is an accuracy problem and requires quantification of the process studied. To answer the question involves mathematics which should mimic the process studied.

To answer that question from for a system of three stars requires to measure the positions of the three stars at regular intervals. The smaller the better. The problem is when there is only one observation point and you measure the positions that what you observe at each interval are not the present positions, but the positions in the past. To solve that a better method is to use a 3D grid of observations points. The accuracy of the solution depends very much about distance between these observations points.. How closer together the better.
Using Newton's Law the first step is to calculate the masses of the objects involved.
For more detail about the problems involved select this link: Reflection 2 - The speed of light.

### Reflection 3 - Physics versus Mathematics

IMO if you want to understand how a process evolves there are two strategies.
From a physical (chemical, mechanical, biological) point of view.
From a mathematical point of view.
• When you start from a physical point of view one strategy is to divide the process in to smaller processes, with the object to understand each. The second step is to understand the interrelations and finally to combine all in order to understand the whole. Understanding each starts by performing observations. A second step is performing experiments to get a better grip about the details of each process. This is specific important if you want to understand the chemical details of the processes happening here on earth.
• Understanding from a mathematical point of view starts with understanding from a physical point of view.
Also here you start with observations. However there is an extra aspect and that is that you want to measure and quatify the observations and parameters of the process. The object is that based on observations presenting the past, you want to predict the future. To do that you try to describe this evolution in mathematical equations. This is the main approach followed in the book "Exploring Complexity". An important consideration is that many processes cannot be understood bij mathematics, specific when human bahaviour becomes involved.
But there is one major gadget. When you consider the universe from a physical point of view, all what exists, is something at the present. There exist no past, there exist no future, nor something what we call time. All what changes, changes at the present throughout the total universe. Visible or invissible for human beings. From all what exists at the presents we can make a 3D picture.
All these concepts are related to the human capability to remember what we see. Humans have brains which allow us to store the pictures we see, of our surroudings. It is important that each of these pictures shows a clock count or time which represents the origin of the picture.
However the 3D pictures we see are different than the 3D pictures of the present. The 3D pictures we see show the objects, we see from a certain location, in the past. This is distance divided by the speed of light. The 3D pictures of the present all the objects are shown at the same clock count or time. It requires a transformation to calculate these positions based on what is observed.
An example of how mathematics can be used is: We start with 4 pictures, each showing the present situation, at 4 different clock counts, with the same interval, and each showing the same object as a red dot. When all all the dots are one behind each other on a straight line, than it easy to calculate where the next dot should be. Again a transformation is required to calculate what is observed. This is important in order to demonstrate that the mathematics used is correct.

A problem with the book "Exploring Complexity" is, that it discusses real examples, which at first sight seem simple, but to study them based on observations and mathematics, is very complex. The book recognizes this problem.

• A typical example is a binary system. The problem at first side is that binary systems are stable. In fact this are idealized physical situations. When you consider a comet around the Sun, such a binary system is not stable, specific because the comet looses mass. This physical behaviour results that the system is irreversible.
• A binary system, anyway, is a too simple physical system too study, it requires at least a third object to perform the observations.
• At page 49 the concept of phase space is discussed with a variable p. Phase space belongs to mathematical space.
P is calculated as m*l*(dtheta/dt). This equation uses the parameter m, which has to defined based on observations in physical space.

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Created: 20 January 2021

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